* algebra/compiler.spad.pamphlet: New.

	(InternalTypeForm): New.
	(Elaboration): New.

9 files changed, 22006 insertions, 21923 deletions

diff --git a/src/ChangeLog b/src/ChangeLog
index 713d7648..4b020b69 100644
--- a/src/ChangeLog
+++ b/src/ChangeLog
@@ -1,6 +1,8 @@
 2010-03-12  Gabriel Dos Reis  <gdr@cs.tamu.edu>
 
 	* algebra/compiler.spad.pamphlet: New.
+	(InternalTypeForm): New.
+	(Elaboration): New.
 
 2010-03-12  Gabriel Dos Reis  <gdr@cs.tamu.edu>
 
diff --git a/src/algebra/Makefile.in b/src/algebra/Makefile.in
index 2a66efb9..75bac0b7 100644
--- a/src/algebra/Makefile.in
+++ b/src/algebra/Makefile.in
@@ -939,7 +939,8 @@ axiom_algebra_layer_user =  \
 	ASP73    ASP27    ASP28    ASP33    ASP49    ASP7     \
         ASP78    ASP9     ASP12    ASP55    ASP8     ASP19    \
         ASP20    ASP30    ASP31    ASP35    ASP41    ASP42    \
-	ASP74    ASP77    ASP80    ASP29    IRFORM   COMPILER
+	ASP74    ASP77    ASP80    ASP29    IRFORM   COMPILER \
+	ITFORM   ELABOR
 
 axiom_algebra_layer_user_nrlibs = \
 	$(addsuffix .NRLIB/code.$(FASLEXT),$(axiom_algebra_layer_user))
@@ -1015,6 +1016,9 @@ $(OUT)/ASP34.$(FASLEXT): $(OUT)/FMC.$(FASLEXT)
 
 $(OUT)/IRFORM.$(FASLEXT): $(OUT)/SYNTAX.$(FASLEXT)
 $(OUT)/COMPILER.$(FASLEXT): $(OUT)/SYNTAX.$(FASLEXT) $(OUT)/ENV.$(FASLEXT)
+$(OUT)/ITFORM.$(FASLEXT): $(OUT)/IRFORM.$(FASLEXT)
+$(OUT)/ELABOR.$(FASLEXT): $(OUT)/ITFORM.$(FASLEXT) $(OUT)/ENV.$(FASLEXT) \
+	$(OUT)/IRFORM.$(FASLEXT)
 
 
 .PHONY: all all-algebra mkdir-output-directory
diff --git a/src/algebra/Makefile.pamphlet b/src/algebra/Makefile.pamphlet
index 60a96eca..ddbb379c 100644
--- a/src/algebra/Makefile.pamphlet
+++ b/src/algebra/Makefile.pamphlet
@@ -1018,7 +1018,8 @@ axiom_algebra_layer_user =  \
 	ASP73    ASP27    ASP28    ASP33    ASP49    ASP7     \
         ASP78    ASP9     ASP12    ASP55    ASP8     ASP19    \
         ASP20    ASP30    ASP31    ASP35    ASP41    ASP42    \
-	ASP74    ASP77    ASP80    ASP29    IRFORM   COMPILER
+	ASP74    ASP77    ASP80    ASP29    IRFORM   COMPILER \
+	ITFORM   ELABOR
 
 axiom_algebra_layer_user_nrlibs = \
 	$(addsuffix .NRLIB/code.$(FASLEXT),$(axiom_algebra_layer_user))
@@ -1094,6 +1095,9 @@ $(OUT)/ASP34.$(FASLEXT): $(OUT)/FMC.$(FASLEXT)
 
 $(OUT)/IRFORM.$(FASLEXT): $(OUT)/SYNTAX.$(FASLEXT)
 $(OUT)/COMPILER.$(FASLEXT): $(OUT)/SYNTAX.$(FASLEXT) $(OUT)/ENV.$(FASLEXT)
+$(OUT)/ITFORM.$(FASLEXT): $(OUT)/IRFORM.$(FASLEXT)
+$(OUT)/ELABOR.$(FASLEXT): $(OUT)/ITFORM.$(FASLEXT) $(OUT)/ENV.$(FASLEXT) \
+	$(OUT)/IRFORM.$(FASLEXT)
 
 @
 
diff --git a/src/algebra/compiler.spad.pamphlet b/src/algebra/compiler.spad.pamphlet
index 00da0a09..24d0b6d8 100644
--- a/src/algebra/compiler.spad.pamphlet
+++ b/src/algebra/compiler.spad.pamphlet
@@ -23,12 +23,53 @@
 ++   This domain provides representations for the intermediate
 ++   form data structure used by the Spad elaborator.
 InternalRepresentationForm(): Public == Private where
-  Public == CoercibleTo OutputForm
+  Public == Join(CoercibleTo OutputForm, HomotopicTo Syntax)
   Private == add
+    coerce(x: %): Syntax == x : Syntax
+    coerce(x: Syntax): % == x : %
     coerce(x: %): OutputForm ==
       (x : Syntax)::OutputForm
+
+@
+
+<<domain ITFORM InternalTypeForm>>=
+)abbrev domain ITFORM InternalTypeForm
+++ Author: Gabriel Dos Reis
+++ Date Created: March 12, 2010
+++ Date Last Modified: March 12, 2010
+++ Description:
+++   This domain provides representations for internal type form.
+InternalTypeForm() == InternalRepresentationForm
+
 @
 
+\section{Elaboration domain}
+<<domain ELABOR Elaboration>>=
+)abbrev domain ELABOR Elaboration
+Elaboration(): Public == Private where
+  Public == CoercibleTo OutputForm with
+    elaboration: (InternalRepresentationForm, InternalTypeForm, Environment) -> %
+      ++ \spad{elaboration(ir,ty,env)} construct an elaboration object for
+      ++ for the internal representation form \spad{ir}, with type \spad{ty},
+      ++ and environment \spad{env}.
+    irForm: % -> InternalRepresentationForm
+      ++ \spad{irForm(x)} returns the internal representation form of
+      ++ the elaboration \spad{x}.
+    typeForm: % -> InternalTypeForm
+      ++ \spad{typeForm(x)} returns the type form of the elaboration \spad{x}.
+    environment: % -> Environment
+      ++ \spad{environment(x)} returns the environment of the
+      ++ elaboration \spad{x}.
+  Private == add
+    Rep == Record(ir: InternalRepresentationForm,
+              type: InternalTypeForm, env: Environment)
+    irForm x == rep(x).ir
+    typeForm x == rep(x).type
+    environment x == rep(x).env
+    coerce(x: %): OutputForm ==
+      bracket([irForm(x)::OutputForm, typeForm(x)::OutputForm,
+                  environment(x)::OutputForm])$OutputForm
+@
 
 \section{A Package for the Spad Compiler}
 
@@ -45,10 +86,14 @@ CompilerPackage(): Public == Private where
        ++ \spad{macroExpand(s,e)} traverses the syntax object \spad{s}
        ++ replacing all (niladic) macro invokations with the
        ++ corresponding substitution.
+     elaborate: Syntax -> Elaboration
+       ++ \spad{elaborate(s)} returns the elaboration of the syntax
+       ++ object \spad{s} in the empty environement.
   Private == add
      macroExpand(s,e) ==
        -- FIXME: this is a short-term stopgap.
        macroExpand(s,e)$Foreign(Builtin)
+
 @
 
 
diff --git a/src/share/algebra/browse.daase b/src/share/algebra/browse.daase
index e000765e..8e837cdc 100644
--- a/src/share/algebra/browse.daase
+++ b/src/share/algebra/browse.daase
@@ -1,12 +1,12 @@
 
-(2263909 . 3477420786)       
+(2264819 . 3477425184)       
 (-18 A S) 
 ((|constructor| (NIL "One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \\spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently grow or shrink on demand as can list structures (see however \\spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result."))) 
 NIL 
 NIL 
 (-19 S) 
 ((|constructor| (NIL "One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \\spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently grow or shrink on demand as can list structures (see however \\spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
 (-20 S) 
 ((|constructor| (NIL "The class of abelian groups,{} \\spadignore{i.e.} additive monoids where each element has an additive inverse. \\blankline")) (- (($ $ $) "\\spad{x-y} is the difference of \\spad{x} and \\spad{y} \\spadignore{i.e.} \\spad{x + (-y)}.") (($ $) "\\spad{-x} is the additive inverse of \\spad{x}"))) 
@@ -38,7 +38,7 @@ NIL
 NIL 
 (-27) 
 ((|constructor| (NIL "Model for algebraically closed fields.")) (|zerosOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{zerosOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) (|Polynomial| $)) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible. Otherwise they are implicit algebraic quantities. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|zeroOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{zeroOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}; if possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity which displays as \\spad{'y}.") (($ (|SparseUnivariatePolynomial| $)) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}; if possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity.") (($ (|Polynomial| $)) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. If possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootsOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note: the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\spad{rootsOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. Note: the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) (|Polynomial| $)) "\\spad{rootsOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. Note: the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{rootOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ (|SparseUnivariatePolynomial| $)) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}.") (($ (|Polynomial| $)) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-28 S R) 
 ((|constructor| (NIL "Model for algebraically closed function spaces.")) (|zerosOf| (((|List| $) $ (|Symbol|)) "\\spad{zerosOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable.")) (|zeroOf| (($ $ (|Symbol|)) "\\spad{zeroOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \\spad{'y}.") (($ $) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \\spad{p} has more than one variable.")) (|rootsOf| (((|List| $) $ (|Symbol|)) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note: the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; Note: the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ $ (|Symbol|)) "\\spad{rootOf(p,y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ $) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) 
@@ -46,7 +46,7 @@ NIL
 NIL 
 (-29 R) 
 ((|constructor| (NIL "Model for algebraically closed function spaces.")) (|zerosOf| (((|List| $) $ (|Symbol|)) "\\spad{zerosOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}\\spad{'s} are expressed in radicals if possible. The returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable.")) (|zeroOf| (($ $ (|Symbol|)) "\\spad{zeroOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \\spad{'y}.") (($ $) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \\spad{p} has more than one variable.")) (|rootsOf| (((|List| $) $ (|Symbol|)) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note: the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; Note: the returned symbols \\spad{y1},{}...,{}\\spad{yn} are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ $ (|Symbol|)) "\\spad{rootOf(p,y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ $) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) 
-((-4424 . T) (-4422 . T) (-4421 . T) ((-4429 "*") . T) (-4420 . T) (-4425 . T) (-4419 . T)) 
+((-4431 . T) (-4429 . T) (-4428 . T) ((-4436 "*") . T) (-4427 . T) (-4432 . T) (-4426 . T)) 
 NIL 
 (-30) 
 ((|constructor| (NIL "\\indented{1}{Plot a NON-SINGULAR plane algebraic curve \\spad{p}(\\spad{x},{}\\spad{y}) = 0.} Author: Clifton \\spad{J}. Williamson Date Created: Fall 1988 Date Last Updated: 27 April 1990 Keywords: algebraic curve,{} non-singular,{} plot Examples: References:")) (|refine| (($ $ (|DoubleFloat|)) "\\spad{refine(p,x)} \\undocumented{}")) (|makeSketch| (($ (|Polynomial| (|Integer|)) (|Symbol|) (|Symbol|) (|Segment| (|Fraction| (|Integer|))) (|Segment| (|Fraction| (|Integer|)))) "\\spad{makeSketch(p,x,y,a..b,c..d)} creates an ACPLOT of the curve \\spad{p = 0} in the region {\\em a <= x <= b, c <= y <= d}. More specifically,{} 'makeSketch' plots a non-singular algebraic curve \\spad{p = 0} in an rectangular region {\\em xMin <= x <= xMax},{} {\\em yMin <= y <= yMax}. The user inputs \\spad{makeSketch(p,x,y,xMin..xMax,yMin..yMax)}. Here \\spad{p} is a polynomial in the variables \\spad{x} and \\spad{y} with integer coefficients (\\spad{p} belongs to the domain \\spad{Polynomial Integer}). The case where \\spad{p} is a polynomial in only one of the variables is allowed. The variables \\spad{x} and \\spad{y} are input to specify the the coordinate axes. The horizontal axis is the \\spad{x}-axis and the vertical axis is the \\spad{y}-axis. The rational numbers xMin,{}...,{}yMax specify the boundaries of the region in which the curve is to be plotted."))) 
@@ -56,14 +56,14 @@ NIL
 ((|constructor| (NIL "This domain represents the syntax for an add-expression.")) (|body| (((|SpadAst|) $) "base(\\spad{d}) returns the actual body of the add-domain expression \\spad{`d'}.")) (|base| (((|SpadAst|) $) "\\spad{base(d)} returns the base domain(\\spad{s}) of the add-domain expression."))) 
 NIL 
 NIL 
-(-32 R -3498) 
+(-32 R -3505) 
 ((|constructor| (NIL "This package provides algebraic functions over an integral domain.")) (|iroot| ((|#2| |#1| (|Integer|)) "\\spad{iroot(p, n)} should be a non-exported function.")) (|definingPolynomial| ((|#2| |#2|) "\\spad{definingPolynomial(f)} returns the defining polynomial of \\spad{f} as an element of \\spad{F}. Error: if \\spad{f} is not a kernel.")) (|minPoly| (((|SparseUnivariatePolynomial| |#2|) (|Kernel| |#2|)) "\\spad{minPoly(k)} returns the defining polynomial of \\spad{k}.")) (** ((|#2| |#2| (|Fraction| (|Integer|))) "\\spad{x ** q} is \\spad{x} raised to the rational power \\spad{q}.")) (|droot| (((|OutputForm|) (|List| |#2|)) "\\spad{droot(l)} should be a non-exported function.")) (|inrootof| ((|#2| (|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{inrootof(p, x)} should be a non-exported function.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}. Error: if \\spad{op} is not an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|rootOf| ((|#2| (|SparseUnivariatePolynomial| |#2|) (|Symbol|)) "\\spad{rootOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}."))) 
 NIL 
-((|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550))))) 
+((|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551))))) 
 (-33 S) 
 ((|constructor| (NIL "The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as \"the set of \\spad{x} satisfying relation {\\em r(x)}\" An attribute \\spadatt{finiteAggregate} is used to assert that a domain has a finite number of elements.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# u} returns the number of items in \\spad{u}.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{size?(u,n)} tests if \\spad{u} has exactly \\spad{n} elements.")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{more?(u,n)} tests if \\spad{u} has greater than \\spad{n} elements.")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{less?(u,n)} tests if \\spad{u} has less than \\spad{n} elements.")) (|empty?| (((|Boolean|) $) "\\spad{empty?(u)} tests if \\spad{u} has 0 elements.")) (|empty| (($) "\\spad{empty()}\\$\\spad{D} creates an aggregate of type \\spad{D} with 0 elements. Note: The {\\em \\$D} can be dropped if understood by context,{} \\spadignore{e.g.} \\axiom{u: \\spad{D} \\spad{:=} empty()}.")) (|copy| (($ $) "\\spad{copy(u)} returns a top-level (non-recursive) copy of \\spad{u}. Note: for collections,{} \\axiom{copy(\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u}]}.")) (|eq?| (((|Boolean|) $ $) "\\spad{eq?(u,v)} tests if \\spad{u} and \\spad{v} are same objects."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4427))) 
+((|HasAttribute| |#1| (QUOTE -4434))) 
 (-34) 
 ((|constructor| (NIL "The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as \"the set of \\spad{x} satisfying relation {\\em r(x)}\" An attribute \\spadatt{finiteAggregate} is used to assert that a domain has a finite number of elements.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# u} returns the number of items in \\spad{u}.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{size?(u,n)} tests if \\spad{u} has exactly \\spad{n} elements.")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{more?(u,n)} tests if \\spad{u} has greater than \\spad{n} elements.")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{less?(u,n)} tests if \\spad{u} has less than \\spad{n} elements.")) (|empty?| (((|Boolean|) $) "\\spad{empty?(u)} tests if \\spad{u} has 0 elements.")) (|empty| (($) "\\spad{empty()}\\$\\spad{D} creates an aggregate of type \\spad{D} with 0 elements. Note: The {\\em \\$D} can be dropped if understood by context,{} \\spadignore{e.g.} \\axiom{u: \\spad{D} \\spad{:=} empty()}.")) (|copy| (($ $) "\\spad{copy(u)} returns a top-level (non-recursive) copy of \\spad{u}. Note: for collections,{} \\axiom{copy(\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u}]}.")) (|eq?| (((|Boolean|) $ $) "\\spad{eq?(u,v)} tests if \\spad{u} and \\spad{v} are same objects."))) 
 NIL 
@@ -74,7 +74,7 @@ NIL
 NIL 
 (-36 |Key| |Entry|) 
 ((|constructor| (NIL "An association list is a list of key entry pairs which may be viewed as a table. It is a poor mans version of a table: searching for a key is a linear operation.")) (|assoc| (((|Union| (|Record| (|:| |key| |#1|) (|:| |entry| |#2|)) "failed") |#1| $) "\\spad{assoc(k,u)} returns the element \\spad{x} in association list \\spad{u} stored with key \\spad{k},{} or \"failed\" if \\spad{u} has no key \\spad{k}."))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
 (-37 S R) 
 ((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline"))) 
@@ -82,20 +82,20 @@ NIL
 NIL 
 (-38 R) 
 ((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-39 UP) 
 ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in \\spadtype{AlgebraicNumber}.")) (|doublyTransitive?| (((|Boolean|) |#1|) "\\spad{doublyTransitive?(p)} is \\spad{true} if \\spad{p} is irreducible over over the field \\spad{K} generated by its coefficients,{} and if \\spad{p(X) / (X - a)} is irreducible over \\spad{K(a)} where \\spad{p(a) = 0}.")) (|split| (((|Factored| |#1|) |#1|) "\\spad{split(p)} returns a prime factorisation of \\spad{p} over its splitting field.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p} over the field generated by its coefficients.") (((|Factored| |#1|) |#1| (|List| (|AlgebraicNumber|))) "\\spad{factor(p, [a1,...,an])} returns a prime factorisation of \\spad{p} over the field generated by its coefficients and a1,{}...,{}an."))) 
 NIL 
 NIL 
-(-40 -3498 UP UPUP -3016) 
+(-40 -3505 UP UPUP -3023) 
 ((|constructor| (NIL "Function field defined by \\spad{f}(\\spad{x},{} \\spad{y}) = 0.")) (|knownInfBasis| (((|Void|) (|NonNegativeInteger|)) "\\spad{knownInfBasis(n)} \\undocumented{}"))) 
-((-4420 |has| (-411 |#2|) (-366)) (-4425 |has| (-411 |#2|) (-366)) (-4419 |has| (-411 |#2|) (-366)) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-411 |#2|) (QUOTE (-145))) (|HasCategory| (-411 |#2|) (QUOTE (-147))) (|HasCategory| (-411 |#2|) (QUOTE (-353))) (-3962 (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (QUOTE (-353)))) (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (QUOTE (-371))) (-3962 (-12 (|HasCategory| (-411 |#2|) (QUOTE (-234))) (|HasCategory| (-411 |#2|) (QUOTE (-366)))) (|HasCategory| (-411 |#2|) (QUOTE (-353)))) (-3962 (-12 (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| (-411 |#2|) (QUOTE (-353))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -904) (QUOTE (-1181)))))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -642) (QUOTE (-550)))) (-3962 (|HasCategory| (-411 |#2|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-411 |#2|) (QUOTE (-366)))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-371))) (-12 (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| (-411 |#2|) (QUOTE (-234))) (|HasCategory| (-411 |#2|) (QUOTE (-366))))) 
-(-41 R -3498) 
+((-4427 |has| (-412 |#2|) (-367)) (-4432 |has| (-412 |#2|) (-367)) (-4426 |has| (-412 |#2|) (-367)) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-412 |#2|) (QUOTE (-145))) (|HasCategory| (-412 |#2|) (QUOTE (-147))) (|HasCategory| (-412 |#2|) (QUOTE (-354))) (-3969 (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (QUOTE (-354)))) (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (QUOTE (-372))) (-3969 (-12 (|HasCategory| (-412 |#2|) (QUOTE (-234))) (|HasCategory| (-412 |#2|) (QUOTE (-367)))) (|HasCategory| (-412 |#2|) (QUOTE (-354)))) (-3969 (-12 (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -906) (QUOTE (-1183))))) (-12 (|HasCategory| (-412 |#2|) (QUOTE (-354))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -906) (QUOTE (-1183)))))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -644) (QUOTE (-551)))) (-3969 (|HasCategory| (-412 |#2|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-412 |#2|) (QUOTE (-367)))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-372))) (-12 (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -906) (QUOTE (-1183))))) (-12 (|HasCategory| (-412 |#2|) (QUOTE (-234))) (|HasCategory| (-412 |#2|) (QUOTE (-367))))) 
+(-41 R -3505) 
 ((|constructor| (NIL "AlgebraicManipulations provides functions to simplify and expand expressions involving algebraic operators.")) (|rootKerSimp| ((|#2| (|BasicOperator|) |#2| (|NonNegativeInteger|)) "\\spad{rootKerSimp(op,f,n)} should be local but conditional.")) (|rootSimp| ((|#2| |#2|) "\\spad{rootSimp(f)} transforms every radical of the form \\spad{(a * b**(q*n+r))**(1/n)} appearing in \\spad{f} into \\spad{b**q * (a * b**r)**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{b}.")) (|rootProduct| ((|#2| |#2|) "\\spad{rootProduct(f)} combines every product of the form \\spad{(a**(1/n))**m * (a**(1/s))**t} into a single power of a root of \\spad{a},{} and transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form.")) (|rootPower| ((|#2| |#2|) "\\spad{rootPower(f)} transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form if \\spad{m} and \\spad{n} have a common factor.")) (|ratPoly| (((|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{ratPoly(f)} returns a polynomial \\spad{p} such that \\spad{p} has no algebraic coefficients,{} and \\spad{p(f) = 0}.")) (|ratDenom| ((|#2| |#2| (|List| (|Kernel| |#2|))) "\\spad{ratDenom(f, [a1,...,an])} removes the \\spad{ai}\\spad{'s} which are algebraic from the denominators in \\spad{f}.") ((|#2| |#2| (|List| |#2|)) "\\spad{ratDenom(f, [a1,...,an])} removes the \\spad{ai}\\spad{'s} which are algebraic kernels from the denominators in \\spad{f}.") ((|#2| |#2| |#2|) "\\spad{ratDenom(f, a)} removes \\spad{a} from the denominators in \\spad{f} if \\spad{a} is an algebraic kernel.") ((|#2| |#2|) "\\spad{ratDenom(f)} rationalizes the denominators appearing in \\spad{f} by moving all the algebraic quantities into the numerators.")) (|rootSplit| ((|#2| |#2|) "\\spad{rootSplit(f)} transforms every radical of the form \\spad{(a/b)**(1/n)} appearing in \\spad{f} into \\spad{a**(1/n) / b**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{a} and \\spad{b}.")) (|coerce| (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{coerce(x)} \\undocumented")) (|denom| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{denom(x)} \\undocumented")) (|numer| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{numer(x)} \\undocumented"))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -425) (|devaluate| |#1|))))) 
+((-12 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -426) (|devaluate| |#1|))))) 
 (-42 OV E P) 
 ((|constructor| (NIL "This package factors multivariate polynomials over the domain of \\spadtype{AlgebraicNumber} by allowing the user to specify a list of algebraic numbers generating the particular extension to factor over.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#3|)) (|SparseUnivariatePolynomial| |#3|) (|List| (|AlgebraicNumber|))) "\\spad{factor(p,lan)} factors the polynomial \\spad{p} over the extension generated by the algebraic numbers given by the list \\spad{lan}. \\spad{p} is presented as a univariate polynomial with multivariate coefficients.") (((|Factored| |#3|) |#3| (|List| (|AlgebraicNumber|))) "\\spad{factor(p,lan)} factors the polynomial \\spad{p} over the extension generated by the algebraic numbers given by the list \\spad{lan}."))) 
 NIL 
@@ -103,34 +103,34 @@ NIL
 (-43 R A) 
 ((|constructor| (NIL "AlgebraPackage assembles a variety of useful functions for general algebras.")) (|basis| (((|Vector| |#2|) (|Vector| |#2|)) "\\spad{basis(va)} selects a basis from the elements of \\spad{va}.")) (|radicalOfLeftTraceForm| (((|List| |#2|)) "\\spad{radicalOfLeftTraceForm()} returns basis for null space of \\spad{leftTraceMatrix()},{} if the algebra is associative,{} alternative or a Jordan algebra,{} then this space equals the radical (maximal nil ideal) of the algebra.")) (|basisOfCentroid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfCentroid()} returns a basis of the centroid,{} \\spadignore{i.e.} the endomorphism ring of \\spad{A} considered as \\spad{(A,A)}-bimodule.")) (|basisOfRightNucloid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfRightNucloid()} returns a basis of the space of endomorphisms of \\spad{A} as left module. Note: right nucloid coincides with right nucleus if \\spad{A} has a unit.")) (|basisOfLeftNucloid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfLeftNucloid()} returns a basis of the space of endomorphisms of \\spad{A} as right module. Note: left nucloid coincides with left nucleus if \\spad{A} has a unit.")) (|basisOfCenter| (((|List| |#2|)) "\\spad{basisOfCenter()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{commutator(x,a) = 0} and \\spad{associator(x,a,b) = associator(a,x,b) = associator(a,b,x) = 0} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfNucleus| (((|List| |#2|)) "\\spad{basisOfNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{associator(x,a,b) = associator(a,x,b) = associator(a,b,x) = 0} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfMiddleNucleus| (((|List| |#2|)) "\\spad{basisOfMiddleNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(a,x,b)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfRightNucleus| (((|List| |#2|)) "\\spad{basisOfRightNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(a,b,x)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfLeftNucleus| (((|List| |#2|)) "\\spad{basisOfLeftNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(x,a,b)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfRightAnnihilator| (((|List| |#2|) |#2|) "\\spad{basisOfRightAnnihilator(a)} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = a*x}.")) (|basisOfLeftAnnihilator| (((|List| |#2|) |#2|) "\\spad{basisOfLeftAnnihilator(a)} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = x*a}.")) (|basisOfCommutingElements| (((|List| |#2|)) "\\spad{basisOfCommutingElements()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = commutator(x,a)} for all \\spad{a} in \\spad{A}.")) (|biRank| (((|NonNegativeInteger|) |#2|) "\\spad{biRank(x)} determines the number of linearly independent elements in \\spad{x},{} \\spad{x*bi},{} \\spad{bi*x},{} \\spad{bi*x*bj},{} \\spad{i,j=1,...,n},{} where \\spad{b=[b1,...,bn]} is a basis. Note: if \\spad{A} has a unit,{} then \\spadfunFrom{doubleRank}{AlgebraPackage},{} \\spadfunFrom{weakBiRank}{AlgebraPackage} and \\spadfunFrom{biRank}{AlgebraPackage} coincide.")) (|weakBiRank| (((|NonNegativeInteger|) |#2|) "\\spad{weakBiRank(x)} determines the number of linearly independent elements in the \\spad{bi*x*bj},{} \\spad{i,j=1,...,n},{} where \\spad{b=[b1,...,bn]} is a basis.")) (|doubleRank| (((|NonNegativeInteger|) |#2|) "\\spad{doubleRank(x)} determines the number of linearly independent elements in \\spad{b1*x},{}...,{}\\spad{x*bn},{} where \\spad{b=[b1,...,bn]} is a basis.")) (|rightRank| (((|NonNegativeInteger|) |#2|) "\\spad{rightRank(x)} determines the number of linearly independent elements in \\spad{b1*x},{}...,{}\\spad{bn*x},{} where \\spad{b=[b1,...,bn]} is a basis.")) (|leftRank| (((|NonNegativeInteger|) |#2|) "\\spad{leftRank(x)} determines the number of linearly independent elements in \\spad{x*b1},{}...,{}\\spad{x*bn},{} where \\spad{b=[b1,...,bn]} is a basis."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-309)))) 
+((|HasCategory| |#1| (QUOTE (-310)))) 
 (-44 R |n| |ls| |gamma|) 
 ((|constructor| (NIL "AlgebraGivenByStructuralConstants implements finite rank algebras over a commutative ring,{} given by the structural constants \\spad{gamma} with respect to a fixed basis \\spad{[a1,..,an]},{} where \\spad{gamma} is an \\spad{n}-vector of \\spad{n} by \\spad{n} matrices \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{ai * aj = gammaij1 * a1 + ... + gammaijn * an}. The symbols for the fixed basis have to be given as a list of symbols.")) (|coerce| (($ (|Vector| |#1|)) "\\spad{coerce(v)} converts a vector to a member of the algebra by forming a linear combination with the basis element. Note: the vector is assumed to have length equal to the dimension of the algebra."))) 
-((-4424 |has| |#1| (-561)) (-4422 . T) (-4421 . T)) 
-((|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) 
+((-4431 |has| |#1| (-562)) (-4429 . T) (-4428 . T)) 
+((|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) 
 (-45 |Key| |Entry|) 
 ((|constructor| (NIL "\\spadtype{AssociationList} implements association lists. These may be viewed as lists of pairs where the first part is a key and the second is the stored value. For example,{} the key might be a string with a persons employee identification number and the value might be a record with personnel data."))) 
-((-4427 . T) (-4428 . T)) 
-((-3962 (-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-853)))) (-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105))))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-853))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-3962 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-853))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (-3962 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105))))) 
+((-4434 . T) (-4435 . T)) 
+((-3969 (-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-855)))) (-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107))))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-855))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-3969 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-855))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) (-3969 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107))))) 
 (-46 S R E) 
 ((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#2|) "\\spad{p/c} divides \\spad{p} by the coefficient \\spad{c}.")) (|coefficient| ((|#2| $ |#3|) "\\spad{coefficient(p,e)} extracts the coefficient of the monomial with exponent \\spad{e} from polynomial \\spad{p},{} or returns zero if exponent is not present.")) (|reductum| (($ $) "\\spad{reductum(u)} returns \\spad{u} minus its leading monomial returns zero if handed the zero element.")) (|monomial| (($ |#2| |#3|) "\\spad{monomial(r,e)} makes a term from a coefficient \\spad{r} and an exponent \\spad{e}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(p)} tests if \\spad{p} is a single monomial.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|degree| ((|#3| $) "\\spad{degree(p)} returns the maximum of the exponents of the terms of \\spad{p}.")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(p)} returns the monomial of \\spad{p} with the highest degree.")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-366)))) 
+((|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-367)))) 
 (-47 R E) 
 ((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#1|) "\\spad{p/c} divides \\spad{p} by the coefficient \\spad{c}.")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(p,e)} extracts the coefficient of the monomial with exponent \\spad{e} from polynomial \\spad{p},{} or returns zero if exponent is not present.")) (|reductum| (($ $) "\\spad{reductum(u)} returns \\spad{u} minus its leading monomial returns zero if handed the zero element.")) (|monomial| (($ |#1| |#2|) "\\spad{monomial(r,e)} makes a term from a coefficient \\spad{r} and an exponent \\spad{e}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(p)} tests if \\spad{p} is a single monomial.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|degree| ((|#2| $) "\\spad{degree(p)} returns the maximum of the exponents of the terms of \\spad{p}.")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(p)} returns the monomial of \\spad{p} with the highest degree.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-48) 
 ((|constructor| (NIL "Algebraic closure of the rational numbers,{} with mathematical =")) (|norm| (($ $ (|List| (|Kernel| $))) "\\spad{norm(f,l)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernels \\spad{l}") (($ $ (|Kernel| $)) "\\spad{norm(f,k)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernel \\spad{k}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|List| (|Kernel| $))) "\\spad{norm(p,l)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernels \\spad{l}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{norm(p,k)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernel \\spad{k}")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \\spad{f} by applying their defining relations.")) (|denom| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|numer| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|coerce| (($ (|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} viewed as an algebraic number."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| $ (QUOTE (-1053))) (|HasCategory| $ (LIST (QUOTE -1042) (QUOTE (-550))))) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| $ (QUOTE (-1055))) (|HasCategory| $ (LIST (QUOTE -1044) (QUOTE (-551))))) 
 (-49) 
 ((|constructor| (NIL "This domain implements anonymous functions")) (|body| (((|Syntax|) $) "\\spad{body(f)} returns the body of the unnamed function \\spad{`f'}.")) (|parameters| (((|List| (|Identifier|)) $) "\\spad{parameters(f)} returns the list of parameters bound by \\spad{`f'}."))) 
 NIL 
 NIL 
 (-50 R |lVar|) 
 ((|constructor| (NIL "The domain of antisymmetric polynomials.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,p)} changes each coefficient of \\spad{p} by the application of \\spad{f}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} returns the homogeneous degree of \\spad{p}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?(p)} tests if \\spad{p} is a 0-form,{} \\spadignore{i.e.} if degree(\\spad{p}) = 0.")) (|homogeneous?| (((|Boolean|) $) "\\spad{homogeneous?(p)} tests if all of the terms of \\spad{p} have the same degree.")) (|exp| (($ (|List| (|Integer|))) "\\spad{exp([i1,...in])} returns \\spad{u_1\\^{i_1} ... u_n\\^{i_n}}")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(n)} returns the \\spad{n}th multiplicative generator,{} a basis term.")) (|coefficient| ((|#1| $ $) "\\spad{coefficient(p,u)} returns the coefficient of the term in \\spad{p} containing the basis term \\spad{u} if such a term exists,{} and 0 otherwise. Error: if the second argument \\spad{u} is not a basis element.")) (|reductum| (($ $) "\\spad{reductum(p)},{} where \\spad{p} is an antisymmetric polynomial,{} returns \\spad{p} minus the leading term of \\spad{p} if \\spad{p} has at least two terms,{} and 0 otherwise.")) (|leadingBasisTerm| (($ $) "\\spad{leadingBasisTerm(p)} returns the leading basis term of antisymmetric polynomial \\spad{p}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(p)} returns the leading coefficient of antisymmetric polynomial \\spad{p}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
 (-51) 
 ((|constructor| (NIL "\\spadtype{Any} implements a type that packages up objects and their types in objects of \\spadtype{Any}. Roughly speaking that means that if \\spad{s : S} then when converted to \\spadtype{Any},{} the new object will include both the original object and its type. This is a way of converting arbitrary objects into a single type without losing any of the original information. Any object can be converted to one of \\spadtype{Any}. The original object can be recovered by `is-case' pattern matching as exemplified here and AnyFunctions1.")) (|obj| (((|None|) $) "\\spad{obj(a)} essentially returns the original object that was converted to \\spadtype{Any} except that the type is forced to be \\spadtype{None}.")) (|dom| (((|SExpression|) $) "\\spad{dom(a)} returns a \\spadgloss{LISP} form of the type of the original object that was converted to \\spadtype{Any}.")) (|any| (($ (|SExpression|) (|None|)) "\\spad{any(type,object)} is a technical function for creating an \\spad{object} of \\spadtype{Any}. Arugment \\spad{type} is a \\spadgloss{LISP} form for the \\spad{type} of \\spad{object}."))) 
@@ -144,7 +144,7 @@ NIL
 ((|constructor| (NIL "\\spad{ApplyUnivariateSkewPolynomial} (internal) allows univariate skew polynomials to be applied to appropriate modules.")) (|apply| ((|#2| |#3| (|Mapping| |#2| |#2|) |#2|) "\\spad{apply(p, f, m)} returns \\spad{p(m)} where the action is given by \\spad{x m = f(m)}. \\spad{f} must be an \\spad{R}-pseudo linear map on \\spad{M}."))) 
 NIL 
 NIL 
-(-54 |Base| R -3498) 
+(-54 |Base| R -3505) 
 ((|constructor| (NIL "This package apply rewrite rules to expressions,{} calling the pattern matcher.")) (|localUnquote| ((|#3| |#3| (|List| (|Symbol|))) "\\spad{localUnquote(f,ls)} is a local function.")) (|applyRules| ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3| (|PositiveInteger|)) "\\spad{applyRules([r1,...,rn], expr, n)} applies the rules \\spad{r1},{}...,{}\\spad{rn} to \\spad{f} a most \\spad{n} times.") ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3|) "\\spad{applyRules([r1,...,rn], expr)} applies the rules \\spad{r1},{}...,{}\\spad{rn} to \\spad{f} an unlimited number of times,{} \\spadignore{i.e.} until none of \\spad{r1},{}...,{}\\spad{rn} is applicable to the expression."))) 
 NIL 
 NIL 
@@ -158,77 +158,77 @@ NIL
 NIL 
 (-57 R |Row| |Col|) 
 ((|constructor| (NIL "\\indented{1}{TwoDimensionalArrayCategory is a general array category which} allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and columns returned as objects of type Col. The index of the 'first' row may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.")) (|map!| (($ (|Mapping| |#1| |#1|) $) "\\spad{map!(f,a)} assign \\spad{a(i,j)} to \\spad{f(a(i,j))} for all \\spad{i, j}")) (|map| (($ (|Mapping| |#1| |#1| |#1|) $ $ |#1|) "\\spad{map(f,a,b,r)} returns \\spad{c},{} where \\spad{c(i,j) = f(a(i,j),b(i,j))} when both \\spad{a(i,j)} and \\spad{b(i,j)} exist; else \\spad{c(i,j) = f(r, b(i,j))} when \\spad{a(i,j)} does not exist; else \\spad{c(i,j) = f(a(i,j),r)} when \\spad{b(i,j)} does not exist; otherwise \\spad{c(i,j) = f(r,r)}.") (($ (|Mapping| |#1| |#1| |#1|) $ $) "\\spad{map(f,a,b)} returns \\spad{c},{} where \\spad{c(i,j) = f(a(i,j),b(i,j))} for all \\spad{i, j}") (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,a)} returns \\spad{b},{} where \\spad{b(i,j) = f(a(i,j))} for all \\spad{i, j}")) (|setColumn!| (($ $ (|Integer|) |#3|) "\\spad{setColumn!(m,j,v)} sets to \\spad{j}th column of \\spad{m} to \\spad{v}")) (|setRow!| (($ $ (|Integer|) |#2|) "\\spad{setRow!(m,i,v)} sets to \\spad{i}th row of \\spad{m} to \\spad{v}")) (|qsetelt!| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\spad{qsetelt!(m,i,j,r)} sets the element in the \\spad{i}th row and \\spad{j}th column of \\spad{m} to \\spad{r} NO error check to determine if indices are in proper ranges")) (|setelt| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\spad{setelt(m,i,j,r)} sets the element in the \\spad{i}th row and \\spad{j}th column of \\spad{m} to \\spad{r} error check to determine if indices are in proper ranges")) (|parts| (((|List| |#1|) $) "\\spad{parts(m)} returns a list of the elements of \\spad{m} in row major order")) (|column| ((|#3| $ (|Integer|)) "\\spad{column(m,j)} returns the \\spad{j}th column of \\spad{m} error check to determine if index is in proper ranges")) (|row| ((|#2| $ (|Integer|)) "\\spad{row(m,i)} returns the \\spad{i}th row of \\spad{m} error check to determine if index is in proper ranges")) (|qelt| ((|#1| $ (|Integer|) (|Integer|)) "\\spad{qelt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the array \\spad{m} NO error check to determine if indices are in proper ranges")) (|elt| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\spad{elt(m,i,j,r)} returns the element in the \\spad{i}th row and \\spad{j}th column of the array \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{} and returns \\spad{r} otherwise") ((|#1| $ (|Integer|) (|Integer|)) "\\spad{elt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the array \\spad{m} error check to determine if indices are in proper ranges")) (|ncols| (((|NonNegativeInteger|) $) "\\spad{ncols(m)} returns the number of columns in the array \\spad{m}")) (|nrows| (((|NonNegativeInteger|) $) "\\spad{nrows(m)} returns the number of rows in the array \\spad{m}")) (|maxColIndex| (((|Integer|) $) "\\spad{maxColIndex(m)} returns the index of the 'last' column of the array \\spad{m}")) (|minColIndex| (((|Integer|) $) "\\spad{minColIndex(m)} returns the index of the 'first' column of the array \\spad{m}")) (|maxRowIndex| (((|Integer|) $) "\\spad{maxRowIndex(m)} returns the index of the 'last' row of the array \\spad{m}")) (|minRowIndex| (((|Integer|) $) "\\spad{minRowIndex(m)} returns the index of the 'first' row of the array \\spad{m}")) (|fill!| (($ $ |#1|) "\\spad{fill!(m,r)} fills \\spad{m} with \\spad{r}\\spad{'s}")) (|new| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) |#1|) "\\spad{new(m,n,r)} is an \\spad{m}-by-\\spad{n} array all of whose entries are \\spad{r}")) (|finiteAggregate| ((|attribute|) "two-dimensional arrays are finite")) (|shallowlyMutable| ((|attribute|) "one may destructively alter arrays"))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
 (-58 S) 
 ((|constructor| (NIL "This is the domain of 1-based one dimensional arrays")) (|oneDimensionalArray| (($ (|NonNegativeInteger|) |#1|) "\\spad{oneDimensionalArray(n,s)} creates an array from \\spad{n} copies of element \\spad{s}") (($ (|List| |#1|)) "\\spad{oneDimensionalArray(l)} creates an array from a list of elements \\spad{l}"))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
 (-59 A B) 
 ((|constructor| (NIL "\\indented{1}{This package provides tools for operating on one-dimensional arrays} with unary and binary functions involving different underlying types")) (|map| (((|OneDimensionalArray| |#2|) (|Mapping| |#2| |#1|) (|OneDimensionalArray| |#1|)) "\\spad{map(f,a)} applies function \\spad{f} to each member of one-dimensional array \\spad{a} resulting in a new one-dimensional array over a possibly different underlying domain.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|OneDimensionalArray| |#1|) |#2|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the one-dimensional array \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as the identity element for the function \\spad{f}.")) (|scan| (((|OneDimensionalArray| |#2|) (|Mapping| |#2| |#1| |#2|) (|OneDimensionalArray| |#1|) |#2|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-arrays \\spad{x} of one-dimensional array \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}."))) 
 NIL 
 NIL 
 (-60 R) 
 ((|constructor| (NIL "\\indented{1}{A TwoDimensionalArray is a two dimensional array with} 1-based indexing for both rows and columns.")) (|shallowlyMutable| ((|attribute|) "One may destructively alter TwoDimensionalArray\\spad{'s}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-61 -3975) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-61 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp1} produces Fortran for Type 1 ASPs,{} needed for various NAG routines. Type 1 ASPs take a univariate expression (in the symbol \\spad{X}) and turn it into a Fortran Function like the following:\\begin{verbatim}      DOUBLE PRECISION FUNCTION F(X)      DOUBLE PRECISION X      F=DSIN(X)      RETURN      END\\end{verbatim}")) (|coerce| (($ (|FortranExpression| (|construct| (QUOTE X)) (|construct|) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
 NIL 
 NIL 
-(-62 -3975) 
+(-62 -3982) 
 ((|constructor| (NIL "\\spadtype{ASP10} produces Fortran for Type 10 ASPs,{} needed for NAG routine \\axiomOpFrom{d02kef}{d02Package}. This ASP computes the values of a set of functions,{} for example:\\begin{verbatim}      SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT)      DOUBLE PRECISION ELAM,P,Q,X,DQDL      INTEGER JINT      P=1.0D0      Q=((-1.0D0*X**3)+ELAM*X*X-2.0D0)/(X*X)      DQDL=1.0D0      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE JINT) (QUOTE X) (QUOTE ELAM)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-63 -3975) 
+(-63 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp12} produces Fortran for Type 12 ASPs,{} needed for NAG routine \\axiomOpFrom{d02kef}{d02Package} etc.,{} for example:\\begin{verbatim}      SUBROUTINE MONIT (MAXIT,IFLAG,ELAM,FINFO)      DOUBLE PRECISION ELAM,FINFO(15)      INTEGER MAXIT,IFLAG      IF(MAXIT.EQ.-1)THEN        PRINT*,\"Output from Monit\"      ENDIF      PRINT*,MAXIT,IFLAG,ELAM,(FINFO(I),I=1,4)      RETURN      END\\end{verbatim}")) (|outputAsFortran| (((|Void|)) "\\spad{outputAsFortran()} generates the default code for \\spadtype{ASP12}."))) 
 NIL 
 NIL 
-(-64 -3975) 
+(-64 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp19} produces Fortran for Type 19 ASPs,{} evaluating a set of functions and their jacobian at a given point,{} for example:\\begin{verbatim}      SUBROUTINE LSFUN2(M,N,XC,FVECC,FJACC,LJC)      DOUBLE PRECISION FVECC(M),FJACC(LJC,N),XC(N)      INTEGER M,N,LJC      INTEGER I,J      DO 25003 I=1,LJC        DO 25004 J=1,N          FJACC(I,J)=0.0D025004   CONTINUE25003 CONTINUE      FVECC(1)=((XC(1)-0.14D0)*XC(3)+(15.0D0*XC(1)-2.1D0)*XC(2)+1.0D0)/(     &XC(3)+15.0D0*XC(2))      FVECC(2)=((XC(1)-0.18D0)*XC(3)+(7.0D0*XC(1)-1.26D0)*XC(2)+1.0D0)/(     &XC(3)+7.0D0*XC(2))      FVECC(3)=((XC(1)-0.22D0)*XC(3)+(4.333333333333333D0*XC(1)-0.953333     &3333333333D0)*XC(2)+1.0D0)/(XC(3)+4.333333333333333D0*XC(2))      FVECC(4)=((XC(1)-0.25D0)*XC(3)+(3.0D0*XC(1)-0.75D0)*XC(2)+1.0D0)/(     &XC(3)+3.0D0*XC(2))      FVECC(5)=((XC(1)-0.29D0)*XC(3)+(2.2D0*XC(1)-0.6379999999999999D0)*     &XC(2)+1.0D0)/(XC(3)+2.2D0*XC(2))      FVECC(6)=((XC(1)-0.32D0)*XC(3)+(1.666666666666667D0*XC(1)-0.533333     &3333333333D0)*XC(2)+1.0D0)/(XC(3)+1.666666666666667D0*XC(2))      FVECC(7)=((XC(1)-0.35D0)*XC(3)+(1.285714285714286D0*XC(1)-0.45D0)*     &XC(2)+1.0D0)/(XC(3)+1.285714285714286D0*XC(2))      FVECC(8)=((XC(1)-0.39D0)*XC(3)+(XC(1)-0.39D0)*XC(2)+1.0D0)/(XC(3)+     &XC(2))      FVECC(9)=((XC(1)-0.37D0)*XC(3)+(XC(1)-0.37D0)*XC(2)+1.285714285714     &286D0)/(XC(3)+XC(2))      FVECC(10)=((XC(1)-0.58D0)*XC(3)+(XC(1)-0.58D0)*XC(2)+1.66666666666     &6667D0)/(XC(3)+XC(2))      FVECC(11)=((XC(1)-0.73D0)*XC(3)+(XC(1)-0.73D0)*XC(2)+2.2D0)/(XC(3)     &+XC(2))      FVECC(12)=((XC(1)-0.96D0)*XC(3)+(XC(1)-0.96D0)*XC(2)+3.0D0)/(XC(3)     &+XC(2))      FVECC(13)=((XC(1)-1.34D0)*XC(3)+(XC(1)-1.34D0)*XC(2)+4.33333333333     &3333D0)/(XC(3)+XC(2))      FVECC(14)=((XC(1)-2.1D0)*XC(3)+(XC(1)-2.1D0)*XC(2)+7.0D0)/(XC(3)+X     &C(2))      FVECC(15)=((XC(1)-4.39D0)*XC(3)+(XC(1)-4.39D0)*XC(2)+15.0D0)/(XC(3     &)+XC(2))      FJACC(1,1)=1.0D0      FJACC(1,2)=-15.0D0/(XC(3)**2+30.0D0*XC(2)*XC(3)+225.0D0*XC(2)**2)      FJACC(1,3)=-1.0D0/(XC(3)**2+30.0D0*XC(2)*XC(3)+225.0D0*XC(2)**2)      FJACC(2,1)=1.0D0      FJACC(2,2)=-7.0D0/(XC(3)**2+14.0D0*XC(2)*XC(3)+49.0D0*XC(2)**2)      FJACC(2,3)=-1.0D0/(XC(3)**2+14.0D0*XC(2)*XC(3)+49.0D0*XC(2)**2)      FJACC(3,1)=1.0D0      FJACC(3,2)=((-0.1110223024625157D-15*XC(3))-4.333333333333333D0)/(     &XC(3)**2+8.666666666666666D0*XC(2)*XC(3)+18.77777777777778D0*XC(2)     &**2)      FJACC(3,3)=(0.1110223024625157D-15*XC(2)-1.0D0)/(XC(3)**2+8.666666     &666666666D0*XC(2)*XC(3)+18.77777777777778D0*XC(2)**2)      FJACC(4,1)=1.0D0      FJACC(4,2)=-3.0D0/(XC(3)**2+6.0D0*XC(2)*XC(3)+9.0D0*XC(2)**2)      FJACC(4,3)=-1.0D0/(XC(3)**2+6.0D0*XC(2)*XC(3)+9.0D0*XC(2)**2)      FJACC(5,1)=1.0D0      FJACC(5,2)=((-0.1110223024625157D-15*XC(3))-2.2D0)/(XC(3)**2+4.399     &999999999999D0*XC(2)*XC(3)+4.839999999999998D0*XC(2)**2)      FJACC(5,3)=(0.1110223024625157D-15*XC(2)-1.0D0)/(XC(3)**2+4.399999     &999999999D0*XC(2)*XC(3)+4.839999999999998D0*XC(2)**2)      FJACC(6,1)=1.0D0      FJACC(6,2)=((-0.2220446049250313D-15*XC(3))-1.666666666666667D0)/(     &XC(3)**2+3.333333333333333D0*XC(2)*XC(3)+2.777777777777777D0*XC(2)     &**2)      FJACC(6,3)=(0.2220446049250313D-15*XC(2)-1.0D0)/(XC(3)**2+3.333333     &333333333D0*XC(2)*XC(3)+2.777777777777777D0*XC(2)**2)      FJACC(7,1)=1.0D0      FJACC(7,2)=((-0.5551115123125783D-16*XC(3))-1.285714285714286D0)/(     &XC(3)**2+2.571428571428571D0*XC(2)*XC(3)+1.653061224489796D0*XC(2)     &**2)      FJACC(7,3)=(0.5551115123125783D-16*XC(2)-1.0D0)/(XC(3)**2+2.571428     &571428571D0*XC(2)*XC(3)+1.653061224489796D0*XC(2)**2)      FJACC(8,1)=1.0D0      FJACC(8,2)=-1.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(8,3)=-1.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(9,1)=1.0D0      FJACC(9,2)=-1.285714285714286D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)*     &*2)      FJACC(9,3)=-1.285714285714286D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)*     &*2)      FJACC(10,1)=1.0D0      FJACC(10,2)=-1.666666666666667D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)     &**2)      FJACC(10,3)=-1.666666666666667D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)     &**2)      FJACC(11,1)=1.0D0      FJACC(11,2)=-2.2D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(11,3)=-2.2D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(12,1)=1.0D0      FJACC(12,2)=-3.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(12,3)=-3.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(13,1)=1.0D0      FJACC(13,2)=-4.333333333333333D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)     &**2)      FJACC(13,3)=-4.333333333333333D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)     &**2)      FJACC(14,1)=1.0D0      FJACC(14,2)=-7.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(14,3)=-7.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(15,1)=1.0D0      FJACC(15,2)=-15.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      FJACC(15,3)=-15.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE XC)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-65 -3975) 
+(-65 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp20} produces Fortran for Type 20 ASPs,{} for example:\\begin{verbatim}      SUBROUTINE QPHESS(N,NROWH,NCOLH,JTHCOL,HESS,X,HX)      DOUBLE PRECISION HX(N),X(N),HESS(NROWH,NCOLH)      INTEGER JTHCOL,N,NROWH,NCOLH      HX(1)=2.0D0*X(1)      HX(2)=2.0D0*X(2)      HX(3)=2.0D0*X(4)+2.0D0*X(3)      HX(4)=2.0D0*X(4)+2.0D0*X(3)      HX(5)=2.0D0*X(5)      HX(6)=(-2.0D0*X(7))+(-2.0D0*X(6))      HX(7)=(-2.0D0*X(7))+(-2.0D0*X(6))      RETURN      END\\end{verbatim}"))) 
 NIL 
 NIL 
-(-66 -3975) 
+(-66 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp24} produces Fortran for Type 24 ASPs which evaluate a multivariate function at a point (needed for NAG routine \\axiomOpFrom{e04jaf}{e04Package}),{} for example:\\begin{verbatim}      SUBROUTINE FUNCT1(N,XC,FC)      DOUBLE PRECISION FC,XC(N)      INTEGER N      FC=10.0D0*XC(4)**4+(-40.0D0*XC(1)*XC(4)**3)+(60.0D0*XC(1)**2+5     &.0D0)*XC(4)**2+((-10.0D0*XC(3))+(-40.0D0*XC(1)**3))*XC(4)+16.0D0*X     &C(3)**4+(-32.0D0*XC(2)*XC(3)**3)+(24.0D0*XC(2)**2+5.0D0)*XC(3)**2+     &(-8.0D0*XC(2)**3*XC(3))+XC(2)**4+100.0D0*XC(2)**2+20.0D0*XC(1)*XC(     &2)+10.0D0*XC(1)**4+XC(1)**2      RETURN      END\\end{verbatim}")) (|coerce| (($ (|FortranExpression| (|construct|) (|construct| (QUOTE XC)) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
 NIL 
 NIL 
-(-67 -3975) 
+(-67 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp27} produces Fortran for Type 27 ASPs,{} needed for NAG routine \\axiomOpFrom{f02fjf}{f02Package} ,{}for example:\\begin{verbatim}      FUNCTION DOT(IFLAG,N,Z,W,RWORK,LRWORK,IWORK,LIWORK)      DOUBLE PRECISION W(N),Z(N),RWORK(LRWORK)      INTEGER N,LIWORK,IFLAG,LRWORK,IWORK(LIWORK)      DOT=(W(16)+(-0.5D0*W(15)))*Z(16)+((-0.5D0*W(16))+W(15)+(-0.5D0*W(1     &4)))*Z(15)+((-0.5D0*W(15))+W(14)+(-0.5D0*W(13)))*Z(14)+((-0.5D0*W(     &14))+W(13)+(-0.5D0*W(12)))*Z(13)+((-0.5D0*W(13))+W(12)+(-0.5D0*W(1     &1)))*Z(12)+((-0.5D0*W(12))+W(11)+(-0.5D0*W(10)))*Z(11)+((-0.5D0*W(     &11))+W(10)+(-0.5D0*W(9)))*Z(10)+((-0.5D0*W(10))+W(9)+(-0.5D0*W(8))     &)*Z(9)+((-0.5D0*W(9))+W(8)+(-0.5D0*W(7)))*Z(8)+((-0.5D0*W(8))+W(7)     &+(-0.5D0*W(6)))*Z(7)+((-0.5D0*W(7))+W(6)+(-0.5D0*W(5)))*Z(6)+((-0.     &5D0*W(6))+W(5)+(-0.5D0*W(4)))*Z(5)+((-0.5D0*W(5))+W(4)+(-0.5D0*W(3     &)))*Z(4)+((-0.5D0*W(4))+W(3)+(-0.5D0*W(2)))*Z(3)+((-0.5D0*W(3))+W(     &2)+(-0.5D0*W(1)))*Z(2)+((-0.5D0*W(2))+W(1))*Z(1)      RETURN      END\\end{verbatim}"))) 
 NIL 
 NIL 
-(-68 -3975) 
+(-68 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp28} produces Fortran for Type 28 ASPs,{} used in NAG routine \\axiomOpFrom{f02fjf}{f02Package},{} for example:\\begin{verbatim}      SUBROUTINE IMAGE(IFLAG,N,Z,W,RWORK,LRWORK,IWORK,LIWORK)      DOUBLE PRECISION Z(N),W(N),IWORK(LRWORK),RWORK(LRWORK)      INTEGER N,LIWORK,IFLAG,LRWORK      W(1)=0.01707454969713436D0*Z(16)+0.001747395874954051D0*Z(15)+0.00     &2106973900813502D0*Z(14)+0.002957434991769087D0*Z(13)+(-0.00700554     &0882865317D0*Z(12))+(-0.01219194009813166D0*Z(11))+0.0037230647365     &3087D0*Z(10)+0.04932374658377151D0*Z(9)+(-0.03586220812223305D0*Z(     &8))+(-0.04723268012114625D0*Z(7))+(-0.02434652144032987D0*Z(6))+0.     &2264766947290192D0*Z(5)+(-0.1385343580686922D0*Z(4))+(-0.116530050     &8238904D0*Z(3))+(-0.2803531651057233D0*Z(2))+1.019463911841327D0*Z     &(1)      W(2)=0.0227345011107737D0*Z(16)+0.008812321197398072D0*Z(15)+0.010     &94012210519586D0*Z(14)+(-0.01764072463999744D0*Z(13))+(-0.01357136     &72105995D0*Z(12))+0.00157466157362272D0*Z(11)+0.05258889186338282D     &0*Z(10)+(-0.01981532388243379D0*Z(9))+(-0.06095390688679697D0*Z(8)     &)+(-0.04153119955569051D0*Z(7))+0.2176561076571465D0*Z(6)+(-0.0532     &5555586632358D0*Z(5))+(-0.1688977368984641D0*Z(4))+(-0.32440166056     &67343D0*Z(3))+0.9128222941872173D0*Z(2)+(-0.2419652703415429D0*Z(1     &))      W(3)=0.03371198197190302D0*Z(16)+0.02021603150122265D0*Z(15)+(-0.0     &06607305534689702D0*Z(14))+(-0.03032392238968179D0*Z(13))+0.002033     &305231024948D0*Z(12)+0.05375944956767728D0*Z(11)+(-0.0163213312502     &9967D0*Z(10))+(-0.05483186562035512D0*Z(9))+(-0.04901428822579872D     &0*Z(8))+0.2091097927887612D0*Z(7)+(-0.05760560341383113D0*Z(6))+(-     &0.1236679206156403D0*Z(5))+(-0.3523683853026259D0*Z(4))+0.88929961     &32269974D0*Z(3)+(-0.2995429545781457D0*Z(2))+(-0.02986582812574917     &D0*Z(1))      W(4)=0.05141563713660119D0*Z(16)+0.005239165960779299D0*Z(15)+(-0.     &01623427735779699D0*Z(14))+(-0.01965809746040371D0*Z(13))+0.054688     &97337339577D0*Z(12)+(-0.014224695935687D0*Z(11))+(-0.0505181779315     &6355D0*Z(10))+(-0.04353074206076491D0*Z(9))+0.2012230497530726D0*Z     &(8)+(-0.06630874514535952D0*Z(7))+(-0.1280829963720053D0*Z(6))+(-0     &.305169742604165D0*Z(5))+0.8600427128450191D0*Z(4)+(-0.32415033802     &68184D0*Z(3))+(-0.09033531980693314D0*Z(2))+0.09089205517109111D0*     &Z(1)      W(5)=0.04556369767776375D0*Z(16)+(-0.001822737697581869D0*Z(15))+(     &-0.002512226501941856D0*Z(14))+0.02947046460707379D0*Z(13)+(-0.014     &45079632086177D0*Z(12))+(-0.05034242196614937D0*Z(11))+(-0.0376966     &3291725935D0*Z(10))+0.2171103102175198D0*Z(9)+(-0.0824949256021352     &4D0*Z(8))+(-0.1473995209288945D0*Z(7))+(-0.315042193418466D0*Z(6))     &+0.9591623347824002D0*Z(5)+(-0.3852396953763045D0*Z(4))+(-0.141718     &5427288274D0*Z(3))+(-0.03423495461011043D0*Z(2))+0.319820917706851     &6D0*Z(1)      W(6)=0.04015147277405744D0*Z(16)+0.01328585741341559D0*Z(15)+0.048     &26082005465965D0*Z(14)+(-0.04319641116207706D0*Z(13))+(-0.04931323     &319055762D0*Z(12))+(-0.03526886317505474D0*Z(11))+0.22295383396730     &01D0*Z(10)+(-0.07375317649315155D0*Z(9))+(-0.1589391311991561D0*Z(     &8))+(-0.328001910890377D0*Z(7))+0.952576555482747D0*Z(6)+(-0.31583     &09975786731D0*Z(5))+(-0.1846882042225383D0*Z(4))+(-0.0703762046700     &4427D0*Z(3))+0.2311852964327382D0*Z(2)+0.04254083491825025D0*Z(1)      W(7)=0.06069778964023718D0*Z(16)+0.06681263884671322D0*Z(15)+(-0.0     &2113506688615768D0*Z(14))+(-0.083996867458326D0*Z(13))+(-0.0329843     &8523869648D0*Z(12))+0.2276878326327734D0*Z(11)+(-0.067356038933017     &95D0*Z(10))+(-0.1559813965382218D0*Z(9))+(-0.3363262957694705D0*Z(     &8))+0.9442791158560948D0*Z(7)+(-0.3199955249404657D0*Z(6))+(-0.136     &2463839920727D0*Z(5))+(-0.1006185171570586D0*Z(4))+0.2057504515015     &423D0*Z(3)+(-0.02065879269286707D0*Z(2))+0.03160990266745513D0*Z(1     &)      W(8)=0.126386868896738D0*Z(16)+0.002563370039476418D0*Z(15)+(-0.05     &581757739455641D0*Z(14))+(-0.07777893205900685D0*Z(13))+0.23117338     &45834199D0*Z(12)+(-0.06031581134427592D0*Z(11))+(-0.14805474755869     &52D0*Z(10))+(-0.3364014128402243D0*Z(9))+0.9364014128402244D0*Z(8)     &+(-0.3269452524413048D0*Z(7))+(-0.1396841886557241D0*Z(6))+(-0.056     &1733845834199D0*Z(5))+0.1777789320590069D0*Z(4)+(-0.04418242260544     &359D0*Z(3))+(-0.02756337003947642D0*Z(2))+0.07361313110326199D0*Z(     &1)      W(9)=0.07361313110326199D0*Z(16)+(-0.02756337003947642D0*Z(15))+(-     &0.04418242260544359D0*Z(14))+0.1777789320590069D0*Z(13)+(-0.056173     &3845834199D0*Z(12))+(-0.1396841886557241D0*Z(11))+(-0.326945252441     &3048D0*Z(10))+0.9364014128402244D0*Z(9)+(-0.3364014128402243D0*Z(8     &))+(-0.1480547475586952D0*Z(7))+(-0.06031581134427592D0*Z(6))+0.23     &11733845834199D0*Z(5)+(-0.07777893205900685D0*Z(4))+(-0.0558175773     &9455641D0*Z(3))+0.002563370039476418D0*Z(2)+0.126386868896738D0*Z(     &1)      W(10)=0.03160990266745513D0*Z(16)+(-0.02065879269286707D0*Z(15))+0     &.2057504515015423D0*Z(14)+(-0.1006185171570586D0*Z(13))+(-0.136246     &3839920727D0*Z(12))+(-0.3199955249404657D0*Z(11))+0.94427911585609     &48D0*Z(10)+(-0.3363262957694705D0*Z(9))+(-0.1559813965382218D0*Z(8     &))+(-0.06735603893301795D0*Z(7))+0.2276878326327734D0*Z(6)+(-0.032     &98438523869648D0*Z(5))+(-0.083996867458326D0*Z(4))+(-0.02113506688     &615768D0*Z(3))+0.06681263884671322D0*Z(2)+0.06069778964023718D0*Z(     &1)      W(11)=0.04254083491825025D0*Z(16)+0.2311852964327382D0*Z(15)+(-0.0     &7037620467004427D0*Z(14))+(-0.1846882042225383D0*Z(13))+(-0.315830     &9975786731D0*Z(12))+0.952576555482747D0*Z(11)+(-0.328001910890377D     &0*Z(10))+(-0.1589391311991561D0*Z(9))+(-0.07375317649315155D0*Z(8)     &)+0.2229538339673001D0*Z(7)+(-0.03526886317505474D0*Z(6))+(-0.0493     &1323319055762D0*Z(5))+(-0.04319641116207706D0*Z(4))+0.048260820054     &65965D0*Z(3)+0.01328585741341559D0*Z(2)+0.04015147277405744D0*Z(1)      W(12)=0.3198209177068516D0*Z(16)+(-0.03423495461011043D0*Z(15))+(-     &0.1417185427288274D0*Z(14))+(-0.3852396953763045D0*Z(13))+0.959162     &3347824002D0*Z(12)+(-0.315042193418466D0*Z(11))+(-0.14739952092889     &45D0*Z(10))+(-0.08249492560213524D0*Z(9))+0.2171103102175198D0*Z(8     &)+(-0.03769663291725935D0*Z(7))+(-0.05034242196614937D0*Z(6))+(-0.     &01445079632086177D0*Z(5))+0.02947046460707379D0*Z(4)+(-0.002512226     &501941856D0*Z(3))+(-0.001822737697581869D0*Z(2))+0.045563697677763     &75D0*Z(1)      W(13)=0.09089205517109111D0*Z(16)+(-0.09033531980693314D0*Z(15))+(     &-0.3241503380268184D0*Z(14))+0.8600427128450191D0*Z(13)+(-0.305169     &742604165D0*Z(12))+(-0.1280829963720053D0*Z(11))+(-0.0663087451453     &5952D0*Z(10))+0.2012230497530726D0*Z(9)+(-0.04353074206076491D0*Z(     &8))+(-0.05051817793156355D0*Z(7))+(-0.014224695935687D0*Z(6))+0.05     &468897337339577D0*Z(5)+(-0.01965809746040371D0*Z(4))+(-0.016234277     &35779699D0*Z(3))+0.005239165960779299D0*Z(2)+0.05141563713660119D0     &*Z(1)      W(14)=(-0.02986582812574917D0*Z(16))+(-0.2995429545781457D0*Z(15))     &+0.8892996132269974D0*Z(14)+(-0.3523683853026259D0*Z(13))+(-0.1236     &679206156403D0*Z(12))+(-0.05760560341383113D0*Z(11))+0.20910979278     &87612D0*Z(10)+(-0.04901428822579872D0*Z(9))+(-0.05483186562035512D     &0*Z(8))+(-0.01632133125029967D0*Z(7))+0.05375944956767728D0*Z(6)+0     &.002033305231024948D0*Z(5)+(-0.03032392238968179D0*Z(4))+(-0.00660     &7305534689702D0*Z(3))+0.02021603150122265D0*Z(2)+0.033711981971903     &02D0*Z(1)      W(15)=(-0.2419652703415429D0*Z(16))+0.9128222941872173D0*Z(15)+(-0     &.3244016605667343D0*Z(14))+(-0.1688977368984641D0*Z(13))+(-0.05325     &555586632358D0*Z(12))+0.2176561076571465D0*Z(11)+(-0.0415311995556     &9051D0*Z(10))+(-0.06095390688679697D0*Z(9))+(-0.01981532388243379D     &0*Z(8))+0.05258889186338282D0*Z(7)+0.00157466157362272D0*Z(6)+(-0.     &0135713672105995D0*Z(5))+(-0.01764072463999744D0*Z(4))+0.010940122     &10519586D0*Z(3)+0.008812321197398072D0*Z(2)+0.0227345011107737D0*Z     &(1)      W(16)=1.019463911841327D0*Z(16)+(-0.2803531651057233D0*Z(15))+(-0.     &1165300508238904D0*Z(14))+(-0.1385343580686922D0*Z(13))+0.22647669     &47290192D0*Z(12)+(-0.02434652144032987D0*Z(11))+(-0.04723268012114     &625D0*Z(10))+(-0.03586220812223305D0*Z(9))+0.04932374658377151D0*Z     &(8)+0.00372306473653087D0*Z(7)+(-0.01219194009813166D0*Z(6))+(-0.0     &07005540882865317D0*Z(5))+0.002957434991769087D0*Z(4)+0.0021069739     &00813502D0*Z(3)+0.001747395874954051D0*Z(2)+0.01707454969713436D0*     &Z(1)      RETURN      END\\end{verbatim}"))) 
 NIL 
 NIL 
-(-69 -3975) 
+(-69 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp29} produces Fortran for Type 29 ASPs,{} needed for NAG routine \\axiomOpFrom{f02fjf}{f02Package},{} for example:\\begin{verbatim}      SUBROUTINE MONIT(ISTATE,NEXTIT,NEVALS,NEVECS,K,F,D)      DOUBLE PRECISION D(K),F(K)      INTEGER K,NEXTIT,NEVALS,NVECS,ISTATE      CALL F02FJZ(ISTATE,NEXTIT,NEVALS,NEVECS,K,F,D)      RETURN      END\\end{verbatim}")) (|outputAsFortran| (((|Void|)) "\\spad{outputAsFortran()} generates the default code for \\spadtype{ASP29}."))) 
 NIL 
 NIL 
-(-70 -3975) 
+(-70 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp30} produces Fortran for Type 30 ASPs,{} needed for NAG routine \\axiomOpFrom{f04qaf}{f04Package},{} for example:\\begin{verbatim}      SUBROUTINE APROD(MODE,M,N,X,Y,RWORK,LRWORK,IWORK,LIWORK)      DOUBLE PRECISION X(N),Y(M),RWORK(LRWORK)      INTEGER M,N,LIWORK,IFAIL,LRWORK,IWORK(LIWORK),MODE      DOUBLE PRECISION A(5,5)      EXTERNAL F06PAF      A(1,1)=1.0D0      A(1,2)=0.0D0      A(1,3)=0.0D0      A(1,4)=-1.0D0      A(1,5)=0.0D0      A(2,1)=0.0D0      A(2,2)=1.0D0      A(2,3)=0.0D0      A(2,4)=0.0D0      A(2,5)=-1.0D0      A(3,1)=0.0D0      A(3,2)=0.0D0      A(3,3)=1.0D0      A(3,4)=-1.0D0      A(3,5)=0.0D0      A(4,1)=-1.0D0      A(4,2)=0.0D0      A(4,3)=-1.0D0      A(4,4)=4.0D0      A(4,5)=-1.0D0      A(5,1)=0.0D0      A(5,2)=-1.0D0      A(5,3)=0.0D0      A(5,4)=-1.0D0      A(5,5)=4.0D0      IF(MODE.EQ.1)THEN        CALL F06PAF('N',M,N,1.0D0,A,M,X,1,1.0D0,Y,1)      ELSEIF(MODE.EQ.2)THEN        CALL F06PAF('T',M,N,1.0D0,A,M,Y,1,1.0D0,X,1)      ENDIF      RETURN      END\\end{verbatim}"))) 
 NIL 
 NIL 
-(-71 -3975) 
+(-71 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp31} produces Fortran for Type 31 ASPs,{} needed for NAG routine \\axiomOpFrom{d02ejf}{d02Package},{} for example:\\begin{verbatim}      SUBROUTINE PEDERV(X,Y,PW)      DOUBLE PRECISION X,Y(*)      DOUBLE PRECISION PW(3,3)      PW(1,1)=-0.03999999999999999D0      PW(1,2)=10000.0D0*Y(3)      PW(1,3)=10000.0D0*Y(2)      PW(2,1)=0.03999999999999999D0      PW(2,2)=(-10000.0D0*Y(3))+(-60000000.0D0*Y(2))      PW(2,3)=-10000.0D0*Y(2)      PW(3,1)=0.0D0      PW(3,2)=60000000.0D0*Y(2)      PW(3,3)=0.0D0      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X)) (|construct| (QUOTE Y)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-72 -3975) 
+(-72 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp33} produces Fortran for Type 33 ASPs,{} needed for NAG routine \\axiomOpFrom{d02kef}{d02Package}. The code is a dummy ASP:\\begin{verbatim}      SUBROUTINE REPORT(X,V,JINT)      DOUBLE PRECISION V(3),X      INTEGER JINT      RETURN      END\\end{verbatim}")) (|outputAsFortran| (((|Void|)) "\\spad{outputAsFortran()} generates the default code for \\spadtype{ASP33}."))) 
 NIL 
 NIL 
-(-73 -3975) 
+(-73 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp34} produces Fortran for Type 34 ASPs,{} needed for NAG routine \\axiomOpFrom{f04mbf}{f04Package},{} for example:\\begin{verbatim}      SUBROUTINE MSOLVE(IFLAG,N,X,Y,RWORK,LRWORK,IWORK,LIWORK)      DOUBLE PRECISION RWORK(LRWORK),X(N),Y(N)      INTEGER I,J,N,LIWORK,IFLAG,LRWORK,IWORK(LIWORK)      DOUBLE PRECISION W1(3),W2(3),MS(3,3)      IFLAG=-1      MS(1,1)=2.0D0      MS(1,2)=1.0D0      MS(1,3)=0.0D0      MS(2,1)=1.0D0      MS(2,2)=2.0D0      MS(2,3)=1.0D0      MS(3,1)=0.0D0      MS(3,2)=1.0D0      MS(3,3)=2.0D0      CALL F04ASF(MS,N,X,N,Y,W1,W2,IFLAG)      IFLAG=-IFLAG      RETURN      END\\end{verbatim}"))) 
 NIL 
 NIL 
-(-74 -3975) 
+(-74 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp35} produces Fortran for Type 35 ASPs,{} needed for NAG routines \\axiomOpFrom{c05pbf}{c05Package},{} \\axiomOpFrom{c05pcf}{c05Package},{} for example:\\begin{verbatim}      SUBROUTINE FCN(N,X,FVEC,FJAC,LDFJAC,IFLAG)      DOUBLE PRECISION X(N),FVEC(N),FJAC(LDFJAC,N)      INTEGER LDFJAC,N,IFLAG      IF(IFLAG.EQ.1)THEN        FVEC(1)=(-1.0D0*X(2))+X(1)        FVEC(2)=(-1.0D0*X(3))+2.0D0*X(2)        FVEC(3)=3.0D0*X(3)      ELSEIF(IFLAG.EQ.2)THEN        FJAC(1,1)=1.0D0        FJAC(1,2)=-1.0D0        FJAC(1,3)=0.0D0        FJAC(2,1)=0.0D0        FJAC(2,2)=2.0D0        FJAC(2,3)=-1.0D0        FJAC(3,1)=0.0D0        FJAC(3,2)=0.0D0        FJAC(3,3)=3.0D0      ENDIF      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-75 -3975) 
+(-75 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp4} produces Fortran for Type 4 ASPs,{} which take an expression in \\spad{X}(1) .. \\spad{X}(NDIM) and produce a real function of the form:\\begin{verbatim}      DOUBLE PRECISION FUNCTION FUNCTN(NDIM,X)      DOUBLE PRECISION X(NDIM)      INTEGER NDIM      FUNCTN=(4.0D0*X(1)*X(3)**2*DEXP(2.0D0*X(1)*X(3)))/(X(4)**2+(2.0D0*     &X(2)+2.0D0)*X(4)+X(2)**2+2.0D0*X(2)+1.0D0)      RETURN      END\\end{verbatim}")) (|coerce| (($ (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
 NIL 
 NIL 
@@ -240,62 +240,62 @@ NIL
 ((|constructor| (NIL "\\spadtype{Asp42} produces Fortran for Type 42 ASPs,{} needed for NAG routines \\axiomOpFrom{d02raf}{d02Package} and \\axiomOpFrom{d02saf}{d02Package} in particular. These ASPs are in fact three Fortran routines which return a vector of functions,{} and their derivatives \\spad{wrt} \\spad{Y}(\\spad{i}) and also a continuation parameter EPS,{} for example:\\begin{verbatim}      SUBROUTINE G(EPS,YA,YB,BC,N)      DOUBLE PRECISION EPS,YA(N),YB(N),BC(N)      INTEGER N      BC(1)=YA(1)      BC(2)=YA(2)      BC(3)=YB(2)-1.0D0      RETURN      END      SUBROUTINE JACOBG(EPS,YA,YB,AJ,BJ,N)      DOUBLE PRECISION EPS,YA(N),AJ(N,N),BJ(N,N),YB(N)      INTEGER N      AJ(1,1)=1.0D0      AJ(1,2)=0.0D0      AJ(1,3)=0.0D0      AJ(2,1)=0.0D0      AJ(2,2)=1.0D0      AJ(2,3)=0.0D0      AJ(3,1)=0.0D0      AJ(3,2)=0.0D0      AJ(3,3)=0.0D0      BJ(1,1)=0.0D0      BJ(1,2)=0.0D0      BJ(1,3)=0.0D0      BJ(2,1)=0.0D0      BJ(2,2)=0.0D0      BJ(2,3)=0.0D0      BJ(3,1)=0.0D0      BJ(3,2)=1.0D0      BJ(3,3)=0.0D0      RETURN      END      SUBROUTINE JACGEP(EPS,YA,YB,BCEP,N)      DOUBLE PRECISION EPS,YA(N),YB(N),BCEP(N)      INTEGER N      BCEP(1)=0.0D0      BCEP(2)=0.0D0      BCEP(3)=0.0D0      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE EPS)) (|construct| (QUOTE YA) (QUOTE YB)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-78 -3975) 
+(-78 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp49} produces Fortran for Type 49 ASPs,{} needed for NAG routines \\axiomOpFrom{e04dgf}{e04Package},{} \\axiomOpFrom{e04ucf}{e04Package},{} for example:\\begin{verbatim}      SUBROUTINE OBJFUN(MODE,N,X,OBJF,OBJGRD,NSTATE,IUSER,USER)      DOUBLE PRECISION X(N),OBJF,OBJGRD(N),USER(*)      INTEGER N,IUSER(*),MODE,NSTATE      OBJF=X(4)*X(9)+((-1.0D0*X(5))+X(3))*X(8)+((-1.0D0*X(3))+X(1))*X(7)     &+(-1.0D0*X(2)*X(6))      OBJGRD(1)=X(7)      OBJGRD(2)=-1.0D0*X(6)      OBJGRD(3)=X(8)+(-1.0D0*X(7))      OBJGRD(4)=X(9)      OBJGRD(5)=-1.0D0*X(8)      OBJGRD(6)=-1.0D0*X(2)      OBJGRD(7)=(-1.0D0*X(3))+X(1)      OBJGRD(8)=(-1.0D0*X(5))+X(3)      OBJGRD(9)=X(4)      RETURN      END\\end{verbatim}")) (|coerce| (($ (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
 NIL 
 NIL 
-(-79 -3975) 
+(-79 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp50} produces Fortran for Type 50 ASPs,{} needed for NAG routine \\axiomOpFrom{e04fdf}{e04Package},{} for example:\\begin{verbatim}      SUBROUTINE LSFUN1(M,N,XC,FVECC)      DOUBLE PRECISION FVECC(M),XC(N)      INTEGER I,M,N      FVECC(1)=((XC(1)-2.4D0)*XC(3)+(15.0D0*XC(1)-36.0D0)*XC(2)+1.0D0)/(     &XC(3)+15.0D0*XC(2))      FVECC(2)=((XC(1)-2.8D0)*XC(3)+(7.0D0*XC(1)-19.6D0)*XC(2)+1.0D0)/(X     &C(3)+7.0D0*XC(2))      FVECC(3)=((XC(1)-3.2D0)*XC(3)+(4.333333333333333D0*XC(1)-13.866666     &66666667D0)*XC(2)+1.0D0)/(XC(3)+4.333333333333333D0*XC(2))      FVECC(4)=((XC(1)-3.5D0)*XC(3)+(3.0D0*XC(1)-10.5D0)*XC(2)+1.0D0)/(X     &C(3)+3.0D0*XC(2))      FVECC(5)=((XC(1)-3.9D0)*XC(3)+(2.2D0*XC(1)-8.579999999999998D0)*XC     &(2)+1.0D0)/(XC(3)+2.2D0*XC(2))      FVECC(6)=((XC(1)-4.199999999999999D0)*XC(3)+(1.666666666666667D0*X     &C(1)-7.0D0)*XC(2)+1.0D0)/(XC(3)+1.666666666666667D0*XC(2))      FVECC(7)=((XC(1)-4.5D0)*XC(3)+(1.285714285714286D0*XC(1)-5.7857142     &85714286D0)*XC(2)+1.0D0)/(XC(3)+1.285714285714286D0*XC(2))      FVECC(8)=((XC(1)-4.899999999999999D0)*XC(3)+(XC(1)-4.8999999999999     &99D0)*XC(2)+1.0D0)/(XC(3)+XC(2))      FVECC(9)=((XC(1)-4.699999999999999D0)*XC(3)+(XC(1)-4.6999999999999     &99D0)*XC(2)+1.285714285714286D0)/(XC(3)+XC(2))      FVECC(10)=((XC(1)-6.8D0)*XC(3)+(XC(1)-6.8D0)*XC(2)+1.6666666666666     &67D0)/(XC(3)+XC(2))      FVECC(11)=((XC(1)-8.299999999999999D0)*XC(3)+(XC(1)-8.299999999999     &999D0)*XC(2)+2.2D0)/(XC(3)+XC(2))      FVECC(12)=((XC(1)-10.6D0)*XC(3)+(XC(1)-10.6D0)*XC(2)+3.0D0)/(XC(3)     &+XC(2))      FVECC(13)=((XC(1)-1.34D0)*XC(3)+(XC(1)-1.34D0)*XC(2)+4.33333333333     &3333D0)/(XC(3)+XC(2))      FVECC(14)=((XC(1)-2.1D0)*XC(3)+(XC(1)-2.1D0)*XC(2)+7.0D0)/(XC(3)+X     &C(2))      FVECC(15)=((XC(1)-4.39D0)*XC(3)+(XC(1)-4.39D0)*XC(2)+15.0D0)/(XC(3     &)+XC(2))      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE XC)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-80 -3975) 
+(-80 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp55} produces Fortran for Type 55 ASPs,{} needed for NAG routines \\axiomOpFrom{e04dgf}{e04Package} and \\axiomOpFrom{e04ucf}{e04Package},{} for example:\\begin{verbatim}      SUBROUTINE CONFUN(MODE,NCNLN,N,NROWJ,NEEDC,X,C,CJAC,NSTATE,IUSER     &,USER)      DOUBLE PRECISION C(NCNLN),X(N),CJAC(NROWJ,N),USER(*)      INTEGER N,IUSER(*),NEEDC(NCNLN),NROWJ,MODE,NCNLN,NSTATE      IF(NEEDC(1).GT.0)THEN        C(1)=X(6)**2+X(1)**2        CJAC(1,1)=2.0D0*X(1)        CJAC(1,2)=0.0D0        CJAC(1,3)=0.0D0        CJAC(1,4)=0.0D0        CJAC(1,5)=0.0D0        CJAC(1,6)=2.0D0*X(6)      ENDIF      IF(NEEDC(2).GT.0)THEN        C(2)=X(2)**2+(-2.0D0*X(1)*X(2))+X(1)**2        CJAC(2,1)=(-2.0D0*X(2))+2.0D0*X(1)        CJAC(2,2)=2.0D0*X(2)+(-2.0D0*X(1))        CJAC(2,3)=0.0D0        CJAC(2,4)=0.0D0        CJAC(2,5)=0.0D0        CJAC(2,6)=0.0D0      ENDIF      IF(NEEDC(3).GT.0)THEN        C(3)=X(3)**2+(-2.0D0*X(1)*X(3))+X(2)**2+X(1)**2        CJAC(3,1)=(-2.0D0*X(3))+2.0D0*X(1)        CJAC(3,2)=2.0D0*X(2)        CJAC(3,3)=2.0D0*X(3)+(-2.0D0*X(1))        CJAC(3,4)=0.0D0        CJAC(3,5)=0.0D0        CJAC(3,6)=0.0D0      ENDIF      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct|) (|construct| (QUOTE X)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-81 -3975) 
+(-81 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp6} produces Fortran for Type 6 ASPs,{} needed for NAG routines \\axiomOpFrom{c05nbf}{c05Package},{} \\axiomOpFrom{c05ncf}{c05Package}. These represent vectors of functions of \\spad{X}(\\spad{i}) and look like:\\begin{verbatim}      SUBROUTINE FCN(N,X,FVEC,IFLAG)      DOUBLE PRECISION X(N),FVEC(N)      INTEGER N,IFLAG      FVEC(1)=(-2.0D0*X(2))+(-2.0D0*X(1)**2)+3.0D0*X(1)+1.0D0      FVEC(2)=(-2.0D0*X(3))+(-2.0D0*X(2)**2)+3.0D0*X(2)+(-1.0D0*X(1))+1.     &0D0      FVEC(3)=(-2.0D0*X(4))+(-2.0D0*X(3)**2)+3.0D0*X(3)+(-1.0D0*X(2))+1.     &0D0      FVEC(4)=(-2.0D0*X(5))+(-2.0D0*X(4)**2)+3.0D0*X(4)+(-1.0D0*X(3))+1.     &0D0      FVEC(5)=(-2.0D0*X(6))+(-2.0D0*X(5)**2)+3.0D0*X(5)+(-1.0D0*X(4))+1.     &0D0      FVEC(6)=(-2.0D0*X(7))+(-2.0D0*X(6)**2)+3.0D0*X(6)+(-1.0D0*X(5))+1.     &0D0      FVEC(7)=(-2.0D0*X(8))+(-2.0D0*X(7)**2)+3.0D0*X(7)+(-1.0D0*X(6))+1.     &0D0      FVEC(8)=(-2.0D0*X(9))+(-2.0D0*X(8)**2)+3.0D0*X(8)+(-1.0D0*X(7))+1.     &0D0      FVEC(9)=(-2.0D0*X(9)**2)+3.0D0*X(9)+(-1.0D0*X(8))+1.0D0      RETURN      END\\end{verbatim}"))) 
 NIL 
 NIL 
-(-82 -3975) 
+(-82 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp7} produces Fortran for Type 7 ASPs,{} needed for NAG routines \\axiomOpFrom{d02bbf}{d02Package},{} \\axiomOpFrom{d02gaf}{d02Package}. These represent a vector of functions of the scalar \\spad{X} and the array \\spad{Z},{} and look like:\\begin{verbatim}      SUBROUTINE FCN(X,Z,F)      DOUBLE PRECISION F(*),X,Z(*)      F(1)=DTAN(Z(3))      F(2)=((-0.03199999999999999D0*DCOS(Z(3))*DTAN(Z(3)))+(-0.02D0*Z(2)     &**2))/(Z(2)*DCOS(Z(3)))      F(3)=-0.03199999999999999D0/(X*Z(2)**2)      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X)) (|construct| (QUOTE Y)) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-83 -3975) 
+(-83 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp73} produces Fortran for Type 73 ASPs,{} needed for NAG routine \\axiomOpFrom{d03eef}{d03Package},{} for example:\\begin{verbatim}      SUBROUTINE PDEF(X,Y,ALPHA,BETA,GAMMA,DELTA,EPSOLN,PHI,PSI)      DOUBLE PRECISION ALPHA,EPSOLN,PHI,X,Y,BETA,DELTA,GAMMA,PSI      ALPHA=DSIN(X)      BETA=Y      GAMMA=X*Y      DELTA=DCOS(X)*DSIN(Y)      EPSOLN=Y+X      PHI=X      PSI=Y      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X) (QUOTE Y)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-84 -3975) 
+(-84 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp74} produces Fortran for Type 74 ASPs,{} needed for NAG routine \\axiomOpFrom{d03eef}{d03Package},{} for example:\\begin{verbatim}      SUBROUTINE BNDY(X,Y,A,B,C,IBND)      DOUBLE PRECISION A,B,C,X,Y      INTEGER IBND      IF(IBND.EQ.0)THEN        A=0.0D0        B=1.0D0        C=-1.0D0*DSIN(X)      ELSEIF(IBND.EQ.1)THEN        A=1.0D0        B=0.0D0        C=DSIN(X)*DSIN(Y)      ELSEIF(IBND.EQ.2)THEN        A=1.0D0        B=0.0D0        C=DSIN(X)*DSIN(Y)      ELSEIF(IBND.EQ.3)THEN        A=0.0D0        B=1.0D0        C=-1.0D0*DSIN(Y)      ENDIF      END\\end{verbatim}")) (|coerce| (($ (|Matrix| (|FortranExpression| (|construct| (QUOTE X) (QUOTE Y)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-85 -3975) 
+(-85 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp77} produces Fortran for Type 77 ASPs,{} needed for NAG routine \\axiomOpFrom{d02gbf}{d02Package},{} for example:\\begin{verbatim}      SUBROUTINE FCNF(X,F)      DOUBLE PRECISION X      DOUBLE PRECISION F(2,2)      F(1,1)=0.0D0      F(1,2)=1.0D0      F(2,1)=0.0D0      F(2,2)=-10.0D0      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Matrix| (|FortranExpression| (|construct| (QUOTE X)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-86 -3975) 
+(-86 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp78} produces Fortran for Type 78 ASPs,{} needed for NAG routine \\axiomOpFrom{d02gbf}{d02Package},{} for example:\\begin{verbatim}      SUBROUTINE FCNG(X,G)      DOUBLE PRECISION G(*),X      G(1)=0.0D0      G(2)=0.0D0      END\\end{verbatim}")) (|coerce| (($ (|Vector| (|FortranExpression| (|construct| (QUOTE X)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-87 -3975) 
+(-87 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp8} produces Fortran for Type 8 ASPs,{} needed for NAG routine \\axiomOpFrom{d02bbf}{d02Package}. This ASP prints intermediate values of the computed solution of an ODE and might look like:\\begin{verbatim}      SUBROUTINE OUTPUT(XSOL,Y,COUNT,M,N,RESULT,FORWRD)      DOUBLE PRECISION Y(N),RESULT(M,N),XSOL      INTEGER M,N,COUNT      LOGICAL FORWRD      DOUBLE PRECISION X02ALF,POINTS(8)      EXTERNAL X02ALF      INTEGER I      POINTS(1)=1.0D0      POINTS(2)=2.0D0      POINTS(3)=3.0D0      POINTS(4)=4.0D0      POINTS(5)=5.0D0      POINTS(6)=6.0D0      POINTS(7)=7.0D0      POINTS(8)=8.0D0      COUNT=COUNT+1      DO 25001 I=1,N        RESULT(COUNT,I)=Y(I)25001 CONTINUE      IF(COUNT.EQ.M)THEN        IF(FORWRD)THEN          XSOL=X02ALF()        ELSE          XSOL=-X02ALF()        ENDIF      ELSE        XSOL=POINTS(COUNT)      ENDIF      END\\end{verbatim}"))) 
 NIL 
 NIL 
-(-88 -3975) 
+(-88 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp80} produces Fortran for Type 80 ASPs,{} needed for NAG routine \\axiomOpFrom{d02kef}{d02Package},{} for example:\\begin{verbatim}      SUBROUTINE BDYVAL(XL,XR,ELAM,YL,YR)      DOUBLE PRECISION ELAM,XL,YL(3),XR,YR(3)      YL(1)=XL      YL(2)=2.0D0      YR(1)=1.0D0      YR(2)=-1.0D0*DSQRT(XR+(-1.0D0*ELAM))      RETURN      END\\end{verbatim}")) (|coerce| (($ (|Matrix| (|FortranExpression| (|construct| (QUOTE XL) (QUOTE XR) (QUOTE ELAM)) (|construct|) (|MachineFloat|)))) "\\spad{coerce(f)} takes objects from the appropriate instantiation of \\spadtype{FortranExpression} and turns them into an ASP."))) 
 NIL 
 NIL 
-(-89 -3975) 
+(-89 -3982) 
 ((|constructor| (NIL "\\spadtype{Asp9} produces Fortran for Type 9 ASPs,{} needed for NAG routines \\axiomOpFrom{d02bhf}{d02Package},{} \\axiomOpFrom{d02cjf}{d02Package},{} \\axiomOpFrom{d02ejf}{d02Package}. These ASPs represent a function of a scalar \\spad{X} and a vector \\spad{Y},{} for example:\\begin{verbatim}      DOUBLE PRECISION FUNCTION G(X,Y)      DOUBLE PRECISION X,Y(*)      G=X+Y(1)      RETURN      END\\end{verbatim} If the user provides a constant value for \\spad{G},{} then extra information is added via COMMON blocks used by certain routines. This specifies that the value returned by \\spad{G} in this case is to be ignored.")) (|coerce| (($ (|FortranExpression| (|construct| (QUOTE X)) (|construct| (QUOTE Y)) (|MachineFloat|))) "\\spad{coerce(f)} takes an object from the appropriate instantiation of \\spadtype{FortranExpression} and turns it into an ASP."))) 
 NIL 
 NIL 
 (-90 R L) 
 ((|constructor| (NIL "\\spadtype{AssociatedEquations} provides functions to compute the associated equations needed for factoring operators")) (|associatedEquations| (((|Record| (|:| |minor| (|List| (|PositiveInteger|))) (|:| |eq| |#2|) (|:| |minors| (|List| (|List| (|PositiveInteger|)))) (|:| |ops| (|List| |#2|))) |#2| (|PositiveInteger|)) "\\spad{associatedEquations(op, m)} returns \\spad{[w, eq, lw, lop]} such that \\spad{eq(w) = 0} where \\spad{w} is the given minor,{} and \\spad{lw_i = lop_i(w)} for all the other minors.")) (|uncouplingMatrices| (((|Vector| (|Matrix| |#1|)) (|Matrix| |#1|)) "\\spad{uncouplingMatrices(M)} returns \\spad{[A_1,...,A_n]} such that if \\spad{y = [y_1,...,y_n]} is a solution of \\spad{y' = M y},{} then \\spad{[\\$y_j',y_j'',...,y_j^{(n)}\\$] = \\$A_j y\\$} for all \\spad{j}\\spad{'s}.")) (|associatedSystem| (((|Record| (|:| |mat| (|Matrix| |#1|)) (|:| |vec| (|Vector| (|List| (|PositiveInteger|))))) |#2| (|PositiveInteger|)) "\\spad{associatedSystem(op, m)} returns \\spad{[M,w]} such that the \\spad{m}-th associated equation system to \\spad{L} is \\spad{w' = M w}."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-366)))) 
+((|HasCategory| |#1| (QUOTE (-367)))) 
 (-91 S) 
 ((|constructor| (NIL "A stack represented as a flexible array.")) (|arrayStack| (($ (|List| |#1|)) "\\spad{arrayStack([x,y,...,z])} creates an array stack with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last element \\spad{z}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-92 S) 
 ((|constructor| (NIL "This is the category of Spad abstract syntax trees."))) 
 NIL 
@@ -318,15 +318,15 @@ NIL
 NIL 
 (-97) 
 ((|constructor| (NIL "\\axiomType{AttributeButtons} implements a database and associated adjustment mechanisms for a set of attributes. \\blankline For ODEs these attributes are \"stiffness\",{} \"stability\" (\\spadignore{i.e.} how much affect the cosine or sine component of the solution has on the stability of the result),{} \"accuracy\" and \"expense\" (\\spadignore{i.e.} how expensive is the evaluation of the ODE). All these have bearing on the cost of calculating the solution given that reducing the step-length to achieve greater accuracy requires considerable number of evaluations and calculations. \\blankline The effect of each of these attributes can be altered by increasing or decreasing the button value. \\blankline For Integration there is a button for increasing and decreasing the preset number of function evaluations for each method. This is automatically used by ANNA when a method fails due to insufficient workspace or where the limit of function evaluations has been reached before the required accuracy is achieved. \\blankline")) (|setButtonValue| (((|Float|) (|String|) (|String|) (|Float|)) "\\axiom{setButtonValue(attributeName,{}routineName,{}\\spad{n})} sets the value of the button of attribute \\spad{attributeName} to routine \\spad{routineName} to \\spad{n}. \\spad{n} must be in the range [0..1]. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".") (((|Float|) (|String|) (|Float|)) "\\axiom{setButtonValue(attributeName,{}\\spad{n})} sets the value of all buttons of attribute \\spad{attributeName} to \\spad{n}. \\spad{n} must be in the range [0..1]. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".")) (|setAttributeButtonStep| (((|Float|) (|Float|)) "\\axiom{setAttributeButtonStep(\\spad{n})} sets the value of the steps for increasing and decreasing the button values. \\axiom{\\spad{n}} must be greater than 0 and less than 1. The preset value is 0.5.")) (|resetAttributeButtons| (((|Void|)) "\\axiom{resetAttributeButtons()} resets the Attribute buttons to a neutral level.")) (|getButtonValue| (((|Float|) (|String|) (|String|)) "\\axiom{getButtonValue(routineName,{}attributeName)} returns the current value for the effect of the attribute \\axiom{attributeName} with routine \\axiom{routineName}. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".")) (|decrease| (((|Float|) (|String|)) "\\axiom{decrease(attributeName)} decreases the value for the effect of the attribute \\axiom{attributeName} with all routines. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".") (((|Float|) (|String|) (|String|)) "\\axiom{decrease(routineName,{}attributeName)} decreases the value for the effect of the attribute \\axiom{attributeName} with routine \\axiom{routineName}. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".")) (|increase| (((|Float|) (|String|)) "\\axiom{increase(attributeName)} increases the value for the effect of the attribute \\axiom{attributeName} with all routines. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\".") (((|Float|) (|String|) (|String|)) "\\axiom{increase(routineName,{}attributeName)} increases the value for the effect of the attribute \\axiom{attributeName} with routine \\axiom{routineName}. \\blankline \\axiom{attributeName} should be one of the values \"stiffness\",{} \"stability\",{} \"accuracy\",{} \"expense\" or \"functionEvaluations\"."))) 
-((-4427 . T)) 
+((-4434 . T)) 
 NIL 
 (-98) 
 ((|constructor| (NIL "This category exports the attributes in the AXIOM Library")) (|canonical| ((|attribute|) "\\spad{canonical} is \\spad{true} if and only if distinct elements have distinct data structures. For example,{} a domain of mathematical objects which has the \\spad{canonical} attribute means that two objects are mathematically equal if and only if their data structures are equal.")) (|multiplicativeValuation| ((|attribute|) "\\spad{multiplicativeValuation} implies \\spad{euclideanSize(a*b)=euclideanSize(a)*euclideanSize(b)}.")) (|additiveValuation| ((|attribute|) "\\spad{additiveValuation} implies \\spad{euclideanSize(a*b)=euclideanSize(a)+euclideanSize(b)}.")) (|noetherian| ((|attribute|) "\\spad{noetherian} is \\spad{true} if all of its ideals are finitely generated.")) (|central| ((|attribute|) "\\spad{central} is \\spad{true} if,{} given an algebra over a ring \\spad{R},{} the image of \\spad{R} is the center of the algebra,{} \\spadignore{i.e.} the set of members of the algebra which commute with all others is precisely the image of \\spad{R} in the algebra.")) (|partiallyOrderedSet| ((|attribute|) "\\spad{partiallyOrderedSet} is \\spad{true} if a set with \\spadop{<} which is transitive,{} but \\spad{not(a < b or a = b)} does not necessarily imply \\spad{b<a}.")) (|arbitraryPrecision| ((|attribute|) "\\spad{arbitraryPrecision} means the user can set the precision for subsequent calculations.")) (|canonicalsClosed| ((|attribute|) "\\spad{canonicalsClosed} is \\spad{true} if \\spad{unitCanonical(a)*unitCanonical(b) = unitCanonical(a*b)}.")) (|canonicalUnitNormal| ((|attribute|) "\\spad{canonicalUnitNormal} is \\spad{true} if we can choose a canonical representative for each class of associate elements,{} that is \\spad{associates?(a,b)} returns \\spad{true} if and only if \\spad{unitCanonical(a) = unitCanonical(b)}.")) (|noZeroDivisors| ((|attribute|) "\\spad{noZeroDivisors} is \\spad{true} if \\spad{x * y \\~~= 0} implies both \\spad{x} and \\spad{y} are non-zero.")) (|rightUnitary| ((|attribute|) "\\spad{rightUnitary} is \\spad{true} if \\spad{x * 1 = x} for all \\spad{x}.")) (|leftUnitary| ((|attribute|) "\\spad{leftUnitary} is \\spad{true} if \\spad{1 * x = x} for all \\spad{x}.")) (|unitsKnown| ((|attribute|) "\\spad{unitsKnown} is \\spad{true} if a monoid (a multiplicative semigroup with a 1) has \\spad{unitsKnown} means that the operation \\spadfun{recip} can only return \"failed\" if its argument is not a unit.")) (|shallowlyMutable| ((|attribute|) "\\spad{shallowlyMutable} is \\spad{true} if its values have immediate components that are updateable (mutable). Note: the properties of any component domain are irrevelant to the \\spad{shallowlyMutable} proper.")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} is \\spad{true} if it has an operation \\spad{\"*\": (D,D) -> D} which is commutative.")) (|finiteAggregate| ((|attribute|) "\\spad{finiteAggregate} is \\spad{true} if it is an aggregate with a finite number of elements."))) 
-((-4427 . T) ((-4429 "*") . T) (-4428 . T) (-4424 . T) (-4422 . T) (-4421 . T) (-4420 . T) (-4425 . T) (-4419 . T) (-4418 . T) (-4417 . T) (-4416 . T) (-4415 . T) (-4423 . T) (-4426 . T) (|NullSquare| . T) (|JacobiIdentity| . T) (-4414 . T)) 
+((-4434 . T) ((-4436 "*") . T) (-4435 . T) (-4431 . T) (-4429 . T) (-4428 . T) (-4427 . T) (-4432 . T) (-4426 . T) (-4425 . T) (-4424 . T) (-4423 . T) (-4422 . T) (-4430 . T) (-4433 . T) (|NullSquare| . T) (|JacobiIdentity| . T) (-4421 . T)) 
 NIL 
 (-99 R) 
 ((|constructor| (NIL "Automorphism \\spad{R} is the multiplicative group of automorphisms of \\spad{R}.")) (|morphism| (($ (|Mapping| |#1| |#1| (|Integer|))) "\\spad{morphism(f)} returns the morphism given by \\spad{f^n(x) = f(x,n)}.") (($ (|Mapping| |#1| |#1|) (|Mapping| |#1| |#1|)) "\\spad{morphism(f, g)} returns the invertible morphism given by \\spad{f},{} where \\spad{g} is the inverse of \\spad{f}..") (($ (|Mapping| |#1| |#1|)) "\\spad{morphism(f)} returns the non-invertible morphism given by \\spad{f}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
 (-100 R UP) 
 ((|constructor| (NIL "This package provides balanced factorisations of polynomials.")) (|balancedFactorisation| (((|Factored| |#2|) |#2| (|List| |#2|)) "\\spad{balancedFactorisation(a, [b1,...,bn])} returns a factorisation \\spad{a = p1^e1 ... pm^em} such that each \\spad{pi} is balanced with respect to \\spad{[b1,...,bm]}.") (((|Factored| |#2|) |#2| |#2|) "\\spad{balancedFactorisation(a, b)} returns a factorisation \\spad{a = p1^e1 ... pm^em} such that each \\spad{pi} is balanced with respect to \\spad{b}."))) 
@@ -342,15 +342,15 @@ NIL
 NIL 
 (-103 S) 
 ((|constructor| (NIL "\\spadtype{BalancedBinaryTree(S)} is the domain of balanced binary trees (bbtree). A balanced binary tree of \\spad{2**k} leaves,{} for some \\spad{k > 0},{} is symmetric,{} that is,{} the left and right subtree of each interior node have identical shape. In general,{} the left and right subtree of a given node can differ by at most leaf node.")) (|mapDown!| (($ $ |#1| (|Mapping| (|List| |#1|) |#1| |#1| |#1|)) "\\spad{mapDown!(t,p,f)} returns \\spad{t} after traversing \\spad{t} in \"preorder\" (node then left then right) fashion replacing the successive interior nodes as follows. Let \\spad{l} and \\spad{r} denote the left and right subtrees of \\spad{t}. The root value \\spad{x} of \\spad{t} is replaced by \\spad{p}. Then \\spad{f}(value \\spad{l},{} value \\spad{r},{} \\spad{p}),{} where \\spad{l} and \\spad{r} denote the left and right subtrees of \\spad{t},{} is evaluated producing two values \\spad{pl} and \\spad{pr}. Then \\spad{mapDown!(l,pl,f)} and \\spad{mapDown!(l,pr,f)} are evaluated.") (($ $ |#1| (|Mapping| |#1| |#1| |#1|)) "\\spad{mapDown!(t,p,f)} returns \\spad{t} after traversing \\spad{t} in \"preorder\" (node then left then right) fashion replacing the successive interior nodes as follows. The root value \\spad{x} is replaced by \\spad{q} \\spad{:=} \\spad{f}(\\spad{p},{}\\spad{x}). The mapDown!(\\spad{l},{}\\spad{q},{}\\spad{f}) and mapDown!(\\spad{r},{}\\spad{q},{}\\spad{f}) are evaluated for the left and right subtrees \\spad{l} and \\spad{r} of \\spad{t}.")) (|mapUp!| (($ $ $ (|Mapping| |#1| |#1| |#1| |#1| |#1|)) "\\spad{mapUp!(t,t1,f)} traverses \\spad{t} in an \"endorder\" (left then right then node) fashion returning \\spad{t} with the value at each successive interior node of \\spad{t} replaced by \\spad{f}(\\spad{l},{}\\spad{r},{}\\spad{l1},{}\\spad{r1}) where \\spad{l} and \\spad{r} are the values at the immediate left and right nodes. Values \\spad{l1} and \\spad{r1} are values at the corresponding nodes of a balanced binary tree \\spad{t1},{} of identical shape at \\spad{t}.") ((|#1| $ (|Mapping| |#1| |#1| |#1|)) "\\spad{mapUp!(t,f)} traverses balanced binary tree \\spad{t} in an \"endorder\" (left then right then node) fashion returning \\spad{t} with the value at each successive interior node of \\spad{t} replaced by \\spad{f}(\\spad{l},{}\\spad{r}) where \\spad{l} and \\spad{r} are the values at the immediate left and right nodes.")) (|setleaves!| (($ $ (|List| |#1|)) "\\spad{setleaves!(t, ls)} sets the leaves of \\spad{t} in left-to-right order to the elements of \\spad{ls}.")) (|balancedBinaryTree| (($ (|NonNegativeInteger|) |#1|) "\\spad{balancedBinaryTree(n, s)} creates a balanced binary tree with \\spad{n} nodes each with value \\spad{s}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-104 R UP M |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{BezoutMatrix} contains functions for computing resultants and discriminants using Bezout matrices.")) (|bezoutDiscriminant| ((|#1| |#2|) "\\spad{bezoutDiscriminant(p)} computes the discriminant of a polynomial \\spad{p} by computing the determinant of a Bezout matrix.")) (|bezoutResultant| ((|#1| |#2| |#2|) "\\spad{bezoutResultant(p,q)} computes the resultant of the two polynomials \\spad{p} and \\spad{q} by computing the determinant of a Bezout matrix.")) (|bezoutMatrix| ((|#3| |#2| |#2|) "\\spad{bezoutMatrix(p,q)} returns the Bezout matrix for the two polynomials \\spad{p} and \\spad{q}.")) (|sylvesterMatrix| ((|#3| |#2| |#2|) "\\spad{sylvesterMatrix(p,q)} returns the Sylvester matrix for the two polynomials \\spad{p} and \\spad{q}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE (-4429 "*")))) 
+((|HasAttribute| |#1| (QUOTE (-4436 "*")))) 
 (-105) 
 ((|bfEntry| (((|Record| (|:| |zeros| (|Stream| (|DoubleFloat|))) (|:| |ones| (|Stream| (|DoubleFloat|))) (|:| |singularities| (|Stream| (|DoubleFloat|)))) (|Symbol|)) "\\spad{bfEntry(k)} returns the entry in the \\axiomType{BasicFunctions} table corresponding to \\spad{k}")) (|bfKeys| (((|List| (|Symbol|))) "\\spad{bfKeys()} returns the names of each function in the \\axiomType{BasicFunctions} table"))) 
-((-4427 . T)) 
+((-4434 . T)) 
 NIL 
 (-106 A S) 
 ((|constructor| (NIL "A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.")) (|inspect| ((|#2| $) "\\spad{inspect(u)} returns an (random) element from a bag.")) (|insert!| (($ |#2| $) "\\spad{insert!(x,u)} inserts item \\spad{x} into bag \\spad{u}.")) (|extract!| ((|#2| $) "\\spad{extract!(u)} destructively removes a (random) item from bag \\spad{u}.")) (|bag| (($ (|List| |#2|)) "\\spad{bag([x,y,...,z])} creates a bag with elements \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.")) (|shallowlyMutable| ((|attribute|) "shallowlyMutable means that elements of bags may be destructively changed."))) 
@@ -358,23 +358,23 @@ NIL
 NIL 
 (-107 S) 
 ((|constructor| (NIL "A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.")) (|inspect| ((|#1| $) "\\spad{inspect(u)} returns an (random) element from a bag.")) (|insert!| (($ |#1| $) "\\spad{insert!(x,u)} inserts item \\spad{x} into bag \\spad{u}.")) (|extract!| ((|#1| $) "\\spad{extract!(u)} destructively removes a (random) item from bag \\spad{u}.")) (|bag| (($ (|List| |#1|)) "\\spad{bag([x,y,...,z])} creates a bag with elements \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.")) (|shallowlyMutable| ((|attribute|) "shallowlyMutable means that elements of bags may be destructively changed."))) 
-((-4428 . T)) 
+((-4435 . T)) 
 NIL 
 (-108) 
 ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating binary expansions.")) (|binary| (($ (|Fraction| (|Integer|))) "\\spad{binary(r)} converts a rational number to a binary expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(b)} returns the fractional part of a binary expansion."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-550) (QUOTE (-914))) (|HasCategory| (-550) (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| (-550) (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-147))) (|HasCategory| (-550) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-550) (QUOTE (-1024))) (|HasCategory| (-550) (QUOTE (-823))) (-3962 (|HasCategory| (-550) (QUOTE (-823))) (|HasCategory| (-550) (QUOTE (-853)))) (|HasCategory| (-550) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-550) (QUOTE (-1155))) (|HasCategory| (-550) (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-550) (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-550) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-550) (QUOTE (-234))) (|HasCategory| (-550) (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| (-550) (LIST (QUOTE -518) (QUOTE (-1181)) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -311) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -288) (QUOTE (-550)) (QUOTE (-550)))) (|HasCategory| (-550) (QUOTE (-309))) (|HasCategory| (-550) (QUOTE (-549))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-550) (LIST (QUOTE -642) (QUOTE (-550)))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-914)))) (-3962 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-914)))) (|HasCategory| (-550) (QUOTE (-145))))) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-551) (QUOTE (-916))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| (-551) (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-147))) (|HasCategory| (-551) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-551) (QUOTE (-1026))) (|HasCategory| (-551) (QUOTE (-825))) (-3969 (|HasCategory| (-551) (QUOTE (-825))) (|HasCategory| (-551) (QUOTE (-855)))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-1157))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-551) (QUOTE (-234))) (|HasCategory| (-551) (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| (-551) (LIST (QUOTE -519) (QUOTE (-1183)) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -312) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -289) (QUOTE (-551)) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-310))) (|HasCategory| (-551) (QUOTE (-550))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-551) (LIST (QUOTE -644) (QUOTE (-551)))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (-3969 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (|HasCategory| (-551) (QUOTE (-145))))) 
 (-109) 
 ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. A `Binding' is a name asosciated with a collection of properties.")) (|binding| (($ (|Identifier|) (|List| (|Property|))) "\\spad{binding(n,props)} constructs a binding with name \\spad{`n'} and property list `props'.")) (|properties| (((|List| (|Property|)) $) "\\spad{properties(b)} returns the properties associated with binding \\spad{b}.")) (|name| (((|Identifier|) $) "\\spad{name(b)} returns the name of binding \\spad{b}"))) 
 NIL 
 NIL 
 (-110) 
 ((|constructor| (NIL "\\spadtype{Bits} provides logical functions for Indexed Bits.")) (|bits| (($ (|NonNegativeInteger|) (|Boolean|)) "\\spad{bits(n,b)} creates bits with \\spad{n} values of \\spad{b}"))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| (-112) (QUOTE (-1105))) (|HasCategory| (-112) (LIST (QUOTE -311) (QUOTE (-112))))) (|HasCategory| (-112) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-112) (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-112) (QUOTE (-1105))) (|HasCategory| (-112) (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| (-112) (QUOTE (-1107))) (|HasCategory| (-112) (LIST (QUOTE -312) (QUOTE (-112))))) (|HasCategory| (-112) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-112) (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-112) (QUOTE (-1107))) (|HasCategory| (-112) (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-111 R S) 
 ((|constructor| (NIL "A \\spadtype{BiModule} is both a left and right module with respect to potentially different rings. \\blankline")) (|rightUnitary| ((|attribute|) "\\spad{x * 1 = x}")) (|leftUnitary| ((|attribute|) "\\spad{1 * x = x}"))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 NIL 
 (-112) 
 ((|constructor| (NIL "\\indented{1}{\\spadtype{Boolean} is the elementary logic with 2 values:} \\spad{true} and \\spad{false}")) (|test| (($ $) "\\spad{test(b)} returns \\spad{b} and is provided for compatibility with the new compiler.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical negation of \\spad{a} or \\spad{b}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical negation of \\spad{a} and \\spad{b}.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical exclusive {\\em or} of Boolean \\spad{a} and \\spad{b}."))) 
@@ -388,22 +388,22 @@ NIL
 ((|constructor| (NIL "This package exports functions to set some commonly used properties of operators,{} including properties which contain functions.")) (|constantOpIfCan| (((|Union| |#1| "failed") (|BasicOperator|)) "\\spad{constantOpIfCan(op)} returns \\spad{a} if \\spad{op} is the constant nullary operator always returning \\spad{a},{} \"failed\" otherwise.")) (|constantOperator| (((|BasicOperator|) |#1|) "\\spad{constantOperator(a)} returns a nullary operator op such that \\spad{op()} always evaluate to \\spad{a}.")) (|derivative| (((|Union| (|List| (|Mapping| |#1| (|List| |#1|))) "failed") (|BasicOperator|)) "\\spad{derivative(op)} returns the value of the \"\\%diff\" property of \\spad{op} if it has one,{} and \"failed\" otherwise.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| |#1|)) "\\spad{derivative(op, foo)} attaches foo as the \"\\%diff\" property of \\spad{op}. If \\spad{op} has an \"\\%diff\" property \\spad{f},{} then applying a derivation \\spad{D} to \\spad{op}(a) returns \\spad{f(a) * D(a)}. Argument \\spad{op} must be unary.") (((|BasicOperator|) (|BasicOperator|) (|List| (|Mapping| |#1| (|List| |#1|)))) "\\spad{derivative(op, [foo1,...,foon])} attaches [foo1,{}...,{}foon] as the \"\\%diff\" property of \\spad{op}. If \\spad{op} has an \"\\%diff\" property \\spad{[f1,...,fn]} then applying a derivation \\spad{D} to \\spad{op(a1,...,an)} returns \\spad{f1(a1,...,an) * D(a1) + ... + fn(a1,...,an) * D(an)}.")) (|evaluate| (((|Union| (|Mapping| |#1| (|List| |#1|)) "failed") (|BasicOperator|)) "\\spad{evaluate(op)} returns the value of the \"\\%eval\" property of \\spad{op} if it has one,{} and \"failed\" otherwise.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| |#1|)) "\\spad{evaluate(op, foo)} attaches foo as the \"\\%eval\" property of \\spad{op}. If \\spad{op} has an \"\\%eval\" property \\spad{f},{} then applying \\spad{op} to a returns the result of \\spad{f(a)}. Argument \\spad{op} must be unary.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| (|List| |#1|))) "\\spad{evaluate(op, foo)} attaches foo as the \"\\%eval\" property of \\spad{op}. If \\spad{op} has an \"\\%eval\" property \\spad{f},{} then applying \\spad{op} to \\spad{(a1,...,an)} returns the result of \\spad{f(a1,...,an)}.") (((|Union| |#1| "failed") (|BasicOperator|) (|List| |#1|)) "\\spad{evaluate(op, [a1,...,an])} checks if \\spad{op} has an \"\\%eval\" property \\spad{f}. If it has,{} then \\spad{f(a1,...,an)} is returned,{} and \"failed\" otherwise."))) 
 NIL 
 NIL 
-(-115 -3498 UP) 
+(-115 -3505 UP) 
 ((|constructor| (NIL "\\spadtype{BoundIntegerRoots} provides functions to find lower bounds on the integer roots of a polynomial.")) (|integerBound| (((|Integer|) |#2|) "\\spad{integerBound(p)} returns a lower bound on the negative integer roots of \\spad{p},{} and 0 if \\spad{p} has no negative integer roots."))) 
 NIL 
 NIL 
 (-116 |p|) 
 ((|constructor| (NIL "Stream-based implementation of \\spad{Zp:} \\spad{p}-adic numbers are represented as sum(\\spad{i} = 0..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in -(\\spad{p} - 1)\\spad{/2},{}...,{}(\\spad{p} - 1)\\spad{/2}."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-117 |p|) 
 ((|constructor| (NIL "Stream-based implementation of \\spad{Qp:} numbers are represented as sum(\\spad{i} = \\spad{k}..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in -(\\spad{p} - 1)\\spad{/2},{}...,{}(\\spad{p} - 1)\\spad{/2}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-116 |#1|) (QUOTE (-914))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| (-116 |#1|) (QUOTE (-145))) (|HasCategory| (-116 |#1|) (QUOTE (-147))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-116 |#1|) (QUOTE (-1024))) (|HasCategory| (-116 |#1|) (QUOTE (-823))) (-3962 (|HasCategory| (-116 |#1|) (QUOTE (-823))) (|HasCategory| (-116 |#1|) (QUOTE (-853)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-116 |#1|) (QUOTE (-1155))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| (-116 |#1|) (QUOTE (-234))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -518) (QUOTE (-1181)) (LIST (QUOTE -116) (|devaluate| |#1|)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -311) (LIST (QUOTE -116) (|devaluate| |#1|)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -288) (LIST (QUOTE -116) (|devaluate| |#1|)) (LIST (QUOTE -116) (|devaluate| |#1|)))) (|HasCategory| (-116 |#1|) (QUOTE (-309))) (|HasCategory| (-116 |#1|) (QUOTE (-549))) (|HasCategory| (-116 |#1|) (QUOTE (-853))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-116 |#1|) (QUOTE (-914)))) (-3962 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-116 |#1|) (QUOTE (-914)))) (|HasCategory| (-116 |#1|) (QUOTE (-145))))) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-116 |#1|) (QUOTE (-916))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| (-116 |#1|) (QUOTE (-145))) (|HasCategory| (-116 |#1|) (QUOTE (-147))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-116 |#1|) (QUOTE (-1026))) (|HasCategory| (-116 |#1|) (QUOTE (-825))) (-3969 (|HasCategory| (-116 |#1|) (QUOTE (-825))) (|HasCategory| (-116 |#1|) (QUOTE (-855)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-116 |#1|) (QUOTE (-1157))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| (-116 |#1|) (QUOTE (-234))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -519) (QUOTE (-1183)) (LIST (QUOTE -116) (|devaluate| |#1|)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -312) (LIST (QUOTE -116) (|devaluate| |#1|)))) (|HasCategory| (-116 |#1|) (LIST (QUOTE -289) (LIST (QUOTE -116) (|devaluate| |#1|)) (LIST (QUOTE -116) (|devaluate| |#1|)))) (|HasCategory| (-116 |#1|) (QUOTE (-310))) (|HasCategory| (-116 |#1|) (QUOTE (-550))) (|HasCategory| (-116 |#1|) (QUOTE (-855))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-116 |#1|) (QUOTE (-916)))) (-3969 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-116 |#1|) (QUOTE (-916)))) (|HasCategory| (-116 |#1|) (QUOTE (-145))))) 
 (-118 A S) 
 ((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "right" $) "\\spad{setelt(a,\"right\",b)} (also written \\axiom{\\spad{b} . right \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setright!(a,{}\\spad{b})}.") (($ $ "left" $) "\\spad{setelt(a,\"left\",b)} (also written \\axiom{a . left \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setleft!(a,{}\\spad{b})}.")) (|right| (($ $) "\\spad{right(a)} returns the right child.")) (|elt| (($ $ "right") "\\spad{elt(a,\"right\")} (also written: \\axiom{a . right}) is equivalent to \\axiom{right(a)}.") (($ $ "left") "\\spad{elt(u,\"left\")} (also written: \\axiom{a . left}) is equivalent to \\axiom{left(a)}.")) (|left| (($ $) "\\spad{left(u)} returns the left child."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4428))) 
+((|HasAttribute| |#1| (QUOTE -4435))) 
 (-119 S) 
 ((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "right" $) "\\spad{setelt(a,\"right\",b)} (also written \\axiom{\\spad{b} . right \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setright!(a,{}\\spad{b})}.") (($ $ "left" $) "\\spad{setelt(a,\"left\",b)} (also written \\axiom{a . left \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setleft!(a,{}\\spad{b})}.")) (|right| (($ $) "\\spad{right(a)} returns the right child.")) (|elt| (($ $ "right") "\\spad{elt(a,\"right\")} (also written: \\axiom{a . right}) is equivalent to \\axiom{right(a)}.") (($ $ "left") "\\spad{elt(u,\"left\")} (also written: \\axiom{a . left}) is equivalent to \\axiom{left(a)}.")) (|left| (($ $) "\\spad{left(u)} returns the left child."))) 
 NIL 
@@ -414,15 +414,15 @@ NIL
 NIL 
 (-121 S) 
 ((|constructor| (NIL "BinarySearchTree(\\spad{S}) is the domain of a binary trees where elements are ordered across the tree. A binary search tree is either empty or has a value which is an \\spad{S},{} and a right and left which are both BinaryTree(\\spad{S}) Elements are ordered across the tree.")) (|split| (((|Record| (|:| |less| $) (|:| |greater| $)) |#1| $) "\\spad{split(x,b)} splits binary tree \\spad{b} into two trees,{} one with elements greater than \\spad{x},{} the other with elements less than \\spad{x}.")) (|insertRoot!| (($ |#1| $) "\\spad{insertRoot!(x,b)} inserts element \\spad{x} as a root of binary search tree \\spad{b}.")) (|insert!| (($ |#1| $) "\\spad{insert!(x,b)} inserts element \\spad{x} as leaves into binary search tree \\spad{b}.")) (|binarySearchTree| (($ (|List| |#1|)) "\\spad{binarySearchTree(l)} \\undocumented"))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-122 S) 
 ((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical {\\em exclusive-or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|or| (($ $ $) "\\spad{a or b} returns the logical {\\em or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|and| (($ $ $) "\\spad{a and b} returns the logical {\\em and} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical {\\em nor} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical {\\em nand} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|not| (($ $) "\\spad{not(b)} returns the logical {\\em not} of bit aggregate \\axiom{\\spad{b}}."))) 
 NIL 
 NIL 
 (-123) 
 ((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical {\\em exclusive-or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|or| (($ $ $) "\\spad{a or b} returns the logical {\\em or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|and| (($ $ $) "\\spad{a and b} returns the logical {\\em and} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical {\\em nor} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical {\\em nand} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|not| (($ $) "\\spad{not(b)} returns the logical {\\em not} of bit aggregate \\axiom{\\spad{b}}."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
 (-124 A S) 
 ((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#2| $) "\\spad{node(left,v,right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}.")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) 
@@ -430,24 +430,24 @@ NIL
 NIL 
 (-125 S) 
 ((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#1| $) "\\spad{node(left,v,right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}.")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
 (-126 S) 
 ((|constructor| (NIL "\\spadtype{BinaryTournament(S)} is the domain of binary trees where elements are ordered down the tree. A binary search tree is either empty or is a node containing a \\spadfun{value} of type \\spad{S},{} and a \\spadfun{right} and a \\spadfun{left} which are both \\spadtype{BinaryTree(S)}")) (|insert!| (($ |#1| $) "\\spad{insert!(x,b)} inserts element \\spad{x} as leaves into binary tournament \\spad{b}.")) (|binaryTournament| (($ (|List| |#1|)) "\\spad{binaryTournament(ls)} creates a binary tournament with the elements of \\spad{ls} as values at the nodes."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-127 S) 
 ((|constructor| (NIL "\\spadtype{BinaryTree(S)} is the domain of all binary trees. A binary tree over \\spad{S} is either empty or has a \\spadfun{value} which is an \\spad{S} and a \\spadfun{right} and \\spadfun{left} which are both binary trees.")) (|binaryTree| (($ $ |#1| $) "\\spad{binaryTree(l,v,r)} creates a binary tree with value \\spad{v} with left subtree \\spad{l} and right subtree \\spad{r}.") (($ |#1|) "\\spad{binaryTree(v)} is an non-empty binary tree with value \\spad{v},{} and left and right empty."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-128) 
 ((|constructor| (NIL "Byte is the datatype of 8-bit sized unsigned integer values.")) (|sample| (($) "\\spad{sample} gives a sample datum of type Byte.")) (|bitior| (($ $ $) "bitor(\\spad{x},{}\\spad{y}) returns the bitwise `inclusive or' of \\spad{`x'} and \\spad{`y'}.")) (|bitand| (($ $ $) "\\spad{bitand(x,y)} returns the bitwise `and' of \\spad{`x'} and \\spad{`y'}.")) (|byte| (($ (|NonNegativeInteger|)) "\\spad{byte(x)} injects the unsigned integer value \\spad{`v'} into the Byte algebra. \\spad{`v'} must be non-negative and less than 256."))) 
 NIL 
 NIL 
 (-129) 
 ((|constructor| (NIL "ByteBuffer provides datatype for buffers of bytes. This domain differs from PrimitiveArray Byte in that it is not as rigid as PrimitiveArray Byte. That is,{} the typical use of ByteBuffer is to pre-allocate a vector of Byte of some capacity \\spad{`n'}. The array can then store up to \\spad{`n'} bytes. The actual interesting bytes count (the length of the buffer) is therefore different from the capacity. The length is no more than the capacity,{} but it can be set dynamically as needed. This functionality is used for example when reading bytes from input/output devices where we use buffers to transfer data in and out of the system. Note: a value of type ByteBuffer is 0-based indexed,{} as opposed \\indented{6}{Vector,{} but not unlike PrimitiveArray Byte.}")) (|finiteAggregate| ((|attribute|) "A ByteBuffer object is a finite aggregate")) (|setLength!| (((|NonNegativeInteger|) $ (|NonNegativeInteger|)) "\\spad{setLength!(buf,n)} sets the number of active bytes in the `buf'. Error if \\spad{`n'} is more than the capacity.")) (|capacity| (((|NonNegativeInteger|) $) "\\spad{capacity(buf)} returns the pre-allocated maximum size of `buf'.")) (|byteBuffer| (($ (|NonNegativeInteger|)) "\\spad{byteBuffer(n)} creates a buffer of capacity \\spad{n},{} and length 0."))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| (-128) (QUOTE (-853))) (|HasCategory| (-128) (LIST (QUOTE -311) (QUOTE (-128))))) (-12 (|HasCategory| (-128) (QUOTE (-1105))) (|HasCategory| (-128) (LIST (QUOTE -311) (QUOTE (-128)))))) (-3962 (-12 (|HasCategory| (-128) (QUOTE (-1105))) (|HasCategory| (-128) (LIST (QUOTE -311) (QUOTE (-128))))) (|HasCategory| (-128) (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| (-128) (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| (-128) (QUOTE (-853))) (|HasCategory| (-128) (QUOTE (-1105)))) (|HasCategory| (-128) (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-128) (QUOTE (-1105))) (|HasCategory| (-128) (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| (-128) (QUOTE (-1105))) (|HasCategory| (-128) (LIST (QUOTE -311) (QUOTE (-128)))))) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| (-128) (QUOTE (-855))) (|HasCategory| (-128) (LIST (QUOTE -312) (QUOTE (-128))))) (-12 (|HasCategory| (-128) (QUOTE (-1107))) (|HasCategory| (-128) (LIST (QUOTE -312) (QUOTE (-128)))))) (-3969 (-12 (|HasCategory| (-128) (QUOTE (-1107))) (|HasCategory| (-128) (LIST (QUOTE -312) (QUOTE (-128))))) (|HasCategory| (-128) (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| (-128) (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| (-128) (QUOTE (-855))) (|HasCategory| (-128) (QUOTE (-1107)))) (|HasCategory| (-128) (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-128) (QUOTE (-1107))) (|HasCategory| (-128) (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| (-128) (QUOTE (-1107))) (|HasCategory| (-128) (LIST (QUOTE -312) (QUOTE (-128)))))) 
 (-130) 
 ((|constructor| (NIL "This datatype describes byte order of machine values stored memory.")) (|unknownEndian| (($) "\\spad{unknownEndian} for none of the above.")) (|bigEndian| (($) "\\spad{bigEndian} describes big endian host")) (|littleEndian| (($) "\\spad{littleEndian} describes little endian host"))) 
 NIL 
@@ -466,13 +466,13 @@ NIL
 NIL 
 (-134) 
 ((|constructor| (NIL "Members of the domain CardinalNumber are values indicating the cardinality of sets,{} both finite and infinite. Arithmetic operations are defined on cardinal numbers as follows. \\blankline If \\spad{x = \\#X} and \\spad{y = \\#Y} then \\indented{2}{\\spad{x+y\\space{2}= \\#(X+Y)}\\space{3}\\tab{30}disjoint union} \\indented{2}{\\spad{x-y\\space{2}= \\#(X-Y)}\\space{3}\\tab{30}relative complement} \\indented{2}{\\spad{x*y\\space{2}= \\#(X*Y)}\\space{3}\\tab{30}cartesian product} \\indented{2}{\\spad{x**y = \\#(X**Y)}\\space{2}\\tab{30}\\spad{X**Y = \\{g| g:Y->X\\}}} \\blankline The non-negative integers have a natural construction as cardinals \\indented{2}{\\spad{0 = \\#\\{\\}},{} \\spad{1 = \\{0\\}},{} \\spad{2 = \\{0, 1\\}},{} ...,{} \\spad{n = \\{i| 0 <= i < n\\}}.} \\blankline That \\spad{0} acts as a zero for the multiplication of cardinals is equivalent to the axiom of choice. \\blankline The generalized continuum hypothesis asserts \\center{\\spad{2**Aleph i = Aleph(i+1)}} and is independent of the axioms of set theory [Goedel 1940]. \\blankline Three commonly encountered cardinal numbers are \\indented{3}{\\spad{a = \\#Z}\\space{7}\\tab{30}countable infinity} \\indented{3}{\\spad{c = \\#R}\\space{7}\\tab{30}the continuum} \\indented{3}{\\spad{f = \\#\\{g| g:[0,1]->R\\}}} \\blankline In this domain,{} these values are obtained using \\indented{3}{\\spad{a := Aleph 0},{} \\spad{c := 2**a},{} \\spad{f := 2**c}.} \\blankline")) (|generalizedContinuumHypothesisAssumed| (((|Boolean|) (|Boolean|)) "\\spad{generalizedContinuumHypothesisAssumed(bool)} is used to dictate whether the hypothesis is to be assumed.")) (|generalizedContinuumHypothesisAssumed?| (((|Boolean|)) "\\spad{generalizedContinuumHypothesisAssumed?()} tests if the hypothesis is currently assumed.")) (|countable?| (((|Boolean|) $) "\\spad{countable?(\\spad{a})} determines whether \\spad{a} is a countable cardinal,{} \\spadignore{i.e.} an integer or \\spad{Aleph 0}.")) (|finite?| (((|Boolean|) $) "\\spad{finite?(\\spad{a})} determines whether \\spad{a} is a finite cardinal,{} \\spadignore{i.e.} an integer.")) (|Aleph| (($ (|NonNegativeInteger|)) "\\spad{Aleph(n)} provides the named (infinite) cardinal number.")) (** (($ $ $) "\\spad{x**y} returns \\spad{\\#(X**Y)} where \\spad{X**Y} is defined \\indented{1}{as \\spad{\\{g| g:Y->X\\}}.}")) (- (((|Union| $ "failed") $ $) "\\spad{x - y} returns an element \\spad{z} such that \\spad{z+y=x} or \"failed\" if no such element exists.")) (|commutative| ((|attribute| "*") "a domain \\spad{D} has \\spad{commutative(\"*\")} if it has an operation \\spad{\"*\": (D,D) -> D} which is commutative."))) 
-(((-4429 "*") . T)) 
+(((-4436 "*") . T)) 
 NIL 
-(-135 |minix| -3023 R) 
+(-135 |minix| -3030 R) 
 ((|constructor| (NIL "CartesianTensor(minix,{}dim,{}\\spad{R}) provides Cartesian tensors with components belonging to a commutative ring \\spad{R}. These tensors can have any number of indices. Each index takes values from \\spad{minix} to \\spad{minix + dim - 1}.")) (|sample| (($) "\\spad{sample()} returns an object of type \\%.")) (|unravel| (($ (|List| |#3|)) "\\spad{unravel(t)} produces a tensor from a list of components such that \\indented{2}{\\spad{unravel(ravel(t)) = t}.}")) (|ravel| (((|List| |#3|) $) "\\spad{ravel(t)} produces a list of components from a tensor such that \\indented{2}{\\spad{unravel(ravel(t)) = t}.}")) (|leviCivitaSymbol| (($) "\\spad{leviCivitaSymbol()} is the rank \\spad{dim} tensor defined by \\spad{leviCivitaSymbol()(i1,...idim) = +1/0/-1} if \\spad{i1,...,idim} is an even/is nota /is an odd permutation of \\spad{minix,...,minix+dim-1}.")) (|kroneckerDelta| (($) "\\spad{kroneckerDelta()} is the rank 2 tensor defined by \\indented{3}{\\spad{kroneckerDelta()(i,j)}} \\indented{6}{\\spad{= 1\\space{2}if i = j}} \\indented{6}{\\spad{= 0 if\\space{2}i \\~= j}}")) (|reindex| (($ $ (|List| (|Integer|))) "\\spad{reindex(t,[i1,...,idim])} permutes the indices of \\spad{t}. For example,{} if \\spad{r = reindex(t, [4,1,2,3])} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank for tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(l,i,j,k)}.}")) (|transpose| (($ $ (|Integer|) (|Integer|)) "\\spad{transpose(t,i,j)} exchanges the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th indices of \\spad{t}. For example,{} if \\spad{r = transpose(t,2,3)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(i,k,j,l)}.}") (($ $) "\\spad{transpose(t)} exchanges the first and last indices of \\spad{t}. For example,{} if \\spad{r = transpose(t)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(l,j,k,i)}.}")) (|contract| (($ $ (|Integer|) (|Integer|)) "\\spad{contract(t,i,j)} is the contraction of tensor \\spad{t} which sums along the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th indices. For example,{} if \\spad{r = contract(t,1,3)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 2 \\spad{(= 4 - 2)} tensor given by \\indented{4}{\\spad{r(i,j) = sum(h=1..dim,t(h,i,h,j))}.}") (($ $ (|Integer|) $ (|Integer|)) "\\spad{contract(t,i,s,j)} is the inner product of tenors \\spad{s} and \\spad{t} which sums along the \\spad{k1}\\spad{-}th index of \\spad{t} and the \\spad{k2}\\spad{-}th index of \\spad{s}. For example,{} if \\spad{r = contract(s,2,t,1)} for rank 3 tensors rank 3 tensors \\spad{s} and \\spad{t},{} then \\spad{r} is the rank 4 \\spad{(= 3 + 3 - 2)} tensor given by \\indented{4}{\\spad{r(i,j,k,l) = sum(h=1..dim,s(i,h,j)*t(h,k,l))}.}")) (* (($ $ $) "\\spad{s*t} is the inner product of the tensors \\spad{s} and \\spad{t} which contracts the last index of \\spad{s} with the first index of \\spad{t},{} \\spadignore{i.e.} \\indented{4}{\\spad{t*s = contract(t,rank t, s, 1)}} \\indented{4}{\\spad{t*s = sum(k=1..N, t[i1,..,iN,k]*s[k,j1,..,jM])}} This is compatible with the use of \\spad{M*v} to denote the matrix-vector inner product.")) (|product| (($ $ $) "\\spad{product(s,t)} is the outer product of the tensors \\spad{s} and \\spad{t}. For example,{} if \\spad{r = product(s,t)} for rank 2 tensors \\spad{s} and \\spad{t},{} then \\spad{r} is a rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = s(i,j)*t(k,l)}.}")) (|elt| ((|#3| $ (|List| (|Integer|))) "\\spad{elt(t,[i1,...,iN])} gives a component of a rank \\spad{N} tensor.") ((|#3| $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{elt(t,i,j,k,l)} gives a component of a rank 4 tensor.") ((|#3| $ (|Integer|) (|Integer|) (|Integer|)) "\\spad{elt(t,i,j,k)} gives a component of a rank 3 tensor.") ((|#3| $ (|Integer|) (|Integer|)) "\\spad{elt(t,i,j)} gives a component of a rank 2 tensor.") ((|#3| $ (|Integer|)) "\\spad{elt(t,i)} gives a component of a rank 1 tensor.") ((|#3| $) "\\spad{elt(t)} gives the component of a rank 0 tensor.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(t)} returns the tensorial rank of \\spad{t} (that is,{} the number of indices). This is the same as the graded module degree.")) (|coerce| (($ (|List| $)) "\\spad{coerce([t_1,...,t_dim])} allows tensors to be constructed using lists.") (($ (|List| |#3|)) "\\spad{coerce([r_1,...,r_dim])} allows tensors to be constructed using lists.") (($ (|SquareMatrix| |#2| |#3|)) "\\spad{coerce(m)} views a matrix as a rank 2 tensor.") (($ (|DirectProduct| |#2| |#3|)) "\\spad{coerce(v)} views a vector as a rank 1 tensor."))) 
 NIL 
 NIL 
-(-136 |minix| -3023 S T$) 
+(-136 |minix| -3030 S T$) 
 ((|constructor| (NIL "This package provides functions to enable conversion of tensors given conversion of the components.")) (|map| (((|CartesianTensor| |#1| |#2| |#4|) (|Mapping| |#4| |#3|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{map(f,ts)} does a componentwise conversion of the tensor \\spad{ts} to a tensor with components of type \\spad{T}.")) (|reshape| (((|CartesianTensor| |#1| |#2| |#4|) (|List| |#4|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{reshape(lt,ts)} organizes the list of components \\spad{lt} into a tensor with the same shape as \\spad{ts}."))) 
 NIL 
 NIL 
@@ -494,8 +494,8 @@ NIL
 NIL 
 (-141) 
 ((|constructor| (NIL "This domain allows classes of characters to be defined and manipulated efficiently.")) (|alphanumeric| (($) "\\spad{alphanumeric()} returns the class of all characters for which \\spadfunFrom{alphanumeric?}{Character} is \\spad{true}.")) (|alphabetic| (($) "\\spad{alphabetic()} returns the class of all characters for which \\spadfunFrom{alphabetic?}{Character} is \\spad{true}.")) (|lowerCase| (($) "\\spad{lowerCase()} returns the class of all characters for which \\spadfunFrom{lowerCase?}{Character} is \\spad{true}.")) (|upperCase| (($) "\\spad{upperCase()} returns the class of all characters for which \\spadfunFrom{upperCase?}{Character} is \\spad{true}.")) (|hexDigit| (($) "\\spad{hexDigit()} returns the class of all characters for which \\spadfunFrom{hexDigit?}{Character} is \\spad{true}.")) (|digit| (($) "\\spad{digit()} returns the class of all characters for which \\spadfunFrom{digit?}{Character} is \\spad{true}.")) (|charClass| (($ (|List| (|Character|))) "\\spad{charClass(l)} creates a character class which contains exactly the characters given in the list \\spad{l}.") (($ (|String|)) "\\spad{charClass(s)} creates a character class which contains exactly the characters given in the string \\spad{s}."))) 
-((-4427 . T) (-4417 . T) (-4428 . T)) 
-((-3962 (-12 (|HasCategory| (-144) (QUOTE (-371))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144))))) (-12 (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144)))))) (|HasCategory| (-144) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-144) (QUOTE (-371))) (|HasCategory| (-144) (QUOTE (-853))) (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144)))))) 
+((-4434 . T) (-4424 . T) (-4435 . T)) 
+((-3969 (-12 (|HasCategory| (-144) (QUOTE (-372))) (|HasCategory| (-144) (LIST (QUOTE -312) (QUOTE (-144))))) (-12 (|HasCategory| (-144) (QUOTE (-1107))) (|HasCategory| (-144) (LIST (QUOTE -312) (QUOTE (-144)))))) (|HasCategory| (-144) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-144) (QUOTE (-372))) (|HasCategory| (-144) (QUOTE (-855))) (|HasCategory| (-144) (QUOTE (-1107))) (|HasCategory| (-144) (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| (-144) (QUOTE (-1107))) (|HasCategory| (-144) (LIST (QUOTE -312) (QUOTE (-144)))))) 
 (-142 R Q A) 
 ((|constructor| (NIL "CommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#3|) "\\spad{splitDenominator([q1,...,qn])} returns \\spad{[[p1,...,pn], d]} such that \\spad{qi = pi/d} and \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|clearDenominator| ((|#3| |#3|) "\\spad{clearDenominator([q1,...,qn])} returns \\spad{[p1,...,pn]} such that \\spad{qi = pi/d} where \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|commonDenominator| ((|#1| |#3|) "\\spad{commonDenominator([q1,...,qn])} returns a common denominator \\spad{d} for \\spad{q1},{}...,{}\\spad{qn}."))) 
 NIL 
@@ -510,7 +510,7 @@ NIL
 NIL 
 (-145) 
 ((|constructor| (NIL "Rings of Characteristic Non Zero")) (|charthRoot| (((|Union| $ "failed") $) "\\spad{charthRoot(x)} returns the \\spad{p}th root of \\spad{x} where \\spad{p} is the characteristic of the ring."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
 (-146 R) 
 ((|constructor| (NIL "This package provides a characteristicPolynomial function for any matrix over a commutative ring.")) (|characteristicPolynomial| ((|#1| (|Matrix| |#1|) |#1|) "\\spad{characteristicPolynomial(m,r)} computes the characteristic polynomial of the matrix \\spad{m} evaluated at the point \\spad{r}. In particular,{} if \\spad{r} is the polynomial \\spad{'x},{} then it returns the characteristic polynomial expressed as a polynomial in \\spad{'x}."))) 
@@ -518,9 +518,9 @@ NIL
 NIL 
 (-147) 
 ((|constructor| (NIL "Rings of Characteristic Zero."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-148 -3498 UP UPUP) 
+(-148 -3505 UP UPUP) 
 ((|constructor| (NIL "Tools to send a point to infinity on an algebraic curve.")) (|chvar| (((|Record| (|:| |func| |#3|) (|:| |poly| |#3|) (|:| |c1| (|Fraction| |#2|)) (|:| |c2| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) |#3| |#3|) "\\spad{chvar(f(x,y), p(x,y))} returns \\spad{[g(z,t), q(z,t), c1(z), c2(z), n]} such that under the change of variable \\spad{x = c1(z)},{} \\spad{y = t * c2(z)},{} one gets \\spad{f(x,y) = g(z,t)}. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x, y) = 0}. The algebraic relation between \\spad{z} and \\spad{t} is \\spad{q(z, t) = 0}.")) (|eval| ((|#3| |#3| (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{eval(p(x,y), f(x), g(x))} returns \\spad{p(f(x), y * g(x))}.")) (|goodPoint| ((|#1| |#3| |#3|) "\\spad{goodPoint(p, q)} returns an integer a such that a is neither a pole of \\spad{p(x,y)} nor a branch point of \\spad{q(x,y) = 0}.")) (|rootPoly| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| (|Fraction| |#2|)) (|:| |radicand| |#2|)) (|Fraction| |#2|) (|NonNegativeInteger|)) "\\spad{rootPoly(g, n)} returns \\spad{[m, c, P]} such that \\spad{c * g ** (1/n) = P ** (1/m)} thus if \\spad{y**n = g},{} then \\spad{z**m = P} where \\spad{z = c * y}.")) (|radPoly| (((|Union| (|Record| (|:| |radicand| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) "failed") |#3|) "\\spad{radPoly(p(x, y))} returns \\spad{[c(x), n]} if \\spad{p} is of the form \\spad{y**n - c(x)},{} \"failed\" otherwise.")) (|mkIntegral| (((|Record| (|:| |coef| (|Fraction| |#2|)) (|:| |poly| |#3|)) |#3|) "\\spad{mkIntegral(p(x,y))} returns \\spad{[c(x), q(x,z)]} such that \\spad{z = c * y} is integral. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x, y) = 0}. The algebraic relation between \\spad{x} and \\spad{z} is \\spad{q(x, z) = 0}."))) 
 NIL 
 NIL 
@@ -531,14 +531,14 @@ NIL
 (-150 A S) 
 ((|constructor| (NIL "A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \\spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \\spadignore{e.g.} \\spadfun{list} for \\spadtype{List},{} \\spadfun{flexibleArray} for \\spadtype{FlexibleArray},{} and so on.")) (|removeDuplicates| (($ $) "\\spad{removeDuplicates(u)} returns a copy of \\spad{u} with all duplicates removed.")) (|select| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select(p,u)} returns a copy of \\spad{u} containing only those elements such \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{select(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})]}.")) (|remove| (($ |#2| $) "\\spad{remove(x,u)} returns a copy of \\spad{u} with all elements \\axiom{\\spad{y} = \\spad{x}} removed. Note: \\axiom{remove(\\spad{y},{}\\spad{c}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{c} | \\spad{x} \\spad{~=} \\spad{y}]}.") (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove(p,u)} returns a copy of \\spad{u} removing all elements \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{remove(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | not \\spad{p}(\\spad{x})]}.")) (|reduce| ((|#2| (|Mapping| |#2| |#2| |#2|) $ |#2| |#2|) "\\spad{reduce(f,u,x,z)} reduces the binary operation \\spad{f} across \\spad{u},{} stopping when an \"absorbing element\" \\spad{z} is encountered. As for \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})},{} \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})} when \\spad{u} contains no element \\spad{z}. Thus the third argument \\spad{x} is returned when \\spad{u} is empty.") ((|#2| (|Mapping| |#2| |#2| |#2|) $ |#2|) "\\spad{reduce(f,u,x)} reduces the binary operation \\spad{f} across \\spad{u},{} where \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u})} if \\spad{u} has 2 or more elements. Returns \\axiom{\\spad{f}(\\spad{x},{}\\spad{y})} if \\spad{u} has one element \\spad{y},{} \\spad{x} if \\spad{u} is empty. For example,{} \\axiom{reduce(+,{}\\spad{u},{}0)} returns the sum of the elements of \\spad{u}.") ((|#2| (|Mapping| |#2| |#2| |#2|) $) "\\spad{reduce(f,u)} reduces the binary operation \\spad{f} across \\spad{u}. For example,{} if \\spad{u} is \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]} then \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\axiom{\\spad{f}(..\\spad{f}(\\spad{f}(\\spad{x},{}\\spad{y}),{}...),{}\\spad{z})}. Note: if \\spad{u} has one element \\spad{x},{} \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\spad{x}. Error: if \\spad{u} is empty.")) (|find| (((|Union| |#2| "failed") (|Mapping| (|Boolean|) |#2|) $) "\\spad{find(p,u)} returns the first \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \"failed\" otherwise.")) (|construct| (($ (|List| |#2|)) "\\axiom{construct(\\spad{x},{}\\spad{y},{}...,{}\\spad{z})} returns the collection of elements \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}} ordered as given. Equivalently written as \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]\\$\\spad{D}},{} where \\spad{D} is the domain. \\spad{D} may be omitted for those of type List."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasAttribute| |#1| (QUOTE -4427))) 
+((|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasAttribute| |#1| (QUOTE -4434))) 
 (-151 S) 
 ((|constructor| (NIL "A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \\spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \\spadignore{e.g.} \\spadfun{list} for \\spadtype{List},{} \\spadfun{flexibleArray} for \\spadtype{FlexibleArray},{} and so on.")) (|removeDuplicates| (($ $) "\\spad{removeDuplicates(u)} returns a copy of \\spad{u} with all duplicates removed.")) (|select| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select(p,u)} returns a copy of \\spad{u} containing only those elements such \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{select(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})]}.")) (|remove| (($ |#1| $) "\\spad{remove(x,u)} returns a copy of \\spad{u} with all elements \\axiom{\\spad{y} = \\spad{x}} removed. Note: \\axiom{remove(\\spad{y},{}\\spad{c}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{c} | \\spad{x} \\spad{~=} \\spad{y}]}.") (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove(p,u)} returns a copy of \\spad{u} removing all elements \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{remove(\\spad{p},{}\\spad{u}) \\spad{==} [\\spad{x} for \\spad{x} in \\spad{u} | not \\spad{p}(\\spad{x})]}.")) (|reduce| ((|#1| (|Mapping| |#1| |#1| |#1|) $ |#1| |#1|) "\\spad{reduce(f,u,x,z)} reduces the binary operation \\spad{f} across \\spad{u},{} stopping when an \"absorbing element\" \\spad{z} is encountered. As for \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})},{} \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})} when \\spad{u} contains no element \\spad{z}. Thus the third argument \\spad{x} is returned when \\spad{u} is empty.") ((|#1| (|Mapping| |#1| |#1| |#1|) $ |#1|) "\\spad{reduce(f,u,x)} reduces the binary operation \\spad{f} across \\spad{u},{} where \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u})} if \\spad{u} has 2 or more elements. Returns \\axiom{\\spad{f}(\\spad{x},{}\\spad{y})} if \\spad{u} has one element \\spad{y},{} \\spad{x} if \\spad{u} is empty. For example,{} \\axiom{reduce(+,{}\\spad{u},{}0)} returns the sum of the elements of \\spad{u}.") ((|#1| (|Mapping| |#1| |#1| |#1|) $) "\\spad{reduce(f,u)} reduces the binary operation \\spad{f} across \\spad{u}. For example,{} if \\spad{u} is \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]} then \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\axiom{\\spad{f}(..\\spad{f}(\\spad{f}(\\spad{x},{}\\spad{y}),{}...),{}\\spad{z})}. Note: if \\spad{u} has one element \\spad{x},{} \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\spad{x}. Error: if \\spad{u} is empty.")) (|find| (((|Union| |#1| "failed") (|Mapping| (|Boolean|) |#1|) $) "\\spad{find(p,u)} returns the first \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \"failed\" otherwise.")) (|construct| (($ (|List| |#1|)) "\\axiom{construct(\\spad{x},{}\\spad{y},{}...,{}\\spad{z})} returns the collection of elements \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}} ordered as given. Equivalently written as \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]\\$\\spad{D}},{} where \\spad{D} is the domain. \\spad{D} may be omitted for those of type List."))) 
 NIL 
 NIL 
 (-152 |n| K Q) 
 ((|constructor| (NIL "CliffordAlgebra(\\spad{n},{} \\spad{K},{} \\spad{Q}) defines a vector space of dimension \\spad{2**n} over \\spad{K},{} given a quadratic form \\spad{Q} on \\spad{K**n}. \\blankline If \\spad{e[i]},{} \\spad{1<=i<=n} is a basis for \\spad{K**n} then \\indented{3}{1,{} \\spad{e[i]} (\\spad{1<=i<=n}),{} \\spad{e[i1]*e[i2]}} (\\spad{1<=i1<i2<=n}),{}...,{}\\spad{e[1]*e[2]*..*e[n]} is a basis for the Clifford Algebra. \\blankline The algebra is defined by the relations \\indented{3}{\\spad{e[i]*e[j] = -e[j]*e[i]}\\space{2}(\\spad{i \\~~= j}),{}} \\indented{3}{\\spad{e[i]*e[i] = Q(e[i])}} \\blankline Examples of Clifford Algebras are: gaussians,{} quaternions,{} exterior algebras and spin algebras.")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} computes the multiplicative inverse of \\spad{x} or \"failed\" if \\spad{x} is not invertible.")) (|coefficient| ((|#2| $ (|List| (|PositiveInteger|))) "\\spad{coefficient(x,[i1,i2,...,iN])} extracts the coefficient of \\spad{e(i1)*e(i2)*...*e(iN)} in \\spad{x}.")) (|monomial| (($ |#2| (|List| (|PositiveInteger|))) "\\spad{monomial(c,[i1,i2,...,iN])} produces the value given by \\spad{c*e(i1)*e(i2)*...*e(iN)}.")) (|e| (($ (|PositiveInteger|)) "\\spad{e(n)} produces the appropriate unit element."))) 
-((-4422 . T) (-4421 . T) (-4424 . T)) 
+((-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
 (-153) 
 ((|constructor| (NIL "\\indented{1}{The purpose of this package is to provide reasonable plots of} functions with singularities.")) (|clipWithRanges| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|)))) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{clipWithRanges(pointLists,xMin,xMax,yMin,yMax)} performs clipping on a list of lists of points,{} \\spad{pointLists}. Clipping is done within the specified ranges of \\spad{xMin},{} \\spad{xMax} and \\spad{yMin},{} \\spad{yMax}. This function is used internally by the \\fakeAxiomFun{iClipParametric} subroutine in this package.")) (|clipParametric| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|) (|Fraction| (|Integer|)) (|Fraction| (|Integer|))) "\\spad{clipParametric(p,frac,sc)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)}; the fraction parameter is specified by \\spad{frac} and the scale parameter is specified by \\spad{sc} for use in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|)) "\\spad{clipParametric(p)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)}; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.")) (|clip| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{clip(ll)} performs two-dimensional clipping on a list of lists of points,{} \\spad{ll}; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|Point| (|DoubleFloat|)))) "\\spad{clip(l)} performs two-dimensional clipping on a curve \\spad{l},{} which is a list of points; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|) (|Fraction| (|Integer|)) (|Fraction| (|Integer|))) "\\spad{clip(p,frac,sc)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the graph of one variable \\spad{y = f(x)}; the fraction parameter is specified by \\spad{frac} and the scale parameter is specified by \\spad{sc} for use in the \\spadfun{clip} function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|)) "\\spad{clip(p)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the graph of one variable,{} \\spad{y = f(x)}; the default parameters \\spad{1/4} for the fraction and \\spad{5/1} for the scale are used in the \\spadfun{clip} function."))) 
@@ -560,7 +560,7 @@ NIL
 ((|constructor| (NIL "Color() specifies a domain of 27 colors provided in the \\Language{} system (the colors mix additively).")) (|color| (($ (|Integer|)) "\\spad{color(i)} returns a color of the indicated hue \\spad{i}.")) (|numberOfHues| (((|PositiveInteger|)) "\\spad{numberOfHues()} returns the number of total hues,{} set in totalHues.")) (|hue| (((|Integer|) $) "\\spad{hue(c)} returns the hue index of the indicated color \\spad{c}.")) (|blue| (($) "\\spad{blue()} returns the position of the blue hue from total hues.")) (|green| (($) "\\spad{green()} returns the position of the green hue from total hues.")) (|yellow| (($) "\\spad{yellow()} returns the position of the yellow hue from total hues.")) (|red| (($) "\\spad{red()} returns the position of the red hue from total hues.")) (+ (($ $ $) "\\spad{c1 + c2} additively mixes the two colors \\spad{c1} and \\spad{c2}.")) (* (($ (|DoubleFloat|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}.") (($ (|PositiveInteger|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}."))) 
 NIL 
 NIL 
-(-158 R -3498) 
+(-158 R -3505) 
 ((|constructor| (NIL "Provides combinatorial functions over an integral domain.")) (|ipow| ((|#2| (|List| |#2|)) "\\spad{ipow(l)} should be local but conditional.")) (|iidprod| ((|#2| (|List| |#2|)) "\\spad{iidprod(l)} should be local but conditional.")) (|iidsum| ((|#2| (|List| |#2|)) "\\spad{iidsum(l)} should be local but conditional.")) (|iipow| ((|#2| (|List| |#2|)) "\\spad{iipow(l)} should be local but conditional.")) (|iiperm| ((|#2| (|List| |#2|)) "\\spad{iiperm(l)} should be local but conditional.")) (|iibinom| ((|#2| (|List| |#2|)) "\\spad{iibinom(l)} should be local but conditional.")) (|iifact| ((|#2| |#2|) "\\spad{iifact(x)} should be local but conditional.")) (|product| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{product(f(n), n = a..b)} returns \\spad{f}(a) * ... * \\spad{f}(\\spad{b}) as a formal product.") ((|#2| |#2| (|Symbol|)) "\\spad{product(f(n), n)} returns the formal product \\spad{P}(\\spad{n}) which verifies \\spad{P}(\\spad{n+1})\\spad{/P}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|summation| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{summation(f(n), n = a..b)} returns \\spad{f}(a) + ... + \\spad{f}(\\spad{b}) as a formal sum.") ((|#2| |#2| (|Symbol|)) "\\spad{summation(f(n), n)} returns the formal sum \\spad{S}(\\spad{n}) which verifies \\spad{S}(\\spad{n+1}) - \\spad{S}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|factorials| ((|#2| |#2| (|Symbol|)) "\\spad{factorials(f, x)} rewrites the permutations and binomials in \\spad{f} involving \\spad{x} in terms of factorials.") ((|#2| |#2|) "\\spad{factorials(f)} rewrites the permutations and binomials in \\spad{f} in terms of factorials.")) (|factorial| ((|#2| |#2|) "\\spad{factorial(n)} returns the factorial of \\spad{n},{} \\spadignore{i.e.} \\spad{n!}.")) (|permutation| ((|#2| |#2| |#2|) "\\spad{permutation(n, r)} returns the number of permutations of \\spad{n} objects taken \\spad{r} at a time,{} \\spadignore{i.e.} \\spad{n!/}(\\spad{n}-\\spad{r})!.")) (|binomial| ((|#2| |#2| |#2|) "\\spad{binomial(n, r)} returns the number of subsets of \\spad{r} objects taken among \\spad{n} objects,{} \\spadignore{i.e.} \\spad{n!/}(\\spad{r!} * (\\spad{n}-\\spad{r})!).")) (** ((|#2| |#2| |#2|) "\\spad{a ** b} is the formal exponential a**b.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}; error if \\spad{op} is not a combinatorial operator.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is a combinatorial operator."))) 
 NIL 
 NIL 
@@ -591,23 +591,23 @@ NIL
 (-165 S R) 
 ((|constructor| (NIL "This category represents the extension of a ring by a square root of \\spad{-1}.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} or \"failed\" if \\spad{x} is not a rational number.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a rational number.")) (|polarCoordinates| (((|Record| (|:| |r| |#2|) (|:| |phi| |#2|)) $) "\\spad{polarCoordinates(x)} returns (\\spad{r},{} phi) such that \\spad{x} = \\spad{r} * exp(\\%\\spad{i} * phi).")) (|argument| ((|#2| $) "\\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\\spad{x}).")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x} = sqrt(norm(\\spad{x})).")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(x, r)} returns the exact quotient of \\spad{x} by \\spad{r},{} or \"failed\" if \\spad{r} does not divide \\spad{x} exactly.")) (|norm| ((|#2| $) "\\spad{norm(x)} returns \\spad{x} * conjugate(\\spad{x})")) (|real| ((|#2| $) "\\spad{real(x)} returns real part of \\spad{x}.")) (|imag| ((|#2| $) "\\spad{imag(x)} returns imaginary part of \\spad{x}.")) (|conjugate| (($ $) "\\spad{conjugate(x + \\%i y)} returns \\spad{x} - \\%\\spad{i} \\spad{y}.")) (|imaginary| (($) "\\spad{imaginary()} = sqrt(\\spad{-1}) = \\%\\spad{i}.")) (|complex| (($ |#2| |#2|) "\\spad{complex(x,y)} constructs \\spad{x} + \\%i*y.") ((|attribute|) "indicates that \\% has sqrt(\\spad{-1})"))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| |#2| (QUOTE (-549))) (|HasCategory| |#2| (QUOTE (-1006))) (|HasCategory| |#2| (QUOTE (-1206))) (|HasCategory| |#2| (QUOTE (-1064))) (|HasCategory| |#2| (QUOTE (-1024))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (QUOTE (-366))) (|HasAttribute| |#2| (QUOTE -4423)) (|HasAttribute| |#2| (QUOTE -4426)) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-561)))) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-1008))) (|HasCategory| |#2| (QUOTE (-1208))) (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (QUOTE (-1026))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (QUOTE (-367))) (|HasAttribute| |#2| (QUOTE -4430)) (|HasAttribute| |#2| (QUOTE -4433)) (|HasCategory| |#2| (QUOTE (-310))) (|HasCategory| |#2| (QUOTE (-562)))) 
 (-166 R) 
 ((|constructor| (NIL "This category represents the extension of a ring by a square root of \\spad{-1}.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} or \"failed\" if \\spad{x} is not a rational number.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a rational number.")) (|polarCoordinates| (((|Record| (|:| |r| |#1|) (|:| |phi| |#1|)) $) "\\spad{polarCoordinates(x)} returns (\\spad{r},{} phi) such that \\spad{x} = \\spad{r} * exp(\\%\\spad{i} * phi).")) (|argument| ((|#1| $) "\\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\\spad{x}).")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x} = sqrt(norm(\\spad{x})).")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(x, r)} returns the exact quotient of \\spad{x} by \\spad{r},{} or \"failed\" if \\spad{r} does not divide \\spad{x} exactly.")) (|norm| ((|#1| $) "\\spad{norm(x)} returns \\spad{x} * conjugate(\\spad{x})")) (|real| ((|#1| $) "\\spad{real(x)} returns real part of \\spad{x}.")) (|imag| ((|#1| $) "\\spad{imag(x)} returns imaginary part of \\spad{x}.")) (|conjugate| (($ $) "\\spad{conjugate(x + \\%i y)} returns \\spad{x} - \\%\\spad{i} \\spad{y}.")) (|imaginary| (($) "\\spad{imaginary()} = sqrt(\\spad{-1}) = \\%\\spad{i}.")) (|complex| (($ |#1| |#1|) "\\spad{complex(x,y)} constructs \\spad{x} + \\%i*y.") ((|attribute|) "indicates that \\% has sqrt(\\spad{-1})"))) 
-((-4420 -3962 (|has| |#1| (-561)) (-12 (|has| |#1| (-309)) (|has| |#1| (-914)))) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4423 |has| |#1| (-6 -4423)) (-4426 |has| |#1| (-6 -4426)) (-1464 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 -3969 (|has| |#1| (-562)) (-12 (|has| |#1| (-310)) (|has| |#1| (-916)))) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4430 |has| |#1| (-6 -4430)) (-4433 |has| |#1| (-6 -4433)) (-1466 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-167 RR PR) 
 ((|constructor| (NIL "\\indented{1}{Author:} Date Created: Date Last Updated: Basic Functions: Related Constructors: Complex,{} UnivariatePolynomial Also See: AMS Classifications: Keywords: complex,{} polynomial factorization,{} factor References:")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} factorizes the polynomial \\spad{p} with complex coefficients."))) 
 NIL 
 NIL 
 (-168) 
-((|constructor| (NIL "This package implements a Spad compiler.")) (|macroExpand| (((|Syntax|) (|Syntax|) (|Environment|)) "\\spad{macroExpand(s,e)} traverses the syntax object \\spad{s} replacing all (niladic) macro invokations with the corresponding substitution."))) 
+((|constructor| (NIL "This package implements a Spad compiler.")) (|elaborate| ((|Elaboration| (|Syntax|)) "\\spad{elaborate(s)} returns the elaboration of the syntax object \\spad{s} in the empty environement.")) (|macroExpand| (((|Syntax|) (|Syntax|) (|Environment|)) "\\spad{macroExpand(s,e)} traverses the syntax object \\spad{s} replacing all (niladic) macro invokations with the corresponding substitution."))) 
 NIL 
 NIL 
 (-169 R) 
 ((|constructor| (NIL "\\spadtype {Complex(R)} creates the domain of elements of the form \\spad{a + b * i} where \\spad{a} and \\spad{b} come from the ring \\spad{R},{} and \\spad{i} is a new element such that \\spad{i**2 = -1}."))) 
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(-916)))) (-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-916)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-916))))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (QUOTE (-1008))) (|HasCategory| |#1| (QUOTE (-1208)))) (|HasCategory| |#1| (QUOTE (-1208))) (|HasCategory| |#1| (QUOTE (-1026))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-562)))) (-3969 (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#1| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -289) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-826))) (|HasCategory| |#1| (QUOTE (-1066))) (-12 (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (QUOTE (-1208)))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-916))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-367)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-234))) (-12 (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasAttribute| |#1| (QUOTE -4430)) (|HasAttribute| |#1| (QUOTE -4433)) (-12 (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (QUOTE (-367)))) (-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-354))))) 
 (-170 R S) 
 ((|constructor| (NIL "This package extends maps from underlying rings to maps between complex over those rings.")) (|map| (((|Complex| |#2|) (|Mapping| |#2| |#1|) (|Complex| |#1|)) "\\spad{map(f,u)} maps \\spad{f} onto real and imaginary parts of \\spad{u}."))) 
 NIL 
@@ -622,7 +622,7 @@ NIL
 NIL 
 (-173) 
 ((|constructor| (NIL "The category of commutative rings with unity,{} \\spadignore{i.e.} rings where \\spadop{*} is commutative,{} and which have a multiplicative identity. element.")) (|commutative| ((|attribute| "*") "multiplication is commutative."))) 
-(((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-174) 
 ((|constructor| (NIL "This category is the root of the I/O conduits.")) (|close!| (($ $) "\\spad{close!(c)} closes the conduit \\spad{c},{} changing its state to one that is invalid for future read or write operations."))) 
@@ -630,7 +630,7 @@ NIL
 NIL 
 (-175 R) 
 ((|constructor| (NIL "\\spadtype{ContinuedFraction} implements general \\indented{1}{continued fractions.\\space{2}This version is not restricted to simple,{}} \\indented{1}{finite fractions and uses the \\spadtype{Stream} as a} \\indented{1}{representation.\\space{2}The arithmetic functions assume that the} \\indented{1}{approximants alternate below/above the convergence point.} \\indented{1}{This is enforced by ensuring the partial numerators and partial} \\indented{1}{denominators are greater than 0 in the Euclidean domain view of \\spad{R}} \\indented{1}{(\\spadignore{i.e.} \\spad{sizeLess?(0, x)}).}")) (|complete| (($ $) "\\spad{complete(x)} causes all entries in \\spadvar{\\spad{x}} to be computed. Normally entries are only computed as needed. If \\spadvar{\\spad{x}} is an infinite continued fraction,{} a user-initiated interrupt is necessary to stop the computation.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(x,n)} causes the first \\spadvar{\\spad{n}} entries in the continued fraction \\spadvar{\\spad{x}} to be computed. Normally entries are only computed as needed.")) (|denominators| (((|Stream| |#1|) $) "\\spad{denominators(x)} returns the stream of denominators of the approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|numerators| (((|Stream| |#1|) $) "\\spad{numerators(x)} returns the stream of numerators of the approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|convergents| (((|Stream| (|Fraction| |#1|)) $) "\\spad{convergents(x)} returns the stream of the convergents of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|approximants| (((|Stream| (|Fraction| |#1|)) $) "\\spad{approximants(x)} returns the stream of approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be infinite and periodic with period 1.")) (|reducedForm| (($ $) "\\spad{reducedForm(x)} puts the continued fraction \\spadvar{\\spad{x}} in reduced form,{} \\spadignore{i.e.} the function returns an equivalent continued fraction of the form \\spad{continuedFraction(b0,[1,1,1,...],[b1,b2,b3,...])}.")) (|wholePart| ((|#1| $) "\\spad{wholePart(x)} extracts the whole part of \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{wholePart(x) = b0}.")) (|partialQuotients| (((|Stream| |#1|) $) "\\spad{partialQuotients(x)} extracts the partial quotients in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{partialQuotients(x) = [b0,b1,b2,b3,...]}.")) (|partialDenominators| (((|Stream| |#1|) $) "\\spad{partialDenominators(x)} extracts the denominators in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{partialDenominators(x) = [b1,b2,b3,...]}.")) (|partialNumerators| (((|Stream| |#1|) $) "\\spad{partialNumerators(x)} extracts the numerators in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{partialNumerators(x) = [a1,a2,a3,...]}.")) (|reducedContinuedFraction| (($ |#1| (|Stream| |#1|)) "\\spad{reducedContinuedFraction(b0,b)} constructs a continued fraction in the following way: if \\spad{b = [b1,b2,...]} then the result is the continued fraction \\spad{b0 + 1/(b1 + 1/(b2 + ...))}. That is,{} the result is the same as \\spad{continuedFraction(b0,[1,1,1,...],[b1,b2,b3,...])}.")) (|continuedFraction| (($ |#1| (|Stream| |#1|) (|Stream| |#1|)) "\\spad{continuedFraction(b0,a,b)} constructs a continued fraction in the following way: if \\spad{a = [a1,a2,...]} and \\spad{b = [b1,b2,...]} then the result is the continued fraction \\spad{b0 + a1/(b1 + a2/(b2 + ...))}.") (($ (|Fraction| |#1|)) "\\spad{continuedFraction(r)} converts the fraction \\spadvar{\\spad{r}} with components of type \\spad{R} to a continued fraction over \\spad{R}."))) 
-(((-4429 "*") . T) (-4420 . T) (-4425 . T) (-4419 . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") . T) (-4427 . T) (-4432 . T) (-4426 . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-176) 
 ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. A `Contour' a list of bindings making up a `virtual scope'.")) (|findBinding| (((|Maybe| (|Binding|)) (|Identifier|) $) "\\spad{findBinding(c,n)} returns the first binding associated with \\spad{`n'}. Otherwise `nothing.")) (|push| (($ (|Binding|) $) "\\spad{push(c,b)} augments the contour with binding \\spad{`b'}.")) (|bindings| (((|List| (|Binding|)) $) "\\spad{bindings(c)} returns the list of bindings in countour \\spad{c}."))) 
@@ -647,7 +647,7 @@ NIL
 (-179 R S CS) 
 ((|constructor| (NIL "This package supports matching patterns involving complex expressions")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(cexpr, pat, res)} matches the pattern \\spad{pat} to the complex expression \\spad{cexpr}. res contains the variables of \\spad{pat} which are already matched and their matches."))) 
 NIL 
-((|HasCategory| (-950 |#2|) (LIST (QUOTE -890) (|devaluate| |#1|)))) 
+((|HasCategory| (-952 |#2|) (LIST (QUOTE -892) (|devaluate| |#1|)))) 
 (-180 R) 
 ((|constructor| (NIL "This package \\undocumented{}")) (|multiEuclideanTree| (((|List| |#1|) (|List| |#1|) |#1|) "\\spad{multiEuclideanTree(l,r)} \\undocumented{}")) (|chineseRemainder| (((|List| |#1|) (|List| (|List| |#1|)) (|List| |#1|)) "\\spad{chineseRemainder(llv,lm)} returns a list of values,{} each of which corresponds to the Chinese remainder of the associated element of \\axiom{\\spad{llv}} and axiom{\\spad{lm}}. This is more efficient than applying chineseRemainder several times.") ((|#1| (|List| |#1|) (|List| |#1|)) "\\spad{chineseRemainder(lv,lm)} returns a value \\axiom{\\spad{v}} such that,{} if \\spad{x} is \\axiom{\\spad{lv}.\\spad{i}} modulo \\axiom{\\spad{lm}.\\spad{i}} for all \\axiom{\\spad{i}},{} then \\spad{x} is \\axiom{\\spad{v}} modulo \\axiom{\\spad{lm}(1)\\spad{*lm}(2)*...\\spad{*lm}(\\spad{n})}.")) (|modTree| (((|List| |#1|) |#1| (|List| |#1|)) "\\spad{modTree(r,l)} \\undocumented{}"))) 
 NIL 
@@ -684,7 +684,7 @@ NIL
 ((|constructor| (NIL "This domain enumerates the three kinds of constructors available in OpenAxiom: category constructors,{} domain constructors,{} and package constructors.")) (|package| (($) "`package' is the kind of package constructors.")) (|domain| (($) "`domain' is the kind of domain constructors")) (|category| (($) "`category' is the kind of category constructors"))) 
 NIL 
 NIL 
-(-189 R -3498) 
+(-189 R -3505) 
 ((|constructor| (NIL "\\spadtype{ComplexTrigonometricManipulations} provides function that compute the real and imaginary parts of complex functions.")) (|complexForm| (((|Complex| (|Expression| |#1|)) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f, imag f]}.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| (((|Expression| |#1|) |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| (((|Expression| |#1|) |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f, x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f, x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) 
 NIL 
 NIL 
@@ -792,23 +792,23 @@ NIL
 ((|constructor| (NIL "\\indented{1}{This domain implements a simple view of a database whose fields are} indexed by symbols")) (- (($ $ $) "\\spad{db1-db2} returns the difference of databases \\spad{db1} and \\spad{db2} \\spadignore{i.e.} consisting of elements in \\spad{db1} but not in \\spad{db2}")) (+ (($ $ $) "\\spad{db1+db2} returns the merge of databases \\spad{db1} and \\spad{db2}")) (|fullDisplay| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{fullDisplay(db,start,end )} prints full details of entries in the range \\axiom{\\spad{start}..end} in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(db)} prints full details of each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(x)} displays \\spad{x} in detail")) (|display| (((|Void|) $) "\\spad{display(db)} prints a summary line for each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{display(x)} displays \\spad{x} in some form")) (|elt| (((|DataList| (|String|)) $ (|Symbol|)) "\\spad{elt(db,s)} returns the \\axiom{\\spad{s}} field of each element of \\axiom{\\spad{db}}.") (($ $ (|QueryEquation|)) "\\spad{elt(db,q)} returns all elements of \\axiom{\\spad{db}} which satisfy \\axiom{\\spad{q}}.") (((|String|) $ (|Symbol|)) "\\spad{elt(x,s)} returns an element of \\spad{x} indexed by \\spad{s}"))) 
 NIL 
 NIL 
-(-216 -3498 UP UPUP R) 
+(-216 -3505 UP UPUP R) 
 ((|constructor| (NIL "This package provides functions for computing the residues of a function on an algebraic curve.")) (|doubleResultant| ((|#2| |#4| (|Mapping| |#2| |#2|)) "\\spad{doubleResultant(f, ')} returns \\spad{p}(\\spad{x}) whose roots are rational multiples of the residues of \\spad{f} at all its finite poles. Argument ' is the derivation to use."))) 
 NIL 
 NIL 
-(-217 -3498 FP) 
+(-217 -3505 FP) 
 ((|constructor| (NIL "Package for the factorization of a univariate polynomial with coefficients in a finite field. The algorithm used is the \"distinct degree\" algorithm of Cantor-Zassenhaus,{} modified to use trace instead of the norm and a table for computing Frobenius as suggested by Naudin and Quitte .")) (|irreducible?| (((|Boolean|) |#2|) "\\spad{irreducible?(p)} tests whether the polynomial \\spad{p} is irreducible.")) (|tracePowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{tracePowMod(u,k,v)} produces the sum of \\spad{u**(q**i)} for \\spad{i} running and \\spad{q=} size \\spad{F}")) (|trace2PowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{trace2PowMod(u,k,v)} produces the sum of \\spad{u**(2**i)} for \\spad{i} running from 1 to \\spad{k} all computed modulo the polynomial \\spad{v}.")) (|exptMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{exptMod(u,k,v)} raises the polynomial \\spad{u} to the \\spad{k}th power modulo the polynomial \\spad{v}.")) (|separateFactors| (((|List| |#2|) (|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|)))) "\\spad{separateFactors(lfact)} takes the list produced by \\spadfunFrom{separateDegrees}{DistinctDegreeFactorization} and produces the complete list of factors.")) (|separateDegrees| (((|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|))) |#2|) "\\spad{separateDegrees(p)} splits the square free polynomial \\spad{p} into factors each of which is a product of irreducibles of the same degree.")) (|distdfact| (((|Record| (|:| |cont| |#1|) (|:| |factors| (|List| (|Record| (|:| |irr| |#2|) (|:| |pow| (|Integer|)))))) |#2| (|Boolean|)) "\\spad{distdfact(p,sqfrflag)} produces the complete factorization of the polynomial \\spad{p} returning an internal data structure. If argument \\spad{sqfrflag} is \\spad{true},{} the polynomial is assumed square free.")) (|factorSquareFree| (((|Factored| |#2|) |#2|) "\\spad{factorSquareFree(p)} produces the complete factorization of the square free polynomial \\spad{p}.")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} produces the complete factorization of the polynomial \\spad{p}."))) 
 NIL 
 NIL 
 (-218) 
 ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating decimal expansions.")) (|decimal| (($ (|Fraction| (|Integer|))) "\\spad{decimal(r)} converts a rational number to a decimal expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(d)} returns the fractional part of a decimal expansion."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-550) (QUOTE (-914))) (|HasCategory| (-550) (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| (-550) (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-147))) (|HasCategory| (-550) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-550) (QUOTE (-1024))) (|HasCategory| (-550) (QUOTE (-823))) (-3962 (|HasCategory| (-550) (QUOTE (-823))) (|HasCategory| (-550) (QUOTE (-853)))) (|HasCategory| (-550) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-550) (QUOTE (-1155))) (|HasCategory| (-550) (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-550) (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-550) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-550) (QUOTE (-234))) (|HasCategory| (-550) (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| (-550) (LIST (QUOTE -518) (QUOTE (-1181)) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -311) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -288) (QUOTE (-550)) (QUOTE (-550)))) (|HasCategory| (-550) (QUOTE (-309))) (|HasCategory| (-550) (QUOTE (-549))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-550) (LIST (QUOTE -642) (QUOTE (-550)))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-914)))) (-3962 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-914)))) (|HasCategory| (-550) (QUOTE (-145))))) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-551) (QUOTE (-916))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| (-551) (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-147))) (|HasCategory| (-551) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-551) (QUOTE (-1026))) (|HasCategory| (-551) (QUOTE (-825))) (-3969 (|HasCategory| (-551) (QUOTE (-825))) (|HasCategory| (-551) (QUOTE (-855)))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-1157))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-551) (QUOTE (-234))) (|HasCategory| (-551) (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| (-551) (LIST (QUOTE -519) (QUOTE (-1183)) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -312) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -289) (QUOTE (-551)) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-310))) (|HasCategory| (-551) (QUOTE (-550))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-551) (LIST (QUOTE -644) (QUOTE (-551)))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (-3969 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (|HasCategory| (-551) (QUOTE (-145))))) 
 (-219) 
 ((|constructor| (NIL "This domain represents the syntax of a definition.")) (|body| (((|SpadAst|) $) "\\spad{body(d)} returns the right hand side of the definition \\spad{`d'}.")) (|signature| (((|Signature|) $) "\\spad{signature(d)} returns the signature of the operation being defined. Note that this list may be partial in that it contains only the types actually specified in the definition.")) (|head| (((|HeadAst|) $) "\\spad{head(d)} returns the head of the definition \\spad{`d'}. This is a list of identifiers starting with the name of the operation followed by the name of the parameters,{} if any."))) 
 NIL 
 NIL 
-(-220 R -3498) 
+(-220 R -3505) 
 ((|constructor| (NIL "\\spadtype{ElementaryFunctionDefiniteIntegration} provides functions to compute definite integrals of elementary functions.")) (|innerint| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| #1="failed") (|:| |pole| #2="potentialPole")) |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{innerint(f, x, a, b, ignore?)} should be local but conditional")) (|integrate| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| #1#) (|:| |pole| #2#)) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|)) (|String|)) "\\spad{integrate(f, x = a..b, \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| #1#) (|:| |pole| #2#)) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|))) "\\spad{integrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}."))) 
 NIL 
 NIL 
@@ -822,19 +822,19 @@ NIL
 NIL 
 (-223 S) 
 ((|constructor| (NIL "Linked list implementation of a Dequeue")) (|dequeue| (($ (|List| |#1|)) "\\spad{dequeue([x,y,...,z])} creates a dequeue with first (top or front) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom or back) element \\spad{z}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-224 |CoefRing| |listIndVar|) 
 ((|constructor| (NIL "The deRham complex of Euclidean space,{} that is,{} the class of differential forms of arbitary degree over a coefficient ring. See Flanders,{} Harley,{} Differential Forms,{} With Applications to the Physical Sciences,{} New York,{} Academic Press,{} 1963.")) (|exteriorDifferential| (($ $) "\\spad{exteriorDifferential(df)} returns the exterior derivative (gradient,{} curl,{} divergence,{} ...) of the differential form \\spad{df}.")) (|totalDifferential| (($ (|Expression| |#1|)) "\\spad{totalDifferential(x)} returns the total differential (gradient) form for element \\spad{x}.")) (|map| (($ (|Mapping| (|Expression| |#1|) (|Expression| |#1|)) $) "\\spad{map(f,df)} replaces each coefficient \\spad{x} of differential form \\spad{df} by \\spad{f(x)}.")) (|degree| (((|Integer|) $) "\\spad{degree(df)} returns the homogeneous degree of differential form \\spad{df}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?(df)} tests if differential form \\spad{df} is a 0-form,{} \\spadignore{i.e.} if degree(\\spad{df}) = 0.")) (|homogeneous?| (((|Boolean|) $) "\\spad{homogeneous?(df)} tests if all of the terms of differential form \\spad{df} have the same degree.")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(n)} returns the \\spad{n}th basis term for a differential form.")) (|coefficient| (((|Expression| |#1|) $ $) "\\spad{coefficient(df,u)},{} where \\spad{df} is a differential form,{} returns the coefficient of \\spad{df} containing the basis term \\spad{u} if such a term exists,{} and 0 otherwise.")) (|reductum| (($ $) "\\spad{reductum(df)},{} where \\spad{df} is a differential form,{} returns \\spad{df} minus the leading term of \\spad{df} if \\spad{df} has two or more terms,{} and 0 otherwise.")) (|leadingBasisTerm| (($ $) "\\spad{leadingBasisTerm(df)} returns the leading basis term of differential form \\spad{df}.")) (|leadingCoefficient| (((|Expression| |#1|) $) "\\spad{leadingCoefficient(df)} returns the leading coefficient of differential form \\spad{df}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-225 R -3498) 
+(-225 R -3505) 
 ((|constructor| (NIL "\\spadtype{DefiniteIntegrationTools} provides common tools used by the definite integration of both rational and elementary functions.")) (|checkForZero| (((|Union| (|Boolean|) "failed") (|SparseUnivariatePolynomial| |#2|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p, a, b, incl?)} is \\spad{true} if \\spad{p} has a zero between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.") (((|Union| (|Boolean|) "failed") (|Polynomial| |#1|) (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p, x, a, b, incl?)} is \\spad{true} if \\spad{p} has a zero for \\spad{x} between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.")) (|computeInt| (((|Union| (|OrderedCompletion| |#2|) "failed") (|Kernel| |#2|) |#2| (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{computeInt(x, g, a, b, eval?)} returns the integral of \\spad{f} for \\spad{x} between a and \\spad{b},{} assuming that \\spad{g} is an indefinite integral of \\spad{f} and \\spad{f} has no pole between a and \\spad{b}. If \\spad{eval?} is \\spad{true},{} then \\spad{g} can be evaluated safely at \\spad{a} and \\spad{b},{} provided that they are finite values. Otherwise,{} limits must be computed.")) (|ignore?| (((|Boolean|) (|String|)) "\\spad{ignore?(s)} is \\spad{true} if \\spad{s} is the string that tells the integrator to assume that the function has no pole in the integration interval."))) 
 NIL 
 NIL 
 (-226) 
 ((|constructor| (NIL "\\indented{1}{\\spadtype{DoubleFloat} is intended to make accessible} hardware floating point arithmetic in \\Language{},{} either native double precision,{} or IEEE. On most machines,{} there will be hardware support for the arithmetic operations: \\spadfunFrom{+}{DoubleFloat},{} \\spadfunFrom{*}{DoubleFloat},{} \\spadfunFrom{/}{DoubleFloat} and possibly also the \\spadfunFrom{sqrt}{DoubleFloat} operation. The operations \\spadfunFrom{exp}{DoubleFloat},{} \\spadfunFrom{log}{DoubleFloat},{} \\spadfunFrom{sin}{DoubleFloat},{} \\spadfunFrom{cos}{DoubleFloat},{} \\spadfunFrom{atan}{DoubleFloat} are normally coded in software based on minimax polynomial/rational approximations. Note that under Lisp/VM,{} \\spadfunFrom{atan}{DoubleFloat} is not available at this time. Some general comments about the accuracy of the operations: the operations \\spadfunFrom{+}{DoubleFloat},{} \\spadfunFrom{*}{DoubleFloat},{} \\spadfunFrom{/}{DoubleFloat} and \\spadfunFrom{sqrt}{DoubleFloat} are expected to be fully accurate. The operations \\spadfunFrom{exp}{DoubleFloat},{} \\spadfunFrom{log}{DoubleFloat},{} \\spadfunFrom{sin}{DoubleFloat},{} \\spadfunFrom{cos}{DoubleFloat} and \\spadfunFrom{atan}{DoubleFloat} are not expected to be fully accurate. In particular,{} \\spadfunFrom{sin}{DoubleFloat} and \\spadfunFrom{cos}{DoubleFloat} will lose all precision for large arguments. \\blankline The \\spadtype{Float} domain provides an alternative to the \\spad{DoubleFloat} domain. It provides an arbitrary precision model of floating point arithmetic. This means that accuracy problems like those above are eliminated by increasing the working precision where necessary. \\spadtype{Float} provides some special functions such as \\spadfunFrom{erf}{DoubleFloat},{} the error function in addition to the elementary functions. The disadvantage of \\spadtype{Float} is that it is much more expensive than small floats when the latter can be used.")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n, b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)} (that is,{} \\spad{|(r-f)/f| < b**(-n)}).") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|Beta| (($ $ $) "\\spad{Beta(x,y)} is \\spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.")) (|Gamma| (($ $) "\\spad{Gamma(x)} is the Euler Gamma function.")) (|atan| (($ $ $) "\\spad{atan(x,y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm with base 10 for \\spad{x}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm with base 2 for \\spad{x}.")) (|exp1| (($) "\\spad{exp1()} returns the natural log base \\spad{2.718281828...}.")) (** (($ $ $) "\\spad{x ** y} returns the \\spad{y}th power of \\spad{x} (equal to \\spad{exp(y log x)}).")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) 
-((-4203 . T) (-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4210 . T) (-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-227) 
 ((|constructor| (NIL "This package provides special functions for double precision real and complex floating point.")) (|hypergeometric0F1| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{hypergeometric0F1(c,z)} is the hypergeometric function \\spad{0F1(; c; z)}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{hypergeometric0F1(c,z)} is the hypergeometric function \\spad{0F1(; c; z)}.")) (|airyBi| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Bi''(x) - x * Bi(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Bi''(x) - x * Bi(x) = 0}.}")) (|airyAi| (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Ai''(x) - x * Ai(x) = 0}.}") (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Ai''(x) - x * Ai(x) = 0}.}")) (|besselK| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselK(v,x)} is the modified Bessel function of the first kind,{} \\spad{K(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{K(v,x) = \\%pi/2*(I(-v,x) - I(v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselK(v,x)} is the modified Bessel function of the first kind,{} \\spad{K(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{K(v,x) = \\%pi/2*(I(-v,x) - I(v,x))/sin(v*\\%pi)}.} so is not valid for integer values of \\spad{v}.")) (|besselI| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselI(v,x)} is the modified Bessel function of the first kind,{} \\spad{I(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselI(v,x)} is the modified Bessel function of the first kind,{} \\spad{I(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}")) (|besselY| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselY(v,x)} is the Bessel function of the second kind,{} \\spad{Y(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{Y(v,x) = (J(v,x) cos(v*\\%pi) - J(-v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselY(v,x)} is the Bessel function of the second kind,{} \\spad{Y(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{Y(v,x) = (J(v,x) cos(v*\\%pi) - J(-v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.")) (|besselJ| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselJ(v,x)} is the Bessel function of the first kind,{} \\spad{J(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselJ(v,x)} is the Bessel function of the first kind,{} \\spad{J(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}")) (|polygamma| (((|Complex| (|DoubleFloat|)) (|NonNegativeInteger|) (|Complex| (|DoubleFloat|))) "\\spad{polygamma(n, x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.") (((|DoubleFloat|) (|NonNegativeInteger|) (|DoubleFloat|)) "\\spad{polygamma(n, x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.")) (|digamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}")) (|logGamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.")) (|Beta| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Beta(x, y)} is the Euler beta function,{} \\spad{B(x,y)},{} defined by \\indented{2}{\\spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{Beta(x, y)} is the Euler beta function,{} \\spad{B(x,y)},{} defined by \\indented{2}{\\spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}")) (|Gamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\\%infinity)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\\%infinity)}.}"))) 
@@ -842,23 +842,23 @@ NIL
 NIL 
 (-228 R) 
 ((|constructor| (NIL "\\indented{1}{A Denavit-Hartenberg Matrix is a 4x4 Matrix of the form:} \\indented{1}{\\spad{nx ox ax px}} \\indented{1}{\\spad{ny oy ay py}} \\indented{1}{\\spad{nz oz az pz}} \\indented{2}{\\spad{0\\space{2}0\\space{2}0\\space{2}1}} (\\spad{n},{} \\spad{o},{} and a are the direction cosines)")) (|translate| (($ |#1| |#1| |#1|) "\\spad{translate(X,Y,Z)} returns a dhmatrix for translation by \\spad{X},{} \\spad{Y},{} and \\spad{Z}")) (|scale| (($ |#1| |#1| |#1|) "\\spad{scale(sx,sy,sz)} returns a dhmatrix for scaling in the \\spad{X},{} \\spad{Y} and \\spad{Z} directions")) (|rotatez| (($ |#1|) "\\spad{rotatez(r)} returns a dhmatrix for rotation about axis \\spad{Z} for \\spad{r} degrees")) (|rotatey| (($ |#1|) "\\spad{rotatey(r)} returns a dhmatrix for rotation about axis \\spad{Y} for \\spad{r} degrees")) (|rotatex| (($ |#1|) "\\spad{rotatex(r)} returns a dhmatrix for rotation about axis \\spad{X} for \\spad{r} degrees")) (|identity| (($) "\\spad{identity()} create the identity dhmatrix")) (* (((|Point| |#1|) $ (|Point| |#1|)) "\\spad{t*p} applies the dhmatrix \\spad{t} to point \\spad{p}"))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-561))) (|HasAttribute| |#1| (QUOTE (-4429 "*"))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4436 "*"))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
 (-229 A S) 
 ((|constructor| (NIL "A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones."))) 
 NIL 
 NIL 
 (-230 S) 
 ((|constructor| (NIL "A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones."))) 
-((-4428 . T)) 
+((-4435 . T)) 
 NIL 
 (-231 S R) 
 ((|constructor| (NIL "Differential extensions of a ring \\spad{R}. Given a differentiation on \\spad{R},{} extend it to a differentiation on \\%.")) (D (($ $ (|Mapping| |#2| |#2|) (|NonNegativeInteger|)) "\\spad{D(x, deriv, n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#2| |#2|)) "\\spad{D(x, deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|) (|NonNegativeInteger|)) "\\spad{differentiate(x, deriv, n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#2| |#2|)) "\\spad{differentiate(x, deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (QUOTE (-234)))) 
+((|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#2| (QUOTE (-234)))) 
 (-232 R) 
 ((|constructor| (NIL "Differential extensions of a ring \\spad{R}. Given a differentiation on \\spad{R},{} extend it to a differentiation on \\%.")) (D (($ $ (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{D(x, deriv, n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#1| |#1|)) "\\spad{D(x, deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}.")) (|differentiate| (($ $ (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{differentiate(x, deriv, n)} differentiate \\spad{x} \\spad{n} times using a derivation which extends \\spad{deriv} on \\spad{R}.") (($ $ (|Mapping| |#1| |#1|)) "\\spad{differentiate(x, deriv)} differentiates \\spad{x} extending the derivation deriv on \\spad{R}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
 (-233 S) 
 ((|constructor| (NIL "An ordinary differential ring,{} that is,{} a ring with an operation \\spadfun{differentiate}. \\blankline")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(x, n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{D(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(x, n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified."))) 
@@ -866,33 +866,33 @@ NIL
 NIL 
 (-234) 
 ((|constructor| (NIL "An ordinary differential ring,{} that is,{} a ring with an operation \\spadfun{differentiate}. \\blankline")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(x, n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{D(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(x, n)} returns the \\spad{n}-th derivative of \\spad{x}.") (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
 (-235 A S) 
 ((|constructor| (NIL "This category is a collection of operations common to both categories \\spadtype{Dictionary} and \\spadtype{MultiDictionary}")) (|select!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select!(p,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is not \\spad{true}.")) (|remove!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove!(p,d)} destructively changes dictionary \\spad{d} by removeing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.") (($ |#2| $) "\\spad{remove!(x,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{y} such that \\axiom{\\spad{y} = \\spad{x}}.")) (|dictionary| (($ (|List| |#2|)) "\\spad{dictionary([x,y,...,z])} creates a dictionary consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{dictionary()}\\$\\spad{D} creates an empty dictionary of type \\spad{D}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4427))) 
+((|HasAttribute| |#1| (QUOTE -4434))) 
 (-236 S) 
 ((|constructor| (NIL "This category is a collection of operations common to both categories \\spadtype{Dictionary} and \\spadtype{MultiDictionary}")) (|select!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select!(p,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is not \\spad{true}.")) (|remove!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove!(p,d)} destructively changes dictionary \\spad{d} by removeing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.") (($ |#1| $) "\\spad{remove!(x,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{y} such that \\axiom{\\spad{y} = \\spad{x}}.")) (|dictionary| (($ (|List| |#1|)) "\\spad{dictionary([x,y,...,z])} creates a dictionary consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{dictionary()}\\$\\spad{D} creates an empty dictionary of type \\spad{D}."))) 
-((-4428 . T)) 
+((-4435 . T)) 
 NIL 
 (-237) 
 ((|constructor| (NIL "any solution of a homogeneous linear Diophantine equation can be represented as a sum of minimal solutions,{} which form a \"basis\" (a minimal solution cannot be represented as a nontrivial sum of solutions) in the case of an inhomogeneous linear Diophantine equation,{} each solution is the sum of a inhomogeneous solution and any number of homogeneous solutions therefore,{} it suffices to compute two sets: \\indented{3}{1. all minimal inhomogeneous solutions} \\indented{3}{2. all minimal homogeneous solutions} the algorithm implemented is a completion procedure,{} which enumerates all solutions in a recursive depth-first-search it can be seen as finding monotone paths in a graph for more details see Reference")) (|dioSolve| (((|Record| (|:| |varOrder| (|List| (|Symbol|))) (|:| |inhom| (|Union| (|List| (|Vector| (|NonNegativeInteger|))) "failed")) (|:| |hom| (|List| (|Vector| (|NonNegativeInteger|))))) (|Equation| (|Polynomial| (|Integer|)))) "\\spad{dioSolve(u)} computes a basis of all minimal solutions for linear homogeneous Diophantine equation \\spad{u},{} then all minimal solutions of inhomogeneous equation"))) 
 NIL 
 NIL 
-(-238 S -3023 R) 
+(-238 S -3030 R) 
 ((|constructor| (NIL "\\indented{2}{This category represents a finite cartesian product of a given type.} Many categorical properties are preserved under this construction.")) (* (($ $ |#3|) "\\spad{y * r} multiplies each component of the vector \\spad{y} by the element \\spad{r}.") (($ |#3| $) "\\spad{r * y} multiplies the element \\spad{r} times each component of the vector \\spad{y}.")) (|dot| ((|#3| $ $) "\\spad{dot(x,y)} computes the inner product of the vectors \\spad{x} and \\spad{y}.")) (|unitVector| (($ (|PositiveInteger|)) "\\spad{unitVector(n)} produces a vector with 1 in position \\spad{n} and zero elsewhere.")) (|directProduct| (($ (|Vector| |#3|)) "\\spad{directProduct(v)} converts the vector \\spad{v} to become a direct product. Error: if the length of \\spad{v} is different from dim.")) (|finiteAggregate| ((|attribute|) "attribute to indicate an aggregate of finite size"))) 
 NIL 
-((|HasCategory| |#3| (QUOTE (-366))) (|HasCategory| |#3| (QUOTE (-796))) (|HasCategory| |#3| (QUOTE (-851))) (|HasAttribute| |#3| (QUOTE -4424)) (|HasCategory| |#3| (QUOTE (-173))) (|HasCategory| |#3| (QUOTE (-371))) (|HasCategory| |#3| (QUOTE (-729))) (|HasCategory| |#3| (QUOTE (-131))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (QUOTE (-1105)))) 
-(-239 -3023 R) 
+((|HasCategory| |#3| (QUOTE (-367))) (|HasCategory| |#3| (QUOTE (-798))) (|HasCategory| |#3| (QUOTE (-853))) (|HasAttribute| |#3| (QUOTE -4431)) (|HasCategory| |#3| (QUOTE (-173))) (|HasCategory| |#3| (QUOTE (-372))) (|HasCategory| |#3| (QUOTE (-731))) (|HasCategory| |#3| (QUOTE (-131))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-1055))) (|HasCategory| |#3| (QUOTE (-1107)))) 
+(-239 -3030 R) 
 ((|constructor| (NIL "\\indented{2}{This category represents a finite cartesian product of a given type.} Many categorical properties are preserved under this construction.")) (* (($ $ |#2|) "\\spad{y * r} multiplies each component of the vector \\spad{y} by the element \\spad{r}.") (($ |#2| $) "\\spad{r * y} multiplies the element \\spad{r} times each component of the vector \\spad{y}.")) (|dot| ((|#2| $ $) "\\spad{dot(x,y)} computes the inner product of the vectors \\spad{x} and \\spad{y}.")) (|unitVector| (($ (|PositiveInteger|)) "\\spad{unitVector(n)} produces a vector with 1 in position \\spad{n} and zero elsewhere.")) (|directProduct| (($ (|Vector| |#2|)) "\\spad{directProduct(v)} converts the vector \\spad{v} to become a direct product. Error: if the length of \\spad{v} is different from dim.")) (|finiteAggregate| ((|attribute|) "attribute to indicate an aggregate of finite size"))) 
-((-4421 |has| |#2| (-1053)) (-4422 |has| |#2| (-1053)) (-4424 |has| |#2| (-6 -4424)) ((-4429 "*") |has| |#2| (-173)) (-4427 . T)) 
+((-4428 |has| |#2| (-1055)) (-4429 |has| |#2| (-1055)) (-4431 |has| |#2| (-6 -4431)) ((-4436 "*") |has| |#2| (-173)) (-4434 . T)) 
 NIL 
-(-240 -3023 R) 
+(-240 -3030 R) 
 ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying component type. This contrasts with simple vectors in that the members can be viewed as having constant length. Thus many categorical properties can by lifted from the underlying component type. Component extraction operations are provided but no updating operations. Thus new direct product elements can either be created by converting vector elements using the \\spadfun{directProduct} function or by taking appropriate linear combinations of basis vectors provided by the \\spad{unitVector} operation."))) 
-((-4421 |has| |#2| (-1053)) (-4422 |has| |#2| (-1053)) (-4424 |has| |#2| (-6 -4424)) ((-4429 "*") |has| |#2| (-173)) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-1053)))) (|HasCategory| 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(|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (|HasCategory| |#2| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 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(QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-1053)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))))) (|HasCategory| (-550) (QUOTE (-853))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-1053)))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-1053)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasAttribute| |#2| (QUOTE -4424)) (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))))) 
-(-241 -3023 A B) 
+((-4428 |has| |#2| (-1055)) (-4429 |has| |#2| (-1055)) (-4431 |has| |#2| (-6 -4431)) ((-4436 "*") |has| |#2| (-173)) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-372))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-731))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-798))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-853))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE 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(|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183))))) (-3969 (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551))))) (|HasCategory| |#2| (QUOTE (-1055)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasAttribute| |#2| (QUOTE -4431)) (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))))) 
+(-241 -3030 A B) 
 ((|constructor| (NIL "\\indented{2}{This package provides operations which all take as arguments} direct products of elements of some type \\spad{A} and functions from \\spad{A} to another type \\spad{B}. The operations all iterate over their vector argument and either return a value of type \\spad{B} or a direct product over \\spad{B}.")) (|map| (((|DirectProduct| |#1| |#3|) (|Mapping| |#3| |#2|) (|DirectProduct| |#1| |#2|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values.")) (|reduce| ((|#3| (|Mapping| |#3| |#2| |#3|) (|DirectProduct| |#1| |#2|) |#3|) "\\spad{reduce(func,vec,ident)} combines the elements in \\spad{vec} using the binary function \\spad{func}. Argument \\spad{ident} is returned if the vector is empty.")) (|scan| (((|DirectProduct| |#1| |#3|) (|Mapping| |#3| |#2| |#3|) (|DirectProduct| |#1| |#2|) |#3|) "\\spad{scan(func,vec,ident)} creates a new vector whose elements are the result of applying reduce to the binary function \\spad{func},{} increasing initial subsequences of the vector \\spad{vec},{} and the element \\spad{ident}."))) 
 NIL 
 NIL 
@@ -906,7 +906,7 @@ NIL
 NIL 
 (-244) 
 ((|constructor| (NIL "A division ring (sometimes called a skew field),{} \\spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.")) (|inv| (($ $) "\\spad{inv x} returns the multiplicative inverse of \\spad{x}. Error: if \\spad{x} is 0.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}."))) 
-((-4420 . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 (-245 S) 
 ((|constructor| (NIL "A doubly-linked aggregate serves as a model for a doubly-linked list,{} that is,{} a list which can has links to both next and previous nodes and thus can be efficiently traversed in both directions.")) (|setnext!| (($ $ $) "\\spad{setnext!(u,v)} destructively sets the next node of doubly-linked aggregate \\spad{u} to \\spad{v},{} returning \\spad{v}.")) (|setprevious!| (($ $ $) "\\spad{setprevious!(u,v)} destructively sets the previous node of doubly-linked aggregate \\spad{u} to \\spad{v},{} returning \\spad{v}.")) (|concat!| (($ $ $) "\\spad{concat!(u,v)} destructively concatenates doubly-linked aggregate \\spad{v} to the end of doubly-linked aggregate \\spad{u}.")) (|next| (($ $) "\\spad{next(l)} returns the doubly-linked aggregate beginning with its next element. Error: if \\spad{l} has no next element. Note: \\axiom{next(\\spad{l}) = rest(\\spad{l})} and \\axiom{previous(next(\\spad{l})) = \\spad{l}}.")) (|previous| (($ $) "\\spad{previous(l)} returns the doubly-link list beginning with its previous element. Error: if \\spad{l} has no previous element. Note: \\axiom{next(previous(\\spad{l})) = \\spad{l}}.")) (|tail| (($ $) "\\spad{tail(l)} returns the doubly-linked aggregate \\spad{l} starting at its second element. Error: if \\spad{l} is empty.")) (|head| (($ $) "\\spad{head(l)} returns the first element of a doubly-linked aggregate \\spad{l}. Error: if \\spad{l} is empty.")) (|last| ((|#1| $) "\\spad{last(l)} returns the last element of a doubly-linked aggregate \\spad{l}. Error: if \\spad{l} is empty."))) 
@@ -914,16 +914,16 @@ NIL
 NIL 
 (-246 S) 
 ((|constructor| (NIL "This domain provides some nice functions on lists")) (|elt| (((|NonNegativeInteger|) $ "count") "\\axiom{\\spad{l}.\"count\"} returns the number of elements in \\axiom{\\spad{l}}.") (($ $ "sort") "\\axiom{\\spad{l}.sort} returns \\axiom{\\spad{l}} with elements sorted. Note: \\axiom{\\spad{l}.sort = sort(\\spad{l})}") (($ $ "unique") "\\axiom{\\spad{l}.unique} returns \\axiom{\\spad{l}} with duplicates removed. Note: \\axiom{\\spad{l}.unique = removeDuplicates(\\spad{l})}.")) (|datalist| (($ (|List| |#1|)) "\\spad{datalist(l)} creates a datalist from \\spad{l}"))) 
-((-4428 . T) (-4427 . T)) 
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+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
 (-247 M) 
 ((|constructor| (NIL "DiscreteLogarithmPackage implements help functions for discrete logarithms in monoids using small cyclic groups.")) (|shanksDiscLogAlgorithm| (((|Union| (|NonNegativeInteger|) "failed") |#1| |#1| (|NonNegativeInteger|)) "\\spad{shanksDiscLogAlgorithm(b,a,p)} computes \\spad{s} with \\spad{b**s = a} for assuming that \\spad{a} and \\spad{b} are elements in a 'small' cyclic group of order \\spad{p} by Shank\\spad{'s} algorithm. Note: this is a subroutine of the function \\spadfun{discreteLog}.")) (** ((|#1| |#1| (|Integer|)) "\\spad{x ** n} returns \\spad{x} raised to the integer power \\spad{n}"))) 
 NIL 
 NIL 
 (-248 |vl| R) 
 ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is lexicographic specified by the variable list parameter with the most significant variable first in the list.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p, perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) 
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 (-249) 
 ((|showSummary| (((|Void|) $) "\\spad{showSummary(d)} prints out implementation detail information of domain \\spad{`d'}.")) (|reflect| (($ (|ConstructorCall| (|DomainConstructor|))) "\\spad{reflect cc} returns the domain object designated by the ConstructorCall syntax `cc'. The constructor implied by `cc' must be known to the system since it is instantiated.")) (|reify| (((|ConstructorCall| (|DomainConstructor|)) $) "\\spad{reify(d)} returns the abstract syntax for the domain \\spad{`x'}.")) (|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Create: October 18,{} 2007. Date Last Updated: December 20,{} 2008. Basic Operations: coerce,{} reify Related Constructors: Type,{} Syntax,{} OutputForm Also See: Type,{} ConstructorCall") (((|DomainConstructor|) $) "\\spad{constructor(d)} returns the domain constructor that is instantiated to the domain object \\spad{`d'}."))) 
 NIL 
@@ -938,23 +938,23 @@ NIL
 NIL 
 (-252 |n| R M S) 
 ((|constructor| (NIL "This constructor provides a direct product type with a left matrix-module view."))) 
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 (-253 |n| R S) 
 ((|constructor| (NIL "This constructor provides a direct product of \\spad{R}-modules with an \\spad{R}-module view."))) 
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(|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-729))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-796))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-851))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550)))))) (|HasCategory| (-550) (QUOTE (-853))) (-12 (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#3| (QUOTE (-234))) (|HasCategory| |#3| (QUOTE (-1053)))) (-3962 (-12 (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#3| (QUOTE (-234))) (|HasCategory| |#3| (QUOTE (-1053)))) (|HasCategory| |#3| (QUOTE (-729)))) (-12 (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (-3962 (-12 (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#3| (QUOTE (-1053)))) (-12 (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-3962 (-12 (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#3| (QUOTE (-1053))) (|HasCategory| |#3| (LIST (QUOTE -904) (QUOTE (-1181))))) (|HasAttribute| |#3| (QUOTE -4424)) (-12 (|HasCategory| |#3| (QUOTE (-234))) (|HasCategory| |#3| (QUOTE (-1053))))) (|HasCategory| |#3| (QUOTE (-131))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (LIST (QUOTE -311) (|devaluate| |#3|))))) 
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(QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (-12 (|HasCategory| |#3| (QUOTE (-1055))) (|HasCategory| |#3| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#3| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#3| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))))) (-3969 (-12 (|HasCategory| |#3| (QUOTE (-173))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-234))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-367))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-372))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| 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(-1183))))) (|HasAttribute| |#3| (QUOTE -4431)) (-12 (|HasCategory| |#3| (QUOTE (-234))) (|HasCategory| |#3| (QUOTE (-1055))))) (|HasCategory| |#3| (QUOTE (-131))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -312) (|devaluate| |#3|))))) 
 (-254 A R S V E) 
 ((|constructor| (NIL "\\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \\spadfun{order} and \\spadfun{weight} are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \\spadfun{leader},{} \\spadfun{initial},{} \\spadfun{separant},{} \\spadfun{differentialVariables},{} and \\spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \\spadfun{eval}. A convenient way of referencing derivatives is provided by the functions \\spadfun{makeVariable}. \\blankline To construct a domain using this constructor,{} one needs to provide a ground ring \\spad{R},{} an ordered set \\spad{S} of differential indeterminates,{} a ranking \\spad{V} on the set of derivatives of the differential indeterminates,{} and a set \\spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \\blankline")) (|separant| (($ $) "\\spad{separant(p)} returns the partial derivative of the differential polynomial \\spad{p} with respect to its leader.")) (|initial| (($ $) "\\spad{initial(p)} returns the leading coefficient when the differential polynomial \\spad{p} is written as a univariate polynomial in its leader.")) (|leader| ((|#4| $) "\\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \\spad{p} Note: an error occurs if \\spad{p} is in the ground ring.")) (|isobaric?| (((|Boolean|) $) "\\spad{isobaric?(p)} returns \\spad{true} if every differential monomial appearing in the differential polynomial \\spad{p} has same weight,{} and returns \\spad{false} otherwise.")) (|weight| (((|NonNegativeInteger|) $ |#3|) "\\spad{weight(p, s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|NonNegativeInteger|) $) "\\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p}.")) (|weights| (((|List| (|NonNegativeInteger|)) $ |#3|) "\\spad{weights(p, s)} returns a list of weights of differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|List| (|NonNegativeInteger|)) $) "\\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $ |#3|) "\\spad{degree(p, s)} returns the maximum degree of the differential polynomial \\spad{p} viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of the differential polynomial \\spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \\spad{p}.") (((|NonNegativeInteger|) $ |#3|) "\\spad{order(p,s)} returns the order of the differential polynomial \\spad{p} in differential indeterminate \\spad{s}.")) (|differentialVariables| (((|List| |#3|) $) "\\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \\spad{p}.")) (|makeVariable| (((|Mapping| $ (|NonNegativeInteger|)) $) "\\spad{makeVariable(p)} views \\spad{p} as an element of a differential ring,{} in such a way that the \\spad{n}-th derivative of \\spad{p} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} \\spad{:=} makeVariable(\\spad{p}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored.") (((|Mapping| $ (|NonNegativeInteger|)) |#3|) "\\spad{makeVariable(s)} views \\spad{s} as a differential indeterminate,{} in such a way that the \\spad{n}-th derivative of \\spad{s} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} :=makeVariable(\\spad{s}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored."))) 
 NIL 
 ((|HasCategory| |#2| (QUOTE (-234)))) 
 (-255 R S V E) 
 ((|constructor| (NIL "\\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \\spadfun{order} and \\spadfun{weight} are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \\spadfun{leader},{} \\spadfun{initial},{} \\spadfun{separant},{} \\spadfun{differentialVariables},{} and \\spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \\spadfun{eval}. A convenient way of referencing derivatives is provided by the functions \\spadfun{makeVariable}. \\blankline To construct a domain using this constructor,{} one needs to provide a ground ring \\spad{R},{} an ordered set \\spad{S} of differential indeterminates,{} a ranking \\spad{V} on the set of derivatives of the differential indeterminates,{} and a set \\spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \\blankline")) (|separant| (($ $) "\\spad{separant(p)} returns the partial derivative of the differential polynomial \\spad{p} with respect to its leader.")) (|initial| (($ $) "\\spad{initial(p)} returns the leading coefficient when the differential polynomial \\spad{p} is written as a univariate polynomial in its leader.")) (|leader| ((|#3| $) "\\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \\spad{p} Note: an error occurs if \\spad{p} is in the ground ring.")) (|isobaric?| (((|Boolean|) $) "\\spad{isobaric?(p)} returns \\spad{true} if every differential monomial appearing in the differential polynomial \\spad{p} has same weight,{} and returns \\spad{false} otherwise.")) (|weight| (((|NonNegativeInteger|) $ |#2|) "\\spad{weight(p, s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|NonNegativeInteger|) $) "\\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p}.")) (|weights| (((|List| (|NonNegativeInteger|)) $ |#2|) "\\spad{weights(p, s)} returns a list of weights of differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|List| (|NonNegativeInteger|)) $) "\\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $ |#2|) "\\spad{degree(p, s)} returns the maximum degree of the differential polynomial \\spad{p} viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of the differential polynomial \\spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \\spad{p}.") (((|NonNegativeInteger|) $ |#2|) "\\spad{order(p,s)} returns the order of the differential polynomial \\spad{p} in differential indeterminate \\spad{s}.")) (|differentialVariables| (((|List| |#2|) $) "\\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \\spad{p}.")) (|makeVariable| (((|Mapping| $ (|NonNegativeInteger|)) $) "\\spad{makeVariable(p)} views \\spad{p} as an element of a differential ring,{} in such a way that the \\spad{n}-th derivative of \\spad{p} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} \\spad{:=} makeVariable(\\spad{p}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored.") (((|Mapping| $ (|NonNegativeInteger|)) |#2|) "\\spad{makeVariable(s)} views \\spad{s} as a differential indeterminate,{} in such a way that the \\spad{n}-th derivative of \\spad{s} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} :=makeVariable(\\spad{s}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
 (-256 S) 
 ((|constructor| (NIL "A dequeue is a doubly ended stack,{} that is,{} a bag where first items inserted are the first items extracted,{} at either the front or the back end of the data structure.")) (|reverse!| (($ $) "\\spad{reverse!(d)} destructively replaces \\spad{d} by its reverse dequeue,{} \\spadignore{i.e.} the top (front) element is now the bottom (back) element,{} and so on.")) (|extractBottom!| ((|#1| $) "\\spad{extractBottom!(d)} destructively extracts the bottom (back) element from the dequeue \\spad{d}. Error: if \\spad{d} is empty.")) (|extractTop!| ((|#1| $) "\\spad{extractTop!(d)} destructively extracts the top (front) element from the dequeue \\spad{d}. Error: if \\spad{d} is empty.")) (|insertBottom!| ((|#1| |#1| $) "\\spad{insertBottom!(x,d)} destructively inserts \\spad{x} into the dequeue \\spad{d} at the bottom (back) of the dequeue.")) (|insertTop!| ((|#1| |#1| $) "\\spad{insertTop!(x,d)} destructively inserts \\spad{x} into the dequeue \\spad{d},{} that is,{} at the top (front) of the dequeue. The element previously at the top of the dequeue becomes the second in the dequeue,{} and so on.")) (|bottom!| ((|#1| $) "\\spad{bottom!(d)} returns the element at the bottom (back) of the dequeue.")) (|top!| ((|#1| $) "\\spad{top!(d)} returns the element at the top (front) of the dequeue.")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(d)} returns the number of elements in dequeue \\spad{d}. Note: \\axiom{height(\\spad{d}) = \\# \\spad{d}}.")) (|dequeue| (($ (|List| |#1|)) "\\spad{dequeue([x,y,...,z])} creates a dequeue with first (top or front) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom or back) element \\spad{z}.") (($) "\\spad{dequeue()}\\$\\spad{D} creates an empty dequeue of type \\spad{D}."))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
 (-257 |Ex|) 
 ((|constructor| (NIL "TopLevelDrawFunctions provides top level functions for drawing graphics of expressions.")) (|makeObject| (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{makeObject(f(x,y),x = a..b,y = c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} appears as the default title.") (((|ThreeSpace| (|DoubleFloat|)) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f(x,y),x = a..b,y = c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{makeObject(curve(f(t),g(t),h(t)),t = a..b)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f(t),g(t),h(t)),t = a..b,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.")) (|draw| (((|ThreeDimensionalViewport|) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d)} draws the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title.") (((|ThreeDimensionalViewport|) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l)} draws the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{draw(f(x,y),x = a..b,y = c..d)} draws the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} appears in the title bar.") (((|ThreeDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f(x,y),x = a..b,y = c..d,l)} draws the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{draw(curve(f(t),g(t),h(t)),t = a..b)} draws the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f(t),g(t),h(t)),t = a..b,l)} draws the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{draw(curve(f(t),g(t)),t = a..b)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{(f(t),g(t))} appears in the title bar.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f(t),g(t)),t = a..b,l)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{(f(t),g(t))} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) |#1| (|SegmentBinding| (|Float|))) "\\spad{draw(f(x),x = a..b)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{f(x)} appears in the title bar.") (((|TwoDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f(x),x = a..b,l)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{f(x)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied."))) 
@@ -994,8 +994,8 @@ NIL
 NIL 
 (-266 R S V) 
 ((|constructor| (NIL "\\spadtype{DifferentialSparseMultivariatePolynomial} implements an ordinary differential polynomial ring by combining a domain belonging to the category \\spadtype{DifferentialVariableCategory} with the domain \\spadtype{SparseMultivariatePolynomial}. \\blankline"))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-914))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-914)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| |#3| (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#3| (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| |#3| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#3| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#3| (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasAttribute| |#1| (QUOTE -4425)) (|HasCategory| |#1| (QUOTE (-456))) (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| |#3| (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#3| (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| |#3| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#3| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#3| (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
 (-267 A S) 
 ((|constructor| (NIL "\\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \\spad{x},{}...,{}\\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \\spad{x},{}\\spad{x'},{}\\spad{x''},{}...,{} \\spad{y},{}\\spad{y'},{}\\spad{y''},{}... (Note: in the interpreter,{} the \\spad{n}-th derivative of \\spad{y} is displayed as \\spad{y} with a subscript \\spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \\blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \\spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \\spadfun{<}. For example,{} one may define one derivative to be less than another by lexicographically comparing first the \\spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \\blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{} this constructor provides a function \\spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \\spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.")) (|coerce| (($ |#2|) "\\spad{coerce(s)} returns \\spad{s},{} viewed as the zero-th order derivative of \\spad{s}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(v, n)} returns the \\spad{n}-th derivative of \\spad{v}.") (($ $) "\\spad{differentiate(v)} returns the derivative of \\spad{v}.")) (|weight| (((|NonNegativeInteger|) $) "\\spad{weight(v)} returns the weight of the derivative \\spad{v}.")) (|variable| ((|#2| $) "\\spad{variable(v)} returns \\spad{s} if \\spad{v} is any derivative of the differential indeterminate \\spad{s}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(v)} returns \\spad{n} if \\spad{v} is the \\spad{n}-th derivative of any differential indeterminate.")) (|makeVariable| (($ |#2| (|NonNegativeInteger|)) "\\spad{makeVariable(s, n)} returns the \\spad{n}-th derivative of a differential indeterminate \\spad{s} as an algebraic indeterminate."))) 
 NIL 
@@ -1040,4093 +1040,4101 @@ NIL
 ((|constructor| (NIL "A domain used in the construction of the exterior algebra on a set \\spad{X} over a ring \\spad{R}. This domain represents the set of all ordered subsets of the set \\spad{X},{} assumed to be in correspondance with {1,{}2,{}3,{} ...}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements which can only be 0 or 1. \\blankline The multiplicative identity element of the exterior algebra corresponds to the empty subset of \\spad{X}. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.")) (|Nul| (($ (|NonNegativeInteger|)) "\\spad{Nul()} gives the basis element 1 for the algebra generated by \\spad{n} generators.")) (|exponents| (((|List| (|Integer|)) $) "\\spad{exponents(x)} converts a domain element into a list of zeros and ones corresponding to the exponents in the basis element that \\spad{x} represents.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(x)} gives the numbers of 1\\spad{'s} in \\spad{x},{} \\spadignore{i.e.} the number of non-zero exponents in the basis element that \\spad{x} represents.")) (|coerce| (($ (|List| (|Integer|))) "\\spad{coerce(l)} converts a list of 0\\spad{'s} and 1\\spad{'s} into a basis element,{} where 1 (respectively 0) designates that the variable of the corresponding index of \\spad{l} is (respectively,{} is not) present. Error: if an element of \\spad{l} is not 0 or 1."))) 
 NIL 
 NIL 
-(-278 R -3498) 
+(-278 R -3505) 
 ((|constructor| (NIL "Provides elementary functions over an integral domain.")) (|localReal?| (((|Boolean|) |#2|) "\\spad{localReal?(x)} should be local but conditional")) (|specialTrigs| (((|Union| |#2| "failed") |#2| (|List| (|Record| (|:| |func| |#2|) (|:| |pole| (|Boolean|))))) "\\spad{specialTrigs(x,l)} should be local but conditional")) (|iiacsch| ((|#2| |#2|) "\\spad{iiacsch(x)} should be local but conditional")) (|iiasech| ((|#2| |#2|) "\\spad{iiasech(x)} should be local but conditional")) (|iiacoth| ((|#2| |#2|) "\\spad{iiacoth(x)} should be local but conditional")) (|iiatanh| ((|#2| |#2|) "\\spad{iiatanh(x)} should be local but conditional")) (|iiacosh| ((|#2| |#2|) "\\spad{iiacosh(x)} should be local but conditional")) (|iiasinh| ((|#2| |#2|) "\\spad{iiasinh(x)} should be local but conditional")) (|iicsch| ((|#2| |#2|) "\\spad{iicsch(x)} should be local but conditional")) (|iisech| ((|#2| |#2|) "\\spad{iisech(x)} should be local but conditional")) (|iicoth| ((|#2| |#2|) "\\spad{iicoth(x)} should be local but conditional")) (|iitanh| ((|#2| |#2|) "\\spad{iitanh(x)} should be local but conditional")) (|iicosh| ((|#2| |#2|) "\\spad{iicosh(x)} should be local but conditional")) (|iisinh| ((|#2| |#2|) "\\spad{iisinh(x)} should be local but conditional")) (|iiacsc| ((|#2| |#2|) "\\spad{iiacsc(x)} should be local but conditional")) (|iiasec| ((|#2| |#2|) "\\spad{iiasec(x)} should be local but conditional")) (|iiacot| ((|#2| |#2|) "\\spad{iiacot(x)} should be local but conditional")) (|iiatan| ((|#2| |#2|) "\\spad{iiatan(x)} should be local but conditional")) (|iiacos| ((|#2| |#2|) "\\spad{iiacos(x)} should be local but conditional")) (|iiasin| ((|#2| |#2|) "\\spad{iiasin(x)} should be local but conditional")) (|iicsc| ((|#2| |#2|) "\\spad{iicsc(x)} should be local but conditional")) (|iisec| ((|#2| |#2|) "\\spad{iisec(x)} should be local but conditional")) (|iicot| ((|#2| |#2|) "\\spad{iicot(x)} should be local but conditional")) (|iitan| ((|#2| |#2|) "\\spad{iitan(x)} should be local but conditional")) (|iicos| ((|#2| |#2|) "\\spad{iicos(x)} should be local but conditional")) (|iisin| ((|#2| |#2|) "\\spad{iisin(x)} should be local but conditional")) (|iilog| ((|#2| |#2|) "\\spad{iilog(x)} should be local but conditional")) (|iiexp| ((|#2| |#2|) "\\spad{iiexp(x)} should be local but conditional")) (|iisqrt3| ((|#2|) "\\spad{iisqrt3()} should be local but conditional")) (|iisqrt2| ((|#2|) "\\spad{iisqrt2()} should be local but conditional")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(p)} returns an elementary operator with the same symbol as \\spad{p}")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(p)} returns \\spad{true} if operator \\spad{p} is elementary")) (|pi| ((|#2|) "\\spad{pi()} returns the \\spad{pi} operator")) (|acsch| ((|#2| |#2|) "\\spad{acsch(x)} applies the inverse hyperbolic cosecant operator to \\spad{x}")) (|asech| ((|#2| |#2|) "\\spad{asech(x)} applies the inverse hyperbolic secant operator to \\spad{x}")) (|acoth| ((|#2| |#2|) "\\spad{acoth(x)} applies the inverse hyperbolic cotangent operator to \\spad{x}")) (|atanh| ((|#2| |#2|) "\\spad{atanh(x)} applies the inverse hyperbolic tangent operator to \\spad{x}")) (|acosh| ((|#2| |#2|) "\\spad{acosh(x)} applies the inverse hyperbolic cosine operator to \\spad{x}")) (|asinh| ((|#2| |#2|) "\\spad{asinh(x)} applies the inverse hyperbolic sine operator to \\spad{x}")) (|csch| ((|#2| |#2|) "\\spad{csch(x)} applies the hyperbolic cosecant operator to \\spad{x}")) (|sech| ((|#2| |#2|) "\\spad{sech(x)} applies the hyperbolic secant operator to \\spad{x}")) (|coth| ((|#2| |#2|) "\\spad{coth(x)} applies the hyperbolic cotangent operator to \\spad{x}")) (|tanh| ((|#2| |#2|) "\\spad{tanh(x)} applies the hyperbolic tangent operator to \\spad{x}")) (|cosh| ((|#2| |#2|) "\\spad{cosh(x)} applies the hyperbolic cosine operator to \\spad{x}")) (|sinh| ((|#2| |#2|) "\\spad{sinh(x)} applies the hyperbolic sine operator to \\spad{x}")) (|acsc| ((|#2| |#2|) "\\spad{acsc(x)} applies the inverse cosecant operator to \\spad{x}")) (|asec| ((|#2| |#2|) "\\spad{asec(x)} applies the inverse secant operator to \\spad{x}")) (|acot| ((|#2| |#2|) "\\spad{acot(x)} applies the inverse cotangent operator to \\spad{x}")) (|atan| ((|#2| |#2|) "\\spad{atan(x)} applies the inverse tangent operator to \\spad{x}")) (|acos| ((|#2| |#2|) "\\spad{acos(x)} applies the inverse cosine operator to \\spad{x}")) (|asin| ((|#2| |#2|) "\\spad{asin(x)} applies the inverse sine operator to \\spad{x}")) (|csc| ((|#2| |#2|) "\\spad{csc(x)} applies the cosecant operator to \\spad{x}")) (|sec| ((|#2| |#2|) "\\spad{sec(x)} applies the secant operator to \\spad{x}")) (|cot| ((|#2| |#2|) "\\spad{cot(x)} applies the cotangent operator to \\spad{x}")) (|tan| ((|#2| |#2|) "\\spad{tan(x)} applies the tangent operator to \\spad{x}")) (|cos| ((|#2| |#2|) "\\spad{cos(x)} applies the cosine operator to \\spad{x}")) (|sin| ((|#2| |#2|) "\\spad{sin(x)} applies the sine operator to \\spad{x}")) (|log| ((|#2| |#2|) "\\spad{log(x)} applies the logarithm operator to \\spad{x}")) (|exp| ((|#2| |#2|) "\\spad{exp(x)} applies the exponential operator to \\spad{x}"))) 
 NIL 
 NIL 
-(-279 R -3498) 
+(-279 R -3505) 
 ((|constructor| (NIL "ElementaryFunctionStructurePackage provides functions to test the algebraic independence of various elementary functions,{} using the Risch structure theorem (real and complex versions). It also provides transformations on elementary functions which are not considered simplifications.")) (|tanQ| ((|#2| (|Fraction| (|Integer|)) |#2|) "\\spad{tanQ(q,a)} is a local function with a conditional implementation.")) (|rootNormalize| ((|#2| |#2| (|Kernel| |#2|)) "\\spad{rootNormalize(f, k)} returns \\spad{f} rewriting either \\spad{k} which must be an \\spad{n}th-root in terms of radicals already in \\spad{f},{} or some radicals in \\spad{f} in terms of \\spad{k}.")) (|validExponential| (((|Union| |#2| "failed") (|List| (|Kernel| |#2|)) |#2| (|Symbol|)) "\\spad{validExponential([k1,...,kn],f,x)} returns \\spad{g} if \\spad{exp(f)=g} and \\spad{g} involves only \\spad{k1...kn},{} and \"failed\" otherwise.")) (|realElementary| ((|#2| |#2| (|Symbol|)) "\\spad{realElementary(f,x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log, exp, tan, atan}.") ((|#2| |#2|) "\\spad{realElementary(f)} rewrites \\spad{f} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log, exp, tan, atan}.")) (|rischNormalize| (((|Record| (|:| |func| |#2|) (|:| |kers| (|List| (|Kernel| |#2|))) (|:| |vals| (|List| |#2|))) |#2| (|Symbol|)) "\\spad{rischNormalize(f, x)} returns \\spad{[g, [k1,...,kn], [h1,...,hn]]} such that \\spad{g = normalize(f, x)} and each \\spad{ki} was rewritten as \\spad{hi} during the normalization.")) (|normalize| ((|#2| |#2| (|Symbol|)) "\\spad{normalize(f, x)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{normalize(f)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels."))) 
 NIL 
 NIL 
 (-280 |Coef| UTS ULS) 
 ((|constructor| (NIL "\\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return \"failed\" when this is not possible.")) (|acsch| ((|#3| |#3|) "\\spad{acsch(z)} returns the inverse hyperbolic cosecant of Laurent series \\spad{z}.")) (|asech| ((|#3| |#3|) "\\spad{asech(z)} returns the inverse hyperbolic secant of Laurent series \\spad{z}.")) (|acoth| ((|#3| |#3|) "\\spad{acoth(z)} returns the inverse hyperbolic cotangent of Laurent series \\spad{z}.")) (|atanh| ((|#3| |#3|) "\\spad{atanh(z)} returns the inverse hyperbolic tangent of Laurent series \\spad{z}.")) (|acosh| ((|#3| |#3|) "\\spad{acosh(z)} returns the inverse hyperbolic cosine of Laurent series \\spad{z}.")) (|asinh| ((|#3| |#3|) "\\spad{asinh(z)} returns the inverse hyperbolic sine of Laurent series \\spad{z}.")) (|csch| ((|#3| |#3|) "\\spad{csch(z)} returns the hyperbolic cosecant of Laurent series \\spad{z}.")) (|sech| ((|#3| |#3|) "\\spad{sech(z)} returns the hyperbolic secant of Laurent series \\spad{z}.")) (|coth| ((|#3| |#3|) "\\spad{coth(z)} returns the hyperbolic cotangent of Laurent series \\spad{z}.")) (|tanh| ((|#3| |#3|) "\\spad{tanh(z)} returns the hyperbolic tangent of Laurent series \\spad{z}.")) (|cosh| ((|#3| |#3|) "\\spad{cosh(z)} returns the hyperbolic cosine of Laurent series \\spad{z}.")) (|sinh| ((|#3| |#3|) "\\spad{sinh(z)} returns the hyperbolic sine of Laurent series \\spad{z}.")) (|acsc| ((|#3| |#3|) "\\spad{acsc(z)} returns the arc-cosecant of Laurent series \\spad{z}.")) (|asec| ((|#3| |#3|) "\\spad{asec(z)} returns the arc-secant of Laurent series \\spad{z}.")) (|acot| ((|#3| |#3|) "\\spad{acot(z)} returns the arc-cotangent of Laurent series \\spad{z}.")) (|atan| ((|#3| |#3|) "\\spad{atan(z)} returns the arc-tangent of Laurent series \\spad{z}.")) (|acos| ((|#3| |#3|) "\\spad{acos(z)} returns the arc-cosine of Laurent series \\spad{z}.")) (|asin| ((|#3| |#3|) "\\spad{asin(z)} returns the arc-sine of Laurent series \\spad{z}.")) (|csc| ((|#3| |#3|) "\\spad{csc(z)} returns the cosecant of Laurent series \\spad{z}.")) (|sec| ((|#3| |#3|) "\\spad{sec(z)} returns the secant of Laurent series \\spad{z}.")) (|cot| ((|#3| |#3|) "\\spad{cot(z)} returns the cotangent of Laurent series \\spad{z}.")) (|tan| ((|#3| |#3|) "\\spad{tan(z)} returns the tangent of Laurent series \\spad{z}.")) (|cos| ((|#3| |#3|) "\\spad{cos(z)} returns the cosine of Laurent series \\spad{z}.")) (|sin| ((|#3| |#3|) "\\spad{sin(z)} returns the sine of Laurent series \\spad{z}.")) (|log| ((|#3| |#3|) "\\spad{log(z)} returns the logarithm of Laurent series \\spad{z}.")) (|exp| ((|#3| |#3|) "\\spad{exp(z)} returns the exponential of Laurent series \\spad{z}.")) (** ((|#3| |#3| (|Fraction| (|Integer|))) "\\spad{s ** r} raises a Laurent series \\spad{s} to a rational power \\spad{r}"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-366)))) 
+((|HasCategory| |#1| (QUOTE (-367)))) 
 (-281 |Coef| ULS UPXS EFULS) 
 ((|constructor| (NIL "\\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return \"failed\" when this is not possible.")) (|acsch| ((|#3| |#3|) "\\spad{acsch(z)} returns the inverse hyperbolic cosecant of a Puiseux series \\spad{z}.")) (|asech| ((|#3| |#3|) "\\spad{asech(z)} returns the inverse hyperbolic secant of a Puiseux series \\spad{z}.")) (|acoth| ((|#3| |#3|) "\\spad{acoth(z)} returns the inverse hyperbolic cotangent of a Puiseux series \\spad{z}.")) (|atanh| ((|#3| |#3|) "\\spad{atanh(z)} returns the inverse hyperbolic tangent of a Puiseux series \\spad{z}.")) (|acosh| ((|#3| |#3|) "\\spad{acosh(z)} returns the inverse hyperbolic cosine of a Puiseux series \\spad{z}.")) (|asinh| ((|#3| |#3|) "\\spad{asinh(z)} returns the inverse hyperbolic sine of a Puiseux series \\spad{z}.")) (|csch| ((|#3| |#3|) "\\spad{csch(z)} returns the hyperbolic cosecant of a Puiseux series \\spad{z}.")) (|sech| ((|#3| |#3|) "\\spad{sech(z)} returns the hyperbolic secant of a Puiseux series \\spad{z}.")) (|coth| ((|#3| |#3|) "\\spad{coth(z)} returns the hyperbolic cotangent of a Puiseux series \\spad{z}.")) (|tanh| ((|#3| |#3|) "\\spad{tanh(z)} returns the hyperbolic tangent of a Puiseux series \\spad{z}.")) (|cosh| ((|#3| |#3|) "\\spad{cosh(z)} returns the hyperbolic cosine of a Puiseux series \\spad{z}.")) (|sinh| ((|#3| |#3|) "\\spad{sinh(z)} returns the hyperbolic sine of a Puiseux series \\spad{z}.")) (|acsc| ((|#3| |#3|) "\\spad{acsc(z)} returns the arc-cosecant of a Puiseux series \\spad{z}.")) (|asec| ((|#3| |#3|) "\\spad{asec(z)} returns the arc-secant of a Puiseux series \\spad{z}.")) (|acot| ((|#3| |#3|) "\\spad{acot(z)} returns the arc-cotangent of a Puiseux series \\spad{z}.")) (|atan| ((|#3| |#3|) "\\spad{atan(z)} returns the arc-tangent of a Puiseux series \\spad{z}.")) (|acos| ((|#3| |#3|) "\\spad{acos(z)} returns the arc-cosine of a Puiseux series \\spad{z}.")) (|asin| ((|#3| |#3|) "\\spad{asin(z)} returns the arc-sine of a Puiseux series \\spad{z}.")) (|csc| ((|#3| |#3|) "\\spad{csc(z)} returns the cosecant of a Puiseux series \\spad{z}.")) (|sec| ((|#3| |#3|) "\\spad{sec(z)} returns the secant of a Puiseux series \\spad{z}.")) (|cot| ((|#3| |#3|) "\\spad{cot(z)} returns the cotangent of a Puiseux series \\spad{z}.")) (|tan| ((|#3| |#3|) "\\spad{tan(z)} returns the tangent of a Puiseux series \\spad{z}.")) (|cos| ((|#3| |#3|) "\\spad{cos(z)} returns the cosine of a Puiseux series \\spad{z}.")) (|sin| ((|#3| |#3|) "\\spad{sin(z)} returns the sine of a Puiseux series \\spad{z}.")) (|log| ((|#3| |#3|) "\\spad{log(z)} returns the logarithm of a Puiseux series \\spad{z}.")) (|exp| ((|#3| |#3|) "\\spad{exp(z)} returns the exponential of a Puiseux series \\spad{z}.")) (** ((|#3| |#3| (|Fraction| (|Integer|))) "\\spad{z ** r} raises a Puiseaux series \\spad{z} to a rational power \\spad{r}"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-366)))) 
+((|HasCategory| |#1| (QUOTE (-367)))) 
 (-282) 
 ((|constructor| (NIL "This domains an expresion as elaborated by the interpreter. See Also:")) (|getOperands| (((|Union| (|List| $) "failed") $) "\\spad{getOperands(e)} returns the list of operands in `e',{} assuming it is a call form.")) (|getOperator| (((|Union| (|Identifier|) "failed") $) "\\spad{getOperator(e)} retrieves the operator being invoked in `e',{} when `e' is an expression.")) (|callForm?| (((|Boolean|) $) "\\spad{callForm?(e)} is \\spad{true} when `e' is a call expression.")) (|getIdentifier| (((|Union| (|Identifier|) "failed") $) "\\spad{getIdentifier(e)} retrieves the name of the variable `e'.")) (|variable?| (((|Boolean|) $) "\\spad{variable?(e)} returns \\spad{true} if `e' is a variable.")) (|getConstant| (((|Union| (|SExpression|) "failed") $) "\\spad{getConstant(e)} retrieves the constant value of `e'e.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(e)} returns \\spad{true} if `e' is a constant.")) (|type| (((|Syntax|) $) "\\spad{type(e)} returns the type of the expression as computed by the interpreter."))) 
 NIL 
 NIL 
-(-283 A S) 
+(-283) 
+((|environment| (((|Environment|) $) "\\spad{environment(x)} returns the environment of the elaboration \\spad{x}.")) (|typeForm| (((|InternalTypeForm|) $) "\\spad{typeForm(x)} returns the type form of the elaboration \\spad{x}.")) (|irForm| (((|InternalRepresentationForm|) $) "\\spad{irForm(x)} returns the internal representation form of the elaboration \\spad{x}.")) (|elaboration| (($ (|InternalRepresentationForm|) (|InternalTypeForm|) (|Environment|)) "\\spad{elaboration(ir,ty,env)} construct an elaboration object for for the internal representation form \\spad{ir},{} with type \\spad{ty},{} and environment \\spad{env}."))) 
+NIL 
+NIL 
+(-284 A S) 
 ((|constructor| (NIL "An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \\spadtype{FlexibleArray} for an exception.")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(u)} destructively removes duplicates from \\spad{u}.")) (|select!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select!(p,u)} destructively changes \\spad{u} by keeping only values \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})}.")) (|merge!| (($ $ $) "\\spad{merge!(u,v)} destructively merges \\spad{u} and \\spad{v} in ascending order.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $ $) "\\spad{merge!(p,u,v)} destructively merges \\spad{u} and \\spad{v} using predicate \\spad{p}.")) (|insert!| (($ $ $ (|Integer|)) "\\spad{insert!(v,u,i)} destructively inserts aggregate \\spad{v} into \\spad{u} at position \\spad{i}.") (($ |#2| $ (|Integer|)) "\\spad{insert!(x,u,i)} destructively inserts \\spad{x} into \\spad{u} at position \\spad{i}.")) (|remove!| (($ |#2| $) "\\spad{remove!(x,u)} destructively removes all values \\spad{x} from \\spad{u}.") (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove!(p,u)} destructively removes all elements \\spad{x} of \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.")) (|delete!| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete!(u,i..j)} destructively deletes elements \\spad{u}.\\spad{i} through \\spad{u}.\\spad{j}.") (($ $ (|Integer|)) "\\spad{delete!(u,i)} destructively deletes the \\axiom{\\spad{i}}th element of \\spad{u}.")) (|concat!| (($ $ $) "\\spad{concat!(u,v)} destructively appends \\spad{v} to the end of \\spad{u}. \\spad{v} is unchanged") (($ $ |#2|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-1105)))) 
-(-284 S) 
+((|HasCategory| |#2| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-1107)))) 
+(-285 S) 
 ((|constructor| (NIL "An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \\spadtype{FlexibleArray} for an exception.")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(u)} destructively removes duplicates from \\spad{u}.")) (|select!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select!(p,u)} destructively changes \\spad{u} by keeping only values \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})}.")) (|merge!| (($ $ $) "\\spad{merge!(u,v)} destructively merges \\spad{u} and \\spad{v} in ascending order.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $ $) "\\spad{merge!(p,u,v)} destructively merges \\spad{u} and \\spad{v} using predicate \\spad{p}.")) (|insert!| (($ $ $ (|Integer|)) "\\spad{insert!(v,u,i)} destructively inserts aggregate \\spad{v} into \\spad{u} at position \\spad{i}.") (($ |#1| $ (|Integer|)) "\\spad{insert!(x,u,i)} destructively inserts \\spad{x} into \\spad{u} at position \\spad{i}.")) (|remove!| (($ |#1| $) "\\spad{remove!(x,u)} destructively removes all values \\spad{x} from \\spad{u}.") (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove!(p,u)} destructively removes all elements \\spad{x} of \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.")) (|delete!| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete!(u,i..j)} destructively deletes elements \\spad{u}.\\spad{i} through \\spad{u}.\\spad{j}.") (($ $ (|Integer|)) "\\spad{delete!(u,i)} destructively deletes the \\axiom{\\spad{i}}th element of \\spad{u}.")) (|concat!| (($ $ $) "\\spad{concat!(u,v)} destructively appends \\spad{v} to the end of \\spad{u}. \\spad{v} is unchanged") (($ $ |#1|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}."))) 
-((-4428 . T)) 
+((-4435 . T)) 
 NIL 
-(-285 S) 
+(-286 S) 
 ((|constructor| (NIL "Category for the elementary functions.")) (** (($ $ $) "\\spad{x**y} returns \\spad{x} to the power \\spad{y}.")) (|exp| (($ $) "\\spad{exp(x)} returns \\%\\spad{e} to the power \\spad{x}.")) (|log| (($ $) "\\spad{log(x)} returns the natural logarithm of \\spad{x}."))) 
 NIL 
 NIL 
-(-286) 
+(-287) 
 ((|constructor| (NIL "Category for the elementary functions.")) (** (($ $ $) "\\spad{x**y} returns \\spad{x} to the power \\spad{y}.")) (|exp| (($ $) "\\spad{exp(x)} returns \\%\\spad{e} to the power \\spad{x}.")) (|log| (($ $) "\\spad{log(x)} returns the natural logarithm of \\spad{x}."))) 
 NIL 
 NIL 
-(-287 |Coef| UTS) 
+(-288 |Coef| UTS) 
 ((|constructor| (NIL "The elliptic functions \\spad{sn},{} \\spad{sc} and \\spad{dn} are expanded as Taylor series.")) (|sncndn| (((|List| (|Stream| |#1|)) (|Stream| |#1|) |#1|) "\\spad{sncndn(s,c)} is used internally.")) (|dn| ((|#2| |#2| |#1|) "\\spad{dn(x,k)} expands the elliptic function \\spad{dn} as a Taylor \\indented{1}{series.}")) (|cn| ((|#2| |#2| |#1|) "\\spad{cn(x,k)} expands the elliptic function \\spad{cn} as a Taylor \\indented{1}{series.}")) (|sn| ((|#2| |#2| |#1|) "\\spad{sn(x,k)} expands the elliptic function \\spad{sn} as a Taylor \\indented{1}{series.}"))) 
 NIL 
 NIL 
-(-288 S |Index|) 
+(-289 S |Index|) 
 ((|constructor| (NIL "An eltable over domains \\spad{D} and \\spad{I} is a structure which can be viewed as a function from \\spad{D} to \\spad{I}. Examples of eltable structures range from data structures,{} \\spadignore{e.g.} those of type \\spadtype{List},{} to algebraic structures,{} \\spadignore{e.g.} \\spadtype{Polynomial}.")) (|elt| ((|#2| $ |#1|) "\\spad{elt(u,i)} (also written: \\spad{u} . \\spad{i}) returns the element of \\spad{u} indexed by \\spad{i}. Error: if \\spad{i} is not an index of \\spad{u}."))) 
 NIL 
 NIL 
-(-289 S |Dom| |Im|) 
+(-290 S |Dom| |Im|) 
 ((|constructor| (NIL "An eltable aggregate is one which can be viewed as a function. For example,{} the list \\axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\\em Dom} to an image domain {\\em Im}.")) (|qsetelt!| ((|#3| $ |#2| |#3|) "\\spad{qsetelt!(u,x,y)} sets the image of \\axiom{\\spad{x}} to be \\axiom{\\spad{y}} under \\axiom{\\spad{u}},{} without checking that \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If such a check is required use the function \\axiom{setelt}.")) (|setelt| ((|#3| $ |#2| |#3|) "\\spad{setelt(u,x,y)} sets the image of \\spad{x} to be \\spad{y} under \\spad{u},{} assuming \\spad{x} is in the domain of \\spad{u}. Error: if \\spad{x} is not in the domain of \\spad{u}.")) (|qelt| ((|#3| $ |#2|) "\\spad{qelt(u, x)} applies \\axiom{\\spad{u}} to \\axiom{\\spad{x}} without checking whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If \\axiom{\\spad{x}} is not in the domain of \\axiom{\\spad{u}} a memory-access violation may occur. If a check on whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}} is required,{} use the function \\axiom{elt}.")) (|elt| ((|#3| $ |#2| |#3|) "\\spad{elt(u, x, y)} applies \\spad{u} to \\spad{x} if \\spad{x} is in the domain of \\spad{u},{} and returns \\spad{y} otherwise. For example,{} if \\spad{u} is a polynomial in \\axiom{\\spad{x}} over the rationals,{} \\axiom{elt(\\spad{u},{}\\spad{n},{}0)} may define the coefficient of \\axiom{\\spad{x}} to the power \\spad{n},{} returning 0 when \\spad{n} is out of range."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4428))) 
-(-290 |Dom| |Im|) 
+((|HasAttribute| |#1| (QUOTE -4435))) 
+(-291 |Dom| |Im|) 
 ((|constructor| (NIL "An eltable aggregate is one which can be viewed as a function. For example,{} the list \\axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\\em Dom} to an image domain {\\em Im}.")) (|qsetelt!| ((|#2| $ |#1| |#2|) "\\spad{qsetelt!(u,x,y)} sets the image of \\axiom{\\spad{x}} to be \\axiom{\\spad{y}} under \\axiom{\\spad{u}},{} without checking that \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If such a check is required use the function \\axiom{setelt}.")) (|setelt| ((|#2| $ |#1| |#2|) "\\spad{setelt(u,x,y)} sets the image of \\spad{x} to be \\spad{y} under \\spad{u},{} assuming \\spad{x} is in the domain of \\spad{u}. Error: if \\spad{x} is not in the domain of \\spad{u}.")) (|qelt| ((|#2| $ |#1|) "\\spad{qelt(u, x)} applies \\axiom{\\spad{u}} to \\axiom{\\spad{x}} without checking whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If \\axiom{\\spad{x}} is not in the domain of \\axiom{\\spad{u}} a memory-access violation may occur. If a check on whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}} is required,{} use the function \\axiom{elt}.")) (|elt| ((|#2| $ |#1| |#2|) "\\spad{elt(u, x, y)} applies \\spad{u} to \\spad{x} if \\spad{x} is in the domain of \\spad{u},{} and returns \\spad{y} otherwise. For example,{} if \\spad{u} is a polynomial in \\axiom{\\spad{x}} over the rationals,{} \\axiom{elt(\\spad{u},{}\\spad{n},{}0)} may define the coefficient of \\axiom{\\spad{x}} to the power \\spad{n},{} returning 0 when \\spad{n} is out of range."))) 
 NIL 
 NIL 
-(-291 S R |Mod| -2217 -3943 |exactQuo|) 
+(-292 S R |Mod| -2224 -3950 |exactQuo|) 
 ((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{ModularRing},{} \\spadtype{ModularField}")) (|elt| ((|#2| $ |#2|) "\\spad{elt(x,r)} or \\spad{x}.\\spad{r} \\undocumented")) (|inv| (($ $) "\\spad{inv(x)} \\undocumented")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} \\undocumented")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#2| |#3|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#2| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#3| $) "\\spad{modulus(x)} \\undocumented"))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-292) 
+(-293) 
 ((|constructor| (NIL "Entire Rings (non-commutative Integral Domains),{} \\spadignore{i.e.} a ring not necessarily commutative which has no zero divisors. \\blankline")) (|noZeroDivisors| ((|attribute|) "if a product is zero then one of the factors must be zero."))) 
-((-4420 . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-293) 
+(-294) 
 ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 19,{} 2008. An `Environment' is a stack of scope.")) (|categoryFrame| (($) "the current category environment in the interpreter.")) (|interactiveEnv| (($) "the current interactive environment in effect.")) (|currentEnv| (($) "the current normal environment in effect.")) (|setProperties!| (($ (|Identifier|) (|List| (|Property|)) $) "setBinding!(\\spad{n},{}props,{}\\spad{e}) set the list of properties of \\spad{`n'} to `props' in `e'.")) (|getProperties| (((|List| (|Property|)) (|Identifier|) $) "getBinding(\\spad{n},{}\\spad{e}) returns the list of properties of \\spad{`n'} in \\spad{e}.")) (|setProperty!| (($ (|Identifier|) (|Identifier|) (|SExpression|) $) "\\spad{setProperty!(n,p,v,e)} binds the property `(\\spad{p},{}\\spad{v})' to \\spad{`n'} in the topmost scope of `e'.")) (|getProperty| (((|Maybe| (|SExpression|)) (|Identifier|) (|Identifier|) $) "\\spad{getProperty(n,p,e)} returns the value of property with name \\spad{`p'} for the symbol \\spad{`n'} in environment `e'. Otherwise,{} `nothing.")) (|scopes| (((|List| (|Scope|)) $) "\\spad{scopes(e)} returns the stack of scopes in environment \\spad{e}.")) (|empty| (($) "\\spad{empty()} constructs an empty environment"))) 
 NIL 
 NIL 
-(-294 R) 
+(-295 R) 
 ((|constructor| (NIL "This is a package for the exact computation of eigenvalues and eigenvectors. This package can be made to work for matrices with coefficients which are rational functions over a ring where we can factor polynomials. Rational eigenvalues are always explicitly computed while the non-rational ones are expressed in terms of their minimal polynomial.")) (|eigenvectors| (((|List| (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |eigmult| (|NonNegativeInteger|)) (|:| |eigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|))))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvectors(m)} returns the eigenvalues and eigenvectors for the matrix \\spad{m}. The rational eigenvalues and the correspondent eigenvectors are explicitely computed,{} while the non rational ones are given via their minimal polynomial and the corresponding eigenvectors are expressed in terms of a \"generic\" root of such a polynomial.")) (|generalizedEigenvectors| (((|List| (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |geneigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|))))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{generalizedEigenvectors(m)} returns the generalized eigenvectors of the matrix \\spad{m}.")) (|generalizedEigenvector| (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |eigmult| (|NonNegativeInteger|)) (|:| |eigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{generalizedEigenvector(eigen,m)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \\spad{eigen},{} as returned by the function eigenvectors.") (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|))) (|Matrix| (|Fraction| (|Polynomial| |#1|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generalizedEigenvector(alpha,m,k,g)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \\spad{alpha}. The integers \\spad{k} and \\spad{g} are respectively the algebraic and the geometric multiplicity of tye eigenvalue \\spad{alpha}. \\spad{alpha} can be either rational or not. In the seconda case apha is the minimal polynomial of the eigenvalue.")) (|eigenvector| (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvector(eigval,m)} returns the eigenvectors belonging to the eigenvalue \\spad{eigval} for the matrix \\spad{m}.")) (|eigenvalues| (((|List| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvalues(m)} returns the eigenvalues of the matrix \\spad{m} which are expressible as rational functions over the rational numbers.")) (|characteristicPolynomial| (((|Polynomial| |#1|) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{characteristicPolynomial(m)} returns the characteristicPolynomial of the matrix \\spad{m} using a new generated symbol symbol as the main variable.") (((|Polynomial| |#1|) (|Matrix| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{characteristicPolynomial(m,var)} returns the characteristicPolynomial of the matrix \\spad{m} using the symbol \\spad{var} as the main variable."))) 
 NIL 
 NIL 
-(-295 S) 
+(-296 S) 
 ((|constructor| (NIL "Equations as mathematical objects. All properties of the basis domain,{} \\spadignore{e.g.} being an abelian group are carried over the equation domain,{} by performing the structural operations on the left and on the right hand side.")) (|subst| (($ $ $) "\\spad{subst(eq1,eq2)} substitutes \\spad{eq2} into both sides of \\spad{eq1} the \\spad{lhs} of \\spad{eq2} should be a kernel")) (|inv| (($ $) "\\spad{inv(x)} returns the multiplicative inverse of \\spad{x}.")) (/ (($ $ $) "\\spad{e1/e2} produces a new equation by dividing the left and right hand sides of equations e1 and e2.")) (|factorAndSplit| (((|List| $) $) "\\spad{factorAndSplit(eq)} make the right hand side 0 and factors the new left hand side. Each factor is equated to 0 and put into the resulting list without repetitions.")) (|rightOne| (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side.") (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side,{} if possible.")) (|leftOne| (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side.") (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side,{} if possible.")) (* (($ $ |#1|) "\\spad{eqn*x} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.") (($ |#1| $) "\\spad{x*eqn} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.")) (- (($ $ |#1|) "\\spad{eqn-x} produces a new equation by subtracting \\spad{x} from both sides of equation eqn.") (($ |#1| $) "\\spad{x-eqn} produces a new equation by subtracting both sides of equation eqn from \\spad{x}.")) (|rightZero| (($ $) "\\spad{rightZero(eq)} subtracts the right hand side.")) (|leftZero| (($ $) "\\spad{leftZero(eq)} subtracts the left hand side.")) (+ (($ $ |#1|) "\\spad{eqn+x} produces a new equation by adding \\spad{x} to both sides of equation eqn.") (($ |#1| $) "\\spad{x+eqn} produces a new equation by adding \\spad{x} to both sides of equation eqn.")) (|eval| (($ $ (|List| $)) "\\spad{eval(eqn, [x1=v1, ... xn=vn])} replaces \\spad{xi} by \\spad{vi} in equation \\spad{eqn}.") (($ $ $) "\\spad{eval(eqn, x=f)} replaces \\spad{x} by \\spad{f} in equation \\spad{eqn}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,eqn)} constructs a new equation by applying \\spad{f} to both sides of \\spad{eqn}.")) (|rhs| ((|#1| $) "\\spad{rhs(eqn)} returns the right hand side of equation \\spad{eqn}.")) (|lhs| ((|#1| $) "\\spad{lhs(eqn)} returns the left hand side of equation \\spad{eqn}.")) (|swap| (($ $) "\\spad{swap(eq)} interchanges left and right hand side of equation \\spad{eq}.")) (|equation| (($ |#1| |#1|) "\\spad{equation(a,b)} creates an equation.")) (= (($ |#1| |#1|) "\\spad{a=b} creates an equation."))) 
-((-4424 -3962 (|has| |#1| (-1053)) (|has| |#1| (-477))) (-4421 |has| |#1| (-1053)) (-4422 |has| |#1| (-1053))) 
-((|HasCategory| |#1| (QUOTE (-366))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-1053)))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (-3962 (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-1053)))) (-3962 (|HasCategory| |#1| (QUOTE (-477))) (|HasCategory| |#1| (QUOTE (-729)))) (|HasCategory| |#1| (QUOTE (-477))) (-3962 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-477))) (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (|HasCategory| |#1| (QUOTE (-477))) (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#1| (QUOTE (-1116)))) (|HasCategory| |#1| (LIST (QUOTE -518) (QUOTE (-1181)) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-300))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-477)))) (-3962 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-729)))) (-3962 (|HasCategory| |#1| (QUOTE (-477))) (|HasCategory| |#1| (QUOTE (-1053)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-729)))) 
-(-296 S R) 
+((-4431 -3969 (|has| |#1| (-1055)) (|has| |#1| (-478))) (-4428 |has| |#1| (-1055)) (-4429 |has| |#1| (-1055))) 
+((|HasCategory| |#1| (QUOTE (-367))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-1055)))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (-3969 (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183))))) (-3969 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183))))) (-3969 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183))))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-1055)))) (-3969 (|HasCategory| |#1| (QUOTE (-478))) (|HasCategory| |#1| (QUOTE (-731)))) (|HasCategory| |#1| (QUOTE (-478))) (-3969 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-478))) (|HasCategory| |#1| (QUOTE (-731))) (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (QUOTE (-1118))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183))))) (-3969 (|HasCategory| |#1| (QUOTE (-478))) (|HasCategory| |#1| (QUOTE (-731))) (|HasCategory| |#1| (QUOTE (-1118)))) (|HasCategory| |#1| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-301))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-478)))) (-3969 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-731)))) (-3969 (|HasCategory| |#1| (QUOTE (-478))) (|HasCategory| |#1| (QUOTE (-1055)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1118))) (|HasCategory| |#1| (QUOTE (-731)))) 
+(-297 S R) 
 ((|constructor| (NIL "This package provides operations for mapping the sides of equations.")) (|map| (((|Equation| |#2|) (|Mapping| |#2| |#1|) (|Equation| |#1|)) "\\spad{map(f,eq)} returns an equation where \\spad{f} is applied to the sides of \\spad{eq}"))) 
 NIL 
 NIL 
-(-297 |Key| |Entry|) 
+(-298 |Key| |Entry|) 
 ((|constructor| (NIL "This domain provides tables where the keys are compared using \\spadfun{eq?}. Thus keys are considered equal only if they are the same instance of a structure."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-1105))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-298) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-1107))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-299) 
 ((|constructor| (NIL "ErrorFunctions implements error functions callable from the system interpreter. Typically,{} these functions would be called in user functions. The simple forms of the functions take one argument which is either a string (an error message) or a list of strings which all together make up a message. The list can contain formatting codes (see below). The more sophisticated versions takes two arguments where the first argument is the name of the function from which the error was invoked and the second argument is either a string or a list of strings,{} as above. When you use the one argument version in an interpreter function,{} the system will automatically insert the name of the function as the new first argument. Thus in the user interpreter function \\indented{2}{\\spad{f x == if x < 0 then error \"negative argument\" else x}} the call to error will actually be of the form \\indented{2}{\\spad{error(\"f\",\"negative argument\")}} because the interpreter will have created a new first argument. \\blankline Formatting codes: error messages may contain the following formatting codes (they should either start or end a string or else have blanks around them): \\indented{3}{\\spad{\\%l}\\space{6}start a new line} \\indented{3}{\\spad{\\%b}\\space{6}start printing in a bold font (where available)} \\indented{3}{\\spad{\\%d}\\space{6}stop\\space{2}printing in a bold font (where available)} \\indented{3}{\\spad{ \\%ceon}\\space{2}start centering message lines} \\indented{3}{\\spad{\\%ceoff}\\space{2}stop\\space{2}centering message lines} \\indented{3}{\\spad{\\%rjon}\\space{3}start displaying lines \"ragged left\"} \\indented{3}{\\spad{\\%rjoff}\\space{2}stop\\space{2}displaying lines \"ragged left\"} \\indented{3}{\\spad{\\%i}\\space{6}indent\\space{3}following lines 3 additional spaces} \\indented{3}{\\spad{\\%u}\\space{6}unindent following lines 3 additional spaces} \\indented{3}{\\spad{\\%xN}\\space{5}insert \\spad{N} blanks (eg,{} \\spad{\\%x10} inserts 10 blanks)} \\blankline")) (|error| (((|Exit|) (|String|) (|List| (|String|))) "\\spad{error(nam,lmsg)} displays error messages \\spad{lmsg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|String|) (|String|)) "\\spad{error(nam,msg)} displays error message \\spad{msg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|List| (|String|))) "\\spad{error(lmsg)} displays error message \\spad{lmsg} and terminates.") (((|Exit|) (|String|)) "\\spad{error(msg)} displays error message \\spad{msg} and terminates."))) 
 NIL 
 NIL 
-(-299 S) 
+(-300 S) 
 ((|constructor| (NIL "An expression space is a set which is closed under certain operators.")) (|odd?| (((|Boolean|) $) "\\spad{odd? x} is \\spad{true} if \\spad{x} is an odd integer.")) (|even?| (((|Boolean|) $) "\\spad{even? x} is \\spad{true} if \\spad{x} is an even integer.")) (|definingPolynomial| (($ $) "\\spad{definingPolynomial(x)} returns an expression \\spad{p} such that \\spad{p(x) = 0}.")) (|minPoly| (((|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{minPoly(k)} returns \\spad{p} such that \\spad{p(k) = 0}.")) (|eval| (($ $ (|BasicOperator|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|BasicOperator|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.")) (|freeOf?| (((|Boolean|) $ (|Symbol|)) "\\spad{freeOf?(x, s)} tests if \\spad{x} does not contain any operator whose name is \\spad{s}.") (((|Boolean|) $ $) "\\spad{freeOf?(x, y)} tests if \\spad{x} does not contain any occurrence of \\spad{y},{} where \\spad{y} is a single kernel.")) (|map| (($ (|Mapping| $ $) (|Kernel| $)) "\\spad{map(f, k)} returns \\spad{op(f(x1),...,f(xn))} where \\spad{k = op(x1,...,xn)}.")) (|kernel| (($ (|BasicOperator|) (|List| $)) "\\spad{kernel(op, [f1,...,fn])} constructs \\spad{op(f1,...,fn)} without evaluating it.") (($ (|BasicOperator|) $) "\\spad{kernel(op, x)} constructs \\spad{op}(\\spad{x}) without evaluating it.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(x, s)} tests if \\spad{x} is a kernel and is the name of its operator is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(x, op)} tests if \\spad{x} is a kernel and is its operator is op.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} tests if \\% accepts \\spad{op} as applicable to its elements.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\%.")) (|operators| (((|List| (|BasicOperator|)) $) "\\spad{operators(f)} returns all the basic operators appearing in \\spad{f},{} no matter what their levels are.")) (|tower| (((|List| (|Kernel| $)) $) "\\spad{tower(f)} returns all the kernels appearing in \\spad{f},{} no matter what their levels are.")) (|kernels| (((|List| (|Kernel| $)) $) "\\spad{kernels(f)} returns the list of all the top-level kernels appearing in \\spad{f},{} but not the ones appearing in the arguments of the top-level kernels.")) (|mainKernel| (((|Union| (|Kernel| $) "failed") $) "\\spad{mainKernel(f)} returns a kernel of \\spad{f} with maximum nesting level,{} or if \\spad{f} has no kernels (\\spadignore{i.e.} \\spad{f} is a constant).")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(f)} returns the highest nesting level appearing in \\spad{f}. Constants have height 0. Symbols have height 1. For any operator op and expressions \\spad{f1},{}...,{}\\spad{fn},{} \\spad{op(f1,...,fn)} has height equal to \\spad{1 + max(height(f1),...,height(fn))}.")) (|distribute| (($ $ $) "\\spad{distribute(f, g)} expands all the kernels in \\spad{f} that contain \\spad{g} in their arguments and that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or a \\spadfunFrom{paren}{ExpressionSpace} expression.") (($ $) "\\spad{distribute(f)} expands all the kernels in \\spad{f} that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or \\spadfunFrom{paren}{ExpressionSpace} expression.")) (|paren| (($ (|List| $)) "\\spad{paren([f1,...,fn])} returns \\spad{(f1,...,fn)}. This prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(paren [x, 2])} returns the formal kernel \\spad{atan((x, 2))}.") (($ $) "\\spad{paren(f)} returns (\\spad{f}). This prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(paren 1)} returns the formal kernel log((1)).")) (|box| (($ (|List| $)) "\\spad{box([f1,...,fn])} returns \\spad{(f1,...,fn)} with a 'box' around them that prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(box [x, 2])} returns the formal kernel \\spad{atan(x, 2)}.") (($ $) "\\spad{box(f)} returns \\spad{f} with a 'box' around it that prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(box 1)} returns the formal kernel log(1).")) (|subst| (($ $ (|List| (|Kernel| $)) (|List| $)) "\\spad{subst(f, [k1...,kn], [g1,...,gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|List| (|Equation| $))) "\\spad{subst(f, [k1 = g1,...,kn = gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|Equation| $)) "\\spad{subst(f, k = g)} replaces the kernel \\spad{k} by \\spad{g} formally in \\spad{f}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op,[x1,...,xn])} or \\spad{op}([\\spad{x1},{}...,{}\\spad{xn}]) applies the \\spad{n}-ary operator \\spad{op} to \\spad{x1},{}...,{}\\spad{xn}.") (($ (|BasicOperator|) $ $ $ $) "\\spad{elt(op,x,y,z,t)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z},{} \\spad{t}) applies the 4-ary operator \\spad{op} to \\spad{x},{} \\spad{y},{} \\spad{z} and \\spad{t}.") (($ (|BasicOperator|) $ $ $) "\\spad{elt(op,x,y,z)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z}) applies the ternary operator \\spad{op} to \\spad{x},{} \\spad{y} and \\spad{z}.") (($ (|BasicOperator|) $ $) "\\spad{elt(op,x,y)} or \\spad{op}(\\spad{x},{} \\spad{y}) applies the binary operator \\spad{op} to \\spad{x} and \\spad{y}.") (($ (|BasicOperator|) $) "\\spad{elt(op,x)} or \\spad{op}(\\spad{x}) applies the unary operator \\spad{op} to \\spad{x}."))) 
 NIL 
-((|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-1053)))) 
-(-300) 
+((|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-1055)))) 
+(-301) 
 ((|constructor| (NIL "An expression space is a set which is closed under certain operators.")) (|odd?| (((|Boolean|) $) "\\spad{odd? x} is \\spad{true} if \\spad{x} is an odd integer.")) (|even?| (((|Boolean|) $) "\\spad{even? x} is \\spad{true} if \\spad{x} is an even integer.")) (|definingPolynomial| (($ $) "\\spad{definingPolynomial(x)} returns an expression \\spad{p} such that \\spad{p(x) = 0}.")) (|minPoly| (((|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{minPoly(k)} returns \\spad{p} such that \\spad{p(k) = 0}.")) (|eval| (($ $ (|BasicOperator|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|BasicOperator|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.")) (|freeOf?| (((|Boolean|) $ (|Symbol|)) "\\spad{freeOf?(x, s)} tests if \\spad{x} does not contain any operator whose name is \\spad{s}.") (((|Boolean|) $ $) "\\spad{freeOf?(x, y)} tests if \\spad{x} does not contain any occurrence of \\spad{y},{} where \\spad{y} is a single kernel.")) (|map| (($ (|Mapping| $ $) (|Kernel| $)) "\\spad{map(f, k)} returns \\spad{op(f(x1),...,f(xn))} where \\spad{k = op(x1,...,xn)}.")) (|kernel| (($ (|BasicOperator|) (|List| $)) "\\spad{kernel(op, [f1,...,fn])} constructs \\spad{op(f1,...,fn)} without evaluating it.") (($ (|BasicOperator|) $) "\\spad{kernel(op, x)} constructs \\spad{op}(\\spad{x}) without evaluating it.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(x, s)} tests if \\spad{x} is a kernel and is the name of its operator is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(x, op)} tests if \\spad{x} is a kernel and is its operator is op.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} tests if \\% accepts \\spad{op} as applicable to its elements.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\%.")) (|operators| (((|List| (|BasicOperator|)) $) "\\spad{operators(f)} returns all the basic operators appearing in \\spad{f},{} no matter what their levels are.")) (|tower| (((|List| (|Kernel| $)) $) "\\spad{tower(f)} returns all the kernels appearing in \\spad{f},{} no matter what their levels are.")) (|kernels| (((|List| (|Kernel| $)) $) "\\spad{kernels(f)} returns the list of all the top-level kernels appearing in \\spad{f},{} but not the ones appearing in the arguments of the top-level kernels.")) (|mainKernel| (((|Union| (|Kernel| $) "failed") $) "\\spad{mainKernel(f)} returns a kernel of \\spad{f} with maximum nesting level,{} or if \\spad{f} has no kernels (\\spadignore{i.e.} \\spad{f} is a constant).")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(f)} returns the highest nesting level appearing in \\spad{f}. Constants have height 0. Symbols have height 1. For any operator op and expressions \\spad{f1},{}...,{}\\spad{fn},{} \\spad{op(f1,...,fn)} has height equal to \\spad{1 + max(height(f1),...,height(fn))}.")) (|distribute| (($ $ $) "\\spad{distribute(f, g)} expands all the kernels in \\spad{f} that contain \\spad{g} in their arguments and that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or a \\spadfunFrom{paren}{ExpressionSpace} expression.") (($ $) "\\spad{distribute(f)} expands all the kernels in \\spad{f} that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or \\spadfunFrom{paren}{ExpressionSpace} expression.")) (|paren| (($ (|List| $)) "\\spad{paren([f1,...,fn])} returns \\spad{(f1,...,fn)}. This prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(paren [x, 2])} returns the formal kernel \\spad{atan((x, 2))}.") (($ $) "\\spad{paren(f)} returns (\\spad{f}). This prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(paren 1)} returns the formal kernel log((1)).")) (|box| (($ (|List| $)) "\\spad{box([f1,...,fn])} returns \\spad{(f1,...,fn)} with a 'box' around them that prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(box [x, 2])} returns the formal kernel \\spad{atan(x, 2)}.") (($ $) "\\spad{box(f)} returns \\spad{f} with a 'box' around it that prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(box 1)} returns the formal kernel log(1).")) (|subst| (($ $ (|List| (|Kernel| $)) (|List| $)) "\\spad{subst(f, [k1...,kn], [g1,...,gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|List| (|Equation| $))) "\\spad{subst(f, [k1 = g1,...,kn = gn])} replaces the kernels \\spad{k1},{}...,{}\\spad{kn} by \\spad{g1},{}...,{}\\spad{gn} formally in \\spad{f}.") (($ $ (|Equation| $)) "\\spad{subst(f, k = g)} replaces the kernel \\spad{k} by \\spad{g} formally in \\spad{f}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op,[x1,...,xn])} or \\spad{op}([\\spad{x1},{}...,{}\\spad{xn}]) applies the \\spad{n}-ary operator \\spad{op} to \\spad{x1},{}...,{}\\spad{xn}.") (($ (|BasicOperator|) $ $ $ $) "\\spad{elt(op,x,y,z,t)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z},{} \\spad{t}) applies the 4-ary operator \\spad{op} to \\spad{x},{} \\spad{y},{} \\spad{z} and \\spad{t}.") (($ (|BasicOperator|) $ $ $) "\\spad{elt(op,x,y,z)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z}) applies the ternary operator \\spad{op} to \\spad{x},{} \\spad{y} and \\spad{z}.") (($ (|BasicOperator|) $ $) "\\spad{elt(op,x,y)} or \\spad{op}(\\spad{x},{} \\spad{y}) applies the binary operator \\spad{op} to \\spad{x} and \\spad{y}.") (($ (|BasicOperator|) $) "\\spad{elt(op,x)} or \\spad{op}(\\spad{x}) applies the unary operator \\spad{op} to \\spad{x}."))) 
 NIL 
 NIL 
-(-301 -3498 S) 
+(-302 -3505 S) 
 ((|constructor| (NIL "This package allows a map from any expression space into any object to be lifted to a kernel over the expression set,{} using a given property of the operator of the kernel.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|String|) (|Kernel| |#1|)) "\\spad{map(f, p, k)} uses the property \\spad{p} of the operator of \\spad{k},{} in order to lift \\spad{f} and apply it to \\spad{k}."))) 
 NIL 
 NIL 
-(-302 E -3498) 
+(-303 E -3505) 
 ((|constructor| (NIL "This package allows a mapping \\spad{E} \\spad{->} \\spad{F} to be lifted to a kernel over \\spad{E}; This lifting can fail if the operator of the kernel cannot be applied in \\spad{F}; Do not use this package with \\spad{E} = \\spad{F},{} since this may drop some properties of the operators.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|Kernel| |#1|)) "\\spad{map(f, k)} returns \\spad{g = op(f(a1),...,f(an))} where \\spad{k = op(a1,...,an)}."))) 
 NIL 
 NIL 
-(-303) 
+(-304) 
 ((|constructor| (NIL "ExpertSystemContinuityPackage is a package of functions for the use of domains belonging to the category \\axiomType{NumericalIntegration}.")) (|sdf2lst| (((|List| (|String|)) (|Stream| (|DoubleFloat|))) "\\spad{sdf2lst(ln)} coerces a Stream of \\axiomType{DoubleFloat} to \\axiomType{List}(\\axiomType{String})")) (|ldf2lst| (((|List| (|String|)) (|List| (|DoubleFloat|))) "\\spad{ldf2lst(ln)} coerces a List of \\axiomType{DoubleFloat} to \\axiomType{List}(\\axiomType{String})")) (|df2st| (((|String|) (|DoubleFloat|)) "\\spad{df2st(n)} coerces a \\axiomType{DoubleFloat} to \\axiomType{String}")) (|polynomialZeros| (((|List| (|DoubleFloat|)) (|Polynomial| (|Fraction| (|Integer|))) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{polynomialZeros(fn,var,range)} calculates the real zeros of the polynomial which are contained in the given interval. It returns a list of points (\\axiomType{Doublefloat}) for which the univariate polynomial \\spad{fn} is zero.")) (|singularitiesOf| (((|Stream| (|DoubleFloat|)) (|Vector| (|Expression| (|DoubleFloat|))) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{singularitiesOf(v,vars,range)} returns a list of points (\\axiomType{Doublefloat}) at which a NAG fortran version of \\spad{v} will most likely produce an error. This includes those points which evaluate to 0/0.") (((|Stream| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{singularitiesOf(e,vars,range)} returns a list of points (\\axiomType{Doublefloat}) at which a NAG fortran version of \\spad{e} will most likely produce an error. This includes those points which evaluate to 0/0.")) (|zerosOf| (((|Stream| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|List| (|Symbol|)) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{zerosOf(e,vars,range)} returns a list of points (\\axiomType{Doublefloat}) at which a NAG fortran version of \\spad{e} will most likely produce an error.")) (|problemPoints| (((|List| (|DoubleFloat|)) (|Expression| (|DoubleFloat|)) (|Symbol|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{problemPoints(f,var,range)} returns a list of possible problem points by looking at the zeros of the denominator of the function \\spad{f} if it can be retracted to \\axiomType{Polynomial(DoubleFloat)}.")) (|functionIsFracPolynomial?| (((|Boolean|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{functionIsFracPolynomial?(args)} tests whether the function can be retracted to \\axiomType{Fraction(Polynomial(DoubleFloat))}")) (|gethi| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{gethi(u)} gets the \\axiomType{DoubleFloat} equivalent of the second endpoint of the range \\axiom{\\spad{u}}")) (|getlo| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{getlo(u)} gets the \\axiomType{DoubleFloat} equivalent of the first endpoint of the range \\axiom{\\spad{u}}"))) 
 NIL 
 NIL 
-(-304 A B) 
+(-305 A B) 
 ((|constructor| (NIL "ExpertSystemContinuityPackage1 exports a function to check range inclusion")) (|in?| (((|Boolean|) (|DoubleFloat|)) "\\spad{in?(p)} tests whether point \\spad{p} is internal to the range [\\spad{A..B}]"))) 
 NIL 
 NIL 
-(-305) 
+(-306) 
 ((|constructor| (NIL "\\axiom{ExpertSystemToolsPackage} contains some useful functions for use by the computational agents of numerical solvers.")) (|mat| (((|Matrix| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|NonNegativeInteger|)) "\\spad{mat(a,n)} constructs a one-dimensional matrix of a.")) (|fi2df| (((|DoubleFloat|) (|Fraction| (|Integer|))) "\\spad{fi2df(f)} coerces a \\axiomType{Fraction Integer} to \\axiomType{DoubleFloat}")) (|df2ef| (((|Expression| (|Float|)) (|DoubleFloat|)) "\\spad{df2ef(a)} coerces a \\axiomType{DoubleFloat} to \\axiomType{Expression Float}")) (|pdf2df| (((|DoubleFloat|) (|Polynomial| (|DoubleFloat|))) "\\spad{pdf2df(p)} coerces a \\axiomType{Polynomial DoubleFloat} to \\axiomType{DoubleFloat}. It is an error if \\axiom{\\spad{p}} is not retractable to DoubleFloat.")) (|pdf2ef| (((|Expression| (|Float|)) (|Polynomial| (|DoubleFloat|))) "\\spad{pdf2ef(p)} coerces a \\axiomType{Polynomial DoubleFloat} to \\axiomType{Expression Float}")) (|iflist2Result| (((|Result|) (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|)))) "\\spad{iflist2Result(m)} converts a attributes record into a \\axiomType{Result}")) (|att2Result| (((|Result|) (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated"))))) "\\spad{att2Result(m)} converts a attributes record into a \\axiomType{Result}")) (|measure2Result| (((|Result|) (|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))) (|:| |extra| (|Result|)))) "\\spad{measure2Result(m)} converts a measure record into a \\axiomType{Result}") (((|Result|) (|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))))) "\\spad{measure2Result(m)} converts a measure record into a \\axiomType{Result}")) (|outputMeasure| (((|String|) (|Float|)) "\\spad{outputMeasure(n)} rounds \\spad{n} to 3 decimal places and outputs it as a string")) (|concat| (((|Result|) (|List| (|Result|))) "\\spad{concat(l)} concatenates a list of aggregates of type \\axiomType{Result}") (((|Result|) (|Result|) (|Result|)) "\\spad{concat(a,b)} adds two aggregates of type \\axiomType{Result}.")) (|gethi| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{gethi(u)} gets the \\axiomType{DoubleFloat} equivalent of the second endpoint of the range \\spad{u}")) (|getlo| (((|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{getlo(u)} gets the \\axiomType{DoubleFloat} equivalent of the first endpoint of the range \\spad{u}")) (|sdf2lst| (((|List| (|String|)) (|Stream| (|DoubleFloat|))) "\\spad{sdf2lst(ln)} coerces a \\axiomType{Stream DoubleFloat} to \\axiomType{String}")) (|ldf2lst| (((|List| (|String|)) (|List| (|DoubleFloat|))) "\\spad{ldf2lst(ln)} coerces a \\axiomType{List DoubleFloat} to \\axiomType{List String}")) (|f2st| (((|String|) (|Float|)) "\\spad{f2st(n)} coerces a \\axiomType{Float} to \\axiomType{String}")) (|df2st| (((|String|) (|DoubleFloat|)) "\\spad{df2st(n)} coerces a \\axiomType{DoubleFloat} to \\axiomType{String}")) (|in?| (((|Boolean|) (|DoubleFloat|) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{in?(p,range)} tests whether point \\spad{p} is internal to the \\spad{range} \\spad{range}")) (|vedf2vef| (((|Vector| (|Expression| (|Float|))) (|Vector| (|Expression| (|DoubleFloat|)))) "\\spad{vedf2vef(v)} maps \\axiomType{Vector Expression DoubleFloat} to \\axiomType{Vector Expression Float}")) (|edf2ef| (((|Expression| (|Float|)) (|Expression| (|DoubleFloat|))) "\\spad{edf2ef(e)} maps \\axiomType{Expression DoubleFloat} to \\axiomType{Expression Float}")) (|ldf2vmf| (((|Vector| (|MachineFloat|)) (|List| (|DoubleFloat|))) "\\spad{ldf2vmf(l)} coerces a \\axiomType{List DoubleFloat} to \\axiomType{List MachineFloat}")) (|df2mf| (((|MachineFloat|) (|DoubleFloat|)) "\\spad{df2mf(n)} coerces a \\axiomType{DoubleFloat} to \\axiomType{MachineFloat}")) (|dflist| (((|List| (|DoubleFloat|)) (|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))))) "\\spad{dflist(l)} returns a list of \\axiomType{DoubleFloat} equivalents of list \\spad{l}")) (|dfRange| (((|Segment| (|OrderedCompletion| (|DoubleFloat|))) (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) "\\spad{dfRange(r)} converts a range including \\inputbitmap{\\htbmdir{}/plusminus.bitmap} \\infty to \\axiomType{DoubleFloat} equavalents.")) (|edf2efi| (((|Expression| (|Fraction| (|Integer|))) (|Expression| (|DoubleFloat|))) "\\spad{edf2efi(e)} coerces \\axiomType{Expression DoubleFloat} into \\axiomType{Expression Fraction Integer}")) (|numberOfOperations| (((|Record| (|:| |additions| (|Integer|)) (|:| |multiplications| (|Integer|)) (|:| |exponentiations| (|Integer|)) (|:| |functionCalls| (|Integer|))) (|Vector| (|Expression| (|DoubleFloat|)))) "\\spad{numberOfOperations(ode)} counts additions,{} multiplications,{} exponentiations and function calls in the input set of expressions.")) (|expenseOfEvaluation| (((|Float|) (|Vector| (|Expression| (|DoubleFloat|)))) "\\spad{expenseOfEvaluation(o)} gives an approximation of the cost of evaluating a list of expressions in terms of the number of basic operations. < 0.3 inexpensive ; 0.5 neutral ; > 0.7 very expensive 400 `operation units' \\spad{->} 0.75 200 `operation units' \\spad{->} 0.5 83 `operation units' \\spad{->} 0.25 \\spad{**} = 4 units ,{} function calls = 10 units.")) (|isQuotient| (((|Union| (|Expression| (|DoubleFloat|)) "failed") (|Expression| (|DoubleFloat|))) "\\spad{isQuotient(expr)} returns the quotient part of the input expression or \\spad{\"failed\"} if the expression is not of that form.")) (|edf2df| (((|DoubleFloat|) (|Expression| (|DoubleFloat|))) "\\spad{edf2df(n)} maps \\axiomType{Expression DoubleFloat} to \\axiomType{DoubleFloat} It is an error if \\spad{n} is not coercible to DoubleFloat")) (|edf2fi| (((|Fraction| (|Integer|)) (|Expression| (|DoubleFloat|))) "\\spad{edf2fi(n)} maps \\axiomType{Expression DoubleFloat} to \\axiomType{Fraction Integer} It is an error if \\spad{n} is not coercible to Fraction Integer")) (|df2fi| (((|Fraction| (|Integer|)) (|DoubleFloat|)) "\\spad{df2fi(n)} is a function to convert a \\axiomType{DoubleFloat} to a \\axiomType{Fraction Integer}")) (|convert| (((|List| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|List| (|Segment| (|OrderedCompletion| (|Float|))))) "\\spad{convert(l)} is a function to convert a \\axiomType{Segment OrderedCompletion Float} to a \\axiomType{Segment OrderedCompletion DoubleFloat}")) (|socf2socdf| (((|Segment| (|OrderedCompletion| (|DoubleFloat|))) (|Segment| (|OrderedCompletion| (|Float|)))) "\\spad{socf2socdf(a)} is a function to convert a \\axiomType{Segment OrderedCompletion Float} to a \\axiomType{Segment OrderedCompletion DoubleFloat}")) (|ocf2ocdf| (((|OrderedCompletion| (|DoubleFloat|)) (|OrderedCompletion| (|Float|))) "\\spad{ocf2ocdf(a)} is a function to convert an \\axiomType{OrderedCompletion Float} to an \\axiomType{OrderedCompletion DoubleFloat}")) (|ef2edf| (((|Expression| (|DoubleFloat|)) (|Expression| (|Float|))) "\\spad{ef2edf(f)} is a function to convert an \\axiomType{Expression Float} to an \\axiomType{Expression DoubleFloat}")) (|f2df| (((|DoubleFloat|) (|Float|)) "\\spad{f2df(f)} is a function to convert a \\axiomType{Float} to a \\axiomType{DoubleFloat}"))) 
 NIL 
 NIL 
-(-306 R1) 
+(-307 R1) 
 ((|constructor| (NIL "\\axiom{ExpertSystemToolsPackage1} contains some useful functions for use by the computational agents of Ordinary Differential Equation solvers.")) (|neglist| (((|List| |#1|) (|List| |#1|)) "\\spad{neglist(l)} returns only the negative elements of the list \\spad{l}"))) 
 NIL 
 NIL 
-(-307 R1 R2) 
+(-308 R1 R2) 
 ((|constructor| (NIL "\\axiom{ExpertSystemToolsPackage2} contains some useful functions for use by the computational agents of Ordinary Differential Equation solvers.")) (|map| (((|Matrix| |#2|) (|Mapping| |#2| |#1|) (|Matrix| |#1|)) "\\spad{map(f,m)} applies a mapping f:R1 \\spad{->} \\spad{R2} onto a matrix \\spad{m} in \\spad{R1} returning a matrix in \\spad{R2}"))) 
 NIL 
 NIL 
-(-308 S) 
+(-309 S) 
 ((|constructor| (NIL "A constructive euclidean domain,{} \\spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\\spadfun{euclideanSize}) than the divisor. \\blankline Conditional attributes: \\indented{2}{multiplicativeValuation\\tab{25}\\spad{Size(a*b)=Size(a)*Size(b)}} \\indented{2}{additiveValuation\\tab{25}\\spad{Size(a*b)=Size(a)+Size(b)}}")) (|multiEuclidean| (((|Union| (|List| $) "failed") (|List| $) $) "\\spad{multiEuclidean([f1,...,fn],z)} returns a list of coefficients \\spad{[a1, ..., an]} such that \\spad{ z / prod fi = sum aj/fj}. If no such list of coefficients exists,{} \"failed\" is returned.")) (|extendedEuclidean| (((|Union| (|Record| (|:| |coef1| $) (|:| |coef2| $)) "failed") $ $ $) "\\spad{extendedEuclidean(x,y,z)} either returns a record rec where \\spad{rec.coef1*x+rec.coef2*y=z} or returns \"failed\" if \\spad{z} cannot be expressed as a linear combination of \\spad{x} and \\spad{y}.") (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{extendedEuclidean(x,y)} returns a record rec where \\spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a \\spad{gcd} of \\spad{x} and \\spad{y}. The \\spad{gcd} is unique only up to associates if \\spadatt{canonicalUnitNormal} is not asserted. \\spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.")) (|rem| (($ $ $) "\\spad{x rem y} is the same as \\spad{divide(x,y).remainder}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|quo| (($ $ $) "\\spad{x quo y} is the same as \\spad{divide(x,y).quotient}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(x,y)} divides \\spad{x} by \\spad{y} producing a record containing a \\spad{quotient} and \\spad{remainder},{} where the remainder is smaller (see \\spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \\spad{y}.")) (|euclideanSize| (((|NonNegativeInteger|) $) "\\spad{euclideanSize(x)} returns the euclidean size of the element \\spad{x}. Error: if \\spad{x} is zero.")) (|sizeLess?| (((|Boolean|) $ $) "\\spad{sizeLess?(x,y)} tests whether \\spad{x} is strictly smaller than \\spad{y} with respect to the \\spadfunFrom{euclideanSize}{EuclideanDomain}."))) 
 NIL 
 NIL 
-(-309) 
+(-310) 
 ((|constructor| (NIL "A constructive euclidean domain,{} \\spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\\spadfun{euclideanSize}) than the divisor. \\blankline Conditional attributes: \\indented{2}{multiplicativeValuation\\tab{25}\\spad{Size(a*b)=Size(a)*Size(b)}} \\indented{2}{additiveValuation\\tab{25}\\spad{Size(a*b)=Size(a)+Size(b)}}")) (|multiEuclidean| (((|Union| (|List| $) "failed") (|List| $) $) "\\spad{multiEuclidean([f1,...,fn],z)} returns a list of coefficients \\spad{[a1, ..., an]} such that \\spad{ z / prod fi = sum aj/fj}. If no such list of coefficients exists,{} \"failed\" is returned.")) (|extendedEuclidean| (((|Union| (|Record| (|:| |coef1| $) (|:| |coef2| $)) "failed") $ $ $) "\\spad{extendedEuclidean(x,y,z)} either returns a record rec where \\spad{rec.coef1*x+rec.coef2*y=z} or returns \"failed\" if \\spad{z} cannot be expressed as a linear combination of \\spad{x} and \\spad{y}.") (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{extendedEuclidean(x,y)} returns a record rec where \\spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a \\spad{gcd} of \\spad{x} and \\spad{y}. The \\spad{gcd} is unique only up to associates if \\spadatt{canonicalUnitNormal} is not asserted. \\spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.")) (|rem| (($ $ $) "\\spad{x rem y} is the same as \\spad{divide(x,y).remainder}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|quo| (($ $ $) "\\spad{x quo y} is the same as \\spad{divide(x,y).quotient}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(x,y)} divides \\spad{x} by \\spad{y} producing a record containing a \\spad{quotient} and \\spad{remainder},{} where the remainder is smaller (see \\spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \\spad{y}.")) (|euclideanSize| (((|NonNegativeInteger|) $) "\\spad{euclideanSize(x)} returns the euclidean size of the element \\spad{x}. Error: if \\spad{x} is zero.")) (|sizeLess?| (((|Boolean|) $ $) "\\spad{sizeLess?(x,y)} tests whether \\spad{x} is strictly smaller than \\spad{y} with respect to the \\spadfunFrom{euclideanSize}{EuclideanDomain}."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-310 S R) 
+(-311 S R) 
 ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions.")) (|eval| (($ $ (|List| (|Equation| |#2|))) "\\spad{eval(f, [x1 = v1,...,xn = vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ (|Equation| |#2|)) "\\spad{eval(f,x = v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) 
 NIL 
 NIL 
-(-311 R) 
+(-312 R) 
 ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions.")) (|eval| (($ $ (|List| (|Equation| |#1|))) "\\spad{eval(f, [x1 = v1,...,xn = vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ (|Equation| |#1|)) "\\spad{eval(f,x = v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) 
 NIL 
 NIL 
-(-312 -3498) 
+(-313 -3505) 
 ((|constructor| (NIL "This package is to be used in conjuction with \\indented{12}{the CycleIndicators package. It provides an evaluation} \\indented{12}{function for SymmetricPolynomials.}")) (|eval| ((|#1| (|Mapping| |#1| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{eval(f,s)} evaluates the cycle index \\spad{s} by applying \\indented{1}{the function \\spad{f} to each integer in a monomial partition,{}} \\indented{1}{forms their product and sums the results over all monomials.}"))) 
 NIL 
 NIL 
-(-313) 
+(-314) 
 ((|constructor| (NIL "A function which does not return directly to its caller should have Exit as its return type. \\blankline Note: It is convenient to have a formal \\spad{coerce} into each type from type Exit. This allows,{} for example,{} errors to be raised in one half of a type-balanced \\spad{if}."))) 
 NIL 
 NIL 
-(-314) 
+(-315) 
 ((|constructor| (NIL "This domain represents exit expressions.")) (|level| (((|Integer|) $) "\\spad{level(e)} returns the nesting exit level of `e'")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the exit expression of `e'."))) 
 NIL 
 NIL 
-(-315 R FE |var| |cen|) 
+(-316 R FE |var| |cen|) 
 ((|constructor| (NIL "UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent essential singularities of functions. Objects in this domain are quotients of sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) "\\spad{coerce(f)} converts a \\spadtype{UnivariatePuiseuxSeries} to an \\spadtype{ExponentialExpansion}.")) (|limitPlus| (((|Union| (|OrderedCompletion| |#2|) "failed") $) "\\spad{limitPlus(f(var))} returns \\spad{limit(var -> a+,f(var))}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
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-(-316 R) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
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+(-317 R) 
 ((|constructor| (NIL "Expressions involving symbolic functions.")) (|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} \\undocumented{}")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} \\undocumented{}")) (|simplifyPower| (($ $ (|Integer|)) "simplifyPower?(\\spad{f},{}\\spad{n}) \\undocumented{}")) (|number?| (((|Boolean|) $) "\\spad{number?(f)} tests if \\spad{f} is rational")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic quantities present in \\spad{f} by applying their defining relations."))) 
-((-4424 -3962 (-3258 (|has| |#1| (-1053)) (|has| |#1| (-642 (-550)))) (-12 (|has| |#1| (-561)) (-3962 (-3258 (|has| |#1| (-1053)) (|has| |#1| (-642 (-550)))) (|has| |#1| (-1053)) (|has| |#1| (-477)))) (|has| |#1| (-1053)) (|has| |#1| (-477))) (-4422 |has| |#1| (-173)) (-4421 |has| |#1| (-173)) ((-4429 "*") |has| |#1| (-561)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-561)) (-4419 |has| |#1| (-561))) 
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-(-317 R S) 
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+(-318 R S) 
 ((|constructor| (NIL "Lifting of maps to Expressions. Date Created: 16 Jan 1989 Date Last Updated: 22 Jan 1990")) (|map| (((|Expression| |#2|) (|Mapping| |#2| |#1|) (|Expression| |#1|)) "\\spad{map(f, e)} applies \\spad{f} to all the constants appearing in \\spad{e}."))) 
 NIL 
 NIL 
-(-318 R FE) 
+(-319 R FE) 
 ((|constructor| (NIL "This package provides functions to convert functional expressions to power series.")) (|series| (((|Any|) |#2| (|Equation| |#2|) (|Fraction| (|Integer|))) "\\spad{series(f,x = a,n)} expands the expression \\spad{f} as a series in powers of (\\spad{x} - a); terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{series(f,x = a)} expands the expression \\spad{f} as a series in powers of (\\spad{x} - a).") (((|Any|) |#2| (|Fraction| (|Integer|))) "\\spad{series(f,n)} returns a series expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{series(f)} returns a series expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{series(x)} returns \\spad{x} viewed as a series.")) (|puiseux| (((|Any|) |#2| (|Equation| |#2|) (|Fraction| (|Integer|))) "\\spad{puiseux(f,x = a,n)} expands the expression \\spad{f} as a Puiseux series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{puiseux(f,x = a)} expands the expression \\spad{f} as a Puiseux series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|Fraction| (|Integer|))) "\\spad{puiseux(f,n)} returns a Puiseux expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{puiseux(f)} returns a Puiseux expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{puiseux(x)} returns \\spad{x} viewed as a Puiseux series.")) (|laurent| (((|Any|) |#2| (|Equation| |#2|) (|Integer|)) "\\spad{laurent(f,x = a,n)} expands the expression \\spad{f} as a Laurent series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{laurent(f,x = a)} expands the expression \\spad{f} as a Laurent series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|Integer|)) "\\spad{laurent(f,n)} returns a Laurent expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{laurent(f)} returns a Laurent expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{laurent(x)} returns \\spad{x} viewed as a Laurent series.")) (|taylor| (((|Any|) |#2| (|Equation| |#2|) (|NonNegativeInteger|)) "\\spad{taylor(f,x = a)} expands the expression \\spad{f} as a Taylor series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{taylor(f,x = a)} expands the expression \\spad{f} as a Taylor series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|NonNegativeInteger|)) "\\spad{taylor(f,n)} returns a Taylor expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{taylor(f)} returns a Taylor expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{taylor(x)} returns \\spad{x} viewed as a Taylor series."))) 
 NIL 
 NIL 
-(-319 R -3498) 
+(-320 R -3505) 
 ((|constructor| (NIL "Taylor series solutions of explicit ODE\\spad{'s}.")) (|seriesSolve| (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq, y, x = a, [b0,...,bn])} is equivalent to \\spad{seriesSolve(eq = 0, y, x = a, [b0,...,b(n-1)])}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq, y, x = a, y a = b)} is equivalent to \\spad{seriesSolve(eq=0, y, x=a, y a = b)}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq, y, x = a, b)} is equivalent to \\spad{seriesSolve(eq = 0, y, x = a, y a = b)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq,y, x=a, b)} is equivalent to \\spad{seriesSolve(eq, y, x=a, y a = b)}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,[y1 a = b1,..., yn a = bn])} is equivalent to \\spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x = a, [y1 a = b1,..., yn a = bn])}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])} is equivalent to \\spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x=a, [b1,...,bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])} is equivalent to \\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a, [y1 a = b1,..., yn a = bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "\\spad{seriesSolve([eq1,...,eqn],[y1,...,yn],x = a,[y1 a = b1,...,yn a = bn])} returns a taylor series solution of \\spad{[eq1,...,eqn]} around \\spad{x = a} with initial conditions \\spad{yi(a) = bi}. Note: eqi must be of the form \\spad{fi(x, y1 x, y2 x,..., yn x) y1'(x) + gi(x, y1 x, y2 x,..., yn x) = h(x, y1 x, y2 x,..., yn x)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq,y,x=a,[b0,...,b(n-1)])} returns a Taylor series solution of \\spad{eq} around \\spad{x = a} with initial conditions \\spad{y(a) = b0},{} \\spad{y'(a) = b1},{} \\spad{y''(a) = b2},{} ...,{}\\spad{y(n-1)(a) = b(n-1)} \\spad{eq} must be of the form \\spad{f(x, y x, y'(x),..., y(n-1)(x)) y(n)(x) + g(x,y x,y'(x),...,y(n-1)(x)) = h(x,y x, y'(x),..., y(n-1)(x))}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq,y,x=a, y a = b)} returns a Taylor series solution of \\spad{eq} around \\spad{x} = a with initial condition \\spad{y(a) = b}. Note: \\spad{eq} must be of the form \\spad{f(x, y x) y'(x) + g(x, y x) = h(x, y x)}."))) 
 NIL 
 NIL 
-(-320) 
+(-321) 
 ((|constructor| (NIL "\\indented{1}{Author: Clifton \\spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.")) (|tubePlot| (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|String|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n,s)} puts a tube of radius \\spad{r} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. If \\spad{s} = \"closed\",{} the tube is considered to be closed; if \\spad{s} = \"open\",{} the tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n)} puts a tube of radius \\spad{r} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. The tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Integer|) (|String|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n,s)} puts a tube of radius \\spad{r(t)} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. If \\spad{s} = \"closed\",{} the tube is considered to be closed; if \\spad{s} = \"open\",{} the tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Integer|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n)} puts a tube of radius \\spad{r}(\\spad{t}) with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. The tube is considered to be open.")) (|constantToUnaryFunction| (((|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|DoubleFloat|)) "\\spad{constantToUnaryFunction(s)} is a local function which takes the value of \\spad{s},{} which may be a function of a constant,{} and returns a function which always returns the value \\spadtype{DoubleFloat} \\spad{s}."))) 
 NIL 
 NIL 
-(-321 FE |var| |cen|) 
+(-322 FE |var| |cen|) 
 ((|constructor| (NIL "ExponentialOfUnivariatePuiseuxSeries is a domain used to represent essential singularities of functions. An object in this domain is a function of the form \\spad{exp(f(x))},{} where \\spad{f(x)} is a Puiseux series with no terms of non-negative degree. Objects are ordered according to order of singularity,{} with functions which tend more rapidly to zero or infinity considered to be larger. Thus,{} if \\spad{order(f(x)) < order(g(x))},{} \\spadignore{i.e.} the first non-zero term of \\spad{f(x)} has lower degree than the first non-zero term of \\spad{g(x)},{} then \\spad{exp(f(x)) > exp(g(x))}. If \\spad{order(f(x)) = order(g(x))},{} then the ordering is essentially random. This domain is used in computing limits involving functions with essential singularities.")) (|exponentialOrder| (((|Fraction| (|Integer|)) $) "\\spad{exponentialOrder(exp(c * x **(-n) + ...))} returns \\spad{-n}. exponentialOrder(0) returns \\spad{0}.")) (|exponent| (((|UnivariatePuiseuxSeries| |#1| |#2| |#3|) $) "\\spad{exponent(exp(f(x)))} returns \\spad{f(x)}")) (|exponential| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{exponential(f(x))} returns \\spad{exp(f(x))}. Note: the function does NOT check that \\spad{f(x)} has no non-negative terms."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|)))) (|HasCategory| (-411 (-550)) (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-964))) (|HasCategory| |#1| (QUOTE (-1206))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4246) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1181))))) (|HasSignature| |#1| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#1|))))))) 
-(-322 M) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|)))) (|HasCategory| (-412 (-551)) (QUOTE (-1118))) (|HasCategory| |#1| (QUOTE (-367))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasSignature| |#1| (LIST (QUOTE -4387) (LIST (|devaluate| |#1|) (QUOTE (-1183)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1208))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasSignature| |#1| (LIST (QUOTE -4253) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1183))))) (|HasSignature| |#1| (LIST (QUOTE -3494) (LIST (LIST (QUOTE -646) (QUOTE (-1183))) (|devaluate| |#1|))))))) 
+(-323 M) 
 ((|constructor| (NIL "computes various functions on factored arguments.")) (|log| (((|List| (|Record| (|:| |coef| (|NonNegativeInteger|)) (|:| |logand| |#1|))) (|Factored| |#1|)) "\\spad{log(f)} returns \\spad{[(a1,b1),...,(am,bm)]} such that the logarithm of \\spad{f} is equal to \\spad{a1*log(b1) + ... + am*log(bm)}.")) (|nthRoot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#1|) (|:| |radicand| (|List| |#1|))) (|Factored| |#1|) (|NonNegativeInteger|)) "\\spad{nthRoot(f, n)} returns \\spad{(p, r, [r1,...,rm])} such that the \\spad{n}th-root of \\spad{f} is equal to \\spad{r * \\spad{p}th-root(r1 * ... * rm)},{} where \\spad{r1},{}...,{}\\spad{rm} are distinct factors of \\spad{f},{} each of which has an exponent smaller than \\spad{p} in \\spad{f}."))) 
 NIL 
 NIL 
-(-323 E OV R P) 
+(-324 E OV R P) 
 ((|constructor| (NIL "This package provides utilities used by the factorizers which operate on polynomials represented as univariate polynomials with multivariate coefficients.")) (|ran| ((|#3| (|Integer|)) "\\spad{ran(k)} computes a random integer between \\spad{-k} and \\spad{k} as a member of \\spad{R}.")) (|normalDeriv| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|Integer|)) "\\spad{normalDeriv(poly,i)} computes the \\spad{i}th derivative of \\spad{poly} divided by i!.")) (|raisePolynomial| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#3|)) "\\spad{raisePolynomial(rpoly)} converts \\spad{rpoly} from a univariate polynomial over \\spad{r} to be a univariate polynomial with polynomial coefficients.")) (|lowerPolynomial| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{lowerPolynomial(upoly)} converts \\spad{upoly} to be a univariate polynomial over \\spad{R}. An error if the coefficients contain variables.")) (|variables| (((|List| |#2|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{variables(upoly)} returns the list of variables for the coefficients of \\spad{upoly}.")) (|degree| (((|List| (|NonNegativeInteger|)) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|)) "\\spad{degree(upoly, lvar)} returns a list containing the maximum degree for each variable in lvar.")) (|completeEval| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| |#3|)) "\\spad{completeEval(upoly, lvar, lval)} evaluates the polynomial \\spad{upoly} with each variable in \\spad{lvar} replaced by the corresponding value in lval. Substitutions are done for all variables in \\spad{upoly} producing a univariate polynomial over \\spad{R}."))) 
 NIL 
 NIL 
-(-324 S) 
+(-325 S) 
 ((|constructor| (NIL "The free abelian group on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,[ni * si])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are integers. The operation is commutative."))) 
-((-4422 . T) (-4421 . T)) 
-((|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-795)))) 
-(-325 S E) 
+((-4429 . T) (-4428 . T)) 
+((|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-797)))) 
+(-326 S E) 
 ((|constructor| (NIL "A free abelian monoid on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,[ni * si])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are in a given abelian monoid. The operation is commutative.")) (|highCommonTerms| (($ $ $) "\\spad{highCommonTerms(e1 a1 + ... + en an, f1 b1 + ... + fm bm)} returns \\indented{2}{\\spad{reduce(+,[max(ei, fi) ci])}} where \\spad{ci} ranges in the intersection of \\spad{{a1,...,an}} and \\spad{{b1,...,bm}}.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, e1 a1 +...+ en an)} returns \\spad{e1 f(a1) +...+ en f(an)}.")) (|mapCoef| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapCoef(f, e1 a1 +...+ en an)} returns \\spad{f(e1) a1 +...+ f(en) an}.")) (|coefficient| ((|#2| |#1| $) "\\spad{coefficient(s, e1 a1 + ... + en an)} returns \\spad{ei} such that \\spad{ai} = \\spad{s},{} or 0 if \\spad{s} is not one of the \\spad{ai}\\spad{'s}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x, n)} returns the factor of the n^th term of \\spad{x}.")) (|nthCoef| ((|#2| $ (|Integer|)) "\\spad{nthCoef(x, n)} returns the coefficient of the n^th term of \\spad{x}.")) (|terms| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| |#2|))) $) "\\spad{terms(e1 a1 + ... + en an)} returns \\spad{[[a1, e1],...,[an, en]]}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of terms in \\spad{x}. mapGen(\\spad{f},{} a1\\spad{\\^}e1 ... an\\spad{\\^}en) returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (* (($ |#2| |#1|) "\\spad{e * s} returns \\spad{e} times \\spad{s}.")) (+ (($ |#1| $) "\\spad{s + x} returns the sum of \\spad{s} and \\spad{x}."))) 
 NIL 
 NIL 
-(-326 S) 
+(-327 S) 
 ((|constructor| (NIL "The free abelian monoid on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,[ni * si])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are non-negative integers. The operation is commutative."))) 
 NIL 
-((|HasCategory| (-774) (QUOTE (-795)))) 
-(-327 S R E) 
+((|HasCategory| (-776) (QUOTE (-797)))) 
+(-328 S R E) 
 ((|constructor| (NIL "This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.")) (|primitivePart| (($ $) "\\spad{primitivePart(p)} returns the unit normalized form of polynomial \\spad{p} divided by the content of \\spad{p}.")) (|content| ((|#2| $) "\\spad{content(p)} gives the \\spad{gcd} of the coefficients of polynomial \\spad{p}.")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(p,r)} returns the exact quotient of polynomial \\spad{p} by \\spad{r},{} or \"failed\" if none exists.")) (|binomThmExpt| (($ $ $ (|NonNegativeInteger|)) "\\spad{binomThmExpt(p,q,n)} returns \\spad{(x+y)^n} by means of the binomial theorem trick.")) (|pomopo!| (($ $ |#2| |#3| $) "\\spad{pomopo!(p1,r,e,p2)} returns \\spad{p1 + monomial(e,r) * p2} and may use \\spad{p1} as workspace. The constaant \\spad{r} is assumed to be nonzero.")) (|mapExponents| (($ (|Mapping| |#3| |#3|) $) "\\spad{mapExponents(fn,u)} maps function \\spad{fn} onto the exponents of the non-zero monomials of polynomial \\spad{u}.")) (|minimumDegree| ((|#3| $) "\\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \\spad{p}. Error: if applied to 0.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \\spad{p}.")) (|coefficients| (((|List| |#2|) $) "\\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \\spad{p}.")) (|ground| ((|#2| $) "\\spad{ground(p)} retracts polynomial \\spad{p} to the coefficient ring.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(p)} tests if polynomial \\spad{p} is a member of the coefficient ring."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173)))) 
-(-328 R E) 
+((|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-173)))) 
+(-329 R E) 
 ((|constructor| (NIL "This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.")) (|primitivePart| (($ $) "\\spad{primitivePart(p)} returns the unit normalized form of polynomial \\spad{p} divided by the content of \\spad{p}.")) (|content| ((|#1| $) "\\spad{content(p)} gives the \\spad{gcd} of the coefficients of polynomial \\spad{p}.")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(p,r)} returns the exact quotient of polynomial \\spad{p} by \\spad{r},{} or \"failed\" if none exists.")) (|binomThmExpt| (($ $ $ (|NonNegativeInteger|)) "\\spad{binomThmExpt(p,q,n)} returns \\spad{(x+y)^n} by means of the binomial theorem trick.")) (|pomopo!| (($ $ |#1| |#2| $) "\\spad{pomopo!(p1,r,e,p2)} returns \\spad{p1 + monomial(e,r) * p2} and may use \\spad{p1} as workspace. The constaant \\spad{r} is assumed to be nonzero.")) (|mapExponents| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapExponents(fn,u)} maps function \\spad{fn} onto the exponents of the non-zero monomials of polynomial \\spad{u}.")) (|minimumDegree| ((|#2| $) "\\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \\spad{p}. Error: if applied to 0.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \\spad{p}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \\spad{p}.")) (|ground| ((|#1| $) "\\spad{ground(p)} retracts polynomial \\spad{p} to the coefficient ring.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(p)} tests if polynomial \\spad{p} is a member of the coefficient ring."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-329 S) 
+(-330 S) 
 ((|constructor| (NIL "\\indented{1}{A FlexibleArray is the notion of an array intended to allow for growth} at the end only. Hence the following efficient operations \\indented{2}{\\spad{append(x,a)} meaning append item \\spad{x} at the end of the array \\spad{a}} \\indented{2}{\\spad{delete(a,n)} meaning delete the last item from the array \\spad{a}} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets."))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-330 S -3498) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-331 S -3505) 
 ((|constructor| (NIL "FiniteAlgebraicExtensionField {\\em F} is the category of fields which are finite algebraic extensions of the field {\\em F}. If {\\em F} is finite then any finite algebraic extension of {\\em F} is finite,{} too. Let {\\em K} be a finite algebraic extension of the finite field {\\em F}. The exponentiation of elements of {\\em K} defines a \\spad{Z}-module structure on the multiplicative group of {\\em K}. The additive group of {\\em K} becomes a module over the ring of polynomials over {\\em F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em K},{} {\\em c,d} from {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)} where {\\em q=size()\\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#2|) "failed") $ $) "\\spad{linearAssociatedLog(b,a)} returns a polynomial {\\em g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals {\\em a}. If there is no such polynomial {\\em g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial {\\em g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals {\\em a}.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\\em g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#2|)) "\\spad{linearAssociatedExp(a,f)} is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em \\$},{} {\\em c,d} form {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)},{} where {\\em q=size()\\$F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: \\spad{trace(a,d) = reduce(+,[a**(q**(d*i)) for i in 0..n/d])}.") ((|#2| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: norm(a,{}\\spad{d}) = reduce(*,{}[a**(\\spad{q**}(d*i)) for \\spad{i} in 0..\\spad{n/d}])") ((|#2| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#2|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\spad{\\$} as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\spad{\\$} as \\spad{F}-vectorspace."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-371)))) 
-(-331 -3498) 
+((|HasCategory| |#2| (QUOTE (-372)))) 
+(-332 -3505) 
 ((|constructor| (NIL "FiniteAlgebraicExtensionField {\\em F} is the category of fields which are finite algebraic extensions of the field {\\em F}. If {\\em F} is finite then any finite algebraic extension of {\\em F} is finite,{} too. Let {\\em K} be a finite algebraic extension of the finite field {\\em F}. The exponentiation of elements of {\\em K} defines a \\spad{Z}-module structure on the multiplicative group of {\\em K}. The additive group of {\\em K} becomes a module over the ring of polynomials over {\\em F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em K},{} {\\em c,d} from {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)} where {\\em q=size()\\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") $ $) "\\spad{linearAssociatedLog(b,a)} returns a polynomial {\\em g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals {\\em a}. If there is no such polynomial {\\em g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial {\\em g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals {\\em a}.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\\em g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#1|)) "\\spad{linearAssociatedExp(a,f)} is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em \\$},{} {\\em c,d} form {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\spad{\\$}SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)},{} where {\\em q=size()\\$F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: \\spad{trace(a,d) = reduce(+,[a**(q**(d*i)) for i in 0..n/d])}.") ((|#1| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: norm(a,{}\\spad{d}) = reduce(*,{}[a**(\\spad{q**}(d*i)) for \\spad{i} in 0..\\spad{n/d}])") ((|#1| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#1|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\spad{\\$} as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\spad{\\$} as \\spad{F}-vectorspace."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-332) 
+(-333) 
 ((|constructor| (NIL "This domain builds representations of program code segments for use with the FortranProgram domain.")) (|setLabelValue| (((|SingleInteger|) (|SingleInteger|)) "\\spad{setLabelValue(i)} resets the counter which produces labels to \\spad{i}")) (|getCode| (((|SExpression|) $) "\\spad{getCode(f)} returns a Lisp list of strings representing \\spad{f} in Fortran notation. This is used by the FortranProgram domain.")) (|printCode| (((|Void|) $) "\\spad{printCode(f)} prints out \\spad{f} in FORTRAN notation.")) (|code| (((|Union| (|:| |nullBranch| "null") (|:| |assignmentBranch| (|Record| (|:| |var| (|Symbol|)) (|:| |arrayIndex| (|List| (|Polynomial| (|Integer|)))) (|:| |rand| (|Record| (|:| |ints2Floats?| (|Boolean|)) (|:| |expr| (|OutputForm|)))))) (|:| |arrayAssignmentBranch| (|Record| (|:| |var| (|Symbol|)) (|:| |rand| (|OutputForm|)) (|:| |ints2Floats?| (|Boolean|)))) (|:| |conditionalBranch| (|Record| (|:| |switch| (|Switch|)) (|:| |thenClause| $) (|:| |elseClause| $))) (|:| |returnBranch| (|Record| (|:| |empty?| (|Boolean|)) (|:| |value| (|Record| (|:| |ints2Floats?| (|Boolean|)) (|:| |expr| (|OutputForm|)))))) (|:| |blockBranch| (|List| $)) (|:| |commentBranch| (|List| (|String|))) (|:| |callBranch| (|String|)) (|:| |forBranch| (|Record| (|:| |range| (|SegmentBinding| (|Polynomial| (|Integer|)))) (|:| |span| (|Polynomial| (|Integer|))) (|:| |body| $))) (|:| |labelBranch| (|SingleInteger|)) (|:| |loopBranch| (|Record| (|:| |switch| (|Switch|)) (|:| |body| $))) (|:| |commonBranch| (|Record| (|:| |name| (|Symbol|)) (|:| |contents| (|List| (|Symbol|))))) (|:| |printBranch| (|List| (|OutputForm|)))) $) "\\spad{code(f)} returns the internal representation of the object represented by \\spad{f}.")) (|operation| (((|Union| (|:| |Null| "null") (|:| |Assignment| "assignment") (|:| |Conditional| "conditional") (|:| |Return| "return") (|:| |Block| "block") (|:| |Comment| "comment") (|:| |Call| "call") (|:| |For| "for") (|:| |While| "while") (|:| |Repeat| "repeat") (|:| |Goto| "goto") (|:| |Continue| "continue") (|:| |ArrayAssignment| "arrayAssignment") (|:| |Save| "save") (|:| |Stop| "stop") (|:| |Common| "common") (|:| |Print| "print")) $) "\\spad{operation(f)} returns the name of the operation represented by \\spad{f}.")) (|common| (($ (|Symbol|) (|List| (|Symbol|))) "\\spad{common(name,contents)} creates a representation a named common block.")) (|printStatement| (($ (|List| (|OutputForm|))) "\\spad{printStatement(l)} creates a representation of a PRINT statement.")) (|save| (($) "\\spad{save()} creates a representation of a SAVE statement.")) (|stop| (($) "\\spad{stop()} creates a representation of a STOP statement.")) (|block| (($ (|List| $)) "\\spad{block(l)} creates a representation of the statements in \\spad{l} as a block.")) (|assign| (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|Complex| (|Float|)))) "\\spad{assign(x,l,y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|Float|))) "\\spad{assign(x,l,y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|Integer|))) "\\spad{assign(x,l,y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|Vector| (|Expression| (|Complex| (|Float|))))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|Float|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|Integer|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|Complex| (|Float|))))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|Float|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|Integer|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|Complex| (|Float|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|Float|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|Integer|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|MachineComplex|))) "\\spad{assign(x,l,y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|MachineFloat|))) "\\spad{assign(x,l,y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|List| (|Polynomial| (|Integer|))) (|Expression| (|MachineInteger|))) "\\spad{assign(x,l,y)} creates a representation of the assignment of \\spad{y} to the \\spad{l}\\spad{'}th element of array \\spad{x} (\\spad{l} is a list of indices).") (($ (|Symbol|) (|Vector| (|Expression| (|MachineComplex|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|MachineFloat|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|Expression| (|MachineInteger|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|MachineComplex|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|MachineFloat|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|Expression| (|MachineInteger|)))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|MachineComplex|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|MachineFloat|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Vector| (|MachineInteger|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|MachineComplex|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|MachineFloat|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Matrix| (|MachineInteger|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|MachineComplex|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|MachineFloat|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|Expression| (|MachineInteger|))) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.") (($ (|Symbol|) (|String|)) "\\spad{assign(x,y)} creates a representation of the FORTRAN expression x=y.")) (|cond| (($ (|Switch|) $ $) "\\spad{cond(s,e,f)} creates a representation of the FORTRAN expression IF (\\spad{s}) THEN \\spad{e} ELSE \\spad{f}.") (($ (|Switch|) $) "\\spad{cond(s,e)} creates a representation of the FORTRAN expression IF (\\spad{s}) THEN \\spad{e}.")) (|returns| (($ (|Expression| (|Complex| (|Float|)))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|Integer|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|Float|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|MachineComplex|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|MachineInteger|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($ (|Expression| (|MachineFloat|))) "\\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.") (($) "\\spad{returns()} creates a representation of a FORTRAN RETURN statement.")) (|call| (($ (|String|)) "\\spad{call(s)} creates a representation of a FORTRAN CALL statement")) (|comment| (($ (|List| (|String|))) "\\spad{comment(s)} creates a representation of the Strings \\spad{s} as a multi-line FORTRAN comment.") (($ (|String|)) "\\spad{comment(s)} creates a representation of the String \\spad{s} as a single FORTRAN comment.")) (|continue| (($ (|SingleInteger|)) "\\spad{continue(l)} creates a representation of a FORTRAN CONTINUE labelled with \\spad{l}")) (|goto| (($ (|SingleInteger|)) "\\spad{goto(l)} creates a representation of a FORTRAN GOTO statement")) (|repeatUntilLoop| (($ (|Switch|) $) "\\spad{repeatUntilLoop(s,c)} creates a repeat ... until loop in FORTRAN.")) (|whileLoop| (($ (|Switch|) $) "\\spad{whileLoop(s,c)} creates a while loop in FORTRAN.")) (|forLoop| (($ (|SegmentBinding| (|Polynomial| (|Integer|))) (|Polynomial| (|Integer|)) $) "\\spad{forLoop(i=1..10,n,c)} creates a representation of a FORTRAN DO loop with \\spad{i} ranging over the values 1 to 10 by \\spad{n}.") (($ (|SegmentBinding| (|Polynomial| (|Integer|))) $) "\\spad{forLoop(i=1..10,c)} creates a representation of a FORTRAN DO loop with \\spad{i} ranging over the values 1 to 10."))) 
 NIL 
 NIL 
-(-333 E) 
+(-334 E) 
 ((|constructor| (NIL "\\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: 12 June 1992 Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:")) (|argument| ((|#1| $) "\\spad{argument(x)} returns the argument of a given sin/cos expressions")) (|sin?| (((|Boolean|) $) "\\spad{sin?(x)} returns \\spad{true} if term is a sin,{} otherwise \\spad{false}")) (|cos| (($ |#1|) "\\spad{cos(x)} makes a cos kernel for use in Fourier series")) (|sin| (($ |#1|) "\\spad{sin(x)} makes a sin kernel for use in Fourier series"))) 
 NIL 
 NIL 
-(-334) 
+(-335) 
 ((|constructor| (NIL "\\spadtype{FortranCodePackage1} provides some utilities for producing useful objects in FortranCode domain. The Package may be used with the FortranCode domain and its \\spad{printCode} or possibly via an outputAsFortran. (The package provides items of use in connection with ASPs in the AXIOM-NAG link and,{} where appropriate,{} naming accords with that in IRENA.) The easy-to-use functions use Fortran loop variables I1,{} I2,{} and it is users' responsibility to check that this is sensible. The advanced functions use SegmentBinding to allow users control over Fortran loop variable names.")) (|identitySquareMatrix| (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|))) "\\spad{identitySquareMatrix(s,p)} \\undocumented{}")) (|zeroSquareMatrix| (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|))) "\\spad{zeroSquareMatrix(s,p)} \\undocumented{}")) (|zeroMatrix| (((|FortranCode|) (|Symbol|) (|SegmentBinding| (|Polynomial| (|Integer|))) (|SegmentBinding| (|Polynomial| (|Integer|)))) "\\spad{zeroMatrix(s,b,d)} in this version gives the user control over names of Fortran variables used in loops.") (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|)) (|Polynomial| (|Integer|))) "\\spad{zeroMatrix(s,p,q)} uses loop variables in the Fortran,{} I1 and I2")) (|zeroVector| (((|FortranCode|) (|Symbol|) (|Polynomial| (|Integer|))) "\\spad{zeroVector(s,p)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-335) 
+(-336) 
 ((|constructor| (NIL "Represntation of data needed to instantiate a domain constructor.")) (|lookupFunction| (((|Identifier|) $) "\\spad{lookupFunction x} returns the name of the lookup function associated with the functor data \\spad{x}.")) (|categories| (((|PrimitiveArray| (|ConstructorCall| (|CategoryConstructor|))) $) "\\spad{categories x} returns the list of categories forms each domain object obtained from the domain data \\spad{x} belongs to.")) (|encodingDirectory| (((|PrimitiveArray| (|NonNegativeInteger|)) $) "\\spad{encodintDirectory x} returns the directory of domain-wide entity description.")) (|attributeData| (((|List| (|Pair| (|Syntax|) (|NonNegativeInteger|))) $) "\\spad{attributeData x} returns the list of attribute-predicate bit vector index pair associated with the functor data \\spad{x}.")) (|domainTemplate| (((|DomainTemplate|) $) "\\spad{domainTemplate x} returns the domain template vector associated with the functor data \\spad{x}."))) 
 NIL 
 NIL 
-(-336 -3498 UP UPUP R) 
+(-337 -3505 UP UPUP R) 
 ((|constructor| (NIL "This domains implements finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|lSpaceBasis| (((|Vector| |#4|) $) "\\spad{lSpaceBasis(d)} returns a basis for \\spad{L(d) = {f | (f) >= -d}} as a module over \\spad{K[x]}.")) (|finiteBasis| (((|Vector| |#4|) $) "\\spad{finiteBasis(d)} returns a basis for \\spad{d} as a module over {\\em K[x]}."))) 
 NIL 
 NIL 
-(-337 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) 
+(-338 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) 
 ((|constructor| (NIL "\\indented{1}{Lift a map to finite divisors.} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 19 May 1993")) (|map| (((|FiniteDivisor| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{map(f,d)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-338 S -3498 UP UPUP R) 
+(-339 S -3505 UP UPUP R) 
 ((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|generator| (((|Union| |#5| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) (|:| |principalPart| |#5|)) $) "\\spad{decompose(d)} returns \\spad{[id, f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#5| |#3| |#3| |#3| |#2|) "\\spad{divisor(h, d, d', g, r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,discriminant)} contains the ramified zeros of \\spad{d}") (($ |#2| |#2| (|Integer|)) "\\spad{divisor(a, b, n)} makes the divisor \\spad{nP} where \\spad{P:} \\spad{(x = a, y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#2| |#2|) "\\spad{divisor(a, b)} makes the divisor \\spad{P:} \\spad{(x = a, y = b)}. Error: if \\spad{P} is singular.") (($ |#5|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) 
 NIL 
 NIL 
-(-339 -3498 UP UPUP R) 
+(-340 -3505 UP UPUP R) 
 ((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve.")) (|generator| (((|Union| |#4| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) (|:| |principalPart| |#4|)) $) "\\spad{decompose(d)} returns \\spad{[id, f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#4| |#2| |#2| |#2| |#1|) "\\spad{divisor(h, d, d', g, r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,discriminant)} contains the ramified zeros of \\spad{d}") (($ |#1| |#1| (|Integer|)) "\\spad{divisor(a, b, n)} makes the divisor \\spad{nP} where \\spad{P:} \\spad{(x = a, y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#1| |#1|) "\\spad{divisor(a, b)} makes the divisor \\spad{P:} \\spad{(x = a, y = b)}. Error: if \\spad{P} is singular.") (($ |#4|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) 
 NIL 
 NIL 
-(-340 S R) 
+(-341 S R) 
 ((|constructor| (NIL "This category provides a selection of evaluation operations depending on what the argument type \\spad{R} provides.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(f, ex)} evaluates ex,{} applying \\spad{f} to values of type \\spad{R} in ex."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -518) (QUOTE (-1181)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -288) (|devaluate| |#2|) (|devaluate| |#2|)))) 
-(-341 R) 
+((|HasCategory| |#2| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -289) (|devaluate| |#2|) (|devaluate| |#2|)))) 
+(-342 R) 
 ((|constructor| (NIL "This category provides a selection of evaluation operations depending on what the argument type \\spad{R} provides.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f, ex)} evaluates ex,{} applying \\spad{f} to values of type \\spad{R} in ex."))) 
 NIL 
 NIL 
-(-342 |basicSymbols| |subscriptedSymbols| R) 
+(-343 |basicSymbols| |subscriptedSymbols| R) 
 ((|constructor| (NIL "A domain of expressions involving functions which can be translated into standard Fortran-77,{} with some extra extensions from the NAG Fortran Library.")) (|useNagFunctions| (((|Boolean|) (|Boolean|)) "\\spad{useNagFunctions(v)} sets the flag which controls whether NAG functions \\indented{1}{are being used for mathematical and machine constants.\\space{2}The previous} \\indented{1}{value is returned.}") (((|Boolean|)) "\\spad{useNagFunctions()} indicates whether NAG functions are being used \\indented{1}{for mathematical and machine constants.}")) (|variables| (((|List| (|Symbol|)) $) "\\spad{variables(e)} return a list of all the variables in \\spad{e}.")) (|pi| (($) "\\spad{pi(x)} represents the NAG Library function X01AAF which returns \\indented{1}{an approximation to the value of \\spad{pi}}")) (|tanh| (($ $) "\\spad{tanh(x)} represents the Fortran intrinsic function TANH")) (|cosh| (($ $) "\\spad{cosh(x)} represents the Fortran intrinsic function COSH")) (|sinh| (($ $) "\\spad{sinh(x)} represents the Fortran intrinsic function SINH")) (|atan| (($ $) "\\spad{atan(x)} represents the Fortran intrinsic function ATAN")) (|acos| (($ $) "\\spad{acos(x)} represents the Fortran intrinsic function ACOS")) (|asin| (($ $) "\\spad{asin(x)} represents the Fortran intrinsic function ASIN")) (|tan| (($ $) "\\spad{tan(x)} represents the Fortran intrinsic function TAN")) (|cos| (($ $) "\\spad{cos(x)} represents the Fortran intrinsic function COS")) (|sin| (($ $) "\\spad{sin(x)} represents the Fortran intrinsic function SIN")) (|log10| (($ $) "\\spad{log10(x)} represents the Fortran intrinsic function LOG10")) (|log| (($ $) "\\spad{log(x)} represents the Fortran intrinsic function LOG")) (|exp| (($ $) "\\spad{exp(x)} represents the Fortran intrinsic function EXP")) (|sqrt| (($ $) "\\spad{sqrt(x)} represents the Fortran intrinsic function SQRT")) (|abs| (($ $) "\\spad{abs(x)} represents the Fortran intrinsic function ABS")) (|coerce| (((|Expression| |#3|) $) "\\spad{coerce(x)} \\undocumented{}")) (|retractIfCan| (((|Union| $ "failed") (|Polynomial| (|Float|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (((|Union| $ "failed") (|Expression| (|Float|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (((|Union| $ "failed") (|Expression| (|Integer|))) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (((|Union| $ "failed") (|Symbol|)) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a FortranExpression \\indented{1}{checking that it is one of the given basic symbols} \\indented{1}{or subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (((|Union| $ "failed") (|Expression| |#3|)) "\\spad{retractIfCan(e)} takes \\spad{e} and tries to transform it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}")) (|retract| (($ (|Polynomial| (|Float|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (($ (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (($ (|Expression| (|Float|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (($ (|Polynomial| (|Integer|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (($ (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (($ (|Expression| (|Integer|))) "\\spad{retract(e)} takes \\spad{e} and transforms it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (($ (|Symbol|)) "\\spad{retract(e)} takes \\spad{e} and transforms it into a FortranExpression \\indented{1}{checking that it is one of the given basic symbols} \\indented{1}{or subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}") (($ (|Expression| |#3|)) "\\spad{retract(e)} takes \\spad{e} and transforms it into a \\indented{1}{FortranExpression checking that it contains no non-Fortran} \\indented{1}{functions,{} and that it only contains the given basic symbols} \\indented{1}{and subscripted symbols which correspond to scalar and array} \\indented{1}{parameters respectively.}"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#3| (LIST (QUOTE -1042) (QUOTE (-381)))) (|HasCategory| $ (QUOTE (-1053))) (|HasCategory| $ (LIST (QUOTE -1042) (QUOTE (-550))))) 
-(-343 |p| |n|) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-382)))) (|HasCategory| $ (QUOTE (-1055))) (|HasCategory| $ (LIST (QUOTE -1044) (QUOTE (-551))))) 
+(-344 |p| |n|) 
 ((|constructor| (NIL "FiniteField(\\spad{p},{}\\spad{n}) implements finite fields with p**n elements. This packages checks that \\spad{p} is prime. For a non-checking version,{} see \\spadtype{InnerFiniteField}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| (-910 |#1|) (QUOTE (-145))) (|HasCategory| (-910 |#1|) (QUOTE (-371)))) (|HasCategory| (-910 |#1|) (QUOTE (-147))) (|HasCategory| (-910 |#1|) (QUOTE (-371))) (|HasCategory| (-910 |#1|) (QUOTE (-145)))) 
-(-344 S -3498 UP UPUP) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| (-912 |#1|) (QUOTE (-145))) (|HasCategory| (-912 |#1|) (QUOTE (-372)))) (|HasCategory| (-912 |#1|) (QUOTE (-147))) (|HasCategory| (-912 |#1|) (QUOTE (-372))) (|HasCategory| (-912 |#1|) (QUOTE (-145)))) 
+(-345 S -3505 UP UPUP) 
 ((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#2|))) "\\spad{rationalPoints()} returns the list of all the affine rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#2|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in u1,{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#3|) (|:| |derivden| |#3|) (|:| |gd| |#3|)) $ (|Mapping| |#3| |#3|)) "\\spad{algSplitSimple(f, D)} returns \\spad{[h,d,d',g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d, discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#3| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#3| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#2| $ |#2| |#2|) "\\spad{elt(f,a,b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a, y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#3| |#3|)) "\\spad{differentiate(x, d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#3|)) (|:| |den| |#3|)) (|Mapping| |#3| |#3|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(wi)} with respect to \\spad{(w1,...,wn)} where \\spad{(w1,...,wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#3|) |#3|) "\\spad{integralRepresents([A1,...,An], D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,...,wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,...,wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#3|) |#3|) "\\spad{represents([A0,...,A(n-1)],D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\spad{inverseIntegralMatrixAtInfinity()} returns \\spad{M} such that \\spad{M (v1,...,vn) = (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\spad{integralMatrixAtInfinity()} returns \\spad{M} such that \\spad{(v1,...,vn) = M (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\spad{inverseIntegralMatrix()} returns \\spad{M} such that \\spad{M (w1,...,wn) = (1, y, ..., y**(n-1))} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|integralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\spad{integralMatrix()} returns \\spad{M} such that \\spad{(w1,...,wn) = M (1, y, ..., y**(n-1))},{} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,...,bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,...,bn)} returns the complementary basis \\spad{(b1',...,bn')} of \\spad{(b1,...,bn)}.")) (|integral?| (((|Boolean|) $ |#3|) "\\spad{integral?(f, p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}.") (((|Boolean|) $ |#2|) "\\spad{integral?(f, a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.")) (|integralBasis| (((|Vector| $)) "\\spad{integralBasis()} returns the integral basis for the curve.")) (|ramified?| (((|Boolean|) |#3|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#2|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#3|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#2|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#3|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#2|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.")) (|rationalPoint?| (((|Boolean|) |#2| |#2|) "\\spad{rationalPoint?(a, b)} tests if \\spad{(x=a,y=b)} is on the curve.")) (|absolutelyIrreducible?| (((|Boolean|)) "\\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?")) (|genus| (((|NonNegativeInteger|)) "\\spad{genus()} returns the genus of one absolutely irreducible component")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\spad{numberOfComponents()} returns the number of absolutely irreducible components."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (QUOTE (-366)))) 
-(-345 -3498 UP UPUP) 
+((|HasCategory| |#2| (QUOTE (-372))) (|HasCategory| |#2| (QUOTE (-367)))) 
+(-346 -3505 UP UPUP) 
 ((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#1|))) "\\spad{rationalPoints()} returns the list of all the affine rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in u1,{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#2|) (|:| |derivden| |#2|) (|:| |gd| |#2|)) $ (|Mapping| |#2| |#2|)) "\\spad{algSplitSimple(f, D)} returns \\spad{[h,d,d',g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d, discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#2| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#2| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#1| $ |#1| |#1|) "\\spad{elt(f,a,b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a, y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|)) "\\spad{differentiate(x, d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#2|)) (|:| |den| |#2|)) (|Mapping| |#2| |#2|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(wi)} with respect to \\spad{(w1,...,wn)} where \\spad{(w1,...,wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#2|) |#2|) "\\spad{integralRepresents([A1,...,An], D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,...,wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,...,wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#2|) |#2|) "\\spad{represents([A0,...,A(n-1)],D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\spad{inverseIntegralMatrixAtInfinity()} returns \\spad{M} such that \\spad{M (v1,...,vn) = (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\spad{integralMatrixAtInfinity()} returns \\spad{M} such that \\spad{(v1,...,vn) = M (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\spad{inverseIntegralMatrix()} returns \\spad{M} such that \\spad{M (w1,...,wn) = (1, y, ..., y**(n-1))} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|integralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\spad{integralMatrix()} returns \\spad{M} such that \\spad{(w1,...,wn) = M (1, y, ..., y**(n-1))},{} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,...,bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,...,bn)} returns the complementary basis \\spad{(b1',...,bn')} of \\spad{(b1,...,bn)}.")) (|integral?| (((|Boolean|) $ |#2|) "\\spad{integral?(f, p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}.") (((|Boolean|) $ |#1|) "\\spad{integral?(f, a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.")) (|integralBasis| (((|Vector| $)) "\\spad{integralBasis()} returns the integral basis for the curve.")) (|ramified?| (((|Boolean|) |#2|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#1|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#2|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#1|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#2|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#1|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.")) (|rationalPoint?| (((|Boolean|) |#1| |#1|) "\\spad{rationalPoint?(a, b)} tests if \\spad{(x=a,y=b)} is on the curve.")) (|absolutelyIrreducible?| (((|Boolean|)) "\\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?")) (|genus| (((|NonNegativeInteger|)) "\\spad{genus()} returns the genus of one absolutely irreducible component")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\spad{numberOfComponents()} returns the number of absolutely irreducible components."))) 
-((-4420 |has| (-411 |#2|) (-366)) (-4425 |has| (-411 |#2|) (-366)) (-4419 |has| (-411 |#2|) (-366)) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 |has| (-412 |#2|) (-367)) (-4432 |has| (-412 |#2|) (-367)) (-4426 |has| (-412 |#2|) (-367)) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-346 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) 
+(-347 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) 
 ((|constructor| (NIL "Lifts a map from rings to function fields over them.")) (|map| ((|#8| (|Mapping| |#5| |#1|) |#4|) "\\spad{map(f, p)} lifts \\spad{f} to \\spad{F1} and applies it to \\spad{p}."))) 
 NIL 
 NIL 
-(-347 |p| |extdeg|) 
+(-348 |p| |extdeg|) 
 ((|constructor| (NIL "FiniteFieldCyclicGroup(\\spad{p},{}\\spad{n}) implements a finite field extension of degee \\spad{n} over the prime field with \\spad{p} elements. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\\em createPrimitivePoly} from \\spadtype{FiniteFieldPolynomialPackage}. The Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| (-910 |#1|) (QUOTE (-145))) (|HasCategory| (-910 |#1|) (QUOTE (-371)))) (|HasCategory| (-910 |#1|) (QUOTE (-147))) (|HasCategory| (-910 |#1|) (QUOTE (-371))) (|HasCategory| (-910 |#1|) (QUOTE (-145)))) 
-(-348 GF |defpol|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| (-912 |#1|) (QUOTE (-145))) (|HasCategory| (-912 |#1|) (QUOTE (-372)))) (|HasCategory| (-912 |#1|) (QUOTE (-147))) (|HasCategory| (-912 |#1|) (QUOTE (-372))) (|HasCategory| (-912 |#1|) (QUOTE (-145)))) 
+(-349 GF |defpol|) 
 ((|constructor| (NIL "FiniteFieldCyclicGroupExtensionByPolynomial(\\spad{GF},{}defpol) implements a finite extension field of the ground field {\\em GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial {\\em defpol},{} which MUST be primitive (user responsibility). Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field it is used to perform additions in the field quickly."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-145)))) 
-(-349 GF |extdeg|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-145)))) 
+(-350 GF |extdeg|) 
 ((|constructor| (NIL "FiniteFieldCyclicGroupExtension(\\spad{GF},{}\\spad{n}) implements a extension of degree \\spad{n} over the ground field {\\em GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\\em createPrimitivePoly} from \\spadtype{FiniteFieldPolynomialPackage}. Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-145)))) 
-(-350 GF) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-145)))) 
+(-351 GF) 
 ((|constructor| (NIL "FiniteFieldFunctions(\\spad{GF}) is a package with functions concerning finite extension fields of the finite ground field {\\em GF},{} \\spadignore{e.g.} Zech logarithms.")) (|createLowComplexityNormalBasis| (((|Union| (|SparseUnivariatePolynomial| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) (|PositiveInteger|)) "\\spad{createLowComplexityNormalBasis(n)} tries to find a a low complexity normal basis of degree {\\em n} over {\\em GF} and returns its multiplication matrix If no low complexity basis is found it calls \\axiomFunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}(\\spad{n}) to produce a normal polynomial of degree {\\em n} over {\\em GF}")) (|createLowComplexityTable| (((|Union| (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) "failed") (|PositiveInteger|)) "\\spad{createLowComplexityTable(n)} tries to find a low complexity normal basis of degree {\\em n} over {\\em GF} and returns its multiplication matrix Fails,{} if it does not find a low complexity basis")) (|sizeMultiplication| (((|NonNegativeInteger|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{sizeMultiplication(m)} returns the number of entries of the multiplication table {\\em m}.")) (|createMultiplicationMatrix| (((|Matrix| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{createMultiplicationMatrix(m)} forms the multiplication table {\\em m} into a matrix over the ground field.")) (|createMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createMultiplicationTable(f)} generates a multiplication table for the normal basis of the field extension determined by {\\em f}. This is needed to perform multiplications between elements represented as coordinate vectors to this basis. See \\spadtype{FFNBP},{} \\spadtype{FFNBX}.")) (|createZechTable| (((|PrimitiveArray| (|SingleInteger|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createZechTable(f)} generates a Zech logarithm table for the cyclic group representation of a extension of the ground field by the primitive polynomial {\\em f(x)},{} \\spadignore{i.e.} \\spad{Z(i)},{} defined by {\\em x**Z(i) = 1+x**i} is stored at index \\spad{i}. This is needed in particular to perform addition of field elements in finite fields represented in this way. See \\spadtype{FFCGP},{} \\spadtype{FFCGX}."))) 
 NIL 
 NIL 
-(-351 F1 GF F2) 
+(-352 F1 GF F2) 
 ((|constructor| (NIL "FiniteFieldHomomorphisms(\\spad{F1},{}\\spad{GF},{}\\spad{F2}) exports coercion functions of elements between the fields {\\em F1} and {\\em F2},{} which both must be finite simple algebraic extensions of the finite ground field {\\em GF}.")) (|coerce| ((|#1| |#3|) "\\spad{coerce(x)} is the homomorphic image of \\spad{x} from {\\em F2} in {\\em F1},{} where {\\em coerce} is a field homomorphism between the fields extensions {\\em F2} and {\\em F1} both over ground field {\\em GF} (the second argument to the package). Error: if the extension degree of {\\em F2} doesn\\spad{'t} divide the extension degree of {\\em F1}. Note that the other coercion function in the \\spadtype{FiniteFieldHomomorphisms} is a left inverse.") ((|#3| |#1|) "\\spad{coerce(x)} is the homomorphic image of \\spad{x} from {\\em F1} in {\\em F2}. Thus {\\em coerce} is a field homomorphism between the fields extensions {\\em F1} and {\\em F2} both over ground field {\\em GF} (the second argument to the package). Error: if the extension degree of {\\em F1} doesn\\spad{'t} divide the extension degree of {\\em F2}. Note that the other coercion function in the \\spadtype{FiniteFieldHomomorphisms} is a left inverse."))) 
 NIL 
 NIL 
-(-352 S) 
+(-353 S) 
 ((|constructor| (NIL "FiniteFieldCategory is the category of finite fields")) (|representationType| (((|Union| "prime" "polynomial" "normal" "cyclic")) "\\spad{representationType()} returns the type of the representation,{} one of: \\spad{prime},{} \\spad{polynomial},{} \\spad{normal},{} or \\spad{cyclic}.")) (|order| (((|PositiveInteger|) $) "\\spad{order(b)} computes the order of an element \\spad{b} in the multiplicative group of the field. Error: if \\spad{b} equals 0.")) (|discreteLog| (((|NonNegativeInteger|) $) "\\spad{discreteLog(a)} computes the discrete logarithm of \\spad{a} with respect to \\spad{primitiveElement()} of the field.")) (|primitive?| (((|Boolean|) $) "\\spad{primitive?(b)} tests whether the element \\spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \\spadignore{i.e.} is a primitive element. Implementation Note: see \\spad{ch}.IX.1.3,{} th.2 in \\spad{D}. Lipson.")) (|primitiveElement| (($) "\\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \\spadfun{createPrimitiveElement}.")) (|createPrimitiveElement| (($) "\\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.")) (|tableForDiscreteLogarithm| (((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|)) "\\spad{tableForDiscreteLogarithm(a,n)} returns a table of the discrete logarithms of \\spad{a**0} up to \\spad{a**(n-1)} which,{} called with key \\spad{lookup(a**i)} returns \\spad{i} for \\spad{i} in \\spad{0..n-1}. Error: if not called for prime divisors of order of \\indented{7}{multiplicative group.}")) (|factorsOfCyclicGroupSize| (((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|))))) "\\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\\spad{-1}")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or \"failed\" if no such vector exists.")) (|charthRoot| (($ $) "\\spad{charthRoot(a)} takes the characteristic'th root of {\\em a}. Note: such a root is alway defined in finite fields."))) 
 NIL 
 NIL 
-(-353) 
+(-354) 
 ((|constructor| (NIL "FiniteFieldCategory is the category of finite fields")) (|representationType| (((|Union| "prime" "polynomial" "normal" "cyclic")) "\\spad{representationType()} returns the type of the representation,{} one of: \\spad{prime},{} \\spad{polynomial},{} \\spad{normal},{} or \\spad{cyclic}.")) (|order| (((|PositiveInteger|) $) "\\spad{order(b)} computes the order of an element \\spad{b} in the multiplicative group of the field. Error: if \\spad{b} equals 0.")) (|discreteLog| (((|NonNegativeInteger|) $) "\\spad{discreteLog(a)} computes the discrete logarithm of \\spad{a} with respect to \\spad{primitiveElement()} of the field.")) (|primitive?| (((|Boolean|) $) "\\spad{primitive?(b)} tests whether the element \\spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \\spadignore{i.e.} is a primitive element. Implementation Note: see \\spad{ch}.IX.1.3,{} th.2 in \\spad{D}. Lipson.")) (|primitiveElement| (($) "\\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \\spadfun{createPrimitiveElement}.")) (|createPrimitiveElement| (($) "\\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.")) (|tableForDiscreteLogarithm| (((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|)) "\\spad{tableForDiscreteLogarithm(a,n)} returns a table of the discrete logarithms of \\spad{a**0} up to \\spad{a**(n-1)} which,{} called with key \\spad{lookup(a**i)} returns \\spad{i} for \\spad{i} in \\spad{0..n-1}. Error: if not called for prime divisors of order of \\indented{7}{multiplicative group.}")) (|factorsOfCyclicGroupSize| (((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|))))) "\\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\\spad{-1}")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or \"failed\" if no such vector exists.")) (|charthRoot| (($ $) "\\spad{charthRoot(a)} takes the characteristic'th root of {\\em a}. Note: such a root is alway defined in finite fields."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-354 R UP -3498) 
+(-355 R UP -3505) 
 ((|constructor| (NIL "In this package \\spad{R} is a Euclidean domain and \\spad{F} is a framed algebra over \\spad{R}. The package provides functions to compute the integral closure of \\spad{R} in the quotient field of \\spad{F}. It is assumed that \\spad{char(R/P) = char(R)} for any prime \\spad{P} of \\spad{R}. A typical instance of this is when \\spad{R = K[x]} and \\spad{F} is a function field over \\spad{R}.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) |#1|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{R} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) 
 NIL 
 NIL 
-(-355 |p| |extdeg|) 
+(-356 |p| |extdeg|) 
 ((|constructor| (NIL "FiniteFieldNormalBasis(\\spad{p},{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the prime field with \\spad{p} elements. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial created by \\spadfunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}.")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: The time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| (|PrimeField| |#1|))) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| (|PrimeField| |#1|)) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| (-910 |#1|) (QUOTE (-145))) (|HasCategory| (-910 |#1|) (QUOTE (-371)))) (|HasCategory| (-910 |#1|) (QUOTE (-147))) (|HasCategory| (-910 |#1|) (QUOTE (-371))) (|HasCategory| (-910 |#1|) (QUOTE (-145)))) 
-(-356 GF |uni|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| (-912 |#1|) (QUOTE (-145))) (|HasCategory| (-912 |#1|) (QUOTE (-372)))) (|HasCategory| (-912 |#1|) (QUOTE (-147))) (|HasCategory| (-912 |#1|) (QUOTE (-372))) (|HasCategory| (-912 |#1|) (QUOTE (-145)))) 
+(-357 GF |uni|) 
 ((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(\\spad{GF},{}uni) implements a finite extension of the ground field {\\em GF}. The elements are represented by coordinate vectors with respect to. a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element,{} where \\spad{q} is the size of {\\em GF}. The normal element is chosen as a root of the extension polynomial,{} which MUST be normal over {\\em GF} (user responsibility)")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-145)))) 
-(-357 GF |extdeg|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-145)))) 
+(-358 GF |extdeg|) 
 ((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(\\spad{GF},{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the ground field {\\em GF}. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial,{} created by {\\em createNormalPoly} from \\spadtype{FiniteFieldPolynomialPackage}")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-145)))) 
-(-358 GF |defpol|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-145)))) 
+(-359 GF |defpol|) 
 ((|constructor| (NIL "FiniteFieldExtensionByPolynomial(\\spad{GF},{} defpol) implements the extension of the finite field {\\em GF} generated by the extension polynomial {\\em defpol} which MUST be irreducible. Note: the user has the responsibility to ensure that {\\em defpol} is irreducible."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-145)))) 
-(-359 GF) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-145)))) 
+(-360 GF) 
 ((|constructor| (NIL "This package provides a number of functions for generating,{} counting and testing irreducible,{} normal,{} primitive,{} random polynomials over finite fields.")) (|reducedQPowers| (((|PrimitiveArray| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{reducedQPowers(f)} generates \\spad{[x,x**q,x**(q**2),...,x**(q**(n-1))]} reduced modulo \\spad{f} where \\spad{q = size()\\$GF} and \\spad{n = degree f}.")) (|leastAffineMultiple| (((|SparseUnivariatePolynomial| |#1|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{leastAffineMultiple(f)} computes the least affine polynomial which is divisible by the polynomial \\spad{f} over the finite field {\\em GF},{} \\spadignore{i.e.} a polynomial whose exponents are 0 or a power of \\spad{q},{} the size of {\\em GF}.")) (|random| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{random(m,n)}\\$FFPOLY(\\spad{GF}) generates a random monic polynomial of degree \\spad{d} over the finite field {\\em GF},{} \\spad{d} between \\spad{m} and \\spad{n}.") (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{random(n)}\\$FFPOLY(\\spad{GF}) generates a random monic polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|nextPrimitiveNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitiveNormalPoly(f)} yields the next primitive normal polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g} or,{} in case these numbers are equal,{} if the {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than this number for \\spad{g}. If these numbers are equals,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g},{} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are coefficients according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}. This operation is equivalent to nextNormalPrimitivePoly(\\spad{f}).")) (|nextNormalPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPrimitivePoly(f)} yields the next normal primitive polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g} or if {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than this number for \\spad{g}. Otherwise,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}. This operation is equivalent to nextPrimitiveNormalPoly(\\spad{f}).")) (|nextNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPoly(f)} yields the next normal polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than that for \\spad{g}. In case these numbers are equal,{} \\spad{f < g} if if the number of monomials of \\spad{f} is less that for \\spad{g} or if the list of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|nextPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitivePoly(f)} yields the next primitive polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g}. If these values are equal,{} then \\spad{f < g} if if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|nextIrreduciblePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextIrreduciblePoly(f)} yields the next monic irreducible polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than this number for \\spad{g}. If \\spad{f} and \\spad{g} have the same number of monomials,{} the lists of exponents are compared lexicographically. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|createPrimitiveNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitiveNormalPoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal and primitive polynomial of degree \\spad{n} over the field {\\em GF}. polynomial of degree \\spad{n} over the field {\\em GF}.")) (|createNormalPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal and primitive polynomial of degree \\spad{n} over the field {\\em GF}. Note: this function is equivalent to createPrimitiveNormalPoly(\\spad{n})")) (|createNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPoly(n)}\\$FFPOLY(\\spad{GF}) generates a normal polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|createPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) generates a primitive polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|createIrreduciblePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createIrreduciblePoly(n)}\\$FFPOLY(\\spad{GF}) generates a monic irreducible univariate polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfNormalPoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfNormalPoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of normal polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfPrimitivePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfPrimitivePoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of primitive polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfIrreduciblePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfIrreduciblePoly(n)}\\$FFPOLY(\\spad{GF}) yields the number of monic irreducible univariate polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|normal?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{normal?(f)} tests whether the polynomial \\spad{f} over a finite field is normal,{} \\spadignore{i.e.} its roots are linearly independent over the field.")) (|primitive?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{primitive?(f)} tests whether the polynomial \\spad{f} over a finite field is primitive,{} \\spadignore{i.e.} all its roots are primitive."))) 
 NIL 
 NIL 
-(-360 -3498 GF) 
+(-361 -3505 GF) 
 ((|constructor| (NIL "FiniteFieldPolynomialPackage2(\\spad{F},{}\\spad{GF}) exports some functions concerning finite fields,{} which depend on a finite field {\\em GF} and an algebraic extension \\spad{F} of {\\em GF},{} \\spadignore{e.g.} a zero of a polynomial over {\\em GF} in \\spad{F}.")) (|rootOfIrreduciblePoly| ((|#1| (|SparseUnivariatePolynomial| |#2|)) "\\spad{rootOfIrreduciblePoly(f)} computes one root of the monic,{} irreducible polynomial \\spad{f},{} which degree must divide the extension degree of {\\em F} over {\\em GF},{} \\spadignore{i.e.} \\spad{f} splits into linear factors over {\\em F}.")) (|Frobenius| ((|#1| |#1|) "\\spad{Frobenius(x)} \\undocumented{}")) (|basis| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{basis(n)} \\undocumented{}")) (|lookup| (((|PositiveInteger|) |#1|) "\\spad{lookup(x)} \\undocumented{}")) (|coerce| ((|#1| |#2|) "\\spad{coerce(x)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-361 -3498 FP FPP) 
+(-362 -3505 FP FPP) 
 ((|constructor| (NIL "This package solves linear diophantine equations for Bivariate polynomials over finite fields")) (|solveLinearPolynomialEquation| (((|Union| (|List| |#3|) "failed") (|List| |#3|) |#3|) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}\\spad{'s} exists."))) 
 NIL 
 NIL 
-(-362 GF |n|) 
+(-363 GF |n|) 
 ((|constructor| (NIL "FiniteFieldExtensionByPolynomial(\\spad{GF},{} \\spad{n}) implements an extension of the finite field {\\em GF} of degree \\spad{n} generated by the extension polynomial constructed by \\spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage} from \\spadtype{FiniteFieldPolynomialPackage}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-145)))) 
-(-363 R |ls|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-145)))) 
+(-364 R |ls|) 
 ((|constructor| (NIL "This is just an interface between several packages and domains. The goal is to compute lexicographical Groebner bases of sets of polynomial with type \\spadtype{Polynomial R} by the {\\em FGLM} algorithm if this is possible (\\spadignore{i.e.} if the input system generates a zero-dimensional ideal).")) (|groebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|))) "\\axiom{groebner(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}}. If \\axiom{\\spad{lq1}} generates a zero-dimensional ideal then the {\\em FGLM} strategy is used,{} otherwise the {\\em Sugar} strategy is used.")) (|fglmIfCan| (((|Union| (|List| (|Polynomial| |#1|)) "failed") (|List| (|Polynomial| |#1|))) "\\axiom{fglmIfCan(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}} by using the {\\em FGLM} strategy,{} if \\axiom{zeroDimensional?(\\spad{lq1})} holds.")) (|zeroDimensional?| (((|Boolean|) (|List| (|Polynomial| |#1|))) "\\axiom{zeroDimensional?(\\spad{lq1})} returns \\spad{true} iff \\axiom{\\spad{lq1}} generates a zero-dimensional ideal \\spad{w}.\\spad{r}.\\spad{t}. the variables of \\axiom{\\spad{ls}}."))) 
 NIL 
 NIL 
-(-364 S) 
+(-365 S) 
 ((|constructor| (NIL "The free group on a set \\spad{S} is the group of finite products of the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are integers. The multiplication is not commutative.")) (|factors| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| (|Integer|)))) $) "\\spad{factors(a1\\^e1,...,an\\^en)} returns \\spad{[[a1, e1],...,[an, en]]}.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| (|Integer|) (|Integer|)) $) "\\spad{mapExpon(f, a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x, n)} returns the factor of the n^th monomial of \\spad{x}.")) (|nthExpon| (((|Integer|) $ (|Integer|)) "\\spad{nthExpon(x, n)} returns the exponent of the n^th monomial of \\spad{x}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of monomials in \\spad{x}.")) (** (($ |#1| (|Integer|)) "\\spad{s ** n} returns the product of \\spad{s} by itself \\spad{n} times.")) (* (($ $ |#1|) "\\spad{x * s} returns the product of \\spad{x} by \\spad{s} on the right.") (($ |#1| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-365 S) 
+(-366 S) 
 ((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) 
 NIL 
 NIL 
-(-366) 
+(-367) 
 ((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-367 S) 
+(-368 S) 
 ((|constructor| (NIL "This domain provides a basic model of files to save arbitrary values. The operations provide sequential access to the contents.")) (|readIfCan!| (((|Union| |#1| "failed") $) "\\spad{readIfCan!(f)} returns a value from the file \\spad{f},{} if possible. If \\spad{f} is not open for reading,{} or if \\spad{f} is at the end of file then \\spad{\"failed\"} is the result."))) 
 NIL 
 NIL 
-(-368 |Name| S) 
+(-369 |Name| S) 
 ((|constructor| (NIL "This category provides an interface to operate on files in the computer\\spad{'s} file system. The precise method of naming files is determined by the Name parameter. The type of the contents of the file is determined by \\spad{S}.")) (|write!| ((|#2| $ |#2|) "\\spad{write!(f,s)} puts the value \\spad{s} into the file \\spad{f}. The state of \\spad{f} is modified so subsequents call to \\spad{write!} will append one after another.")) (|read!| ((|#2| $) "\\spad{read!(f)} extracts a value from file \\spad{f}. The state of \\spad{f} is modified so a subsequent call to \\spadfun{read!} will return the next element.")) (|iomode| (((|String|) $) "\\spad{iomode(f)} returns the status of the file \\spad{f}. The input/output status of \\spad{f} may be \"input\",{} \"output\" or \"closed\" mode.")) (|name| ((|#1| $) "\\spad{name(f)} returns the external name of the file \\spad{f}.")) (|close!| (($ $) "\\spad{close!(f)} returns the file \\spad{f} closed to input and output.")) (|reopen!| (($ $ (|String|)) "\\spad{reopen!(f,mode)} returns a file \\spad{f} reopened for operation in the indicated mode: \"input\" or \"output\". \\spad{reopen!(f,\"input\")} will reopen the file \\spad{f} for input.")) (|open| (($ |#1| (|String|)) "\\spad{open(s,mode)} returns a file \\spad{s} open for operation in the indicated mode: \"input\" or \"output\".") (($ |#1|) "\\spad{open(s)} returns the file \\spad{s} open for input."))) 
 NIL 
 NIL 
-(-369 S R) 
+(-370 S R) 
 ((|constructor| (NIL "A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|unitsKnown| ((|attribute|) "unitsKnown means that \\spadfun{recip} truly yields reciprocal or \\spad{\"failed\"} if not a unit,{} similarly for \\spadfun{leftRecip} and \\spadfun{rightRecip}. The reason is that we use left,{} respectively right,{} minimal polynomials to decide this question.")) (|unit| (((|Union| $ "failed")) "\\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnit| (((|Union| $ "failed")) "\\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|leftUnit| (((|Union| $ "failed")) "\\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|rightMinimalPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|leftMinimalPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|associatorDependence| (((|List| (|Vector| |#2|))) "\\spad{associatorDependence()} looks for the associator identities,{} \\spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \\spad{associator(a,b,c)} which yield 0,{} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. The order of the permutations is \\spad{123 231 312 132 321 213}.")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|lieAlgebra?| (((|Boolean|)) "\\spad{lieAlgebra?()} tests if the algebra is anticommutative and \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Lie algebra \\spad{(A,+,*)},{} where \\spad{a*b := a@b-b@a}.")) (|jordanAlgebra?| (((|Boolean|)) "\\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \\spad{(a*b)*a**2 - a*(b*a**2) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Jordan algebra \\spad{(A,+,*)},{} where \\spad{a*b := (a@b+b@a)/2}.")) (|noncommutativeJordanAlgebra?| (((|Boolean|)) "\\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.")) (|jordanAdmissible?| (((|Boolean|)) "\\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \\spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \\spadignore{i.e.} satisfies the Jordan identity. The property of \\spadatt{commutative(\\spad{\"*\"})} follows from by definition.")) (|lieAdmissible?| (((|Boolean|)) "\\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \\spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.")) (|jacobiIdentity?| (((|Boolean|)) "\\spad{jacobiIdentity?()} tests if \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.")) (|powerAssociative?| (((|Boolean|)) "\\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.")) (|alternative?| (((|Boolean|)) "\\spad{alternative?()} tests if \\spad{2*associator(a,a,b) = 0 = 2*associator(a,b,b)} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|flexible?| (((|Boolean|)) "\\spad{flexible?()} tests if \\spad{2*associator(a,b,a) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|rightAlternative?| (((|Boolean|)) "\\spad{rightAlternative?()} tests if \\spad{2*associator(a,b,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|leftAlternative?| (((|Boolean|)) "\\spad{leftAlternative?()} tests if \\spad{2*associator(a,a,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|antiAssociative?| (((|Boolean|)) "\\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \\spadignore{i.e.} \\spad{(a*b)*c + a*(b*c) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra.")) (|associative?| (((|Boolean|)) "\\spad{associative?()} tests if multiplication in algebra is associative.")) (|antiCommutative?| (((|Boolean|)) "\\spad{antiCommutative?()} tests if \\spad{a*a = 0} for all \\spad{a} in the algebra. Note: this implies \\spad{a*b + b*a = 0} for all \\spad{a} and \\spad{b}.")) (|commutative?| (((|Boolean|)) "\\spad{commutative?()} tests if multiplication in the algebra is commutative.")) (|rightCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \\spad{a} with respect to any basis.")) (|leftCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \\spad{a} with respect to any basis.")) (|rightTraceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{rightTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}.")) (|leftTraceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{leftTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}.")) (|rightDiscriminant| ((|#2| (|Vector| $)) "\\spad{rightDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(rightTraceMatrix([v1,...,vn]))}.")) (|leftDiscriminant| ((|#2| (|Vector| $)) "\\spad{leftDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(leftTraceMatrix([v1,...,vn]))}.")) (|represents| (($ (|Vector| |#2|) (|Vector| $)) "\\spad{represents([a1,...,am],[v1,...,vm])} returns the linear combination \\spad{a1*vm + ... + an*vm}.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([a1,...,am],[v1,...,vn])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.") (((|Vector| |#2|) $ (|Vector| $)) "\\spad{coordinates(a,[v1,...,vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rightNorm| ((|#2| $) "\\spad{rightNorm(a)} returns the determinant of the right regular representation of \\spad{a}.")) (|leftNorm| ((|#2| $) "\\spad{leftNorm(a)} returns the determinant of the left regular representation of \\spad{a}.")) (|rightTrace| ((|#2| $) "\\spad{rightTrace(a)} returns the trace of the right regular representation of \\spad{a}.")) (|leftTrace| ((|#2| $) "\\spad{leftTrace(a)} returns the trace of the left regular representation of \\spad{a}.")) (|rightRegularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{rightRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|leftRegularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{leftRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|structuralConstants| (((|Vector| (|Matrix| |#2|)) (|Vector| $)) "\\spad{structuralConstants([v1,v2,...,vm])} calculates the structural constants \\spad{[(gammaijk) for k in 1..m]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \\spad{[v1,...,vm]} is an \\spad{R}-module basis of a subalgebra.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#2|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,...,vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra as \\spad{R}-module.")) (|someBasis| (((|Vector| $)) "\\spad{someBasis()} returns some \\spad{R}-module basis."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-561)))) 
-(-370 R) 
+((|HasCategory| |#2| (QUOTE (-562)))) 
+(-371 R) 
 ((|constructor| (NIL "A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|unitsKnown| ((|attribute|) "unitsKnown means that \\spadfun{recip} truly yields reciprocal or \\spad{\"failed\"} if not a unit,{} similarly for \\spadfun{leftRecip} and \\spadfun{rightRecip}. The reason is that we use left,{} respectively right,{} minimal polynomials to decide this question.")) (|unit| (((|Union| $ "failed")) "\\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnit| (((|Union| $ "failed")) "\\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|leftUnit| (((|Union| $ "failed")) "\\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|rightMinimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|leftMinimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|associatorDependence| (((|List| (|Vector| |#1|))) "\\spad{associatorDependence()} looks for the associator identities,{} \\spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \\spad{associator(a,b,c)} which yield 0,{} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. The order of the permutations is \\spad{123 231 312 132 321 213}.")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|lieAlgebra?| (((|Boolean|)) "\\spad{lieAlgebra?()} tests if the algebra is anticommutative and \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Lie algebra \\spad{(A,+,*)},{} where \\spad{a*b := a@b-b@a}.")) (|jordanAlgebra?| (((|Boolean|)) "\\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \\spad{(a*b)*a**2 - a*(b*a**2) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Jordan algebra \\spad{(A,+,*)},{} where \\spad{a*b := (a@b+b@a)/2}.")) (|noncommutativeJordanAlgebra?| (((|Boolean|)) "\\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.")) (|jordanAdmissible?| (((|Boolean|)) "\\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \\spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \\spadignore{i.e.} satisfies the Jordan identity. The property of \\spadatt{commutative(\\spad{\"*\"})} follows from by definition.")) (|lieAdmissible?| (((|Boolean|)) "\\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \\spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.")) (|jacobiIdentity?| (((|Boolean|)) "\\spad{jacobiIdentity?()} tests if \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.")) (|powerAssociative?| (((|Boolean|)) "\\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.")) (|alternative?| (((|Boolean|)) "\\spad{alternative?()} tests if \\spad{2*associator(a,a,b) = 0 = 2*associator(a,b,b)} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|flexible?| (((|Boolean|)) "\\spad{flexible?()} tests if \\spad{2*associator(a,b,a) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|rightAlternative?| (((|Boolean|)) "\\spad{rightAlternative?()} tests if \\spad{2*associator(a,b,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|leftAlternative?| (((|Boolean|)) "\\spad{leftAlternative?()} tests if \\spad{2*associator(a,a,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don\\spad{'t} know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|antiAssociative?| (((|Boolean|)) "\\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \\spadignore{i.e.} \\spad{(a*b)*c + a*(b*c) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra.")) (|associative?| (((|Boolean|)) "\\spad{associative?()} tests if multiplication in algebra is associative.")) (|antiCommutative?| (((|Boolean|)) "\\spad{antiCommutative?()} tests if \\spad{a*a = 0} for all \\spad{a} in the algebra. Note: this implies \\spad{a*b + b*a = 0} for all \\spad{a} and \\spad{b}.")) (|commutative?| (((|Boolean|)) "\\spad{commutative?()} tests if multiplication in the algebra is commutative.")) (|rightCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \\spad{a} with respect to any basis.")) (|leftCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \\spad{a} with respect to any basis.")) (|rightTraceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{rightTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}.")) (|leftTraceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{leftTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}.")) (|rightDiscriminant| ((|#1| (|Vector| $)) "\\spad{rightDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(rightTraceMatrix([v1,...,vn]))}.")) (|leftDiscriminant| ((|#1| (|Vector| $)) "\\spad{leftDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(leftTraceMatrix([v1,...,vn]))}.")) (|represents| (($ (|Vector| |#1|) (|Vector| $)) "\\spad{represents([a1,...,am],[v1,...,vm])} returns the linear combination \\spad{a1*vm + ... + an*vm}.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([a1,...,am],[v1,...,vn])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.") (((|Vector| |#1|) $ (|Vector| $)) "\\spad{coordinates(a,[v1,...,vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rightNorm| ((|#1| $) "\\spad{rightNorm(a)} returns the determinant of the right regular representation of \\spad{a}.")) (|leftNorm| ((|#1| $) "\\spad{leftNorm(a)} returns the determinant of the left regular representation of \\spad{a}.")) (|rightTrace| ((|#1| $) "\\spad{rightTrace(a)} returns the trace of the right regular representation of \\spad{a}.")) (|leftTrace| ((|#1| $) "\\spad{leftTrace(a)} returns the trace of the left regular representation of \\spad{a}.")) (|rightRegularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{rightRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|leftRegularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{leftRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|structuralConstants| (((|Vector| (|Matrix| |#1|)) (|Vector| $)) "\\spad{structuralConstants([v1,v2,...,vm])} calculates the structural constants \\spad{[(gammaijk) for k in 1..m]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \\spad{[v1,...,vm]} is an \\spad{R}-module basis of a subalgebra.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#1|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,...,vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra as \\spad{R}-module.")) (|someBasis| (((|Vector| $)) "\\spad{someBasis()} returns some \\spad{R}-module basis."))) 
-((-4424 |has| |#1| (-561)) (-4422 . T) (-4421 . T)) 
+((-4431 |has| |#1| (-562)) (-4429 . T) (-4428 . T)) 
 NIL 
-(-371) 
+(-372) 
 ((|constructor| (NIL "The category of domains composed of a finite set of elements. We include the functions \\spadfun{lookup} and \\spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \\blankline")) (|random| (($) "\\spad{random()} returns a random element from the set.")) (|lookup| (((|PositiveInteger|) $) "\\spad{lookup(x)} returns a positive integer such that \\spad{x = index lookup x}.")) (|index| (($ (|PositiveInteger|)) "\\spad{index(i)} takes a positive integer \\spad{i} less than or equal to \\spad{size()} and returns the \\spad{i}\\spad{-}th element of the set. This operation establishs a bijection between the elements of the finite set and \\spad{1..size()}.")) (|size| (((|NonNegativeInteger|)) "\\spad{size()} returns the number of elements in the set."))) 
 NIL 
 NIL 
-(-372 S R UP) 
+(-373 S R UP) 
 ((|constructor| (NIL "A FiniteRankAlgebra is an algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|minimalPolynomial| ((|#3| $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of \\spad{a}.")) (|characteristicPolynomial| ((|#3| $) "\\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \\spad{a} with respect to any basis.")) (|traceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{traceMatrix([v1,..,vn])} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr}(\\spad{vi} * \\spad{vj}) )")) (|discriminant| ((|#2| (|Vector| $)) "\\spad{discriminant([v1,..,vn])} returns \\spad{determinant(traceMatrix([v1,..,vn]))}.")) (|represents| (($ (|Vector| |#2|) (|Vector| $)) "\\spad{represents([a1,..,an],[v1,..,vn])} returns \\spad{a1*v1 + ... + an*vn}.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([v1,...,vm], basis)} returns the coordinates of the \\spad{vi}\\spad{'s} with to the basis \\spad{basis}. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $ (|Vector| $)) "\\spad{coordinates(a,basis)} returns the coordinates of \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|norm| ((|#2| $) "\\spad{norm(a)} returns the determinant of the regular representation of \\spad{a} with respect to any basis.")) (|trace| ((|#2| $) "\\spad{trace(a)} returns the trace of the regular representation of \\spad{a} with respect to any basis.")) (|regularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{regularRepresentation(a,basis)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-366)))) 
-(-373 R UP) 
+((|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-367)))) 
+(-374 R UP) 
 ((|constructor| (NIL "A FiniteRankAlgebra is an algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|minimalPolynomial| ((|#2| $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of \\spad{a}.")) (|characteristicPolynomial| ((|#2| $) "\\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \\spad{a} with respect to any basis.")) (|traceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{traceMatrix([v1,..,vn])} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr}(\\spad{vi} * \\spad{vj}) )")) (|discriminant| ((|#1| (|Vector| $)) "\\spad{discriminant([v1,..,vn])} returns \\spad{determinant(traceMatrix([v1,..,vn]))}.")) (|represents| (($ (|Vector| |#1|) (|Vector| $)) "\\spad{represents([a1,..,an],[v1,..,vn])} returns \\spad{a1*v1 + ... + an*vn}.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([v1,...,vm], basis)} returns the coordinates of the \\spad{vi}\\spad{'s} with to the basis \\spad{basis}. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $ (|Vector| $)) "\\spad{coordinates(a,basis)} returns the coordinates of \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|norm| ((|#1| $) "\\spad{norm(a)} returns the determinant of the regular representation of \\spad{a} with respect to any basis.")) (|trace| ((|#1| $) "\\spad{trace(a)} returns the trace of the regular representation of \\spad{a} with respect to any basis.")) (|regularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{regularRepresentation(a,basis)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-374 A S) 
+(-375 A S) 
 ((|constructor| (NIL "A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \\spadtype{LinearAggregate} such as \\spadfun{reverse},{} \\spadfun{sort},{} and so on.")) (|sort!| (($ $) "\\spad{sort!(u)} returns \\spad{u} with its elements in ascending order.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sort!(p,u)} returns \\spad{u} with its elements ordered by \\spad{p}.")) (|reverse!| (($ $) "\\spad{reverse!(u)} returns \\spad{u} with its elements in reverse order.")) (|copyInto!| (($ $ $ (|Integer|)) "\\spad{copyInto!(u,v,i)} returns aggregate \\spad{u} containing a copy of \\spad{v} inserted at element \\spad{i}.")) (|position| (((|Integer|) |#2| $ (|Integer|)) "\\spad{position(x,a,n)} returns the index \\spad{i} of the first occurrence of \\spad{x} in \\axiom{a} where \\axiom{\\spad{i} \\spad{>=} \\spad{n}},{} and \\axiom{minIndex(a) - 1} if no such \\spad{x} is found.") (((|Integer|) |#2| $) "\\spad{position(x,a)} returns the index \\spad{i} of the first occurrence of \\spad{x} in a,{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.") (((|Integer|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{position(p,a)} returns the index \\spad{i} of the first \\spad{x} in \\axiom{a} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.")) (|sorted?| (((|Boolean|) $) "\\spad{sorted?(u)} tests if the elements of \\spad{u} are in ascending order.") (((|Boolean|) (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sorted?(p,a)} tests if \\axiom{a} is sorted according to predicate \\spad{p}.")) (|sort| (($ $) "\\spad{sort(u)} returns an \\spad{u} with elements in ascending order. Note: \\axiom{sort(\\spad{u}) = sort(\\spad{<=},{}\\spad{u})}.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sort(p,a)} returns a copy of \\axiom{a} sorted using total ordering predicate \\spad{p}.")) (|reverse| (($ $) "\\spad{reverse(a)} returns a copy of \\axiom{a} with elements in reverse order.")) (|merge| (($ $ $) "\\spad{merge(u,v)} merges \\spad{u} and \\spad{v} in ascending order. Note: \\axiom{merge(\\spad{u},{}\\spad{v}) = merge(\\spad{<=},{}\\spad{u},{}\\spad{v})}.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $ $) "\\spad{merge(p,a,b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4428)) (|HasCategory| |#2| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-1105)))) 
-(-375 S) 
+((|HasAttribute| |#1| (QUOTE -4435)) (|HasCategory| |#2| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-1107)))) 
+(-376 S) 
 ((|constructor| (NIL "A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \\spadtype{LinearAggregate} such as \\spadfun{reverse},{} \\spadfun{sort},{} and so on.")) (|sort!| (($ $) "\\spad{sort!(u)} returns \\spad{u} with its elements in ascending order.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sort!(p,u)} returns \\spad{u} with its elements ordered by \\spad{p}.")) (|reverse!| (($ $) "\\spad{reverse!(u)} returns \\spad{u} with its elements in reverse order.")) (|copyInto!| (($ $ $ (|Integer|)) "\\spad{copyInto!(u,v,i)} returns aggregate \\spad{u} containing a copy of \\spad{v} inserted at element \\spad{i}.")) (|position| (((|Integer|) |#1| $ (|Integer|)) "\\spad{position(x,a,n)} returns the index \\spad{i} of the first occurrence of \\spad{x} in \\axiom{a} where \\axiom{\\spad{i} \\spad{>=} \\spad{n}},{} and \\axiom{minIndex(a) - 1} if no such \\spad{x} is found.") (((|Integer|) |#1| $) "\\spad{position(x,a)} returns the index \\spad{i} of the first occurrence of \\spad{x} in a,{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.") (((|Integer|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{position(p,a)} returns the index \\spad{i} of the first \\spad{x} in \\axiom{a} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.")) (|sorted?| (((|Boolean|) $) "\\spad{sorted?(u)} tests if the elements of \\spad{u} are in ascending order.") (((|Boolean|) (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sorted?(p,a)} tests if \\axiom{a} is sorted according to predicate \\spad{p}.")) (|sort| (($ $) "\\spad{sort(u)} returns an \\spad{u} with elements in ascending order. Note: \\axiom{sort(\\spad{u}) = sort(\\spad{<=},{}\\spad{u})}.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sort(p,a)} returns a copy of \\axiom{a} sorted using total ordering predicate \\spad{p}.")) (|reverse| (($ $) "\\spad{reverse(a)} returns a copy of \\axiom{a} with elements in reverse order.")) (|merge| (($ $ $) "\\spad{merge(u,v)} merges \\spad{u} and \\spad{v} in ascending order. Note: \\axiom{merge(\\spad{u},{}\\spad{v}) = merge(\\spad{<=},{}\\spad{u},{}\\spad{v})}.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $ $) "\\spad{merge(p,a,b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) 
-((-4427 . T)) 
+((-4434 . T)) 
 NIL 
-(-376 S A R B) 
+(-377 S A R B) 
 ((|constructor| (NIL "FiniteLinearAggregateFunctions2 provides functions involving two FiniteLinearAggregates where the underlying domains might be different. An example of this might be creating a list of rational numbers by mapping a function across a list of integers where the function divides each integer by 1000.")) (|scan| ((|#4| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-aggregates \\spad{x} of aggregrate \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}.")) (|reduce| ((|#3| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the aggregate \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as the identity element for the function \\spad{f}.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f,a)} applies function \\spad{f} to each member of aggregate \\spad{a} resulting in a new aggregate over a possibly different underlying domain."))) 
 NIL 
 NIL 
-(-377 |VarSet| R) 
+(-378 |VarSet| R) 
 ((|constructor| (NIL "The category of free Lie algebras. It is used by domains of non-commutative algebra: \\spadtype{LiePolynomial} and \\spadtype{XPBWPolynomial}. \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr})")) (|eval| (($ $ (|List| |#1|) (|List| $)) "\\axiom{eval(\\spad{p},{} [\\spad{x1},{}...,{}\\spad{xn}],{} [\\spad{v1},{}...,{}\\spad{vn}])} replaces \\axiom{\\spad{xi}} by \\axiom{\\spad{vi}} in \\axiom{\\spad{p}}.") (($ $ |#1| $) "\\axiom{eval(\\spad{p},{} \\spad{x},{} \\spad{v})} replaces \\axiom{\\spad{x}} by \\axiom{\\spad{v}} in \\axiom{\\spad{p}}.")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{x})} returns the list of distinct entries of \\axiom{\\spad{x}}.")) (|trunc| (($ $ (|NonNegativeInteger|)) "\\axiom{trunc(\\spad{p},{}\\spad{n})} returns the polynomial \\axiom{\\spad{p}} truncated at order \\axiom{\\spad{n}}.")) (|mirror| (($ $) "\\axiom{mirror(\\spad{x})} returns \\axiom{Sum(r_i mirror(w_i))} if \\axiom{\\spad{x}} is \\axiom{Sum(r_i w_i)}.")) (|LiePoly| (($ (|LyndonWord| |#1|)) "\\axiom{LiePoly(\\spad{l})} returns the bracketed form of \\axiom{\\spad{l}} as a Lie polynomial.")) (|rquo| (((|XRecursivePolynomial| |#1| |#2|) (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{rquo(\\spad{x},{}\\spad{y})} returns the right simplification of \\axiom{\\spad{x}} by \\axiom{\\spad{y}}.")) (|lquo| (((|XRecursivePolynomial| |#1| |#2|) (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{lquo(\\spad{x},{}\\spad{y})} returns the left simplification of \\axiom{\\spad{x}} by \\axiom{\\spad{y}}.")) (|degree| (((|NonNegativeInteger|) $) "\\axiom{degree(\\spad{x})} returns the greatest length of a word in the support of \\axiom{\\spad{x}}.")) (|coerce| (((|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as a recursive polynomial.") (((|XDistributedPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as distributed polynomial.") (($ |#1|) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as a Lie polynomial.")) (|coef| ((|#2| (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coef(\\spad{x},{}\\spad{y})} returns the scalar product of \\axiom{\\spad{x}} by \\axiom{\\spad{y}},{} the set of words being regarded as an orthogonal basis."))) 
-((|JacobiIdentity| . T) (|NullSquare| . T) (-4422 . T) (-4421 . T)) 
+((|JacobiIdentity| . T) (|NullSquare| . T) (-4429 . T) (-4428 . T)) 
 NIL 
-(-378 S V) 
+(-379 S V) 
 ((|constructor| (NIL "This package exports 3 sorting algorithms which work over FiniteLinearAggregates.")) (|shellSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{shellSort(f, agg)} sorts the aggregate agg with the ordering function \\spad{f} using the shellSort algorithm.")) (|heapSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{heapSort(f, agg)} sorts the aggregate agg with the ordering function \\spad{f} using the heapsort algorithm.")) (|quickSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{quickSort(f, agg)} sorts the aggregate agg with the ordering function \\spad{f} using the quicksort algorithm."))) 
 NIL 
 NIL 
-(-379 S R) 
+(-380 S R) 
 ((|constructor| (NIL "\\spad{S} is \\spadtype{FullyLinearlyExplicitRingOver R} means that \\spad{S} is a \\spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \\spad{R} is a \\spadtype{LinearlyExplicitRingOver Integer},{} then so is \\spad{S}"))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) 
-(-380 R) 
+((|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551))))) 
+(-381 R) 
 ((|constructor| (NIL "\\spad{S} is \\spadtype{FullyLinearlyExplicitRingOver R} means that \\spad{S} is a \\spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \\spad{R} is a \\spadtype{LinearlyExplicitRingOver Integer},{} then so is \\spad{S}"))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-381) 
+(-382) 
 ((|constructor| (NIL "\\spadtype{Float} implements arbitrary precision floating point arithmetic. The number of significant digits of each operation can be set to an arbitrary value (the default is 20 decimal digits). The operation \\spad{float(mantissa,exponent,\\spadfunFrom{base}{FloatingPointSystem})} for integer \\spad{mantissa},{} \\spad{exponent} specifies the number \\spad{mantissa * \\spadfunFrom{base}{FloatingPointSystem} ** exponent} The underlying representation for floats is binary not decimal. The implications of this are described below. \\blankline The model adopted is that arithmetic operations are rounded to to nearest unit in the last place,{} that is,{} accurate to within \\spad{2**(-\\spadfunFrom{bits}{FloatingPointSystem})}. Also,{} the elementary functions and constants are accurate to one unit in the last place. A float is represented as a record of two integers,{} the mantissa and the exponent. The \\spadfunFrom{base}{FloatingPointSystem} of the representation is binary,{} hence a \\spad{Record(m:mantissa,e:exponent)} represents the number \\spad{m * 2 ** e}. Though it is not assumed that the underlying integers are represented with a binary \\spadfunFrom{base}{FloatingPointSystem},{} the code will be most efficient when this is the the case (this is \\spad{true} in most implementations of Lisp). The decision to choose the \\spadfunFrom{base}{FloatingPointSystem} to be binary has some unfortunate consequences. First,{} decimal numbers like 0.3 cannot be represented exactly. Second,{} there is a further loss of accuracy during conversion to decimal for output. To compensate for this,{} if \\spad{d} digits of precision are specified,{} \\spad{1 + ceiling(log2 d)} bits are used. Two numbers that are displayed identically may therefore be not equal. On the other hand,{} a significant efficiency loss would be incurred if we chose to use a decimal \\spadfunFrom{base}{FloatingPointSystem} when the underlying integer base is binary. \\blankline Algorithms used: For the elementary functions,{} the general approach is to apply identities so that the taylor series can be used,{} and,{} so that it will converge within \\spad{O( sqrt n )} steps. For example,{} using the identity \\spad{exp(x) = exp(x/2)**2},{} we can compute \\spad{exp(1/3)} to \\spad{n} digits of precision as follows. We have \\spad{exp(1/3) = exp(2 ** (-sqrt s) / 3) ** (2 ** sqrt s)}. The taylor series will converge in less than sqrt \\spad{n} steps and the exponentiation requires sqrt \\spad{n} multiplications for a total of \\spad{2 sqrt n} multiplications. Assuming integer multiplication costs \\spad{O( n**2 )} the overall running time is \\spad{O( sqrt(n) n**2 )}. This approach is the best known approach for precisions up to about 10,{}000 digits at which point the methods of Brent which are \\spad{O( log(n) n**2 )} become competitive. Note also that summing the terms of the taylor series for the elementary functions is done using integer operations. This avoids the overhead of floating point operations and results in efficient code at low precisions. This implementation makes no attempt to reuse storage,{} relying on the underlying system to do \\spadgloss{garbage collection}. \\spad{I} estimate that the efficiency of this package at low precisions could be improved by a factor of 2 if in-place operations were available. \\blankline Running times: in the following,{} \\spad{n} is the number of bits of precision \\indented{5}{\\spad{*},{} \\spad{/},{} \\spad{sqrt},{} \\spad{pi},{} \\spad{exp1},{} \\spad{log2},{} \\spad{log10}: \\spad{ O( n**2 )}} \\indented{5}{\\spad{exp},{} \\spad{log},{} \\spad{sin},{} \\spad{atan}:\\space{2}\\spad{ O( sqrt(n) n**2 )}} The other elementary functions are coded in terms of the ones above.")) (|outputSpacing| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputSpacing(n)} inserts a space after \\spad{n} (default 10) digits on output; outputSpacing(0) means no spaces are inserted.")) (|outputGeneral| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputGeneral(n)} sets the output mode to general notation with \\spad{n} significant digits displayed.") (((|Void|)) "\\spad{outputGeneral()} sets the output mode (default mode) to general notation; numbers will be displayed in either fixed or floating (scientific) notation depending on the magnitude.")) (|outputFixed| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFixed(n)} sets the output mode to fixed point notation,{} with \\spad{n} digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFixed()} sets the output mode to fixed point notation; the output will contain a decimal point.")) (|outputFloating| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFloating(n)} sets the output mode to floating (scientific) notation with \\spad{n} significant digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFloating()} sets the output mode to floating (scientific) notation,{} \\spadignore{i.e.} \\spad{mantissa * 10 exponent} is displayed as \\spad{0.mantissa E exponent}.")) (|atan| (($ $ $) "\\spad{atan(x,y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|exp1| (($) "\\spad{exp1()} returns exp 1: \\spad{2.7182818284...}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm for \\spad{x} to base 10.") (($) "\\spad{log10()} returns \\spad{ln 10}: \\spad{2.3025809299...}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm for \\spad{x} to base 2.") (($) "\\spad{log2()} returns \\spad{ln 2},{} \\spadignore{i.e.} \\spad{0.6931471805...}.")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n, b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)},{} that is \\spad{|(r-f)/f| < b**(-n)}.") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|shift| (($ $ (|Integer|)) "\\spad{shift(x,n)} adds \\spad{n} to the exponent of float \\spad{x}.")) (|relerror| (((|Integer|) $ $) "\\spad{relerror(x,y)} computes the absolute value of \\spad{x - y} divided by \\spad{y},{} when \\spad{y \\~= 0}.")) (|normalize| (($ $) "\\spad{normalize(x)} normalizes \\spad{x} at current precision.")) (** (($ $ $) "\\spad{x ** y} computes \\spad{exp(y log x)} where \\spad{x >= 0}.")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) 
-((-4410 . T) (-4418 . T) (-4203 . T) (-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4417 . T) (-4425 . T) (-4210 . T) (-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-382 |Par|) 
+(-383 |Par|) 
 ((|constructor| (NIL "\\indented{3}{This is a package for the approximation of complex solutions for} systems of equations of rational functions with complex rational coefficients. The results are expressed as either complex rational numbers or complex floats depending on the type of the precision parameter which can be either a rational number or a floating point number.")) (|complexRoots| (((|List| (|List| (|Complex| |#1|))) (|List| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) (|List| (|Symbol|)) |#1|) "\\spad{complexRoots(lrf, lv, eps)} finds all the complex solutions of a list of rational functions with rational number coefficients with respect the the variables appearing in \\spad{lv}. Each solution is computed to precision eps and returned as list corresponding to the order of variables in \\spad{lv}.") (((|List| (|Complex| |#1|)) (|Fraction| (|Polynomial| (|Complex| (|Integer|)))) |#1|) "\\spad{complexRoots(rf, eps)} finds all the complex solutions of a univariate rational function with rational number coefficients. The solutions are computed to precision eps.")) (|complexSolve| (((|List| (|Equation| (|Polynomial| (|Complex| |#1|)))) (|Equation| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) |#1|) "\\spad{complexSolve(eq,eps)} finds all the complex solutions of the equation \\spad{eq} of rational functions with rational rational coefficients with respect to all the variables appearing in \\spad{eq},{} with precision \\spad{eps}.") (((|List| (|Equation| (|Polynomial| (|Complex| |#1|)))) (|Fraction| (|Polynomial| (|Complex| (|Integer|)))) |#1|) "\\spad{complexSolve(p,eps)} find all the complex solutions of the rational function \\spad{p} with complex rational coefficients with respect to all the variables appearing in \\spad{p},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| (|Complex| |#1|))))) (|List| (|Equation| (|Fraction| (|Polynomial| (|Complex| (|Integer|)))))) |#1|) "\\spad{complexSolve(leq,eps)} finds all the complex solutions to precision \\spad{eps} of the system \\spad{leq} of equations of rational functions over complex rationals with respect to all the variables appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Polynomial| (|Complex| |#1|))))) (|List| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) |#1|) "\\spad{complexSolve(lp,eps)} finds all the complex solutions to precision \\spad{eps} of the system \\spad{lp} of rational functions over the complex rationals with respect to all the variables appearing in \\spad{lp}."))) 
 NIL 
 NIL 
-(-383 |Par|) 
+(-384 |Par|) 
 ((|constructor| (NIL "\\indented{3}{This is a package for the approximation of real solutions for} systems of polynomial equations over the rational numbers. The results are expressed as either rational numbers or floats depending on the type of the precision parameter which can be either a rational number or a floating point number.")) (|realRoots| (((|List| |#1|) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{realRoots(rf, eps)} finds the real zeros of a univariate rational function with precision given by eps.") (((|List| (|List| |#1|)) (|List| (|Fraction| (|Polynomial| (|Integer|)))) (|List| (|Symbol|)) |#1|) "\\spad{realRoots(lp,lv,eps)} computes the list of the real solutions of the list \\spad{lp} of rational functions with rational coefficients with respect to the variables in \\spad{lv},{} with precision \\spad{eps}. Each solution is expressed as a list of numbers in order corresponding to the variables in \\spad{lv}.")) (|solve| (((|List| (|Equation| (|Polynomial| |#1|))) (|Equation| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(eq,eps)} finds all of the real solutions of the univariate equation \\spad{eq} of rational functions with respect to the unique variables appearing in \\spad{eq},{} with precision \\spad{eps}.") (((|List| (|Equation| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{solve(p,eps)} finds all of the real solutions of the univariate rational function \\spad{p} with rational coefficients with respect to the unique variable appearing in \\spad{p},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| (|Integer|))))) |#1|) "\\spad{solve(leq,eps)} finds all of the real solutions of the system \\spad{leq} of equationas of rational functions with respect to all the variables appearing in \\spad{lp},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(lp,eps)} finds all of the real solutions of the system \\spad{lp} of rational functions over the rational numbers with respect to all the variables appearing in \\spad{lp},{} with precision \\spad{eps}."))) 
 NIL 
 NIL 
-(-384 R S) 
+(-385 R S) 
 ((|constructor| (NIL "A \\spad{bi}-module is a free module over a ring with generators indexed by an ordered set. Each element can be expressed as a finite linear combination of generators. Only non-zero terms are stored."))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 ((|HasCategory| |#1| (QUOTE (-173)))) 
-(-385 R S) 
+(-386 R S) 
 ((|constructor| (NIL "This domain implements linear combinations of elements from the domain \\spad{S} with coefficients in the domain \\spad{R} where \\spad{S} is an ordered set and \\spad{R} is a ring (which may be non-commutative). This domain is used by domains of non-commutative algebra such as: \\indented{4}{\\spadtype{XDistributedPolynomial},{}} \\indented{4}{\\spadtype{XRecursivePolynomial}.} Author: Michel Petitot (petitot@lifl.\\spad{fr})")) (* (($ |#2| |#1|) "\\spad{s*r} returns the product \\spad{r*s} used by \\spadtype{XRecursivePolynomial}"))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 ((|HasCategory| |#1| (QUOTE (-173)))) 
-(-386) 
+(-387) 
 ((|constructor| (NIL "\\axiomType{FortranMatrixCategory} provides support for producing Functions and Subroutines when the input to these is an AXIOM object of type \\axiomType{Matrix} or in domains involving \\axiomType{FortranCode}.")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|Matrix| (|MachineFloat|))) "\\spad{coerce(v)} produces an ASP which returns the value of \\spad{v}."))) 
 NIL 
 NIL 
-(-387 R |Basis|) 
+(-388 R |Basis|) 
 ((|constructor| (NIL "A domain of this category implements formal linear combinations of elements from a domain \\spad{Basis} with coefficients in a domain \\spad{R}. The domain \\spad{Basis} needs only to belong to the category \\spadtype{SetCategory} and \\spad{R} to the category \\spadtype{Ring}. Thus the coefficient ring may be non-commutative. See the \\spadtype{XDistributedPolynomial} constructor for examples of domains built with the \\spadtype{FreeModuleCat} category constructor. Author: Michel Petitot (petitot@lifl.\\spad{fr})")) (|reductum| (($ $) "\\spad{reductum(x)} returns \\spad{x} minus its leading term.")) (|leadingTerm| (((|Record| (|:| |k| |#2|) (|:| |c| |#1|)) $) "\\spad{leadingTerm(x)} returns the first term which appears in \\spad{ListOfTerms(x)}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(x)} returns the first coefficient which appears in \\spad{ListOfTerms(x)}.")) (|leadingMonomial| ((|#2| $) "\\spad{leadingMonomial(x)} returns the first element from \\spad{Basis} which appears in \\spad{ListOfTerms(x)}.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(x)} returns the number of monomials of \\spad{x}.")) (|monomials| (((|List| $) $) "\\spad{monomials(x)} returns the list of \\spad{r_i*b_i} whose sum is \\spad{x}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(x)} returns the list of coefficients of \\spad{x}.")) (|ListOfTerms| (((|List| (|Record| (|:| |k| |#2|) (|:| |c| |#1|))) $) "\\spad{ListOfTerms(x)} returns a list \\spad{lt} of terms with type \\spad{Record(k: Basis, c: R)} such that \\spad{x} equals \\spad{reduce(+, map(x +-> monom(x.k, x.c), lt))}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} returns \\spad{true} if \\spad{x} contains a single monomial.")) (|monom| (($ |#2| |#1|) "\\spad{monom(b,r)} returns the element with the single monomial \\indented{1}{\\spad{b} and coefficient \\spad{r}.}")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients \\indented{1}{of the non-zero monomials of \\spad{u}.}")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(x,b)} returns the coefficient of \\spad{b} in \\spad{x}.")) (* (($ |#1| |#2|) "\\spad{r*b} returns the product of \\spad{r} by \\spad{b}."))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 NIL 
-(-388) 
+(-389) 
 ((|constructor| (NIL "\\axiomType{FortranMatrixFunctionCategory} provides support for producing Functions and Subroutines representing matrices of expressions.")) (|retractIfCan| (((|Union| $ "failed") (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Fraction| (|Polynomial| (|Float|))))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Polynomial| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Polynomial| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Expression| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Matrix| (|Expression| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|retract| (($ (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Fraction| (|Polynomial| (|Float|))))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Polynomial| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Polynomial| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Expression| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Matrix| (|Expression| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}"))) 
 NIL 
 NIL 
-(-389 S) 
+(-390 S) 
 ((|constructor| (NIL "A free monoid on a set \\spad{S} is the monoid of finite products of the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are nonnegative integers. The multiplication is not commutative.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| (|NonNegativeInteger|) (|NonNegativeInteger|)) $) "\\spad{mapExpon(f, a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x, n)} returns the factor of the n^th monomial of \\spad{x}.")) (|nthExpon| (((|NonNegativeInteger|) $ (|Integer|)) "\\spad{nthExpon(x, n)} returns the exponent of the n^th monomial of \\spad{x}.")) (|factors| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| (|NonNegativeInteger|)))) $) "\\spad{factors(a1\\^e1,...,an\\^en)} returns \\spad{[[a1, e1],...,[an, en]]}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of monomials in \\spad{x}.")) (|overlap| (((|Record| (|:| |lm| $) (|:| |mm| $) (|:| |rm| $)) $ $) "\\spad{overlap(x, y)} returns \\spad{[l, m, r]} such that \\spad{x = l * m},{} \\spad{y = m * r} and \\spad{l} and \\spad{r} have no overlap,{} \\spadignore{i.e.} \\spad{overlap(l, r) = [l, 1, r]}.")) (|divide| (((|Union| (|Record| (|:| |lm| $) (|:| |rm| $)) "failed") $ $) "\\spad{divide(x, y)} returns the left and right exact quotients of \\spad{x} by \\spad{y},{} \\spadignore{i.e.} \\spad{[l, r]} such that \\spad{x = l * y * r},{} \"failed\" if \\spad{x} is not of the form \\spad{l * y * r}.")) (|rquo| (((|Union| $ "failed") $ $) "\\spad{rquo(x, y)} returns the exact right quotient of \\spad{x} by \\spad{y} \\spadignore{i.e.} \\spad{q} such that \\spad{x = q * y},{} \"failed\" if \\spad{x} is not of the form \\spad{q * y}.")) (|lquo| (((|Union| $ "failed") $ $) "\\spad{lquo(x, y)} returns the exact left quotient of \\spad{x} by \\spad{y} \\spadignore{i.e.} \\spad{q} such that \\spad{x = y * q},{} \"failed\" if \\spad{x} is not of the form \\spad{y * q}.")) (|hcrf| (($ $ $) "\\spad{hcrf(x, y)} returns the highest common right factor of \\spad{x} and \\spad{y},{} \\spadignore{i.e.} the largest \\spad{d} such that \\spad{x = a d} and \\spad{y = b d}.")) (|hclf| (($ $ $) "\\spad{hclf(x, y)} returns the highest common left factor of \\spad{x} and \\spad{y},{} \\spadignore{i.e.} the largest \\spad{d} such that \\spad{x = d a} and \\spad{y = d b}.")) (** (($ |#1| (|NonNegativeInteger|)) "\\spad{s ** n} returns the product of \\spad{s} by itself \\spad{n} times.")) (* (($ $ |#1|) "\\spad{x * s} returns the product of \\spad{x} by \\spad{s} on the right.") (($ |#1| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) 
 NIL 
 NIL 
-(-390 S) 
+(-391 S) 
 ((|constructor| (NIL "The free monoid on a set \\spad{S} is the monoid of finite products of the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are nonnegative integers. The multiplication is not commutative."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-853)))) 
-(-391) 
+((|HasCategory| |#1| (QUOTE (-855)))) 
+(-392) 
 ((|constructor| (NIL "A category of domains which model machine arithmetic used by machines in the AXIOM-NAG link."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-392) 
+(-393) 
 ((|constructor| (NIL "This domain provides an interface to names in the file system."))) 
 NIL 
 NIL 
-(-393) 
+(-394) 
 ((|constructor| (NIL "This category provides an interface to names in the file system.")) (|new| (($ (|String|) (|String|) (|String|)) "\\spad{new(d,pref,e)} constructs the name of a new writable file with \\spad{d} as its directory,{} \\spad{pref} as a prefix of its name and \\spad{e} as its extension. When \\spad{d} or \\spad{t} is the empty string,{} a default is used. An error occurs if a new file cannot be written in the given directory.")) (|writable?| (((|Boolean|) $) "\\spad{writable?(f)} tests if the named file be opened for writing. The named file need not already exist.")) (|readable?| (((|Boolean|) $) "\\spad{readable?(f)} tests if the named file exist and can it be opened for reading.")) (|exists?| (((|Boolean|) $) "\\spad{exists?(f)} tests if the file exists in the file system.")) (|extension| (((|String|) $) "\\spad{extension(f)} returns the type part of the file name.")) (|name| (((|String|) $) "\\spad{name(f)} returns the name part of the file name.")) (|directory| (((|String|) $) "\\spad{directory(f)} returns the directory part of the file name.")) (|filename| (($ (|String|) (|String|) (|String|)) "\\spad{filename(d,n,e)} creates a file name with \\spad{d} as its directory,{} \\spad{n} as its name and \\spad{e} as its extension. This is a portable way to create file names. When \\spad{d} or \\spad{t} is the empty string,{} a default is used."))) 
 NIL 
 NIL 
-(-394 |n| |class| R) 
+(-395 |n| |class| R) 
 ((|constructor| (NIL "Generate the Free Lie Algebra over a ring \\spad{R} with identity; A \\spad{P}. Hall basis is generated by a package call to HallBasis.")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(i)} is the \\spad{i}th Hall Basis element")) (|shallowExpand| (((|OutputForm|) $) "\\spad{shallowExpand(x)} \\undocumented{}")) (|deepExpand| (((|OutputForm|) $) "\\spad{deepExpand(x)} \\undocumented{}")) (|dimension| (((|NonNegativeInteger|)) "\\spad{dimension()} is the rank of this Lie algebra"))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 NIL 
-(-395) 
+(-396) 
 ((|constructor| (NIL "Code to manipulate Fortran Output Stack")) (|topFortranOutputStack| (((|String|)) "\\spad{topFortranOutputStack()} returns the top element of the Fortran output stack")) (|pushFortranOutputStack| (((|Void|) (|String|)) "\\spad{pushFortranOutputStack(f)} pushes \\spad{f} onto the Fortran output stack") (((|Void|) (|FileName|)) "\\spad{pushFortranOutputStack(f)} pushes \\spad{f} onto the Fortran output stack")) (|popFortranOutputStack| (((|Void|)) "\\spad{popFortranOutputStack()} pops the Fortran output stack")) (|showFortranOutputStack| (((|Stack| (|String|))) "\\spad{showFortranOutputStack()} returns the Fortran output stack")) (|clearFortranOutputStack| (((|Stack| (|String|))) "\\spad{clearFortranOutputStack()} clears the Fortran output stack"))) 
 NIL 
 NIL 
-(-396 -3498 UP UPUP R) 
+(-397 -3505 UP UPUP R) 
 ((|constructor| (NIL "\\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 11 Jul 1990")) (|order| (((|NonNegativeInteger|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{order(x)} \\undocumented"))) 
 NIL 
 NIL 
-(-397) 
+(-398) 
 ((|constructor| (NIL "\\spadtype{ScriptFormulaFormat} provides a coercion from \\spadtype{OutputForm} to IBM SCRIPT/VS Mathematical Formula Format. The basic SCRIPT formula format object consists of three parts: a prologue,{} a formula part and an epilogue. The functions \\spadfun{prologue},{} \\spadfun{formula} and \\spadfun{epilogue} extract these parts,{} respectively. The central parts of the expression go into the formula part. The other parts can be set (\\spadfun{setPrologue!},{} \\spadfun{setEpilogue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain \":df.\" and \":edf.\" so that the formula section will be printed in display math mode.")) (|setPrologue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setPrologue!(t,strings)} sets the prologue section of a formatted object \\spad{t} to \\spad{strings}.")) (|setFormula!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setFormula!(t,strings)} sets the formula section of a formatted object \\spad{t} to \\spad{strings}.")) (|setEpilogue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setEpilogue!(t,strings)} sets the epilogue section of a formatted object \\spad{t} to \\spad{strings}.")) (|prologue| (((|List| (|String|)) $) "\\spad{prologue(t)} extracts the prologue section of a formatted object \\spad{t}.")) (|new| (($) "\\spad{new()} create a new,{} empty object. Use \\spadfun{setPrologue!},{} \\spadfun{setFormula!} and \\spadfun{setEpilogue!} to set the various components of this object.")) (|formula| (((|List| (|String|)) $) "\\spad{formula(t)} extracts the formula section of a formatted object \\spad{t}.")) (|epilogue| (((|List| (|String|)) $) "\\spad{epilogue(t)} extracts the epilogue section of a formatted object \\spad{t}.")) (|display| (((|Void|) $) "\\spad{display(t)} outputs the formatted code \\spad{t} so that each line has length less than or equal to the value set by the system command \\spadsyscom{set output length}.") (((|Void|) $ (|Integer|)) "\\spad{display(t,width)} outputs the formatted code \\spad{t} so that each line has length less than or equal to \\spadvar{\\spad{width}}.")) (|convert| (($ (|OutputForm|) (|Integer|)) "\\spad{convert(o,step)} changes \\spad{o} in standard output format to SCRIPT formula format and also adds the given \\spad{step} number. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \\spad{step} numbers."))) 
 NIL 
 NIL 
-(-398 S) 
+(-399 S) 
 ((|constructor| (NIL "\\spadtype{ScriptFormulaFormat1} provides a utility coercion for changing to SCRIPT formula format anything that has a coercion to the standard output format.")) (|coerce| (((|ScriptFormulaFormat|) |#1|) "\\spad{coerce(s)} provides a direct coercion from an expression \\spad{s} of domain \\spad{S} to SCRIPT formula format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to SCRIPT formula format."))) 
 NIL 
 NIL 
-(-399) 
+(-400) 
 ((|constructor| (NIL "provides an interface to the boot code for calling Fortran")) (|setLegalFortranSourceExtensions| (((|List| (|String|)) (|List| (|String|))) "\\spad{setLegalFortranSourceExtensions(l)} \\undocumented{}")) (|outputAsFortran| (((|Void|) (|FileName|)) "\\spad{outputAsFortran(fn)} \\undocumented{}")) (|linkToFortran| (((|SExpression|) (|Symbol|) (|List| (|Symbol|)) (|TheSymbolTable|) (|List| (|Symbol|))) "\\spad{linkToFortran(s,l,t,lv)} \\undocumented{}") (((|SExpression|) (|Symbol|) (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|)))) (|List| (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|))))) (|List| (|Symbol|)) (|Symbol|)) "\\spad{linkToFortran(s,l,ll,lv,t)} \\undocumented{}") (((|SExpression|) (|Symbol|) (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|)))) (|List| (|List| (|Union| (|:| |array| (|List| (|Symbol|))) (|:| |scalar| (|Symbol|))))) (|List| (|Symbol|))) "\\spad{linkToFortran(s,l,ll,lv)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-400) 
+(-401) 
 ((|constructor| (NIL "\\axiomType{FortranProgramCategory} provides various models of FORTRAN subprograms. These can be transformed into actual FORTRAN code.")) (|outputAsFortran| (((|Void|) $) "\\axiom{outputAsFortran(\\spad{u})} translates \\axiom{\\spad{u}} into a legal FORTRAN subprogram."))) 
 NIL 
 NIL 
-(-401) 
+(-402) 
 ((|constructor| (NIL "\\axiomType{FortranFunctionCategory} is the category of arguments to NAG Library routines which return (sets of) function values.")) (|retractIfCan| (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Polynomial| (|Float|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Expression| (|Integer|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Expression| (|Float|))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|retract| (($ (|Fraction| (|Polynomial| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Fraction| (|Polynomial| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Polynomial| (|Integer|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Polynomial| (|Float|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Expression| (|Integer|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Expression| (|Float|))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}"))) 
 NIL 
 NIL 
-(-402 -3975 |returnType| -1509 |symbols|) 
+(-403 -3982 |returnType| -1512 |symbols|) 
 ((|constructor| (NIL "\\axiomType{FortranProgram} allows the user to build and manipulate simple models of FORTRAN subprograms. These can then be transformed into actual FORTRAN notation.")) (|coerce| (($ (|Equation| (|Expression| (|Complex| (|Float|))))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|Float|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|Integer|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Expression| (|Complex| (|Float|)))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|Float|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|Integer|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Equation| (|Expression| (|MachineComplex|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|MachineFloat|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Equation| (|Expression| (|MachineInteger|)))) "\\spad{coerce(eq)} \\undocumented{}") (($ (|Expression| (|MachineComplex|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|MachineFloat|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Expression| (|MachineInteger|))) "\\spad{coerce(e)} \\undocumented{}") (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(r)} \\undocumented{}") (($ (|List| (|FortranCode|))) "\\spad{coerce(lfc)} \\undocumented{}") (($ (|FortranCode|)) "\\spad{coerce(fc)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-403 -3498 UP) 
+(-404 -3505 UP) 
 ((|constructor| (NIL "\\indented{1}{Full partial fraction expansion of rational functions} Author: Manuel Bronstein Date Created: 9 December 1992 Date Last Updated: 6 October 1993 References: \\spad{M}.Bronstein & \\spad{B}.Salvy,{} \\indented{12}{Full Partial Fraction Decomposition of Rational Functions,{}} \\indented{12}{in Proceedings of ISSAC'93,{} Kiev,{} ACM Press.}")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(f, n)} returns the \\spad{n}-th derivative of \\spad{f}.") (($ $) "\\spad{D(f)} returns the derivative of \\spad{f}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(f, n)} returns the \\spad{n}-th derivative of \\spad{f}.") (($ $) "\\spad{differentiate(f)} returns the derivative of \\spad{f}.")) (|construct| (($ (|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|)))) "\\spad{construct(l)} is the inverse of fracPart.")) (|fracPart| (((|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|))) $) "\\spad{fracPart(f)} returns the list of summands of the fractional part of \\spad{f}.")) (|polyPart| ((|#2| $) "\\spad{polyPart(f)} returns the polynomial part of \\spad{f}.")) (|fullPartialFraction| (($ (|Fraction| |#2|)) "\\spad{fullPartialFraction(f)} returns \\spad{[p, [[j, Dj, Hj]...]]} such that \\spad{f = p(x) + \\sum_{[j,Dj,Hj] in l} \\sum_{Dj(a)=0} Hj(a)/(x - a)\\^j}.")) (+ (($ |#2| $) "\\spad{p + x} returns the sum of \\spad{p} and \\spad{x}"))) 
 NIL 
 NIL 
-(-404 R) 
+(-405 R) 
 ((|constructor| (NIL "A set \\spad{S} is PatternMatchable over \\spad{R} if \\spad{S} can lift the pattern-matching functions of \\spad{S} over the integers and float to itself (necessary for matching in towers)."))) 
 NIL 
 NIL 
-(-405 S) 
+(-406 S) 
 ((|constructor| (NIL "FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \\spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.")) (|primeFrobenius| (($ $ (|NonNegativeInteger|)) "\\spad{primeFrobenius(a,s)} returns \\spad{a**(p**s)} where \\spad{p} is the characteristic.") (($ $) "\\spad{primeFrobenius(a)} returns \\spad{a ** p} where \\spad{p} is the characteristic.")) (|discreteLog| (((|Union| (|NonNegativeInteger|) "failed") $ $) "\\spad{discreteLog(b,a)} computes \\spad{s} with \\spad{b**s = a} if such an \\spad{s} exists.")) (|order| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \\spad{a} is 0."))) 
 NIL 
 NIL 
-(-406) 
+(-407) 
 ((|constructor| (NIL "FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \\spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.")) (|primeFrobenius| (($ $ (|NonNegativeInteger|)) "\\spad{primeFrobenius(a,s)} returns \\spad{a**(p**s)} where \\spad{p} is the characteristic.") (($ $) "\\spad{primeFrobenius(a)} returns \\spad{a ** p} where \\spad{p} is the characteristic.")) (|discreteLog| (((|Union| (|NonNegativeInteger|) "failed") $ $) "\\spad{discreteLog(b,a)} computes \\spad{s} with \\spad{b**s = a} if such an \\spad{s} exists.")) (|order| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \\spad{a} is 0."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-407 S) 
+(-408 S) 
 ((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline \\indented{2}{1: \\spadfunFrom{base}{FloatingPointSystem} of the \\spadfunFrom{exponent}{FloatingPointSystem}.} \\indented{9}{(actual implemenations are usually binary or decimal)} \\indented{2}{2: \\spadfunFrom{precision}{FloatingPointSystem} of the \\spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \\indented{2}{3: rounding error for operations} \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\\spad{\"+\"}) does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\spad{max()} returns the maximum floating point number.")) (|min| (($) "\\spad{min()} returns the minimum floating point number.")) (|decreasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{decreasePrecision(n)} decreases the current \\spadfunFrom{precision}{FloatingPointSystem} precision by \\spad{n} decimal digits.")) (|increasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{increasePrecision(n)} increases the current \\spadfunFrom{precision}{FloatingPointSystem} by \\spad{n} decimal digits.")) (|precision| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(n)} set the precision in the base to \\spad{n} decimal digits.") (((|PositiveInteger|)) "\\spad{precision()} returns the precision in digits base.")) (|digits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{digits(d)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{d} digits.") (((|PositiveInteger|)) "\\spad{digits()} returns ceiling\\spad{'s} precision in decimal digits.")) (|bits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{bits(n)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{n} bits.") (((|PositiveInteger|)) "\\spad{bits()} returns ceiling\\spad{'s} precision in bits.")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(x)} returns the mantissa part of \\spad{x}.")) (|exponent| (((|Integer|) $) "\\spad{exponent(x)} returns the \\spadfunFrom{exponent}{FloatingPointSystem} part of \\spad{x}.")) (|base| (((|PositiveInteger|)) "\\spad{base()} returns the base of the \\spadfunFrom{exponent}{FloatingPointSystem}.")) (|order| (((|Integer|) $) "\\spad{order x} is the order of magnitude of \\spad{x}. Note: \\spad{base ** order x <= |x| < base ** (1 + order x)}.")) (|float| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{float(a,e,b)} returns \\spad{a * b ** e}.") (($ (|Integer|) (|Integer|)) "\\spad{float(a,e)} returns \\spad{a * base() ** e}.")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\"."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4410)) (|HasAttribute| |#1| (QUOTE -4418))) 
-(-408) 
+((|HasAttribute| |#1| (QUOTE -4417)) (|HasAttribute| |#1| (QUOTE -4425))) 
+(-409) 
 ((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline \\indented{2}{1: \\spadfunFrom{base}{FloatingPointSystem} of the \\spadfunFrom{exponent}{FloatingPointSystem}.} \\indented{9}{(actual implemenations are usually binary or decimal)} \\indented{2}{2: \\spadfunFrom{precision}{FloatingPointSystem} of the \\spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \\indented{2}{3: rounding error for operations} \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\\spad{\"+\"}) does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\spad{max()} returns the maximum floating point number.")) (|min| (($) "\\spad{min()} returns the minimum floating point number.")) (|decreasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{decreasePrecision(n)} decreases the current \\spadfunFrom{precision}{FloatingPointSystem} precision by \\spad{n} decimal digits.")) (|increasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{increasePrecision(n)} increases the current \\spadfunFrom{precision}{FloatingPointSystem} by \\spad{n} decimal digits.")) (|precision| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(n)} set the precision in the base to \\spad{n} decimal digits.") (((|PositiveInteger|)) "\\spad{precision()} returns the precision in digits base.")) (|digits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{digits(d)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{d} digits.") (((|PositiveInteger|)) "\\spad{digits()} returns ceiling\\spad{'s} precision in decimal digits.")) (|bits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{bits(n)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{n} bits.") (((|PositiveInteger|)) "\\spad{bits()} returns ceiling\\spad{'s} precision in bits.")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(x)} returns the mantissa part of \\spad{x}.")) (|exponent| (((|Integer|) $) "\\spad{exponent(x)} returns the \\spadfunFrom{exponent}{FloatingPointSystem} part of \\spad{x}.")) (|base| (((|PositiveInteger|)) "\\spad{base()} returns the base of the \\spadfunFrom{exponent}{FloatingPointSystem}.")) (|order| (((|Integer|) $) "\\spad{order x} is the order of magnitude of \\spad{x}. Note: \\spad{base ** order x <= |x| < base ** (1 + order x)}.")) (|float| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{float(a,e,b)} returns \\spad{a * b ** e}.") (($ (|Integer|) (|Integer|)) "\\spad{float(a,e)} returns \\spad{a * base() ** e}.")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\"."))) 
-((-4203 . T) (-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4210 . T) (-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-409 R) 
+(-410 R) 
 ((|constructor| (NIL "\\spadtype{Factored} creates a domain whose objects are kept in factored form as long as possible. Thus certain operations like multiplication and \\spad{gcd} are relatively easy to do. Others,{} like addition require somewhat more work,{} and unless the argument domain provides a factor function,{} the result may not be completely factored. Each object consists of a unit and a list of factors,{} where a factor has a member of \\spad{R} (the \"base\"),{} and exponent and a flag indicating what is known about the base. A flag may be one of \"nil\",{} \"sqfr\",{} \"irred\" or \"prime\",{} which respectively mean that nothing is known about the base,{} it is square-free,{} it is irreducible,{} or it is prime. The current restriction to integral domains allows simplification to be performed without worrying about multiplication order.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(u)} returns a rational number if \\spad{u} really is one,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(u)} assumes spadvar{\\spad{u}} is actually a rational number and does the conversion to rational number (see \\spadtype{Fraction Integer}).")) (|rational?| (((|Boolean|) $) "\\spad{rational?(u)} tests if \\spadvar{\\spad{u}} is actually a rational number (see \\spadtype{Fraction Integer}).")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps the function \\userfun{\\spad{fn}} across the factors of \\spadvar{\\spad{u}} and creates a new factored object. Note: this clears the information flags (sets them to \"nil\") because the effect of \\userfun{\\spad{fn}} is clearly not known in general.")) (|unitNormalize| (($ $) "\\spad{unitNormalize(u)} normalizes the unit part of the factorization. For example,{} when working with factored integers,{} this operation will ensure that the bases are all positive integers.")) (|unit| ((|#1| $) "\\spad{unit(u)} extracts the unit part of the factorization.")) (|flagFactor| (($ |#1| (|Integer|) (|Union| #1="nil" #2="sqfr" #3="irred" #4="prime")) "\\spad{flagFactor(base,exponent,flag)} creates a factored object with a single factor whose \\spad{base} is asserted to be properly described by the information \\spad{flag}.")) (|sqfrFactor| (($ |#1| (|Integer|)) "\\spad{sqfrFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be square-free (flag = \"sqfr\").")) (|primeFactor| (($ |#1| (|Integer|)) "\\spad{primeFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be prime (flag = \"prime\").")) (|numberOfFactors| (((|NonNegativeInteger|) $) "\\spad{numberOfFactors(u)} returns the number of factors in \\spadvar{\\spad{u}}.")) (|nthFlag| (((|Union| #1# #2# #3# #4#) $ (|Integer|)) "\\spad{nthFlag(u,n)} returns the information flag of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} \"nil\" is returned.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(u,n)} returns the base of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 1 is returned. If \\spadvar{\\spad{u}} consists only of a unit,{} the unit is returned.")) (|nthExponent| (((|Integer|) $ (|Integer|)) "\\spad{nthExponent(u,n)} returns the exponent of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 0 is returned.")) (|irreducibleFactor| (($ |#1| (|Integer|)) "\\spad{irreducibleFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be irreducible (flag = \"irred\").")) (|factors| (((|List| (|Record| (|:| |factor| |#1|) (|:| |exponent| (|Integer|)))) $) "\\spad{factors(u)} returns a list of the factors in a form suitable for iteration. That is,{} it returns a list where each element is a record containing a base and exponent. The original object is the product of all the factors and the unit (which can be extracted by \\axiom{unit(\\spad{u})}).")) (|nilFactor| (($ |#1| (|Integer|)) "\\spad{nilFactor(base,exponent)} creates a factored object with a single factor with no information about the kind of \\spad{base} (flag = \"nil\").")) (|factorList| (((|List| (|Record| (|:| |flg| (|Union| #1# #2# #3# #4#)) (|:| |fctr| |#1|) (|:| |xpnt| (|Integer|)))) $) "\\spad{factorList(u)} returns the list of factors with flags (for use by factoring code).")) (|makeFR| (($ |#1| (|List| (|Record| (|:| |flg| (|Union| #1# #2# #3# #4#)) (|:| |fctr| |#1|) (|:| |xpnt| (|Integer|))))) "\\spad{makeFR(unit,listOfFactors)} creates a factored object (for use by factoring code).")) (|exponent| (((|Integer|) $) "\\spad{exponent(u)} returns the exponent of the first factor of \\spadvar{\\spad{u}},{} or 0 if the factored form consists solely of a unit.")) (|expand| ((|#1| $) "\\spad{expand(f)} multiplies the unit and factors together,{} yielding an \"unfactored\" object. Note: this is purposely not called \\spadfun{coerce} which would cause the interpreter to do this automatically."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
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-(-410 R S) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -519) (QUOTE (-1183)) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -312) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -289) (QUOTE $) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (QUOTE (-1227))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-1227)))) (|HasCategory| |#1| (QUOTE (-1026))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -289) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-457)))) 
+(-411 R S) 
 ((|constructor| (NIL "\\spadtype{FactoredFunctions2} contains functions that involve factored objects whose underlying domains may not be the same. For example,{} \\spadfun{map} might be used to coerce an object of type \\spadtype{Factored(Integer)} to \\spadtype{Factored(Complex(Integer))}.")) (|map| (((|Factored| |#2|) (|Mapping| |#2| |#1|) (|Factored| |#1|)) "\\spad{map(fn,u)} is used to apply the function \\userfun{\\spad{fn}} to every factor of \\spadvar{\\spad{u}}. The new factored object will have all its information flags set to \"nil\". This function is used,{} for example,{} to coerce every factor base to another type."))) 
 NIL 
 NIL 
-(-411 S) 
+(-412 S) 
 ((|constructor| (NIL "Fraction takes an IntegralDomain \\spad{S} and produces the domain of Fractions with numerators and denominators from \\spad{S}. If \\spad{S} is also a GcdDomain,{} then \\spad{gcd}\\spad{'s} between numerator and denominator will be cancelled during all operations.")) (|canonical| ((|attribute|) "\\spad{canonical} means that equal elements are in fact identical."))) 
-((-4414 -12 (|has| |#1| (-6 -4425)) (|has| |#1| (-456)) (|has| |#1| (-6 -4414))) (-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-549))) (|HasCategory| |#1| (QUOTE (-824)))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#1| (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-823))) (-3962 (|HasCategory| |#1| (QUOTE (-823))) (|HasCategory| |#1| (QUOTE (-853)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-549))) (|HasCategory| |#1| (QUOTE (-824)))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-1155))) (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-381)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-549))) (|HasCategory| |#1| (QUOTE (-824)))) (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-549))) (|HasCategory| |#1| (QUOTE (-824)))) (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-549))) (|HasCategory| |#1| (QUOTE (-824)))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#1| (LIST (QUOTE -518) (QUOTE (-1181)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -288) (|devaluate| |#1|) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-549))) (|HasCategory| |#1| (QUOTE (-824)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-549))) (-12 (|HasAttribute| |#1| (QUOTE -4414)) (|HasAttribute| |#1| (QUOTE -4425)) (|HasCategory| |#1| (QUOTE (-456)))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
-(-412 A B) 
+((-4421 -12 (|has| |#1| (-6 -4432)) (|has| |#1| (-457)) (|has| |#1| (-6 -4421))) (-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-826)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (QUOTE (-1026))) (|HasCategory| |#1| (QUOTE (-825))) (-3969 (|HasCategory| |#1| (QUOTE (-825))) (|HasCategory| |#1| (QUOTE (-855)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-826)))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-1157))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-826)))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-826)))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-826)))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -289) (|devaluate| |#1|) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-826)))) (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-550))) (-12 (|HasAttribute| |#1| (QUOTE -4421)) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(-413 A B) 
 ((|constructor| (NIL "This package extends a map between integral domains to a map between Fractions over those domains by applying the map to the numerators and denominators.")) (|map| (((|Fraction| |#2|) (|Mapping| |#2| |#1|) (|Fraction| |#1|)) "\\spad{map(func,frac)} applies the function \\spad{func} to the numerator and denominator of the fraction \\spad{frac}."))) 
 NIL 
 NIL 
-(-413 S R UP) 
+(-414 S R UP) 
 ((|constructor| (NIL "A \\spadtype{FramedAlgebra} is a \\spadtype{FiniteRankAlgebra} together with a fixed \\spad{R}-module basis.")) (|regularRepresentation| (((|Matrix| |#2|) $) "\\spad{regularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed basis.")) (|discriminant| ((|#2|) "\\spad{discriminant()} = determinant(traceMatrix()).")) (|traceMatrix| (((|Matrix| |#2|)) "\\spad{traceMatrix()} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr(vi * vj)} ),{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.")) (|convert| (($ (|Vector| |#2|)) "\\spad{convert([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.") (((|Vector| |#2|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) 
 NIL 
 NIL 
-(-414 R UP) 
+(-415 R UP) 
 ((|constructor| (NIL "A \\spadtype{FramedAlgebra} is a \\spadtype{FiniteRankAlgebra} together with a fixed \\spad{R}-module basis.")) (|regularRepresentation| (((|Matrix| |#1|) $) "\\spad{regularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed basis.")) (|discriminant| ((|#1|) "\\spad{discriminant()} = determinant(traceMatrix()).")) (|traceMatrix| (((|Matrix| |#1|)) "\\spad{traceMatrix()} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr(vi * vj)} ),{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.")) (|convert| (($ (|Vector| |#1|)) "\\spad{convert([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.") (((|Vector| |#1|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}\\spad{'s} with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-415 A S) 
+(-416 A S) 
 ((|constructor| (NIL "\\indented{2}{A is fully retractable to \\spad{B} means that A is retractable to \\spad{B},{} and,{}} \\indented{2}{in addition,{} if \\spad{B} is retractable to the integers or rational} \\indented{2}{numbers then so is A.} \\indented{2}{In particular,{} what we are asserting is that there are no integers} \\indented{2}{(rationals) in A which don\\spad{'t} retract into \\spad{B}.} Date Created: March 1990 Date Last Updated: 9 April 1991"))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) 
-(-416 S) 
+((|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551))))) 
+(-417 S) 
 ((|constructor| (NIL "\\indented{2}{A is fully retractable to \\spad{B} means that A is retractable to \\spad{B},{} and,{}} \\indented{2}{in addition,{} if \\spad{B} is retractable to the integers or rational} \\indented{2}{numbers then so is A.} \\indented{2}{In particular,{} what we are asserting is that there are no integers} \\indented{2}{(rationals) in A which don\\spad{'t} retract into \\spad{B}.} Date Created: March 1990 Date Last Updated: 9 April 1991"))) 
 NIL 
 NIL 
-(-417 R -3498 UP A) 
+(-418 R -3505 UP A) 
 ((|constructor| (NIL "Fractional ideals in a framed algebra.")) (|randomLC| ((|#4| (|NonNegativeInteger|) (|Vector| |#4|)) "\\spad{randomLC(n,x)} should be local but conditional.")) (|minimize| (($ $) "\\spad{minimize(I)} returns a reduced set of generators for \\spad{I}.")) (|denom| ((|#1| $) "\\spad{denom(1/d * (f1,...,fn))} returns \\spad{d}.")) (|numer| (((|Vector| |#4|) $) "\\spad{numer(1/d * (f1,...,fn))} = the vector \\spad{[f1,...,fn]}.")) (|norm| ((|#2| $) "\\spad{norm(I)} returns the norm of the ideal \\spad{I}.")) (|basis| (((|Vector| |#4|) $) "\\spad{basis((f1,...,fn))} returns the vector \\spad{[f1,...,fn]}.")) (|ideal| (($ (|Vector| |#4|)) "\\spad{ideal([f1,...,fn])} returns the ideal \\spad{(f1,...,fn)}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-418 R1 F1 U1 A1 R2 F2 U2 A2) 
+(-419 R1 F1 U1 A1 R2 F2 U2 A2) 
 ((|constructor| (NIL "\\indented{1}{Lifting of morphisms to fractional ideals.} Author: Manuel Bronstein Date Created: 1 Feb 1989 Date Last Updated: 27 Feb 1990 Keywords: ideal,{} algebra,{} module.")) (|map| (((|FractionalIdeal| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FractionalIdeal| |#1| |#2| |#3| |#4|)) "\\spad{map(f,i)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-419 R -3498 UP A |ibasis|) 
+(-420 R -3505 UP A |ibasis|) 
 ((|constructor| (NIL "Module representation of fractional ideals.")) (|module| (($ (|FractionalIdeal| |#1| |#2| |#3| |#4|)) "\\spad{module(I)} returns \\spad{I} viewed has a module over \\spad{R}.") (($ (|Vector| |#4|)) "\\spad{module([f1,...,fn])} = the module generated by \\spad{(f1,...,fn)} over \\spad{R}.")) (|norm| ((|#2| $) "\\spad{norm(f)} returns the norm of the module \\spad{f}.")) (|basis| (((|Vector| |#4|) $) "\\spad{basis((f1,...,fn))} = the vector \\spad{[f1,...,fn]}."))) 
 NIL 
-((|HasCategory| |#4| (LIST (QUOTE -1042) (|devaluate| |#2|)))) 
-(-420 AR R AS S) 
+((|HasCategory| |#4| (LIST (QUOTE -1044) (|devaluate| |#2|)))) 
+(-421 AR R AS S) 
 ((|constructor| (NIL "FramedNonAssociativeAlgebraFunctions2 implements functions between two framed non associative algebra domains defined over different rings. The function map is used to coerce between algebras over different domains having the same structural constants.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the coordinates of \\spad{u} to get an element in \\spad{AS} via identification of the basis of \\spad{AR} as beginning part of the basis of \\spad{AS}."))) 
 NIL 
 NIL 
-(-421 S R) 
+(-422 S R) 
 ((|constructor| (NIL "FramedNonAssociativeAlgebra(\\spad{R}) is a \\spadtype{FiniteRankNonAssociativeAlgebra} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank) over a commutative ring \\spad{R} together with a fixed \\spad{R}-module basis.")) (|apply| (($ (|Matrix| |#2|) $) "\\spad{apply(m,a)} defines a left operation of \\spad{n} by \\spad{n} matrices where \\spad{n} is the rank of the algebra in terms of matrix-vector multiplication,{} this is a substitute for a left module structure. Error: if shape of matrix doesn\\spad{'t} fit.")) (|rightRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#2|))) "\\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|leftRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#2|))) "\\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|rightRegularRepresentation| (((|Matrix| |#2|) $) "\\spad{rightRegularRepresentation(a)} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|leftRegularRepresentation| (((|Matrix| |#2|) $) "\\spad{leftRegularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|rightTraceMatrix| (((|Matrix| |#2|)) "\\spad{rightTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|leftTraceMatrix| (((|Matrix| |#2|)) "\\spad{leftTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|rightDiscriminant| ((|#2|) "\\spad{rightDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(rightTraceMatrix())}.")) (|leftDiscriminant| ((|#2|) "\\spad{leftDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(leftTraceMatrix())}.")) (|convert| (($ (|Vector| |#2|)) "\\spad{convert([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.") (((|Vector| |#2|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#2|))) "\\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \\spad{R}-module basis.")) (|structuralConstants| (((|Vector| (|Matrix| |#2|))) "\\spad{structuralConstants()} calculates the structural constants \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \\spad{v1},{}...,{}\\spad{vn} is the fixed \\spad{R}-module basis.")) (|elt| ((|#2| $ (|Integer|)) "\\spad{elt(a,i)} returns the \\spad{i}-th coefficient of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([a1,...,am])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the fixed \\spad{R}-module basis.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-366)))) 
-(-422 R) 
+((|HasCategory| |#2| (QUOTE (-367)))) 
+(-423 R) 
 ((|constructor| (NIL "FramedNonAssociativeAlgebra(\\spad{R}) is a \\spadtype{FiniteRankNonAssociativeAlgebra} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank) over a commutative ring \\spad{R} together with a fixed \\spad{R}-module basis.")) (|apply| (($ (|Matrix| |#1|) $) "\\spad{apply(m,a)} defines a left operation of \\spad{n} by \\spad{n} matrices where \\spad{n} is the rank of the algebra in terms of matrix-vector multiplication,{} this is a substitute for a left module structure. Error: if shape of matrix doesn\\spad{'t} fit.")) (|rightRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#1|))) "\\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|leftRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#1|))) "\\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|rightRegularRepresentation| (((|Matrix| |#1|) $) "\\spad{rightRegularRepresentation(a)} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|leftRegularRepresentation| (((|Matrix| |#1|) $) "\\spad{leftRegularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|rightTraceMatrix| (((|Matrix| |#1|)) "\\spad{rightTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|leftTraceMatrix| (((|Matrix| |#1|)) "\\spad{leftTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|rightDiscriminant| ((|#1|) "\\spad{rightDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(rightTraceMatrix())}.")) (|leftDiscriminant| ((|#1|) "\\spad{leftDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(leftTraceMatrix())}.")) (|convert| (($ (|Vector| |#1|)) "\\spad{convert([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.") (((|Vector| |#1|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#1|))) "\\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \\spad{R}-module basis.")) (|structuralConstants| (((|Vector| (|Matrix| |#1|))) "\\spad{structuralConstants()} calculates the structural constants \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \\spad{v1},{}...,{}\\spad{vn} is the fixed \\spad{R}-module basis.")) (|elt| ((|#1| $ (|Integer|)) "\\spad{elt(a,i)} returns the \\spad{i}-th coefficient of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([a1,...,am])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the fixed \\spad{R}-module basis.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) 
-((-4424 |has| |#1| (-561)) (-4422 . T) (-4421 . T)) 
+((-4431 |has| |#1| (-562)) (-4429 . T) (-4428 . T)) 
 NIL 
-(-423 R) 
+(-424 R) 
 ((|constructor| (NIL "\\spadtype{FactoredFunctionUtilities} implements some utility functions for manipulating factored objects.")) (|mergeFactors| (((|Factored| |#1|) (|Factored| |#1|) (|Factored| |#1|)) "\\spad{mergeFactors(u,v)} is used when the factorizations of \\spadvar{\\spad{u}} and \\spadvar{\\spad{v}} are known to be disjoint,{} \\spadignore{e.g.} resulting from a content/primitive part split. Essentially,{} it creates a new factored object by multiplying the units together and appending the lists of factors.")) (|refine| (((|Factored| |#1|) (|Factored| |#1|) (|Mapping| (|Factored| |#1|) |#1|)) "\\spad{refine(u,fn)} is used to apply the function \\userfun{\\spad{fn}} to each factor of \\spadvar{\\spad{u}} and then build a new factored object from the results. For example,{} if \\spadvar{\\spad{u}} were created by calling \\spad{nilFactor(10,2)} then \\spad{refine(u,factor)} would create a factored object equal to that created by \\spad{factor(100)} or \\spad{primeFactor(2,2) * primeFactor(5,2)}."))) 
 NIL 
 NIL 
-(-424 S R) 
+(-425 S R) 
 ((|constructor| (NIL "A space of formal functions with arguments in an arbitrary ordered set.")) (|univariate| (((|Fraction| (|SparseUnivariatePolynomial| $)) $ (|Kernel| $)) "\\spad{univariate(f, k)} returns \\spad{f} viewed as a univariate fraction in \\spad{k}.")) (/ (($ (|SparseMultivariatePolynomial| |#2| (|Kernel| $)) (|SparseMultivariatePolynomial| |#2| (|Kernel| $))) "\\spad{p1/p2} returns the quotient of \\spad{p1} and \\spad{p2} as an element of \\%.")) (|denominator| (($ $) "\\spad{denominator(f)} returns the denominator of \\spad{f} converted to \\%.")) (|denom| (((|SparseMultivariatePolynomial| |#2| (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|convert| (($ (|Factored| $)) "\\spad{convert(f1\\^e1 ... fm\\^em)} returns \\spad{(f1)\\^e1 ... (fm)\\^em} as an element of \\%,{} using formal kernels created using a \\spadfunFrom{paren}{ExpressionSpace}.")) (|isPower| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|numerator| (($ $) "\\spad{numerator(f)} returns the numerator of \\spad{f} converted to \\%.")) (|numer| (((|SparseMultivariatePolynomial| |#2| (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R} if \\spad{R} is an integral domain. If not,{} then numer(\\spad{f}) = \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|coerce| (($ (|Fraction| (|Polynomial| (|Fraction| |#2|)))) "\\spad{coerce(f)} returns \\spad{f} as an element of \\%.") (($ (|Polynomial| (|Fraction| |#2|))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.") (($ (|Fraction| |#2|)) "\\spad{coerce(q)} returns \\spad{q} as an element of \\%.") (($ (|SparseMultivariatePolynomial| |#2| (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.")) (|isMult| (((|Union| (|Record| (|:| |coef| (|Integer|)) (|:| |var| (|Kernel| $))) "failed") $) "\\spad{isMult(p)} returns \\spad{[n, x]} if \\spad{p = n * x} and \\spad{n <> 0}.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if \\spad{p = m1 +...+ mn} and \\spad{n > 1}.")) (|isExpt| (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|Symbol|)) "\\spad{isExpt(p,f)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = f(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|BasicOperator|)) "\\spad{isExpt(p,op)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = op(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1*...*an} and \\spad{n > 1}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns \\spad{x} * \\spad{x} * \\spad{x} * ... * \\spad{x} (\\spad{n} times).")) (|eval| (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ $)) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a)**n} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a1,...,am)**n} in \\spad{x} by \\spad{f(a1,...,am)} for any a1,{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)**ni} in \\spad{x} by \\spad{fi(a1,...,an)} for any a1,{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a)**ni} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|List| (|BasicOperator|)) (|List| $) (|Symbol|)) "\\spad{eval(x, [s1,...,sm], [f1,...,fm], y)} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $ (|BasicOperator|) $ (|Symbol|)) "\\spad{eval(x, s, f, y)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $) "\\spad{eval(f)} unquotes all the quoted operators in \\spad{f}.") (($ $ (|List| (|Symbol|))) "\\spad{eval(f, [foo1,...,foon])} unquotes all the \\spad{fooi}\\spad{'s} in \\spad{f}.") (($ $ (|Symbol|)) "\\spad{eval(f, foo)} unquotes all the foo\\spad{'s} in \\spad{f}.")) (|applyQuote| (($ (|Symbol|) (|List| $)) "\\spad{applyQuote(foo, [x1,...,xn])} returns \\spad{'foo(x1,...,xn)}.") (($ (|Symbol|) $ $ $ $) "\\spad{applyQuote(foo, x, y, z, t)} returns \\spad{'foo(x,y,z,t)}.") (($ (|Symbol|) $ $ $) "\\spad{applyQuote(foo, x, y, z)} returns \\spad{'foo(x,y,z)}.") (($ (|Symbol|) $ $) "\\spad{applyQuote(foo, x, y)} returns \\spad{'foo(x,y)}.") (($ (|Symbol|) $) "\\spad{applyQuote(foo, x)} returns \\spad{'foo(x)}.")) (|variables| (((|List| (|Symbol|)) $) "\\spad{variables(f)} returns the list of all the variables of \\spad{f}.")) (|ground| ((|#2| $) "\\spad{ground(f)} returns \\spad{f} as an element of \\spad{R}. An error occurs if \\spad{f} is not an element of \\spad{R}.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(f)} tests if \\spad{f} is an element of \\spad{R}."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-477))) (|HasCategory| |#2| (QUOTE (-1116))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539))))) 
-(-425 R) 
+((|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-1055))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-478))) (|HasCategory| |#2| (QUOTE (-1118))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540))))) 
+(-426 R) 
 ((|constructor| (NIL "A space of formal functions with arguments in an arbitrary ordered set.")) (|univariate| (((|Fraction| (|SparseUnivariatePolynomial| $)) $ (|Kernel| $)) "\\spad{univariate(f, k)} returns \\spad{f} viewed as a univariate fraction in \\spad{k}.")) (/ (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $)) (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{p1/p2} returns the quotient of \\spad{p1} and \\spad{p2} as an element of \\%.")) (|denominator| (($ $) "\\spad{denominator(f)} returns the denominator of \\spad{f} converted to \\%.")) (|denom| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|convert| (($ (|Factored| $)) "\\spad{convert(f1\\^e1 ... fm\\^em)} returns \\spad{(f1)\\^e1 ... (fm)\\^em} as an element of \\%,{} using formal kernels created using a \\spadfunFrom{paren}{ExpressionSpace}.")) (|isPower| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|numerator| (($ $) "\\spad{numerator(f)} returns the numerator of \\spad{f} converted to \\%.")) (|numer| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R} if \\spad{R} is an integral domain. If not,{} then numer(\\spad{f}) = \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|coerce| (($ (|Fraction| (|Polynomial| (|Fraction| |#1|)))) "\\spad{coerce(f)} returns \\spad{f} as an element of \\%.") (($ (|Polynomial| (|Fraction| |#1|))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.") (($ (|Fraction| |#1|)) "\\spad{coerce(q)} returns \\spad{q} as an element of \\%.") (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.")) (|isMult| (((|Union| (|Record| (|:| |coef| (|Integer|)) (|:| |var| (|Kernel| $))) "failed") $) "\\spad{isMult(p)} returns \\spad{[n, x]} if \\spad{p = n * x} and \\spad{n <> 0}.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if \\spad{p = m1 +...+ mn} and \\spad{n > 1}.")) (|isExpt| (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|Symbol|)) "\\spad{isExpt(p,f)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = f(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|BasicOperator|)) "\\spad{isExpt(p,op)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = op(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1*...*an} and \\spad{n > 1}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns \\spad{x} * \\spad{x} * \\spad{x} * ... * \\spad{x} (\\spad{n} times).")) (|eval| (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ $)) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a)**n} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a1,...,am)**n} in \\spad{x} by \\spad{f(a1,...,am)} for any a1,{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)**ni} in \\spad{x} by \\spad{fi(a1,...,an)} for any a1,{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a)**ni} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|List| (|BasicOperator|)) (|List| $) (|Symbol|)) "\\spad{eval(x, [s1,...,sm], [f1,...,fm], y)} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $ (|BasicOperator|) $ (|Symbol|)) "\\spad{eval(x, s, f, y)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $) "\\spad{eval(f)} unquotes all the quoted operators in \\spad{f}.") (($ $ (|List| (|Symbol|))) "\\spad{eval(f, [foo1,...,foon])} unquotes all the \\spad{fooi}\\spad{'s} in \\spad{f}.") (($ $ (|Symbol|)) "\\spad{eval(f, foo)} unquotes all the foo\\spad{'s} in \\spad{f}.")) (|applyQuote| (($ (|Symbol|) (|List| $)) "\\spad{applyQuote(foo, [x1,...,xn])} returns \\spad{'foo(x1,...,xn)}.") (($ (|Symbol|) $ $ $ $) "\\spad{applyQuote(foo, x, y, z, t)} returns \\spad{'foo(x,y,z,t)}.") (($ (|Symbol|) $ $ $) "\\spad{applyQuote(foo, x, y, z)} returns \\spad{'foo(x,y,z)}.") (($ (|Symbol|) $ $) "\\spad{applyQuote(foo, x, y)} returns \\spad{'foo(x,y)}.") (($ (|Symbol|) $) "\\spad{applyQuote(foo, x)} returns \\spad{'foo(x)}.")) (|variables| (((|List| (|Symbol|)) $) "\\spad{variables(f)} returns the list of all the variables of \\spad{f}.")) (|ground| ((|#1| $) "\\spad{ground(f)} returns \\spad{f} as an element of \\spad{R}. An error occurs if \\spad{f} is not an element of \\spad{R}.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(f)} tests if \\spad{f} is an element of \\spad{R}."))) 
-((-4424 -3962 (|has| |#1| (-1053)) (|has| |#1| (-477))) (-4422 |has| |#1| (-173)) (-4421 |has| |#1| (-173)) ((-4429 "*") |has| |#1| (-561)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-561)) (-4419 |has| |#1| (-561))) 
+((-4431 -3969 (|has| |#1| (-1055)) (|has| |#1| (-478))) (-4429 |has| |#1| (-173)) (-4428 |has| |#1| (-173)) ((-4436 "*") |has| |#1| (-562)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-562)) (-4426 |has| |#1| (-562))) 
 NIL 
-(-426 R A S B) 
+(-427 R A S B) 
 ((|constructor| (NIL "This package allows a mapping \\spad{R} \\spad{->} \\spad{S} to be lifted to a mapping from a function space over \\spad{R} to a function space over \\spad{S}.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f, a)} applies \\spad{f} to all the constants in \\spad{R} appearing in \\spad{a}."))) 
 NIL 
 NIL 
-(-427 R FE |x| |cen|) 
+(-428 R FE |x| |cen|) 
 ((|constructor| (NIL "This package converts expressions in some function space to exponential expansions.")) (|localAbs| ((|#2| |#2|) "\\spad{localAbs(fcn)} = \\spad{abs(fcn)} or \\spad{sqrt(fcn**2)} depending on whether or not FE has a function \\spad{abs}. This should be a local function,{} but the compiler won\\spad{'t} allow it.")) (|exprToXXP| (((|Union| (|:| |%expansion| (|ExponentialExpansion| |#1| |#2| |#3| |#4|)) (|:| |%problem| (|Record| (|:| |func| (|String|)) (|:| |prob| (|String|))))) |#2| (|Boolean|)) "\\spad{exprToXXP(fcn,posCheck?)} converts the expression \\spad{fcn} to an exponential expansion. If \\spad{posCheck?} is \\spad{true},{} log\\spad{'s} of negative numbers are not allowed nor are \\spad{n}th roots of negative numbers with \\spad{n} even. If \\spad{posCheck?} is \\spad{false},{} these are allowed."))) 
 NIL 
 NIL 
-(-428 R FE |Expon| UPS TRAN |x|) 
+(-429 R FE |Expon| UPS TRAN |x|) 
 ((|constructor| (NIL "This package converts expressions in some function space to power series in a variable \\spad{x} with coefficients in that function space. The function \\spadfun{exprToUPS} converts expressions to power series whose coefficients do not contain the variable \\spad{x}. The function \\spadfun{exprToGenUPS} converts functional expressions to power series whose coefficients may involve functions of \\spad{log(x)}.")) (|localAbs| ((|#2| |#2|) "\\spad{localAbs(fcn)} = \\spad{abs(fcn)} or \\spad{sqrt(fcn**2)} depending on whether or not FE has a function \\spad{abs}. This should be a local function,{} but the compiler won\\spad{'t} allow it.")) (|exprToGenUPS| (((|Union| (|:| |%series| |#4|) (|:| |%problem| (|Record| (|:| |func| (|String|)) (|:| |prob| (|String|))))) |#2| (|Boolean|) (|String|)) "\\spad{exprToGenUPS(fcn,posCheck?,atanFlag)} converts the expression \\spad{fcn} to a generalized power series. If \\spad{posCheck?} is \\spad{true},{} log\\spad{'s} of negative numbers are not allowed nor are \\spad{n}th roots of negative numbers with \\spad{n} even. If \\spad{posCheck?} is \\spad{false},{} these are allowed. \\spad{atanFlag} determines how the case \\spad{atan(f(x))},{} where \\spad{f(x)} has a pole,{} will be treated. The possible values of \\spad{atanFlag} are \\spad{\"complex\"},{} \\spad{\"real: two sides\"},{} \\spad{\"real: left side\"},{} \\spad{\"real: right side\"},{} and \\spad{\"just do it\"}. If \\spad{atanFlag} is \\spad{\"complex\"},{} then no series expansion will be computed because,{} viewed as a function of a complex variable,{} \\spad{atan(f(x))} has an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \\spad{f(x)} determines the constant coefficient in the series expansion of \\spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \\spad{atanFlag} is \\spad{\"just do it\"}. When the leading term in the series expansion of \\spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficient in the series expansion of \\spad{atan(f(x))} for values to the left differs from that for values to the right. If \\spad{atanFlag} is \\spad{\"real: two sides\"},{} no series expansion will be computed. If \\spad{atanFlag} is \\spad{\"real: left side\"} the constant coefficient for values to the left will be used and if \\spad{atanFlag} \\spad{\"real: right side\"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} we return a record containing the name of the function that caused the problem and a brief description of the problem. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \\spad{x -> x + a} before calling this function.")) (|exprToUPS| (((|Union| (|:| |%series| |#4|) (|:| |%problem| (|Record| (|:| |func| (|String|)) (|:| |prob| (|String|))))) |#2| (|Boolean|) (|String|)) "\\spad{exprToUPS(fcn,posCheck?,atanFlag)} converts the expression \\spad{fcn} to a power series. If \\spad{posCheck?} is \\spad{true},{} log\\spad{'s} of negative numbers are not allowed nor are \\spad{n}th roots of negative numbers with \\spad{n} even. If \\spad{posCheck?} is \\spad{false},{} these are allowed. \\spad{atanFlag} determines how the case \\spad{atan(f(x))},{} where \\spad{f(x)} has a pole,{} will be treated. The possible values of \\spad{atanFlag} are \\spad{\"complex\"},{} \\spad{\"real: two sides\"},{} \\spad{\"real: left side\"},{} \\spad{\"real: right side\"},{} and \\spad{\"just do it\"}. If \\spad{atanFlag} is \\spad{\"complex\"},{} then no series expansion will be computed because,{} viewed as a function of a complex variable,{} \\spad{atan(f(x))} has an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \\spad{f(x)} determines the constant coefficient in the series expansion of \\spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \\spad{atanFlag} is \\spad{\"just do it\"}. When the leading term in the series expansion of \\spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficient in the series expansion of \\spad{atan(f(x))} for values to the left differs from that for values to the right. If \\spad{atanFlag} is \\spad{\"real: two sides\"},{} no series expansion will be computed. If \\spad{atanFlag} is \\spad{\"real: left side\"} the constant coefficient for values to the left will be used and if \\spad{atanFlag} \\spad{\"real: right side\"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} a record containing the name of the function that caused the problem and a brief description of the problem is returned. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \\spad{x -> x + a} before calling this function.")) (|integrate| (($ $) "\\spad{integrate(x)} returns the integral of \\spad{x} since we need to be able to integrate a power series")) (|differentiate| (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x} since we need to be able to differentiate a power series"))) 
 NIL 
 NIL 
-(-429 A S) 
+(-430 A S) 
 ((|constructor| (NIL "A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \\spadtype{Set} for an example.")) (|min| ((|#2| $) "\\spad{min(u)} returns the smallest element of aggregate \\spad{u}.")) (|max| ((|#2| $) "\\spad{max(u)} returns the largest element of aggregate \\spad{u}.")) (|universe| (($) "\\spad{universe()}\\$\\spad{D} returns the universal set for finite set aggregate \\spad{D}.")) (|complement| (($ $) "\\spad{complement(u)} returns the complement of the set \\spad{u},{} \\spadignore{i.e.} the set of all values not in \\spad{u}.")) (|cardinality| (((|NonNegativeInteger|) $) "\\spad{cardinality(u)} returns the number of elements of \\spad{u}. Note: \\axiom{cardinality(\\spad{u}) = \\#u}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-371)))) 
-(-430 S) 
+((|HasCategory| |#2| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-372)))) 
+(-431 S) 
 ((|constructor| (NIL "A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \\spadtype{Set} for an example.")) (|min| ((|#1| $) "\\spad{min(u)} returns the smallest element of aggregate \\spad{u}.")) (|max| ((|#1| $) "\\spad{max(u)} returns the largest element of aggregate \\spad{u}.")) (|universe| (($) "\\spad{universe()}\\$\\spad{D} returns the universal set for finite set aggregate \\spad{D}.")) (|complement| (($ $) "\\spad{complement(u)} returns the complement of the set \\spad{u},{} \\spadignore{i.e.} the set of all values not in \\spad{u}.")) (|cardinality| (((|NonNegativeInteger|) $) "\\spad{cardinality(u)} returns the number of elements of \\spad{u}. Note: \\axiom{cardinality(\\spad{u}) = \\#u}."))) 
-((-4427 . T) (-4417 . T) (-4428 . T)) 
+((-4434 . T) (-4424 . T) (-4435 . T)) 
 NIL 
-(-431 S A R B) 
+(-432 S A R B) 
 ((|constructor| (NIL "FiniteSetAggregateFunctions2 provides functions involving two finite set aggregates where the underlying domains might be different. An example of this is to create a set of rational numbers by mapping a function across a set of integers,{} where the function divides each integer by 1000.")) (|scan| ((|#4| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-aggregates \\spad{x} of aggregate \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad {[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}.")) (|reduce| ((|#3| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the aggregate \\spad{a} and an accumulant initialised to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does a \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as an identity element for the function.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f,a)} applies function \\spad{f} to each member of aggregate \\spad{a},{} creating a new aggregate with a possibly different underlying domain."))) 
 NIL 
 NIL 
-(-432 R -3498) 
+(-433 R -3505) 
 ((|constructor| (NIL "\\spadtype{FunctionSpaceComplexIntegration} provides functions for the indefinite integration of complex-valued functions.")) (|complexIntegrate| ((|#2| |#2| (|Symbol|)) "\\spad{complexIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable.")) (|internalIntegrate0| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate0 should} be a local function,{} but is conditional.")) (|internalIntegrate| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable."))) 
 NIL 
 NIL 
-(-433 R E) 
+(-434 R E) 
 ((|constructor| (NIL "\\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:")) (|makeCos| (($ |#2| |#1|) "\\spad{makeCos(e,r)} makes a sin expression with given argument and coefficient")) (|makeSin| (($ |#2| |#1|) "\\spad{makeSin(e,r)} makes a sin expression with given argument and coefficient")) (|coerce| (($ (|FourierComponent| |#2|)) "\\spad{coerce(c)} converts sin/cos terms into Fourier Series") (($ |#1|) "\\spad{coerce(r)} converts coefficients into Fourier Series"))) 
-((-4414 -12 (|has| |#1| (-6 -4414)) (|has| |#2| (-6 -4414))) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-12 (|HasAttribute| |#1| (QUOTE -4414)) (|HasAttribute| |#2| (QUOTE -4414)))) 
-(-434 R -3498) 
+((-4421 -12 (|has| |#1| (-6 -4421)) (|has| |#2| (-6 -4421))) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-12 (|HasAttribute| |#1| (QUOTE -4421)) (|HasAttribute| |#2| (QUOTE -4421)))) 
+(-435 R -3505) 
 ((|constructor| (NIL "\\spadtype{FunctionSpaceIntegration} provides functions for the indefinite integration of real-valued functions.")) (|integrate| (((|Union| |#2| (|List| |#2|)) |#2| (|Symbol|)) "\\spad{integrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a real variable."))) 
 NIL 
 NIL 
-(-435 R -3498) 
+(-436 R -3505) 
 ((|constructor| (NIL "Provides some special functions over an integral domain.")) (|iiabs| ((|#2| |#2|) "\\spad{iiabs(x)} should be local but conditional.")) (|iiGamma| ((|#2| |#2|) "\\spad{iiGamma(x)} should be local but conditional.")) (|airyBi| ((|#2| |#2|) "\\spad{airyBi(x)} returns the airybi function applied to \\spad{x}")) (|airyAi| ((|#2| |#2|) "\\spad{airyAi(x)} returns the airyai function applied to \\spad{x}")) (|besselK| ((|#2| |#2| |#2|) "\\spad{besselK(x,y)} returns the besselk function applied to \\spad{x} and \\spad{y}")) (|besselI| ((|#2| |#2| |#2|) "\\spad{besselI(x,y)} returns the besseli function applied to \\spad{x} and \\spad{y}")) (|besselY| ((|#2| |#2| |#2|) "\\spad{besselY(x,y)} returns the bessely function applied to \\spad{x} and \\spad{y}")) (|besselJ| ((|#2| |#2| |#2|) "\\spad{besselJ(x,y)} returns the besselj function applied to \\spad{x} and \\spad{y}")) (|polygamma| ((|#2| |#2| |#2|) "\\spad{polygamma(x,y)} returns the polygamma function applied to \\spad{x} and \\spad{y}")) (|digamma| ((|#2| |#2|) "\\spad{digamma(x)} returns the digamma function applied to \\spad{x}")) (|Beta| ((|#2| |#2| |#2|) "\\spad{Beta(x,y)} returns the beta function applied to \\spad{x} and \\spad{y}")) (|Gamma| ((|#2| |#2| |#2|) "\\spad{Gamma(a,x)} returns the incomplete Gamma function applied to a and \\spad{x}") ((|#2| |#2|) "\\spad{Gamma(f)} returns the formal Gamma function applied to \\spad{f}")) (|abs| ((|#2| |#2|) "\\spad{abs(f)} returns the absolute value operator applied to \\spad{f}")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}; error if \\spad{op} is not a special function operator")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is a special function operator."))) 
 NIL 
 NIL 
-(-436 R -3498) 
+(-437 R -3505) 
 ((|constructor| (NIL "FunctionsSpacePrimitiveElement provides functions to compute primitive elements in functions spaces.")) (|primitiveElement| (((|Record| (|:| |primelt| |#2|) (|:| |pol1| (|SparseUnivariatePolynomial| |#2|)) (|:| |pol2| (|SparseUnivariatePolynomial| |#2|)) (|:| |prim| (|SparseUnivariatePolynomial| |#2|))) |#2| |#2|) "\\spad{primitiveElement(a1, a2)} returns \\spad{[a, q1, q2, q]} such that \\spad{k(a1, a2) = k(a)},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The minimal polynomial for a2 may involve \\spad{a1},{} but the minimal polynomial for \\spad{a1} may not involve a2; This operations uses \\spadfun{resultant}.") (((|Record| (|:| |primelt| |#2|) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#2|))) (|:| |prim| (|SparseUnivariatePolynomial| |#2|))) (|List| |#2|)) "\\spad{primitiveElement([a1,...,an])} returns \\spad{[a, [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}."))) 
 NIL 
 ((|HasCategory| |#2| (QUOTE (-27)))) 
-(-437 R -3498) 
+(-438 R -3505) 
 ((|constructor| (NIL "This package provides function which replaces transcendental kernels in a function space by random integers. The correspondence between the kernels and the integers is fixed between calls to new().")) (|newReduc| (((|Void|)) "\\spad{newReduc()} \\undocumented")) (|bringDown| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) |#2| (|Kernel| |#2|)) "\\spad{bringDown(f,k)} \\undocumented") (((|Fraction| (|Integer|)) |#2|) "\\spad{bringDown(f)} \\undocumented"))) 
 NIL 
 NIL 
-(-438) 
+(-439) 
 ((|constructor| (NIL "Creates and manipulates objects which correspond to the basic FORTRAN data types: REAL,{} INTEGER,{} COMPLEX,{} LOGICAL and CHARACTER")) (= (((|Boolean|) $ $) "\\spad{x=y} tests for equality")) (|logical?| (((|Boolean|) $) "\\spad{logical?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type LOGICAL.")) (|character?| (((|Boolean|) $) "\\spad{character?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type CHARACTER.")) (|doubleComplex?| (((|Boolean|) $) "\\spad{doubleComplex?(t)} tests whether \\spad{t} is equivalent to the (non-standard) FORTRAN type DOUBLE COMPLEX.")) (|complex?| (((|Boolean|) $) "\\spad{complex?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type COMPLEX.")) (|integer?| (((|Boolean|) $) "\\spad{integer?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type INTEGER.")) (|double?| (((|Boolean|) $) "\\spad{double?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type DOUBLE PRECISION")) (|real?| (((|Boolean|) $) "\\spad{real?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type REAL.")) (|coerce| (((|SExpression|) $) "\\spad{coerce(x)} returns the \\spad{s}-expression associated with \\spad{x}") (((|Symbol|) $) "\\spad{coerce(x)} returns the symbol associated with \\spad{x}") (($ (|Symbol|)) "\\spad{coerce(s)} transforms the symbol \\spad{s} into an element of FortranScalarType provided \\spad{s} is one of real,{} complex,{}double precision,{} logical,{} integer,{} character,{} REAL,{} COMPLEX,{} LOGICAL,{} INTEGER,{} CHARACTER,{} DOUBLE PRECISION") (($ (|String|)) "\\spad{coerce(s)} transforms the string \\spad{s} into an element of FortranScalarType provided \\spad{s} is one of \"real\",{} \"double precision\",{} \"complex\",{} \"logical\",{} \"integer\",{} \"character\",{} \"REAL\",{} \"COMPLEX\",{} \"LOGICAL\",{} \"INTEGER\",{} \"CHARACTER\",{} \"DOUBLE PRECISION\""))) 
 NIL 
 NIL 
-(-439 R -3498 UP) 
+(-440 R -3505 UP) 
 ((|constructor| (NIL "\\indented{1}{Used internally by IR2F} Author: Manuel Bronstein Date Created: 12 May 1988 Date Last Updated: 22 September 1993 Keywords: function,{} space,{} polynomial,{} factoring")) (|anfactor| (((|Union| (|Factored| (|SparseUnivariatePolynomial| (|AlgebraicNumber|))) "failed") |#3|) "\\spad{anfactor(p)} tries to factor \\spad{p} over algebraic numbers,{} returning \"failed\" if it cannot")) (|UP2ifCan| (((|Union| (|:| |overq| (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) (|:| |overan| (|SparseUnivariatePolynomial| (|AlgebraicNumber|))) (|:| |failed| (|Boolean|))) |#3|) "\\spad{UP2ifCan(x)} should be local but conditional.")) (|qfactor| (((|Union| (|Factored| (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "failed") |#3|) "\\spad{qfactor(p)} tries to factor \\spad{p} over fractions of integers,{} returning \"failed\" if it cannot")) (|ffactor| (((|Factored| |#3|) |#3|) "\\spad{ffactor(p)} tries to factor a univariate polynomial \\spad{p} over \\spad{F}"))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-48))))) 
-(-440) 
+((|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-48))))) 
+(-441) 
 ((|constructor| (NIL "Creates and manipulates objects which correspond to FORTRAN data types,{} including array dimensions.")) (|fortranCharacter| (($) "\\spad{fortranCharacter()} returns CHARACTER,{} an element of FortranType")) (|fortranDoubleComplex| (($) "\\spad{fortranDoubleComplex()} returns DOUBLE COMPLEX,{} an element of FortranType")) (|fortranComplex| (($) "\\spad{fortranComplex()} returns COMPLEX,{} an element of FortranType")) (|fortranLogical| (($) "\\spad{fortranLogical()} returns LOGICAL,{} an element of FortranType")) (|fortranInteger| (($) "\\spad{fortranInteger()} returns INTEGER,{} an element of FortranType")) (|fortranDouble| (($) "\\spad{fortranDouble()} returns DOUBLE PRECISION,{} an element of FortranType")) (|fortranReal| (($) "\\spad{fortranReal()} returns REAL,{} an element of FortranType")) (|construct| (($ (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1="void")) (|List| (|Polynomial| (|Integer|))) (|Boolean|)) "\\spad{construct(type,dims)} creates an element of FortranType") (($ (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#)) (|List| (|Symbol|)) (|Boolean|)) "\\spad{construct(type,dims)} creates an element of FortranType")) (|external?| (((|Boolean|) $) "\\spad{external?(u)} returns \\spad{true} if \\spad{u} is declared to be EXTERNAL")) (|dimensionsOf| (((|List| (|Polynomial| (|Integer|))) $) "\\spad{dimensionsOf(t)} returns the dimensions of \\spad{t}")) (|scalarTypeOf| (((|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#)) $) "\\spad{scalarTypeOf(t)} returns the FORTRAN data type of \\spad{t}")) (|coerce| (($ (|FortranScalarType|)) "\\spad{coerce(t)} creates an element from a scalar type"))) 
 NIL 
 NIL 
-(-441) 
+(-442) 
 ((|constructor| (NIL "Code to manipulate Fortran templates")) (|fortranCarriageReturn| (((|Void|)) "\\spad{fortranCarriageReturn()} produces a carriage return on the current Fortran output stream")) (|fortranLiteral| (((|Void|) (|String|)) "\\spad{fortranLiteral(s)} writes \\spad{s} to the current Fortran output stream")) (|fortranLiteralLine| (((|Void|) (|String|)) "\\spad{fortranLiteralLine(s)} writes \\spad{s} to the current Fortran output stream,{} followed by a carriage return")) (|processTemplate| (((|FileName|) (|FileName|)) "\\spad{processTemplate(tp)} processes the template \\spad{tp},{} writing the result to the current FORTRAN output stream.") (((|FileName|) (|FileName|) (|FileName|)) "\\spad{processTemplate(tp,fn)} processes the template \\spad{tp},{} writing the result out to \\spad{fn}."))) 
 NIL 
 NIL 
-(-442 |f|) 
+(-443 |f|) 
 ((|constructor| (NIL "This domain implements named functions")) (|name| (((|Symbol|) $) "\\spad{name(x)} returns the symbol"))) 
 NIL 
 NIL 
-(-443) 
+(-444) 
 ((|constructor| (NIL "This is the datatype for exported function descriptor. A function descriptor consists of: (1) a signature; (2) a predicate; and (3) a slot into the scope object.")) (|signature| (((|Signature|) $) "\\spad{signature(x)} returns the signature of function described by \\spad{x}."))) 
 NIL 
 NIL 
-(-444) 
+(-445) 
 ((|constructor| (NIL "\\axiomType{FortranVectorCategory} provides support for producing Functions and Subroutines when the input to these is an AXIOM object of type \\axiomType{Vector} or in domains involving \\axiomType{FortranCode}.")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|Vector| (|MachineFloat|))) "\\spad{coerce(v)} produces an ASP which returns the value of \\spad{v}."))) 
 NIL 
 NIL 
-(-445) 
+(-446) 
 ((|constructor| (NIL "\\axiomType{FortranVectorFunctionCategory} is the catagory of arguments to NAG Library routines which return the values of vectors of functions.")) (|retractIfCan| (((|Union| $ "failed") (|Vector| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Vector| (|Fraction| (|Polynomial| (|Float|))))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Vector| (|Polynomial| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Vector| (|Polynomial| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Vector| (|Expression| (|Integer|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (((|Union| $ "failed") (|Vector| (|Expression| (|Float|)))) "\\spad{retractIfCan(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|retract| (($ (|Vector| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Vector| (|Fraction| (|Polynomial| (|Float|))))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Vector| (|Polynomial| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Vector| (|Polynomial| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Vector| (|Expression| (|Integer|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}") (($ (|Vector| (|Expression| (|Float|)))) "\\spad{retract(e)} tries to convert \\spad{e} into an ASP,{} checking that \\indented{1}{legal Fortran-77 is produced.}")) (|coerce| (($ (|Record| (|:| |localSymbols| (|SymbolTable|)) (|:| |code| (|List| (|FortranCode|))))) "\\spad{coerce(e)} takes the component of \\spad{e} from \\spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \\spadtype{SymbolTable} component.") (($ (|FortranCode|)) "\\spad{coerce(e)} takes an object from \\spadtype{FortranCode} and \\indented{1}{uses it as the body of an ASP.}") (($ (|List| (|FortranCode|))) "\\spad{coerce(e)} takes an object from \\spadtype{List FortranCode} and \\indented{1}{uses it as the body of an ASP.}"))) 
 NIL 
 NIL 
-(-446 UP) 
+(-447 UP) 
 ((|constructor| (NIL "\\spadtype{GaloisGroupFactorizer} provides functions to factor resolvents.")) (|btwFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|) (|Set| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{btwFact(p,sqf,pd,r)} returns the factorization of \\spad{p},{} the result is a Record such that \\spad{contp=}content \\spad{p},{} \\spad{factors=}List of irreducible factors of \\spad{p} with exponent. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors). \\spad{pd} is the \\spadtype{Set} of possible degrees. \\spad{r} is a lower bound for the number of factors of \\spad{p}. Please do not use this function in your code because its design may change.")) (|henselFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|)) "\\spad{henselFact(p,sqf)} returns the factorization of \\spad{p},{} the result is a Record such that \\spad{contp=}content \\spad{p},{} \\spad{factors=}List of irreducible factors of \\spad{p} with exponent. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors).")) (|factorOfDegree| (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|) (|Boolean|)) "\\spad{factorOfDegree(d,p,listOfDegrees,r,sqf)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees},{} and that \\spad{p} has at least \\spad{r} factors. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors).") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factorOfDegree(d,p,listOfDegrees,r)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees},{} and that \\spad{p} has at least \\spad{r} factors.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factorOfDegree(d,p,listOfDegrees)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|NonNegativeInteger|)) "\\spad{factorOfDegree(d,p,r)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has at least \\spad{r} factors.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1|) "\\spad{factorOfDegree(d,p)} returns a factor of \\spad{p} of degree \\spad{d}.")) (|factorSquareFree| (((|Factored| |#1|) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,d,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{d} divides the degree of all factors of \\spad{p} and that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,listOfDegrees,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees} and that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factorSquareFree(p,listOfDegrees)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(p)} returns the factorization of \\spad{p} which is supposed not having any repeated factor (this is not checked).")) (|factor| (((|Factored| |#1|) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factor(p,d,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{d} divides the degree of all factors of \\spad{p} and that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factor(p,listOfDegrees,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees} and that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factor(p,listOfDegrees)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}.") (((|Factored| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{factor(p,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns the factorization of \\spad{p} over the integers.")) (|tryFunctionalDecomposition| (((|Boolean|) (|Boolean|)) "\\spad{tryFunctionalDecomposition(b)} chooses whether factorizers have to look for functional decomposition of polynomials (\\spad{true}) or not (\\spad{false}). Returns the previous value.")) (|tryFunctionalDecomposition?| (((|Boolean|)) "\\spad{tryFunctionalDecomposition?()} returns \\spad{true} if factorizers try functional decomposition of polynomials before factoring them.")) (|eisensteinIrreducible?| (((|Boolean|) |#1|) "\\spad{eisensteinIrreducible?(p)} returns \\spad{true} if \\spad{p} can be shown to be irreducible by Eisenstein\\spad{'s} criterion,{} \\spad{false} is inconclusive.")) (|useEisensteinCriterion| (((|Boolean|) (|Boolean|)) "\\spad{useEisensteinCriterion(b)} chooses whether factorizers check Eisenstein\\spad{'s} criterion before factoring: \\spad{true} for using it,{} \\spad{false} else. Returns the previous value.")) (|useEisensteinCriterion?| (((|Boolean|)) "\\spad{useEisensteinCriterion?()} returns \\spad{true} if factorizers check Eisenstein\\spad{'s} criterion before factoring.")) (|useSingleFactorBound| (((|Boolean|) (|Boolean|)) "\\spad{useSingleFactorBound(b)} chooses the algorithm to be used by the factorizers: \\spad{true} for algorithm with single factor bound,{} \\spad{false} for algorithm with overall bound. Returns the previous value.")) (|useSingleFactorBound?| (((|Boolean|)) "\\spad{useSingleFactorBound?()} returns \\spad{true} if algorithm with single factor bound is used for factorization,{} \\spad{false} for algorithm with overall bound.")) (|modularFactor| (((|Record| (|:| |prime| (|Integer|)) (|:| |factors| (|List| |#1|))) |#1|) "\\spad{modularFactor(f)} chooses a \"good\" prime and returns the factorization of \\spad{f} modulo this prime in a form that may be used by \\spadfunFrom{completeHensel}{GeneralHenselPackage}. If prime is zero it means that \\spad{f} has been proved to be irreducible over the integers or that \\spad{f} is a unit (\\spadignore{i.e.} 1 or \\spad{-1}). \\spad{f} shall be primitive (\\spadignore{i.e.} content(\\spad{p})\\spad{=1}) and square free (\\spadignore{i.e.} without repeated factors).")) (|numberOfFactors| (((|NonNegativeInteger|) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|))))) "\\spad{numberOfFactors(ddfactorization)} returns the number of factors of the polynomial \\spad{f} modulo \\spad{p} where \\spad{ddfactorization} is the distinct degree factorization of \\spad{f} computed by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \\spad{p}.")) (|stopMusserTrials| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{stopMusserTrials(n)} sets to \\spad{n} the bound on the number of factors for which \\spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \\spad{2**n} trials. Returns the previous value.") (((|PositiveInteger|)) "\\spad{stopMusserTrials()} returns the bound on the number of factors for which \\spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \\spad{2**stopMusserTrials()} trials.")) (|musserTrials| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{musserTrials(n)} sets to \\spad{n} the number of primes to be tried in \\spadfun{modularFactor} and returns the previous value.") (((|PositiveInteger|)) "\\spad{musserTrials()} returns the number of primes that are tried in \\spadfun{modularFactor}.")) (|degreePartition| (((|Multiset| (|NonNegativeInteger|)) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|))))) "\\spad{degreePartition(ddfactorization)} returns the degree partition of the polynomial \\spad{f} modulo \\spad{p} where \\spad{ddfactorization} is the distinct degree factorization of \\spad{f} computed by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \\spad{p}.")) (|makeFR| (((|Factored| |#1|) (|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|))))))) "\\spad{makeFR(flist)} turns the final factorization of henselFact into a \\spadtype{Factored} object."))) 
 NIL 
 NIL 
-(-447 R UP -3498) 
+(-448 R UP -3505) 
 ((|constructor| (NIL "\\spadtype{GaloisGroupFactorizationUtilities} provides functions that will be used by the factorizer.")) (|length| ((|#3| |#2|) "\\spad{length(p)} returns the sum of the absolute values of the coefficients of the polynomial \\spad{p}.")) (|height| ((|#3| |#2|) "\\spad{height(p)} returns the maximal absolute value of the coefficients of the polynomial \\spad{p}.")) (|infinityNorm| ((|#3| |#2|) "\\spad{infinityNorm(f)} returns the maximal absolute value of the coefficients of the polynomial \\spad{f}.")) (|quadraticNorm| ((|#3| |#2|) "\\spad{quadraticNorm(f)} returns the \\spad{l2} norm of the polynomial \\spad{f}.")) (|norm| ((|#3| |#2| (|PositiveInteger|)) "\\spad{norm(f,p)} returns the \\spad{lp} norm of the polynomial \\spad{f}.")) (|singleFactorBound| (((|Integer|) |#2|) "\\spad{singleFactorBound(p,r)} returns a bound on the infinite norm of the factor of \\spad{p} with smallest Bombieri\\spad{'s} norm. \\spad{p} shall be of degree higher or equal to 2.") (((|Integer|) |#2| (|NonNegativeInteger|)) "\\spad{singleFactorBound(p,r)} returns a bound on the infinite norm of the factor of \\spad{p} with smallest Bombieri\\spad{'s} norm. \\spad{r} is a lower bound for the number of factors of \\spad{p}. \\spad{p} shall be of degree higher or equal to 2.")) (|rootBound| (((|Integer|) |#2|) "\\spad{rootBound(p)} returns a bound on the largest norm of the complex roots of \\spad{p}.")) (|bombieriNorm| ((|#3| |#2| (|PositiveInteger|)) "\\spad{bombieriNorm(p,n)} returns the \\spad{n}th Bombieri\\spad{'s} norm of \\spad{p}.") ((|#3| |#2|) "\\spad{bombieriNorm(p)} returns quadratic Bombieri\\spad{'s} norm of \\spad{p}.")) (|beauzamyBound| (((|Integer|) |#2|) "\\spad{beauzamyBound(p)} returns a bound on the larger coefficient of any factor of \\spad{p}."))) 
 NIL 
 NIL 
-(-448 R UP) 
+(-449 R UP) 
 ((|constructor| (NIL "\\spadtype{GaloisGroupPolynomialUtilities} provides useful functions for univariate polynomials which should be added to \\spadtype{UnivariatePolynomialCategory} or to \\spadtype{Factored} (July 1994).")) (|factorsOfDegree| (((|List| |#2|) (|PositiveInteger|) (|Factored| |#2|)) "\\spad{factorsOfDegree(d,f)} returns the factors of degree \\spad{d} of the factored polynomial \\spad{f}.")) (|factorOfDegree| ((|#2| (|PositiveInteger|) (|Factored| |#2|)) "\\spad{factorOfDegree(d,f)} returns a factor of degree \\spad{d} of the factored polynomial \\spad{f}. Such a factor shall exist.")) (|degreePartition| (((|Multiset| (|NonNegativeInteger|)) (|Factored| |#2|)) "\\spad{degreePartition(f)} returns the degree partition (\\spadignore{i.e.} the multiset of the degrees of the irreducible factors) of the polynomial \\spad{f}.")) (|shiftRoots| ((|#2| |#2| |#1|) "\\spad{shiftRoots(p,c)} returns the polynomial which has for roots \\spad{c} added to the roots of \\spad{p}.")) (|scaleRoots| ((|#2| |#2| |#1|) "\\spad{scaleRoots(p,c)} returns the polynomial which has \\spad{c} times the roots of \\spad{p}.")) (|reverse| ((|#2| |#2|) "\\spad{reverse(p)} returns the reverse polynomial of \\spad{p}.")) (|unvectorise| ((|#2| (|Vector| |#1|)) "\\spad{unvectorise(v)} returns the polynomial which has for coefficients the entries of \\spad{v} in the increasing order.")) (|monic?| (((|Boolean|) |#2|) "\\spad{monic?(p)} tests if \\spad{p} is monic (\\spadignore{i.e.} leading coefficient equal to 1)."))) 
 NIL 
 NIL 
-(-449 R) 
+(-450 R) 
 ((|constructor| (NIL "\\spadtype{GaloisGroupUtilities} provides several useful functions.")) (|safetyMargin| (((|NonNegativeInteger|)) "\\spad{safetyMargin()} returns the number of low weight digits we do not trust in the floating point representation (used by \\spadfun{safeCeiling}).") (((|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{safetyMargin(n)} sets to \\spad{n} the number of low weight digits we do not trust in the floating point representation and returns the previous value (for use by \\spadfun{safeCeiling}).")) (|safeFloor| (((|Integer|) |#1|) "\\spad{safeFloor(x)} returns the integer which is lower or equal to the largest integer which has the same floating point number representation.")) (|safeCeiling| (((|Integer|) |#1|) "\\spad{safeCeiling(x)} returns the integer which is greater than any integer with the same floating point number representation.")) (|fillPascalTriangle| (((|Void|)) "\\spad{fillPascalTriangle()} fills the stored table.")) (|sizePascalTriangle| (((|NonNegativeInteger|)) "\\spad{sizePascalTriangle()} returns the number of entries currently stored in the table.")) (|rangePascalTriangle| (((|NonNegativeInteger|)) "\\spad{rangePascalTriangle()} returns the maximal number of lines stored.") (((|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rangePascalTriangle(n)} sets the maximal number of lines which are stored and returns the previous value.")) (|pascalTriangle| ((|#1| (|NonNegativeInteger|) (|Integer|)) "\\spad{pascalTriangle(n,r)} returns the binomial coefficient \\spad{C(n,r)=n!/(r! (n-r)!)} and stores it in a table to prevent recomputation."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-408)))) 
-(-450) 
+((|HasCategory| |#1| (QUOTE (-409)))) 
+(-451) 
 ((|constructor| (NIL "Package for the factorization of complex or gaussian integers.")) (|prime?| (((|Boolean|) (|Complex| (|Integer|))) "\\spad{prime?(zi)} tests if the complex integer \\spad{zi} is prime.")) (|sumSquares| (((|List| (|Integer|)) (|Integer|)) "\\spad{sumSquares(p)} construct \\spad{a} and \\spad{b} such that \\spad{a**2+b**2} is equal to the integer prime \\spad{p},{} and otherwise returns an error. It will succeed if the prime number \\spad{p} is 2 or congruent to 1 mod 4.")) (|factor| (((|Factored| (|Complex| (|Integer|))) (|Complex| (|Integer|))) "\\spad{factor(zi)} produces the complete factorization of the complex integer \\spad{zi}."))) 
 NIL 
 NIL 
-(-451 |Dom| |Expon| |VarSet| |Dpol|) 
+(-452 |Dom| |Expon| |VarSet| |Dpol|) 
 ((|constructor| (NIL "\\spadtype{GroebnerPackage} computes groebner bases for polynomial ideals. The basic computation provides a distinguished set of generators for polynomial ideals over fields. This basis allows an easy test for membership: the operation \\spadfun{normalForm} returns zero on ideal members. When the provided coefficient domain,{} Dom,{} is not a field,{} the result is equivalent to considering the extended ideal with \\spadtype{Fraction(Dom)} as coefficients,{} but considerably more efficient since all calculations are performed in Dom. Additional argument \"info\" and \"redcrit\" can be given to provide incremental information during computation. Argument \"info\" produces a computational summary for each \\spad{s}-polynomial. Argument \"redcrit\" prints out the reduced critical pairs. The term ordering is determined by the polynomial type used. Suggested types include \\spadtype{DistributedMultivariatePolynomial},{} \\spadtype{HomogeneousDistributedMultivariatePolynomial},{} \\spadtype{GeneralDistributedMultivariatePolynomial}.")) (|normalForm| ((|#4| |#4| (|List| |#4|)) "\\spad{normalForm(poly,gb)} reduces the polynomial \\spad{poly} modulo the precomputed groebner basis \\spad{gb} giving a canonical representative of the residue class.")) (|groebner| (((|List| |#4|) (|List| |#4|) (|String|) (|String|)) "\\spad{groebner(lp, \"info\", \"redcrit\")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp},{} displaying both a summary of the critical pairs considered (\\spad{\"info\"}) and the result of reducing each critical pair (\"redcrit\"). If the second or third arguments have any other string value,{} the indicated information is suppressed.") (((|List| |#4|) (|List| |#4|) (|String|)) "\\spad{groebner(lp, infoflag)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp}. Argument infoflag is used to get information on the computation. If infoflag is \"info\",{} then summary information is displayed for each \\spad{s}-polynomial generated. If infoflag is \"redcrit\",{} the reduced critical pairs are displayed. If infoflag is any other string,{} no information is printed during computation.") (((|List| |#4|) (|List| |#4|)) "\\spad{groebner(lp)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp}."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-366)))) 
-(-452 |Dom| |Expon| |VarSet| |Dpol|) 
+((|HasCategory| |#1| (QUOTE (-367)))) 
+(-453 |Dom| |Expon| |VarSet| |Dpol|) 
 ((|constructor| (NIL "\\spadtype{EuclideanGroebnerBasisPackage} computes groebner bases for polynomial ideals over euclidean domains. The basic computation provides a distinguished set of generators for these ideals. This basis allows an easy test for membership: the operation \\spadfun{euclideanNormalForm} returns zero on ideal members. The string \"info\" and \"redcrit\" can be given as additional args to provide incremental information during the computation. If \"info\" is given,{} \\indented{1}{a computational summary is given for each \\spad{s}-polynomial. If \"redcrit\"} is given,{} the reduced critical pairs are printed. The term ordering is determined by the polynomial type used. Suggested types include \\spadtype{DistributedMultivariatePolynomial},{} \\spadtype{HomogeneousDistributedMultivariatePolynomial},{} \\spadtype{GeneralDistributedMultivariatePolynomial}.")) (|euclideanGroebner| (((|List| |#4|) (|List| |#4|) (|String|) (|String|)) "\\spad{euclideanGroebner(lp, \"info\", \"redcrit\")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp}. If the second argument is \\spad{\"info\"},{} a summary is given of the critical pairs. If the third argument is \"redcrit\",{} critical pairs are printed.") (((|List| |#4|) (|List| |#4|) (|String|)) "\\spad{euclideanGroebner(lp, infoflag)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \\spad{lp}. During computation,{} additional information is printed out if infoflag is given as either \"info\" (for summary information) or \"redcrit\" (for reduced critical pairs)") (((|List| |#4|) (|List| |#4|)) "\\spad{euclideanGroebner(lp)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \\spad{lp}.")) (|euclideanNormalForm| ((|#4| |#4| (|List| |#4|)) "\\spad{euclideanNormalForm(poly,gb)} reduces the polynomial \\spad{poly} modulo the precomputed groebner basis \\spad{gb} giving a canonical representative of the residue class."))) 
 NIL 
 NIL 
-(-453 |Dom| |Expon| |VarSet| |Dpol|) 
+(-454 |Dom| |Expon| |VarSet| |Dpol|) 
 ((|constructor| (NIL "\\spadtype{GroebnerFactorizationPackage} provides the function groebnerFactor\" which uses the factorization routines of \\Language{} to factor each polynomial under consideration while doing the groebner basis algorithm. Then it writes the ideal as an intersection of ideals determined by the irreducible factors. Note that the whole ring may occur as well as other redundancies. We also use the fact,{} that from the second factor on we can assume that the preceding factors are not equal to 0 and we divide all polynomials under considerations by the elements of this list of \"nonZeroRestrictions\". The result is a list of groebner bases,{} whose union of solutions of the corresponding systems of equations is the solution of the system of equation corresponding to the input list. The term ordering is determined by the polynomial type used. Suggested types include \\spadtype{DistributedMultivariatePolynomial},{} \\spadtype{HomogeneousDistributedMultivariatePolynomial},{} \\spadtype{GeneralDistributedMultivariatePolynomial}.")) (|groebnerFactorize| (((|List| (|List| |#4|)) (|List| |#4|) (|Boolean|)) "\\spad{groebnerFactorize(listOfPolys, info)} returns a list of groebner bases. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys}. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p},{} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}. If {\\em info} is \\spad{true},{} information is printed about partial results.") (((|List| (|List| |#4|)) (|List| |#4|)) "\\spad{groebnerFactorize(listOfPolys)} returns a list of groebner bases. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys}. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p},{} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}.") (((|List| (|List| |#4|)) (|List| |#4|) (|List| |#4|) (|Boolean|)) "\\spad{groebnerFactorize(listOfPolys, nonZeroRestrictions, info)} returns a list of groebner basis. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys} under the restriction that the polynomials of {\\em nonZeroRestrictions} don\\spad{'t} vanish. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}. If argument {\\em info} is \\spad{true},{} information is printed about partial results.") (((|List| (|List| |#4|)) (|List| |#4|) (|List| |#4|)) "\\spad{groebnerFactorize(listOfPolys, nonZeroRestrictions)} returns a list of groebner basis. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys} under the restriction that the polynomials of {\\em nonZeroRestrictions} don\\spad{'t} vanish. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p},{} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}.")) (|factorGroebnerBasis| (((|List| (|List| |#4|)) (|List| |#4|) (|Boolean|)) "\\spad{factorGroebnerBasis(basis,info)} checks whether the \\spad{basis} contains reducible polynomials and uses these to split the \\spad{basis}. If argument {\\em info} is \\spad{true},{} information is printed about partial results.") (((|List| (|List| |#4|)) (|List| |#4|)) "\\spad{factorGroebnerBasis(basis)} checks whether the \\spad{basis} contains reducible polynomials and uses these to split the \\spad{basis}."))) 
 NIL 
 NIL 
-(-454 |Dom| |Expon| |VarSet| |Dpol|) 
+(-455 |Dom| |Expon| |VarSet| |Dpol|) 
 ((|constructor| (NIL "\\indented{1}{Author:} Date Created: Date Last Updated: Keywords: Description This package provides low level tools for Groebner basis computations")) (|virtualDegree| (((|NonNegativeInteger|) |#4|) "\\spad{virtualDegree }\\undocumented")) (|makeCrit| (((|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) (|Record| (|:| |totdeg| (|NonNegativeInteger|)) (|:| |pol| |#4|)) |#4| (|NonNegativeInteger|)) "\\spad{makeCrit }\\undocumented")) (|critpOrder| (((|Boolean|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) "\\spad{critpOrder }\\undocumented")) (|prinb| (((|Void|) (|Integer|)) "\\spad{prinb }\\undocumented")) (|prinpolINFO| (((|Void|) (|List| |#4|)) "\\spad{prinpolINFO }\\undocumented")) (|fprindINFO| (((|Integer|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) |#4| |#4| (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{fprindINFO }\\undocumented")) (|prindINFO| (((|Integer|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) |#4| |#4| (|Integer|) (|Integer|) (|Integer|)) "\\spad{prindINFO }\\undocumented")) (|prinshINFO| (((|Void|) |#4|) "\\spad{prinshINFO }\\undocumented")) (|lepol| (((|Integer|) |#4|) "\\spad{lepol }\\undocumented")) (|minGbasis| (((|List| |#4|) (|List| |#4|)) "\\spad{minGbasis }\\undocumented")) (|updatD| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{updatD }\\undocumented")) (|sPol| ((|#4| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) "\\spad{sPol }\\undocumented")) (|updatF| (((|List| (|Record| (|:| |totdeg| (|NonNegativeInteger|)) (|:| |pol| |#4|))) |#4| (|NonNegativeInteger|) (|List| (|Record| (|:| |totdeg| (|NonNegativeInteger|)) (|:| |pol| |#4|)))) "\\spad{updatF }\\undocumented")) (|hMonic| ((|#4| |#4|) "\\spad{hMonic }\\undocumented")) (|redPo| (((|Record| (|:| |poly| |#4|) (|:| |mult| |#1|)) |#4| (|List| |#4|)) "\\spad{redPo }\\undocumented")) (|critMonD1| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) |#2| (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{critMonD1 }\\undocumented")) (|critMTonD1| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{critMTonD1 }\\undocumented")) (|critBonD| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) |#4| (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{critBonD }\\undocumented")) (|critB| (((|Boolean|) |#2| |#2| |#2| |#2|) "\\spad{critB }\\undocumented")) (|critM| (((|Boolean|) |#2| |#2|) "\\spad{critM }\\undocumented")) (|critT| (((|Boolean|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) "\\spad{critT }\\undocumented")) (|gbasis| (((|List| |#4|) (|List| |#4|) (|Integer|) (|Integer|)) "\\spad{gbasis }\\undocumented")) (|redPol| ((|#4| |#4| (|List| |#4|)) "\\spad{redPol }\\undocumented")) (|credPol| ((|#4| |#4| (|List| |#4|)) "\\spad{credPol }\\undocumented"))) 
 NIL 
 NIL 
-(-455 S) 
+(-456 S) 
 ((|constructor| (NIL "This category describes domains where \\spadfun{\\spad{gcd}} can be computed but where there is no guarantee of the existence of \\spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \\spadfun{factor} operation exist,{} factorization will be unique up to order and units.")) (|lcm| (($ (|List| $)) "\\spad{lcm(l)} returns the least common multiple of the elements of the list \\spad{l}.") (($ $ $) "\\spad{lcm(x,y)} returns the least common multiple of \\spad{x} and \\spad{y}.")) (|gcd| (($ (|List| $)) "\\spad{gcd(l)} returns the common \\spad{gcd} of the elements in the list \\spad{l}.") (($ $ $) "\\spad{gcd(x,y)} returns the greatest common divisor of \\spad{x} and \\spad{y}."))) 
 NIL 
 NIL 
-(-456) 
+(-457) 
 ((|constructor| (NIL "This category describes domains where \\spadfun{\\spad{gcd}} can be computed but where there is no guarantee of the existence of \\spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \\spadfun{factor} operation exist,{} factorization will be unique up to order and units.")) (|lcm| (($ (|List| $)) "\\spad{lcm(l)} returns the least common multiple of the elements of the list \\spad{l}.") (($ $ $) "\\spad{lcm(x,y)} returns the least common multiple of \\spad{x} and \\spad{y}.")) (|gcd| (($ (|List| $)) "\\spad{gcd(l)} returns the common \\spad{gcd} of the elements in the list \\spad{l}.") (($ $ $) "\\spad{gcd(x,y)} returns the greatest common divisor of \\spad{x} and \\spad{y}."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-457 R |n| |ls| |gamma|) 
+(-458 R |n| |ls| |gamma|) 
 ((|constructor| (NIL "AlgebraGenericElementPackage allows you to create generic elements of an algebra,{} \\spadignore{i.e.} the scalars are extended to include symbolic coefficients")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#1|))) "\\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \\spad{R}-module basis") (((|List| (|Polynomial| |#1|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,...,vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}")) (|genericRightDiscriminant| (((|Fraction| (|Polynomial| |#1|))) "\\spad{genericRightDiscriminant()} is the determinant of the generic left trace forms of all products of basis element,{} if the generic left trace form is associative,{} an algebra is separable if the generic left discriminant is invertible,{} if it is non-zero,{} there is some ring extension which makes the algebra separable")) (|genericRightTraceForm| (((|Fraction| (|Polynomial| |#1|)) $ $) "\\spad{genericRightTraceForm (a,b)} is defined to be \\spadfun{genericRightTrace (a*b)},{} this defines a symmetric bilinear form on the algebra")) (|genericLeftDiscriminant| (((|Fraction| (|Polynomial| |#1|))) "\\spad{genericLeftDiscriminant()} is the determinant of the generic left trace forms of all products of basis element,{} if the generic left trace form is associative,{} an algebra is separable if the generic left discriminant is invertible,{} if it is non-zero,{} there is some ring extension which makes the algebra separable")) (|genericLeftTraceForm| (((|Fraction| (|Polynomial| |#1|)) $ $) "\\spad{genericLeftTraceForm (a,b)} is defined to be \\spad{genericLeftTrace (a*b)},{} this defines a symmetric bilinear form on the algebra")) (|genericRightNorm| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericRightNorm(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the constant term in \\spadfun{rightRankPolynomial} and changes the sign if the degree of this polynomial is odd")) (|genericRightTrace| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericRightTrace(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the second highest term in \\spadfun{rightRankPolynomial} and changes the sign")) (|genericRightMinimalPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|))) $) "\\spad{genericRightMinimalPolynomial(a)} substitutes the coefficients of \\spad{a} for the generic coefficients in \\spadfun{rightRankPolynomial}")) (|rightRankPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) "\\spad{rightRankPolynomial()} returns the right minimimal polynomial of the generic element")) (|genericLeftNorm| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericLeftNorm(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the constant term in \\spadfun{leftRankPolynomial} and changes the sign if the degree of this polynomial is odd. This is a form of degree \\spad{k}")) (|genericLeftTrace| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericLeftTrace(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the second highest term in \\spadfun{leftRankPolynomial} and changes the sign. \\indented{1}{This is a linear form}")) (|genericLeftMinimalPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|))) $) "\\spad{genericLeftMinimalPolynomial(a)} substitutes the coefficients of {em a} for the generic coefficients in \\spad{leftRankPolynomial()}")) (|leftRankPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) "\\spad{leftRankPolynomial()} returns the left minimimal polynomial of the generic element")) (|generic| (($ (|Vector| (|Symbol|)) (|Vector| $)) "\\spad{generic(vs,ve)} returns a generic element,{} \\spadignore{i.e.} the linear combination of \\spad{ve} with the symbolic coefficients \\spad{vs} error,{} if the vector of symbols is shorter than the vector of elements") (($ (|Symbol|) (|Vector| $)) "\\spad{generic(s,v)} returns a generic element,{} \\spadignore{i.e.} the linear combination of \\spad{v} with the symbolic coefficients \\spad{s1,s2,..}") (($ (|Vector| $)) "\\spad{generic(ve)} returns a generic element,{} \\spadignore{i.e.} the linear combination of \\spad{ve} basis with the symbolic coefficients \\spad{\\%x1,\\%x2,..}") (($ (|Vector| (|Symbol|))) "\\spad{generic(vs)} returns a generic element,{} \\spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \\spad{vs}; error,{} if the vector of symbols is too short") (($ (|Symbol|)) "\\spad{generic(s)} returns a generic element,{} \\spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \\spad{s1,s2,..}") (($) "\\spad{generic()} returns a generic element,{} \\spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \\spad{\\%x1,\\%x2,..}")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none")) (|coerce| (($ (|Vector| (|Fraction| (|Polynomial| |#1|)))) "\\spad{coerce(v)} assumes that it is called with a vector of length equal to the dimension of the algebra,{} then a linear combination with the basis element is formed"))) 
-((-4424 |has| (-411 (-950 |#1|)) (-561)) (-4422 . T) (-4421 . T)) 
-((|HasCategory| (-411 (-950 |#1|)) (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| (-411 (-950 |#1|)) (QUOTE (-561)))) 
-(-458 |vl| R E) 
+((-4431 |has| (-412 (-952 |#1|)) (-562)) (-4429 . T) (-4428 . T)) 
+((|HasCategory| (-412 (-952 |#1|)) (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| (-412 (-952 |#1|)) (QUOTE (-562)))) 
+(-459 |vl| R E) 
 ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is specified by its third parameter. Suggested types which define term orderings include: \\spadtype{DirectProduct},{} \\spadtype{HomogeneousDirectProduct},{} \\spadtype{SplitHomogeneousDirectProduct} and finally \\spadtype{OrderedDirectProduct} which accepts an arbitrary user function to define a term ordering.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p, perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) 
-(((-4429 "*") |has| |#2| (-173)) (-4420 |has| |#2| (-561)) (-4425 |has| |#2| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#2| (QUOTE (-914))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-914)))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-561)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-366))) (|HasAttribute| |#2| (QUOTE -4425)) (|HasCategory| |#2| (QUOTE (-456))) (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
-(-459 R BP) 
+(((-4436 "*") |has| |#2| (-173)) (-4427 |has| |#2| (-562)) (-4432 |has| |#2| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (-3969 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-3969 (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-3969 (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-173))) (-3969 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (QUOTE (-367))) (|HasAttribute| |#2| (QUOTE -4432)) (|HasCategory| |#2| (QUOTE (-457))) (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
+(-460 R BP) 
 ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni.} January 1990 The equation \\spad{Af+Bg=h} and its generalization to \\spad{n} polynomials is solved for solutions over the \\spad{R},{} euclidean domain. A table containing the solutions of \\spad{Af+Bg=x**k} is used. The operations are performed modulus a prime which are in principle big enough,{} but the solutions are tested and,{} in case of failure,{} a hensel lifting process is used to get to the right solutions. It will be used in the factorization of multivariate polynomials over finite field,{} with \\spad{R=F[x]}.")) (|testModulus| (((|Boolean|) |#1| (|List| |#2|)) "\\spad{testModulus(p,lp)} returns \\spad{true} if the the prime \\spad{p} is valid for the list of polynomials \\spad{lp},{} \\spadignore{i.e.} preserves the degree and they remain relatively prime.")) (|solveid| (((|Union| (|List| |#2|) "failed") |#2| |#1| (|Vector| (|List| |#2|))) "\\spad{solveid(h,table)} computes the coefficients of the extended euclidean algorithm for a list of polynomials whose tablePow is \\spad{table} and with right side \\spad{h}.")) (|tablePow| (((|Union| (|Vector| (|List| |#2|)) "failed") (|NonNegativeInteger|) |#1| (|List| |#2|)) "\\spad{tablePow(maxdeg,prime,lpol)} constructs the table with the coefficients of the Extended Euclidean Algorithm for \\spad{lpol}. Here the right side is \\spad{x**k},{} for \\spad{k} less or equal to \\spad{maxdeg}. The operation returns \"failed\" when the elements are not coprime modulo \\spad{prime}.")) (|compBound| (((|NonNegativeInteger|) |#2| (|List| |#2|)) "\\spad{compBound(p,lp)} computes a bound for the coefficients of the solution polynomials. Given a polynomial right hand side \\spad{p},{} and a list \\spad{lp} of left hand side polynomials. Exported because it depends on the valuation.")) (|reduction| ((|#2| |#2| |#1|) "\\spad{reduction(p,prime)} reduces the polynomial \\spad{p} modulo \\spad{prime} of \\spad{R}. Note: this function is exported only because it\\spad{'s} conditional."))) 
 NIL 
 NIL 
-(-460 OV E S R P) 
+(-461 OV E S R P) 
 ((|constructor| (NIL "\\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \\spadtype{Integer} or \\spadtype{Fraction Integer}.")) (|factor| (((|Factored| |#5|) |#5|) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} over its coefficient domain")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} makes an element from symbol \\spad{s} or fails.")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) 
 NIL 
 NIL 
-(-461 E OV R P) 
+(-462 E OV R P) 
 ((|constructor| (NIL "This package provides operations for \\spad{GCD} computations on polynomials")) (|randomR| ((|#3|) "\\spad{randomR()} should be local but conditional")) (|gcdPolynomial| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{gcdPolynomial(p,q)} returns the \\spad{GCD} of \\spad{p} and \\spad{q}"))) 
 NIL 
 NIL 
-(-462 R) 
+(-463 R) 
 ((|constructor| (NIL "\\indented{1}{Description} This package provides operations for the factorization of univariate polynomials with integer coefficients. The factorization is done by \"lifting\" the finite \"berlekamp's\" factorization")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{factor(p)} returns the factorisation of \\spad{p}"))) 
 NIL 
 NIL 
-(-463 R FE) 
+(-464 R FE) 
 ((|constructor| (NIL "\\spadtype{GenerateUnivariatePowerSeries} provides functions that create power series from explicit formulas for their \\spad{n}th coefficient.")) (|series| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{series(a(n),n,x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{series(a(n),n,x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Fraction| (|Integer|))) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{series(n +-> a(n),x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{series(n +-> a(n),x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.") (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{series(a(n),n,x=a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{series(a(n),n,x=a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{series(n +-> a(n),x = a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{series(n +-> a(n),x = a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.") (((|Any|) |#2| (|Symbol|) (|Equation| |#2|)) "\\spad{series(a(n),n,x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|)) "\\spad{series(n +-> a(n),x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}.")) (|puiseux| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{puiseux(a(n),n,x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{puiseux(a(n),n,x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Fraction| (|Integer|))) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{puiseux(n +-> a(n),x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{puiseux(n +-> a(n),x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.")) (|laurent| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{laurent(a(n),n,x=a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{laurent(a(n),n,x=a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{laurent(n +-> a(n),x = a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{laurent(n +-> a(n),x = a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.")) (|taylor| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|NonNegativeInteger|))) "\\spad{taylor(a(n),n,x = a,n0..)} returns \\spad{sum(n = n0..,a(n)*(x-a)**n)}; \\spad{taylor(a(n),n,x = a,n0..n1)} returns \\spad{sum(n = n0..,a(n)*(x-a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|) (|UniversalSegment| (|NonNegativeInteger|))) "\\spad{taylor(n +-> a(n),x = a,n0..)} returns \\spad{sum(n=n0..,a(n)*(x-a)**n)}; \\spad{taylor(n +-> a(n),x = a,n0..n1)} returns \\spad{sum(n = n0..,a(n)*(x-a)**n)}.") (((|Any|) |#2| (|Symbol|) (|Equation| |#2|)) "\\spad{taylor(a(n),n,x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|)) "\\spad{taylor(n +-> a(n),x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}."))) 
 NIL 
 NIL 
-(-464 RP TP) 
+(-465 RP TP) 
 ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni} General Hensel Lifting Used for Factorization of bivariate polynomials over a finite field.")) (|reduction| ((|#2| |#2| |#1|) "\\spad{reduction(u,pol)} computes the symmetric reduction of \\spad{u} mod \\spad{pol}")) (|completeHensel| (((|List| |#2|) |#2| (|List| |#2|) |#1| (|PositiveInteger|)) "\\spad{completeHensel(pol,lfact,prime,bound)} lifts \\spad{lfact},{} the factorization mod \\spad{prime} of \\spad{pol},{} to the factorization mod prime**k>bound. Factors are recombined on the way.")) (|HenselLift| (((|Record| (|:| |plist| (|List| |#2|)) (|:| |modulo| |#1|)) |#2| (|List| |#2|) |#1| (|PositiveInteger|)) "\\spad{HenselLift(pol,lfacts,prime,bound)} lifts \\spad{lfacts},{} that are the factors of \\spad{pol} mod \\spad{prime},{} to factors of \\spad{pol} mod prime**k > \\spad{bound}. No recombining is done ."))) 
 NIL 
 NIL 
-(-465 |vl| R IS E |ff| P) 
+(-466 |vl| R IS E |ff| P) 
 ((|constructor| (NIL "This package \\undocumented")) (* (($ |#6| $) "\\spad{p*x} \\undocumented")) (|multMonom| (($ |#2| |#4| $) "\\spad{multMonom(r,e,x)} \\undocumented")) (|build| (($ |#2| |#3| |#4|) "\\spad{build(r,i,e)} \\undocumented")) (|unitVector| (($ |#3|) "\\spad{unitVector(x)} \\undocumented")) (|monomial| (($ |#2| (|ModuleMonomial| |#3| |#4| |#5|)) "\\spad{monomial(r,x)} \\undocumented")) (|reductum| (($ $) "\\spad{reductum(x)} \\undocumented")) (|leadingIndex| ((|#3| $) "\\spad{leadingIndex(x)} \\undocumented")) (|leadingExponent| ((|#4| $) "\\spad{leadingExponent(x)} \\undocumented")) (|leadingMonomial| (((|ModuleMonomial| |#3| |#4| |#5|) $) "\\spad{leadingMonomial(x)} \\undocumented")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(x)} \\undocumented"))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 NIL 
-(-466 E V R P Q) 
+(-467 E V R P Q) 
 ((|constructor| (NIL "Gosper\\spad{'s} summation algorithm.")) (|GospersMethod| (((|Union| |#5| "failed") |#5| |#2| (|Mapping| |#2|)) "\\spad{GospersMethod(b, n, new)} returns a rational function \\spad{rf(n)} such that \\spad{a(n) * rf(n)} is the indefinite sum of \\spad{a(n)} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{a(n+1) * rf(n+1) - a(n) * rf(n) = a(n)},{} where \\spad{b(n) = a(n)/a(n-1)} is a rational function. Returns \"failed\" if no such rational function \\spad{rf(n)} exists. Note: \\spad{new} is a nullary function returning a new \\spad{V} every time. The condition on \\spad{a(n)} is that \\spad{a(n)/a(n-1)} is a rational function of \\spad{n}."))) 
 NIL 
 NIL 
-(-467 R E |VarSet| P) 
+(-468 R E |VarSet| P) 
 ((|constructor| (NIL "A domain for polynomial sets.")) (|convert| (($ (|List| |#4|)) "\\axiom{convert(\\spad{lp})} returns the polynomial set whose members are the polynomials of \\axiom{\\spad{lp}}."))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#4| (LIST (QUOTE -311) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#4| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-468 S R E) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#4| (LIST (QUOTE -312) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#4| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-469 S R E) 
 ((|constructor| (NIL "GradedAlgebra(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-algebra\\spad{''}. A graded algebra is a graded module together with a degree preserving \\spad{R}-linear map,{} called the {\\em product}. \\blankline The name ``product\\spad{''} is written out in full so inner and outer products with the same mapping type can be distinguished by name.")) (|product| (($ $ $) "\\spad{product(a,b)} is the degree-preserving \\spad{R}-linear product: \\blankline \\indented{2}{\\spad{degree product(a,b) = degree a + degree b}} \\indented{2}{\\spad{product(a1+a2,b) = product(a1,b) + product(a2,b)}} \\indented{2}{\\spad{product(a,b1+b2) = product(a,b1) + product(a,b2)}} \\indented{2}{\\spad{product(r*a,b) = product(a,r*b) = r*product(a,b)}} \\indented{2}{\\spad{product(a,product(b,c)) = product(product(a,b),c)}}")) ((|One|) (($) "1 is the identity for \\spad{product}."))) 
 NIL 
 NIL 
-(-469 R E) 
+(-470 R E) 
 ((|constructor| (NIL "GradedAlgebra(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-algebra\\spad{''}. A graded algebra is a graded module together with a degree preserving \\spad{R}-linear map,{} called the {\\em product}. \\blankline The name ``product\\spad{''} is written out in full so inner and outer products with the same mapping type can be distinguished by name.")) (|product| (($ $ $) "\\spad{product(a,b)} is the degree-preserving \\spad{R}-linear product: \\blankline \\indented{2}{\\spad{degree product(a,b) = degree a + degree b}} \\indented{2}{\\spad{product(a1+a2,b) = product(a1,b) + product(a2,b)}} \\indented{2}{\\spad{product(a,b1+b2) = product(a,b1) + product(a,b2)}} \\indented{2}{\\spad{product(r*a,b) = product(a,r*b) = r*product(a,b)}} \\indented{2}{\\spad{product(a,product(b,c)) = product(product(a,b),c)}}")) ((|One|) (($) "1 is the identity for \\spad{product}."))) 
 NIL 
 NIL 
-(-470) 
+(-471) 
 ((|constructor| (NIL "GrayCode provides a function for efficiently running through all subsets of a finite set,{} only changing one element by another one.")) (|firstSubsetGray| (((|Vector| (|Vector| (|Integer|))) (|PositiveInteger|)) "\\spad{firstSubsetGray(n)} creates the first vector {\\em ww} to start a loop using {\\em nextSubsetGray(ww,n)}")) (|nextSubsetGray| (((|Vector| (|Vector| (|Integer|))) (|Vector| (|Vector| (|Integer|))) (|PositiveInteger|)) "\\spad{nextSubsetGray(ww,n)} returns a vector {\\em vv} whose components have the following meanings:\\begin{items} \\item {\\em vv.1}: a vector of length \\spad{n} whose entries are 0 or 1. This \\indented{3}{can be interpreted as a code for a subset of the set 1,{}...,{}\\spad{n};} \\indented{3}{{\\em vv.1} differs from {\\em ww.1} by exactly one entry;} \\item {\\em vv.2.1} is the number of the entry of {\\em vv.1} which \\indented{3}{will be changed next time;} \\item {\\em vv.2.1 = n+1} means that {\\em vv.1} is the last subset; \\indented{3}{trying to compute nextSubsetGray(\\spad{vv}) if {\\em vv.2.1 = n+1}} \\indented{3}{will produce an error!} \\end{items} The other components of {\\em vv.2} are needed to compute nextSubsetGray efficiently. Note: this is an implementation of [Williamson,{} Topic II,{} 3.54,{} \\spad{p}. 112] for the special case {\\em r1 = r2 = ... = rn = 2}; Note: nextSubsetGray produces a side-effect,{} \\spadignore{i.e.} {\\em nextSubsetGray(vv)} and {\\em vv := nextSubsetGray(vv)} will have the same effect."))) 
 NIL 
 NIL 
-(-471) 
+(-472) 
 ((|constructor| (NIL "TwoDimensionalPlotSettings sets global flags and constants for 2-dimensional plotting.")) (|screenResolution| (((|Integer|) (|Integer|)) "\\spad{screenResolution(n)} sets the screen resolution to \\spad{n}.") (((|Integer|)) "\\spad{screenResolution()} returns the screen resolution \\spad{n}.")) (|minPoints| (((|Integer|) (|Integer|)) "\\spad{minPoints()} sets the minimum number of points in a plot.") (((|Integer|)) "\\spad{minPoints()} returns the minimum number of points in a plot.")) (|maxPoints| (((|Integer|) (|Integer|)) "\\spad{maxPoints()} sets the maximum number of points in a plot.") (((|Integer|)) "\\spad{maxPoints()} returns the maximum number of points in a plot.")) (|adaptive| (((|Boolean|) (|Boolean|)) "\\spad{adaptive(true)} turns adaptive plotting on; \\spad{adaptive(false)} turns adaptive plotting off.") (((|Boolean|)) "\\spad{adaptive()} determines whether plotting will be done adaptively.")) (|drawToScale| (((|Boolean|) (|Boolean|)) "\\spad{drawToScale(true)} causes plots to be drawn to scale. \\spad{drawToScale(false)} causes plots to be drawn so that they fill up the viewport window. The default setting is \\spad{false}.") (((|Boolean|)) "\\spad{drawToScale()} determines whether or not plots are to be drawn to scale.")) (|clipPointsDefault| (((|Boolean|) (|Boolean|)) "\\spad{clipPointsDefault(true)} turns on automatic clipping; \\spad{clipPointsDefault(false)} turns off automatic clipping. The default setting is \\spad{true}.") (((|Boolean|)) "\\spad{clipPointsDefault()} determines whether or not automatic clipping is to be done."))) 
 NIL 
 NIL 
-(-472) 
+(-473) 
 ((|constructor| (NIL "TwoDimensionalGraph creates virtual two dimensional graphs (to be displayed on TwoDimensionalViewports).")) (|putColorInfo| (((|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|Palette|))) "\\spad{putColorInfo(llp,lpal)} takes a list of list of points,{} \\spad{llp},{} and returns the points with their hue and shade components set according to the list of palette colors,{} \\spad{lpal}.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(gi)} returns the indicated graph,{} \\spad{gi},{} of domain \\spadtype{GraphImage} as output of the domain \\spadtype{OutputForm}.") (($ (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{coerce(llp)} component(\\spad{gi},{}\\spad{pt}) creates and returns a graph of the domain \\spadtype{GraphImage} which is composed of the list of list of points given by \\spad{llp},{} and whose point colors,{} line colors and point sizes are determined by the default functions \\spadfun{pointColorDefault},{} \\spadfun{lineColorDefault},{} and \\spadfun{pointSizeDefault}. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.")) (|point| (((|Void|) $ (|Point| (|DoubleFloat|)) (|Palette|)) "\\spad{point(gi,pt,pal)} modifies the graph \\spad{gi} of the domain \\spadtype{GraphImage} to contain one point component,{} \\spad{pt} whose point color is set to be the palette color \\spad{pal},{} and whose line color and point size are determined by the default functions \\spadfun{lineColorDefault} and \\spadfun{pointSizeDefault}.")) (|appendPoint| (((|Void|) $ (|Point| (|DoubleFloat|))) "\\spad{appendPoint(gi,pt)} appends the point \\spad{pt} to the end of the list of points component for the graph,{} \\spad{gi},{} which is of the domain \\spadtype{GraphImage}.")) (|component| (((|Void|) $ (|Point| (|DoubleFloat|)) (|Palette|) (|Palette|) (|PositiveInteger|)) "\\spad{component(gi,pt,pal1,pal2,ps)} modifies the graph \\spad{gi} of the domain \\spadtype{GraphImage} to contain one point component,{} \\spad{pt} whose point color is set to the palette color \\spad{pal1},{} line color is set to the palette color \\spad{pal2},{} and point size is set to the positive integer \\spad{ps}.") (((|Void|) $ (|Point| (|DoubleFloat|))) "\\spad{component(gi,pt)} modifies the graph \\spad{gi} of the domain \\spadtype{GraphImage} to contain one point component,{} \\spad{pt} whose point color,{} line color and point size are determined by the default functions \\spadfun{pointColorDefault},{} \\spadfun{lineColorDefault},{} and \\spadfun{pointSizeDefault}.") (((|Void|) $ (|List| (|Point| (|DoubleFloat|))) (|Palette|) (|Palette|) (|PositiveInteger|)) "\\spad{component(gi,lp,pal1,pal2,p)} sets the components of the graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} to the values given. The point list for \\spad{gi} is set to the list \\spad{lp},{} the color of the points in \\spad{lp} is set to the palette color \\spad{pal1},{} the color of the lines which connect the points \\spad{lp} is set to the palette color \\spad{pal2},{} and the size of the points in \\spad{lp} is given by the integer \\spad{p}.")) (|units| (((|List| (|Float|)) $ (|List| (|Float|))) "\\spad{units(gi,lu)} modifies the list of unit increments for the \\spad{x} and \\spad{y} axes of the given graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} to be that of the list of unit increments,{} \\spad{lu},{} and returns the new list of units for \\spad{gi}.") (((|List| (|Float|)) $) "\\spad{units(gi)} returns the list of unit increments for the \\spad{x} and \\spad{y} axes of the indicated graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|ranges| (((|List| (|Segment| (|Float|))) $ (|List| (|Segment| (|Float|)))) "\\spad{ranges(gi,lr)} modifies the list of ranges for the given graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} to be that of the list of range segments,{} \\spad{lr},{} and returns the new range list for \\spad{gi}.") (((|List| (|Segment| (|Float|))) $) "\\spad{ranges(gi)} returns the list of ranges of the point components from the indicated graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|key| (((|Integer|) $) "\\spad{key(gi)} returns the process ID of the given graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|pointLists| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{pointLists(gi)} returns the list of lists of points which compose the given graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|makeGraphImage| (($ (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|Palette|)) (|List| (|Palette|)) (|List| (|PositiveInteger|)) (|List| (|DrawOption|))) "\\spad{makeGraphImage(llp,lpal1,lpal2,lp,lopt)} returns a graph of the domain \\spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \\spad{llp},{} whose point colors are indicated by the list of palette colors,{} \\spad{lpal1},{} and whose lines are colored according to the list of palette colors,{} \\spad{lpal2}. The paramater \\spad{lp} is a list of integers which denote the size of the data points,{} and \\spad{lopt} is the list of draw command options. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.") (($ (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|Palette|)) (|List| (|Palette|)) (|List| (|PositiveInteger|))) "\\spad{makeGraphImage(llp,lpal1,lpal2,lp)} returns a graph of the domain \\spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \\spad{llp},{} whose point colors are indicated by the list of palette colors,{} \\spad{lpal1},{} and whose lines are colored according to the list of palette colors,{} \\spad{lpal2}. The paramater \\spad{lp} is a list of integers which denote the size of the data points. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.") (($ (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{makeGraphImage(llp)} returns a graph of the domain \\spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \\spad{llp},{} with default point size and default point and line colours. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.") (($ $) "\\spad{makeGraphImage(gi)} takes the given graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} and sends it\\spad{'s} data to the viewport manager where it waits to be included in a two-dimensional viewport window. \\spad{gi} cannot be an empty graph,{} and it\\spad{'s} elements must have been created using the \\spadfun{point} or \\spadfun{component} functions,{} not by a previous \\spadfun{makeGraphImage}.")) (|graphImage| (($) "\\spad{graphImage()} returns an empty graph with 0 point lists of the domain \\spadtype{GraphImage}. A graph image contains the graph data component of a two dimensional viewport."))) 
 NIL 
 NIL 
-(-473 S R E) 
+(-474 S R E) 
 ((|constructor| (NIL "GradedModule(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-module\\spad{''},{} \\spadignore{i.e.} collection of \\spad{R}-modules indexed by an abelian monoid \\spad{E}. An element \\spad{g} of \\spad{G[s]} for some specific \\spad{s} in \\spad{E} is said to be an element of \\spad{G} with {\\em degree} \\spad{s}. Sums are defined in each module \\spad{G[s]} so two elements of \\spad{G} have a sum if they have the same degree. \\blankline Morphisms can be defined and composed by degree to give the mathematical category of graded modules.")) (+ (($ $ $) "\\spad{g+h} is the sum of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.")) (- (($ $ $) "\\spad{g-h} is the difference of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.") (($ $) "\\spad{-g} is the additive inverse of \\spad{g} in the module of elements of the same grade as \\spad{g}.")) (* (($ $ |#2|) "\\spad{g*r} is right module multiplication.") (($ |#2| $) "\\spad{r*g} is left module multiplication.")) ((|Zero|) (($) "0 denotes the zero of degree 0.")) (|degree| ((|#3| $) "\\spad{degree(g)} names the degree of \\spad{g}. The set of all elements of a given degree form an \\spad{R}-module."))) 
 NIL 
 NIL 
-(-474 R E) 
+(-475 R E) 
 ((|constructor| (NIL "GradedModule(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-module\\spad{''},{} \\spadignore{i.e.} collection of \\spad{R}-modules indexed by an abelian monoid \\spad{E}. An element \\spad{g} of \\spad{G[s]} for some specific \\spad{s} in \\spad{E} is said to be an element of \\spad{G} with {\\em degree} \\spad{s}. Sums are defined in each module \\spad{G[s]} so two elements of \\spad{G} have a sum if they have the same degree. \\blankline Morphisms can be defined and composed by degree to give the mathematical category of graded modules.")) (+ (($ $ $) "\\spad{g+h} is the sum of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.")) (- (($ $ $) "\\spad{g-h} is the difference of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.") (($ $) "\\spad{-g} is the additive inverse of \\spad{g} in the module of elements of the same grade as \\spad{g}.")) (* (($ $ |#1|) "\\spad{g*r} is right module multiplication.") (($ |#1| $) "\\spad{r*g} is left module multiplication.")) ((|Zero|) (($) "0 denotes the zero of degree 0.")) (|degree| ((|#2| $) "\\spad{degree(g)} names the degree of \\spad{g}. The set of all elements of a given degree form an \\spad{R}-module."))) 
 NIL 
 NIL 
-(-475 |lv| -3498 R) 
+(-476 |lv| -3505 R) 
 ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni,{} Summer \\spad{'88},{} revised November \\spad{'89}} Solve systems of polynomial equations using Groebner bases Total order Groebner bases are computed and then converted to lex ones This package is mostly intended for internal use.")) (|genericPosition| (((|Record| (|:| |dpolys| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |coords| (|List| (|Integer|)))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{genericPosition(lp,lv)} puts a radical zero dimensional ideal in general position,{} for system \\spad{lp} in variables \\spad{lv}.")) (|testDim| (((|Union| (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "failed") (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{testDim(lp,lv)} tests if the polynomial system \\spad{lp} in variables \\spad{lv} is zero dimensional.")) (|groebSolve| (((|List| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{groebSolve(lp,lv)} reduces the polynomial system \\spad{lp} in variables \\spad{lv} to triangular form. Algorithm based on groebner bases algorithm with linear algebra for change of ordering. Preprocessing for the general solver. The polynomials in input are of type \\spadtype{DMP}."))) 
 NIL 
 NIL 
-(-476 S) 
+(-477 S) 
 ((|constructor| (NIL "The class of multiplicative groups,{} \\spadignore{i.e.} monoids with multiplicative inverses. \\blankline")) (|commutator| (($ $ $) "\\spad{commutator(p,q)} computes \\spad{inv(p) * inv(q) * p * q}.")) (|conjugate| (($ $ $) "\\spad{conjugate(p,q)} computes \\spad{inv(q) * p * q}; this is 'right action by conjugation'.")) (|unitsKnown| ((|attribute|) "unitsKnown asserts that recip only returns \"failed\" for non-units.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}.")) (/ (($ $ $) "\\spad{x/y} is the same as \\spad{x} times the inverse of \\spad{y}.")) (|inv| (($ $) "\\spad{inv(x)} returns the inverse of \\spad{x}."))) 
 NIL 
 NIL 
-(-477) 
+(-478) 
 ((|constructor| (NIL "The class of multiplicative groups,{} \\spadignore{i.e.} monoids with multiplicative inverses. \\blankline")) (|commutator| (($ $ $) "\\spad{commutator(p,q)} computes \\spad{inv(p) * inv(q) * p * q}.")) (|conjugate| (($ $ $) "\\spad{conjugate(p,q)} computes \\spad{inv(q) * p * q}; this is 'right action by conjugation'.")) (|unitsKnown| ((|attribute|) "unitsKnown asserts that recip only returns \"failed\" for non-units.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}.")) (/ (($ $ $) "\\spad{x/y} is the same as \\spad{x} times the inverse of \\spad{y}.")) (|inv| (($ $) "\\spad{inv(x)} returns the inverse of \\spad{x}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-478 |Coef| |var| |cen|) 
+(-479 |Coef| |var| |cen|) 
 ((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x\\^r)}.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{coerce(f)} converts a Puiseux series to a general power series.") (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a Puiseux series."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|)))) (|HasCategory| (-411 (-550)) (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-964))) (|HasCategory| |#1| (QUOTE (-1206))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4246) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1181))))) (|HasSignature| |#1| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#1|))))))) 
-(-479 |Key| |Entry| |Tbl| |dent|) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|)))) (|HasCategory| (-412 (-551)) (QUOTE (-1118))) (|HasCategory| |#1| (QUOTE (-367))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasSignature| |#1| (LIST (QUOTE -4387) (LIST (|devaluate| |#1|) (QUOTE (-1183)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1208))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasSignature| |#1| (LIST (QUOTE -4253) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1183))))) (|HasSignature| |#1| (LIST (QUOTE -3494) (LIST (LIST (QUOTE -646) (QUOTE (-1183))) (|devaluate| |#1|))))))) 
+(-480 |Key| |Entry| |Tbl| |dent|) 
 ((|constructor| (NIL "A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key."))) 
-((-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-853))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) 
-(-480 R E V P) 
+((-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-855))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) 
+(-481 R E V P) 
 ((|constructor| (NIL "A domain constructor of the category \\axiomType{TriangularSetCategory}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. Triangular sets are stored as sorted lists \\spad{w}.\\spad{r}.\\spad{t}. the main variables of their members but they are displayed in reverse order.\\newline References : \\indented{1}{[1] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)}"))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#4| (LIST (QUOTE -311) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#3| (QUOTE (-371))) (|HasCategory| |#4| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-481) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#4| (LIST (QUOTE -312) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-372))) (|HasCategory| |#4| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-482) 
 ((|constructor| (NIL "\\indented{1}{Symbolic fractions in \\%\\spad{pi} with integer coefficients;} \\indented{1}{The point for using \\spad{Pi} as the default domain for those fractions} \\indented{1}{is that \\spad{Pi} is coercible to the float types,{} and not Expression.} Date Created: 21 Feb 1990 Date Last Updated: 12 Mai 1992")) (|pi| (($) "\\spad{pi()} returns the symbolic \\%\\spad{pi}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-482) 
+(-483) 
 ((|constructor| (NIL "This domain represents a `has' expression.")) (|rhs| (((|SpadAst|) $) "\\spad{rhs(e)} returns the right hand side of the case expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the has expression `e'."))) 
 NIL 
 NIL 
-(-483 |Key| |Entry| |hashfn|) 
+(-484 |Key| |Entry| |hashfn|) 
 ((|constructor| (NIL "This domain provides access to the underlying Lisp hash tables. By varying the hashfn parameter,{} tables suited for different purposes can be obtained."))) 
-((-4427 . T) (-4428 . T)) 
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-(-484) 
+((-4434 . T) (-4435 . T)) 
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+(-485) 
 ((|constructor| (NIL "\\indented{1}{Author : Larry Lambe} Date Created : August 1988 Date Last Updated : March 9 1990 Related Constructors: OrderedSetInts,{} Commutator,{} FreeNilpotentLie AMS Classification: Primary 17B05,{} 17B30; Secondary 17A50 Keywords: free Lie algebra,{} Hall basis,{} basic commutators Description : Generate a basis for the free Lie algebra on \\spad{n} generators over a ring \\spad{R} with identity up to basic commutators of length \\spad{c} using the algorithm of \\spad{P}. Hall as given in Serre\\spad{'s} book Lie Groups \\spad{--} Lie Algebras")) (|generate| (((|Vector| (|List| (|Integer|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generate(numberOfGens, maximalWeight)} generates a vector of elements of the form [left,{}weight,{}right] which represents a \\spad{P}. Hall basis element for the free lie algebra on \\spad{numberOfGens} generators. We only generate those basis elements of weight less than or equal to maximalWeight")) (|inHallBasis?| (((|Boolean|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{inHallBasis?(numberOfGens, leftCandidate, rightCandidate, left)} tests to see if a new element should be added to the \\spad{P}. Hall basis being constructed. The list \\spad{[leftCandidate,wt,rightCandidate]} is included in the basis if in the unique factorization of \\spad{rightCandidate},{} we have left factor leftOfRight,{} and leftOfRight \\spad{<=} \\spad{leftCandidate}")) (|lfunc| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{lfunc(d,n)} computes the rank of the \\spad{n}th factor in the lower central series of the free \\spad{d}-generated free Lie algebra; This rank is \\spad{d} if \\spad{n} = 1 and binom(\\spad{d},{}2) if \\spad{n} = 2"))) 
 NIL 
 NIL 
-(-485 |vl| R) 
+(-486 |vl| R) 
 ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is total degree ordering refined by reverse lexicographic ordering with respect to the position that the variables appear in the list of variables parameter.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p, perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) 
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-((|HasCategory| |#2| (QUOTE (-914))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-914)))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-561)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-366))) (|HasAttribute| |#2| (QUOTE -4425)) (|HasCategory| |#2| (QUOTE (-456))) (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
-(-486 -3023 S) 
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+(-487 -3030 S) 
 ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The vectors are ordered first by the sum of their components,{} and then refined using a reverse lexicographic ordering. This type is a suitable third argument for \\spadtype{GeneralDistributedMultivariatePolynomial}."))) 
-((-4421 |has| |#2| (-1053)) (-4422 |has| |#2| (-1053)) (-4424 |has| |#2| (-6 -4424)) ((-4429 "*") |has| |#2| (-173)) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE 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(|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (|HasCategory| |#2| (QUOTE (-234))) (-3962 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (|HasCategory| |#2| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 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 ((|constructor| (NIL "This domain represents the header of a definition.")) (|parameters| (((|List| (|ParameterAst|)) $) "\\spad{parameters(h)} gives the parameters specified in the definition header \\spad{`h'}.")) (|name| (((|Identifier|) $) "\\spad{name(h)} returns the name of the operation defined defined.")) (|headAst| (($ (|Identifier|) (|List| (|ParameterAst|))) "\\spad{headAst(f,[x1,..,xn])} constructs a function definition header."))) 
 NIL 
 NIL 
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 ((|constructor| (NIL "Heap implemented in a flexible array to allow for insertions")) (|heap| (($ (|List| |#1|)) "\\spad{heap(ls)} creates a heap of elements consisting of the elements of \\spad{ls}."))) 
-((-4427 . T) (-4428 . T)) 
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-(-489 -3498 UP UPUP R) 
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 ((|constructor| (NIL "This domains implements finite rational divisors on an hyperelliptic curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}\\spad{'s} are integers and the \\spad{P}\\spad{'s} are finite rational points on the curve. The equation of the curve must be \\spad{y^2} = \\spad{f}(\\spad{x}) and \\spad{f} must have odd degree."))) 
 NIL 
 NIL 
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 ((|constructor| (NIL "This package provides the functions for the heuristic integer \\spad{gcd}. Geddes\\spad{'s} algorithm,{}for univariate polynomials with integer coefficients")) (|lintgcd| (((|Integer|) (|List| (|Integer|))) "\\spad{lintgcd([a1,..,ak])} = \\spad{gcd} of a list of integers")) (|content| (((|List| (|Integer|)) (|List| |#1|)) "\\spad{content([f1,..,fk])} = content of a list of univariate polynonials")) (|gcdcofactprim| (((|List| |#1|) (|List| |#1|)) "\\spad{gcdcofactprim([f1,..fk])} = \\spad{gcd} and cofactors of \\spad{k} primitive polynomials.")) (|gcdcofact| (((|List| |#1|) (|List| |#1|)) "\\spad{gcdcofact([f1,..fk])} = \\spad{gcd} and cofactors of \\spad{k} univariate polynomials.")) (|gcdprim| ((|#1| (|List| |#1|)) "\\spad{gcdprim([f1,..,fk])} = \\spad{gcd} of \\spad{k} PRIMITIVE univariate polynomials")) (|gcd| ((|#1| (|List| |#1|)) "\\spad{gcd([f1,..,fk])} = \\spad{gcd} of the polynomials \\spad{fi}."))) 
 NIL 
 NIL 
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 ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating hexadecimal expansions.")) (|hex| (($ (|Fraction| (|Integer|))) "\\spad{hex(r)} converts a rational number to a hexadecimal expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(h)} returns the fractional part of a hexadecimal expansion."))) 
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-(-492 A S) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-551) (QUOTE (-916))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| (-551) (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-147))) (|HasCategory| (-551) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-551) (QUOTE (-1026))) (|HasCategory| (-551) (QUOTE (-825))) (-3969 (|HasCategory| (-551) (QUOTE (-825))) (|HasCategory| (-551) (QUOTE (-855)))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-1157))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-551) (QUOTE (-234))) (|HasCategory| (-551) (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| (-551) (LIST (QUOTE -519) (QUOTE (-1183)) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -312) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -289) (QUOTE (-551)) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-310))) (|HasCategory| (-551) (QUOTE (-550))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-551) (LIST (QUOTE -644) (QUOTE (-551)))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (-3969 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (|HasCategory| (-551) (QUOTE (-145))))) 
+(-493 A S) 
 ((|constructor| (NIL "A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \\spadatt{finiteAggregate} have a finite number of members. Those with attribute \\spadatt{shallowlyMutable} allow an element to be modified or updated without changing its overall value.")) (|member?| (((|Boolean|) |#2| $) "\\spad{member?(x,u)} tests if \\spad{x} is a member of \\spad{u}. For collections,{} \\axiom{member?(\\spad{x},{}\\spad{u}) = reduce(or,{}[x=y for \\spad{y} in \\spad{u}],{}\\spad{false})}.")) (|members| (((|List| |#2|) $) "\\spad{members(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|parts| (((|List| |#2|) $) "\\spad{parts(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|count| (((|NonNegativeInteger|) |#2| $) "\\spad{count(x,u)} returns the number of occurrences of \\spad{x} in \\spad{u}. For collections,{} \\axiom{count(\\spad{x},{}\\spad{u}) = reduce(+,{}[x=y for \\spad{y} in \\spad{u}],{}0)}.") (((|NonNegativeInteger|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{count(p,u)} returns the number of elements \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. For collections,{} \\axiom{count(\\spad{p},{}\\spad{u}) = reduce(+,{}[1 for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})],{}0)}.")) (|every?| (((|Boolean|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{every?(f,u)} tests if \\spad{p}(\\spad{x}) is \\spad{true} for all elements \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{every?(\\spad{p},{}\\spad{u}) = reduce(and,{}map(\\spad{f},{}\\spad{u}),{}\\spad{true},{}\\spad{false})}.")) (|any?| (((|Boolean|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{any?(p,u)} tests if \\axiom{\\spad{p}(\\spad{x})} is \\spad{true} for any element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{any?(\\spad{p},{}\\spad{u}) = reduce(or,{}map(\\spad{f},{}\\spad{u}),{}\\spad{false},{}\\spad{true})}.")) (|map!| (($ (|Mapping| |#2| |#2|) $) "\\spad{map!(f,u)} destructively replaces each element \\spad{x} of \\spad{u} by \\axiom{\\spad{f}(\\spad{x})}.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(f,u)} returns a copy of \\spad{u} with each element \\spad{x} replaced by \\spad{f}(\\spad{x}). For collections,{} \\axiom{map(\\spad{f},{}\\spad{u}) = [\\spad{f}(\\spad{x}) for \\spad{x} in \\spad{u}]}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4427)) (|HasAttribute| |#1| (QUOTE -4428)) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-493 S) 
+((|HasAttribute| |#1| (QUOTE -4434)) (|HasAttribute| |#1| (QUOTE -4435)) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-494 S) 
 ((|constructor| (NIL "A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \\spadatt{finiteAggregate} have a finite number of members. Those with attribute \\spadatt{shallowlyMutable} allow an element to be modified or updated without changing its overall value.")) (|member?| (((|Boolean|) |#1| $) "\\spad{member?(x,u)} tests if \\spad{x} is a member of \\spad{u}. For collections,{} \\axiom{member?(\\spad{x},{}\\spad{u}) = reduce(or,{}[x=y for \\spad{y} in \\spad{u}],{}\\spad{false})}.")) (|members| (((|List| |#1|) $) "\\spad{members(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|parts| (((|List| |#1|) $) "\\spad{parts(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|count| (((|NonNegativeInteger|) |#1| $) "\\spad{count(x,u)} returns the number of occurrences of \\spad{x} in \\spad{u}. For collections,{} \\axiom{count(\\spad{x},{}\\spad{u}) = reduce(+,{}[x=y for \\spad{y} in \\spad{u}],{}0)}.") (((|NonNegativeInteger|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{count(p,u)} returns the number of elements \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. For collections,{} \\axiom{count(\\spad{p},{}\\spad{u}) = reduce(+,{}[1 for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})],{}0)}.")) (|every?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{every?(f,u)} tests if \\spad{p}(\\spad{x}) is \\spad{true} for all elements \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{every?(\\spad{p},{}\\spad{u}) = reduce(and,{}map(\\spad{f},{}\\spad{u}),{}\\spad{true},{}\\spad{false})}.")) (|any?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{any?(p,u)} tests if \\axiom{\\spad{p}(\\spad{x})} is \\spad{true} for any element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{any?(\\spad{p},{}\\spad{u}) = reduce(or,{}map(\\spad{f},{}\\spad{u}),{}\\spad{false},{}\\spad{true})}.")) (|map!| (($ (|Mapping| |#1| |#1|) $) "\\spad{map!(f,u)} destructively replaces each element \\spad{x} of \\spad{u} by \\axiom{\\spad{f}(\\spad{x})}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,u)} returns a copy of \\spad{u} with each element \\spad{x} replaced by \\spad{f}(\\spad{x}). For collections,{} \\axiom{map(\\spad{f},{}\\spad{u}) = [\\spad{f}(\\spad{x}) for \\spad{x} in \\spad{u}]}."))) 
 NIL 
 NIL 
-(-494 S) 
+(-495 S) 
 ((|constructor| (NIL "A is homotopic to \\spad{B} iff any element of domain \\spad{B} can be automically converted into an element of domain \\spad{B},{} and nay element of domain \\spad{B} can be automatically converted into an A."))) 
 NIL 
 NIL 
-(-495) 
+(-496) 
 ((|constructor| (NIL "This domain represents hostnames on computer network.")) (|host| (($ (|String|)) "\\spad{host(n)} constructs a Hostname from the name \\spad{`n'}."))) 
 NIL 
 NIL 
-(-496 S) 
+(-497 S) 
 ((|constructor| (NIL "Category for the hyperbolic trigonometric functions.")) (|tanh| (($ $) "\\spad{tanh(x)} returns the hyperbolic tangent of \\spad{x}.")) (|sinh| (($ $) "\\spad{sinh(x)} returns the hyperbolic sine of \\spad{x}.")) (|sech| (($ $) "\\spad{sech(x)} returns the hyperbolic secant of \\spad{x}.")) (|csch| (($ $) "\\spad{csch(x)} returns the hyperbolic cosecant of \\spad{x}.")) (|coth| (($ $) "\\spad{coth(x)} returns the hyperbolic cotangent of \\spad{x}.")) (|cosh| (($ $) "\\spad{cosh(x)} returns the hyperbolic cosine of \\spad{x}."))) 
 NIL 
 NIL 
-(-497) 
+(-498) 
 ((|constructor| (NIL "Category for the hyperbolic trigonometric functions.")) (|tanh| (($ $) "\\spad{tanh(x)} returns the hyperbolic tangent of \\spad{x}.")) (|sinh| (($ $) "\\spad{sinh(x)} returns the hyperbolic sine of \\spad{x}.")) (|sech| (($ $) "\\spad{sech(x)} returns the hyperbolic secant of \\spad{x}.")) (|csch| (($ $) "\\spad{csch(x)} returns the hyperbolic cosecant of \\spad{x}.")) (|coth| (($ $) "\\spad{coth(x)} returns the hyperbolic cotangent of \\spad{x}.")) (|cosh| (($ $) "\\spad{cosh(x)} returns the hyperbolic cosine of \\spad{x}."))) 
 NIL 
 NIL 
-(-498 -3498 UP |AlExt| |AlPol|) 
+(-499 -3505 UP |AlExt| |AlPol|) 
 ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of a field over which we can factor UP\\spad{'s}.")) (|factor| (((|Factored| |#4|) |#4| (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{factor(p, f)} returns a prime factorisation of \\spad{p}; \\spad{f} is a factorisation map for elements of UP."))) 
 NIL 
 NIL 
-(-499) 
+(-500) 
 ((|constructor| (NIL "Algebraic closure of the rational numbers.")) (|norm| (($ $ (|List| (|Kernel| $))) "\\spad{norm(f,l)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernels \\spad{l}") (($ $ (|Kernel| $)) "\\spad{norm(f,k)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernel \\spad{k}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|List| (|Kernel| $))) "\\spad{norm(p,l)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernels \\spad{l}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{norm(p,k)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernel \\spad{k}")) (|trueEqual| (((|Boolean|) $ $) "\\spad{trueEqual(x,y)} tries to determine if the two numbers are equal")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \\spad{f} by applying their defining relations.")) (|denom| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|numer| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|coerce| (($ (|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} viewed as an algebraic number."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| $ (QUOTE (-1053))) (|HasCategory| $ (LIST (QUOTE -1042) (QUOTE (-550))))) 
-(-500 S |mn|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| $ (QUOTE (-1055))) (|HasCategory| $ (LIST (QUOTE -1044) (QUOTE (-551))))) 
+(-501 S |mn|) 
 ((|constructor| (NIL "\\indented{1}{Author Micheal Monagan Aug/87} This is the basic one dimensional array data type."))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-501 R |mnRow| |mnCol|) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-502 R |mnRow| |mnCol|) 
 ((|constructor| (NIL "\\indented{1}{An IndexedTwoDimensionalArray is a 2-dimensional array where} the minimal row and column indices are parameters of the type. Rows and columns are returned as IndexedOneDimensionalArray\\spad{'s} with minimal indices matching those of the IndexedTwoDimensionalArray. The index of the 'first' row may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-502 K R UP) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-503 K R UP) 
 ((|constructor| (NIL "\\indented{1}{Author: Clifton Williamson} Date Created: 9 August 1993 Date Last Updated: 3 December 1993 Basic Operations: chineseRemainder,{} factorList Related Domains: PAdicWildFunctionFieldIntegralBasis(\\spad{K},{}\\spad{R},{}UP,{}\\spad{F}) Also See: WildFunctionFieldIntegralBasis,{} FunctionFieldIntegralBasis AMS Classifications: Keywords: function field,{} finite field,{} integral basis Examples: References: Description:")) (|chineseRemainder| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) (|List| |#3|) (|List| (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) (|NonNegativeInteger|)) "\\spad{chineseRemainder(lu,lr,n)} \\undocumented")) (|listConjugateBases| (((|List| (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{listConjugateBases(bas,q,n)} returns the list \\spad{[bas,bas^Frob,bas^(Frob^2),...bas^(Frob^(n-1))]},{} where \\spad{Frob} raises the coefficients of all polynomials appearing in the basis \\spad{bas} to the \\spad{q}th power.")) (|factorList| (((|List| (|SparseUnivariatePolynomial| |#1|)) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factorList(k,n,m,j)} \\undocumented"))) 
 NIL 
 NIL 
-(-503 R UP -3498) 
+(-504 R UP -3505) 
 ((|constructor| (NIL "This package contains functions used in the packages FunctionFieldIntegralBasis and NumberFieldIntegralBasis.")) (|moduleSum| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) (|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) (|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|)))) "\\spad{moduleSum(m1,m2)} returns the sum of two modules in the framed algebra \\spad{F}. Each module \\spad{mi} is represented as follows: \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn} and \\spad{mi} is a record \\spad{[basis,basisDen,basisInv]}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then a basis \\spad{v1,...,vn} for \\spad{mi} is given by \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|idealiserMatrix| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{idealiserMatrix(m1, m2)} returns the matrix representing the linear conditions on the Ring associatied with an ideal defined by \\spad{m1} and \\spad{m2}.")) (|idealiser| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{idealiser(m1,m2,d)} computes the order of an ideal defined by \\spad{m1} and \\spad{m2} where \\spad{d} is the known part of the denominator") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{idealiser(m1,m2)} computes the order of an ideal defined by \\spad{m1} and \\spad{m2}")) (|leastPower| (((|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{leastPower(p,n)} returns \\spad{e},{} where \\spad{e} is the smallest integer such that \\spad{p **e >= n}")) (|divideIfCan!| ((|#1| (|Matrix| |#1|) (|Matrix| |#1|) |#1| (|Integer|)) "\\spad{divideIfCan!(matrix,matrixOut,prime,n)} attempts to divide the entries of \\spad{matrix} by \\spad{prime} and store the result in \\spad{matrixOut}. If it is successful,{} 1 is returned and if not,{} \\spad{prime} is returned. Here both \\spad{matrix} and \\spad{matrixOut} are \\spad{n}-by-\\spad{n} upper triangular matrices.")) (|matrixGcd| ((|#1| (|Matrix| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{matrixGcd(mat,sing,n)} is \\spad{gcd(sing,g)} where \\spad{g} is the \\spad{gcd} of the entries of the \\spad{n}-by-\\spad{n} upper-triangular matrix \\spad{mat}.")) (|diagonalProduct| ((|#1| (|Matrix| |#1|)) "\\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \\spad{m}")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) 
 NIL 
 NIL 
-(-504 |mn|) 
+(-505 |mn|) 
 ((|constructor| (NIL "\\spadtype{IndexedBits} is a domain to compactly represent large quantities of Boolean data.")) (|And| (($ $ $) "\\spad{And(n,m)} returns the bit-by-bit logical {\\em And} of \\spad{n} and \\spad{m}.")) (|Or| (($ $ $) "\\spad{Or(n,m)} returns the bit-by-bit logical {\\em Or} of \\spad{n} and \\spad{m}.")) (|Not| (($ $) "\\spad{Not(n)} returns the bit-by-bit logical {\\em Not} of \\spad{n}."))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| (-112) (QUOTE (-1105))) (|HasCategory| (-112) (LIST (QUOTE -311) (QUOTE (-112))))) (|HasCategory| (-112) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-112) (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-112) (QUOTE (-1105))) (|HasCategory| (-112) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-505 K R UP L) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| (-112) (QUOTE (-1107))) (|HasCategory| (-112) (LIST (QUOTE -312) (QUOTE (-112))))) (|HasCategory| (-112) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-112) (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-112) (QUOTE (-1107))) (|HasCategory| (-112) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-506 K R UP L) 
 ((|constructor| (NIL "IntegralBasisPolynomialTools provides functions for \\indented{1}{mapping functions on the coefficients of univariate and bivariate} \\indented{1}{polynomials.}")) (|mapBivariate| (((|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#4|)) (|Mapping| |#4| |#1|) |#3|) "\\spad{mapBivariate(f,p(x,y))} applies the function \\spad{f} to the coefficients of \\spad{p(x,y)}.")) (|mapMatrixIfCan| (((|Union| (|Matrix| |#2|) "failed") (|Mapping| (|Union| |#1| "failed") |#4|) (|Matrix| (|SparseUnivariatePolynomial| |#4|))) "\\spad{mapMatrixIfCan(f,mat)} applies the function \\spad{f} to the coefficients of the entries of \\spad{mat} if possible,{} and returns \\spad{\"failed\"} otherwise.")) (|mapUnivariateIfCan| (((|Union| |#2| "failed") (|Mapping| (|Union| |#1| "failed") |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{mapUnivariateIfCan(f,p(x))} applies the function \\spad{f} to the coefficients of \\spad{p(x)},{} if possible,{} and returns \\spad{\"failed\"} otherwise.")) (|mapUnivariate| (((|SparseUnivariatePolynomial| |#4|) (|Mapping| |#4| |#1|) |#2|) "\\spad{mapUnivariate(f,p(x))} applies the function \\spad{f} to the coefficients of \\spad{p(x)}.") ((|#2| (|Mapping| |#1| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{mapUnivariate(f,p(x))} applies the function \\spad{f} to the coefficients of \\spad{p(x)}."))) 
 NIL 
 NIL 
-(-506) 
+(-507) 
 ((|constructor| (NIL "\\indented{1}{This domain implements a container of information} about the AXIOM library")) (|coerce| (($ (|String|)) "\\spad{coerce(s)} converts \\axiom{\\spad{s}} into an \\axiom{IndexCard}. Warning: if \\axiom{\\spad{s}} is not of the right format then an error will occur when using it.")) (|fullDisplay| (((|Void|) $) "\\spad{fullDisplay(ic)} prints all of the information contained in \\axiom{\\spad{ic}}.")) (|display| (((|Void|) $) "\\spad{display(ic)} prints a summary of the information contained in \\axiom{\\spad{ic}}.")) (|elt| (((|String|) $ (|Symbol|)) "\\spad{elt(ic,s)} selects a particular field from \\axiom{\\spad{ic}}. Valid fields are \\axiom{name,{} nargs,{} exposed,{} type,{} abbreviation,{} kind,{} origin,{} params,{} condition,{} doc}."))) 
 NIL 
 NIL 
-(-507 R Q A B) 
+(-508 R Q A B) 
 ((|constructor| (NIL "InnerCommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#4|) "\\spad{splitDenominator([q1,...,qn])} returns \\spad{[[p1,...,pn], d]} such that \\spad{qi = pi/d} and \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|clearDenominator| ((|#3| |#4|) "\\spad{clearDenominator([q1,...,qn])} returns \\spad{[p1,...,pn]} such that \\spad{qi = pi/d} where \\spad{d} is a common denominator for the \\spad{qi}\\spad{'s}.")) (|commonDenominator| ((|#1| |#4|) "\\spad{commonDenominator([q1,...,qn])} returns a common denominator \\spad{d} for \\spad{q1},{}...,{}\\spad{qn}."))) 
 NIL 
 NIL 
-(-508 -3498 |Expon| |VarSet| |DPoly|) 
+(-509 -3505 |Expon| |VarSet| |DPoly|) 
 ((|constructor| (NIL "This domain represents polynomial ideals with coefficients in any field and supports the basic ideal operations,{} including intersection sum and quotient. An ideal is represented by a list of polynomials (the generators of the ideal) and a boolean that is \\spad{true} if the generators are a Groebner basis. The algorithms used are based on Groebner basis computations. The ordering is determined by the datatype of the input polynomials. Users may use refinements of total degree orderings.")) (|relationsIdeal| (((|SuchThat| (|List| (|Polynomial| |#1|)) (|List| (|Equation| (|Polynomial| |#1|)))) (|List| |#4|)) "\\spad{relationsIdeal(polyList)} returns the ideal of relations among the polynomials in \\spad{polyList}.")) (|saturate| (($ $ |#4| (|List| |#3|)) "\\spad{saturate(I,f,lvar)} is the saturation with respect to the prime principal ideal which is generated by \\spad{f} in the polynomial ring \\spad{F[lvar]}.") (($ $ |#4|) "\\spad{saturate(I,f)} is the saturation of the ideal \\spad{I} with respect to the multiplicative set generated by the polynomial \\spad{f}.")) (|coerce| (($ (|List| |#4|)) "\\spad{coerce(polyList)} converts the list of polynomials \\spad{polyList} to an ideal.")) (|generators| (((|List| |#4|) $) "\\spad{generators(I)} returns a list of generators for the ideal \\spad{I}.")) (|groebner?| (((|Boolean|) $) "\\spad{groebner?(I)} tests if the generators of the ideal \\spad{I} are a Groebner basis.")) (|groebnerIdeal| (($ (|List| |#4|)) "\\spad{groebnerIdeal(polyList)} constructs the ideal generated by the list of polynomials \\spad{polyList} which are assumed to be a Groebner basis. Note: this operation avoids a Groebner basis computation.")) (|ideal| (($ (|List| |#4|)) "\\spad{ideal(polyList)} constructs the ideal generated by the list of polynomials \\spad{polyList}.")) (|leadingIdeal| (($ $) "\\spad{leadingIdeal(I)} is the ideal generated by the leading terms of the elements of the ideal \\spad{I}.")) (|dimension| (((|Integer|) $) "\\spad{dimension(I)} gives the dimension of the ideal \\spad{I}. in the ring \\spad{F[lvar]},{} where lvar are the variables appearing in \\spad{I}") (((|Integer|) $ (|List| |#3|)) "\\spad{dimension(I,lvar)} gives the dimension of the ideal \\spad{I},{} in the ring \\spad{F[lvar]}")) (|backOldPos| (($ (|Record| (|:| |mval| (|Matrix| |#1|)) (|:| |invmval| (|Matrix| |#1|)) (|:| |genIdeal| $))) "\\spad{backOldPos(genPos)} takes the result produced by \\spadfunFrom{generalPosition}{PolynomialIdeals} and performs the inverse transformation,{} returning the original ideal \\spad{backOldPos(generalPosition(I,listvar))} = \\spad{I}.")) (|generalPosition| (((|Record| (|:| |mval| (|Matrix| |#1|)) (|:| |invmval| (|Matrix| |#1|)) (|:| |genIdeal| $)) $ (|List| |#3|)) "\\spad{generalPosition(I,listvar)} perform a random linear transformation on the variables in \\spad{listvar} and returns the transformed ideal along with the change of basis matrix.")) (|groebner| (($ $) "\\spad{groebner(I)} returns a set of generators of \\spad{I} that are a Groebner basis for \\spad{I}.")) (|quotient| (($ $ |#4|) "\\spad{quotient(I,f)} computes the quotient of the ideal \\spad{I} by the principal ideal generated by the polynomial \\spad{f},{} \\spad{(I:(f))}.") (($ $ $) "\\spad{quotient(I,J)} computes the quotient of the ideals \\spad{I} and \\spad{J},{} \\spad{(I:J)}.")) (|intersect| (($ (|List| $)) "\\spad{intersect(LI)} computes the intersection of the list of ideals \\spad{LI}.") (($ $ $) "\\spad{intersect(I,J)} computes the intersection of the ideals \\spad{I} and \\spad{J}.")) (|zeroDim?| (((|Boolean|) $) "\\spad{zeroDim?(I)} tests if the ideal \\spad{I} is zero dimensional,{} \\spadignore{i.e.} all its associated primes are maximal,{} in the ring \\spad{F[lvar]},{} where lvar are the variables appearing in \\spad{I}") (((|Boolean|) $ (|List| |#3|)) "\\spad{zeroDim?(I,lvar)} tests if the ideal \\spad{I} is zero dimensional,{} \\spadignore{i.e.} all its associated primes are maximal,{} in the ring \\spad{F[lvar]}")) (|inRadical?| (((|Boolean|) |#4| $) "\\spad{inRadical?(f,I)} tests if some power of the polynomial \\spad{f} belongs to the ideal \\spad{I}.")) (|in?| (((|Boolean|) $ $) "\\spad{in?(I,J)} tests if the ideal \\spad{I} is contained in the ideal \\spad{J}.")) (|element?| (((|Boolean|) |#4| $) "\\spad{element?(f,I)} tests whether the polynomial \\spad{f} belongs to the ideal \\spad{I}.")) (|zero?| (((|Boolean|) $) "\\spad{zero?(I)} tests whether the ideal \\spad{I} is the zero ideal")) (|one?| (((|Boolean|) $) "\\spad{one?(I)} tests whether the ideal \\spad{I} is the unit ideal,{} \\spadignore{i.e.} contains 1.")) (+ (($ $ $) "\\spad{I+J} computes the ideal generated by the union of \\spad{I} and \\spad{J}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{I**n} computes the \\spad{n}th power of the ideal \\spad{I}.")) (* (($ $ $) "\\spad{I*J} computes the product of the ideal \\spad{I} and \\spad{J}."))) 
 NIL 
-((|HasCategory| |#3| (LIST (QUOTE -617) (QUOTE (-1181))))) 
-(-509 |vl| |nv|) 
+((|HasCategory| |#3| (LIST (QUOTE -619) (QUOTE (-1183))))) 
+(-510 |vl| |nv|) 
 ((|constructor| (NIL "\\indented{2}{This package provides functions for the primary decomposition of} polynomial ideals over the rational numbers. The ideals are members of the \\spadtype{PolynomialIdeals} domain,{} and the polynomial generators are required to be from the \\spadtype{DistributedMultivariatePolynomial} domain.")) (|contract| (((|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|List| (|OrderedVariableList| |#1|))) "\\spad{contract(I,lvar)} contracts the ideal \\spad{I} to the polynomial ring \\spad{F[lvar]}.")) (|primaryDecomp| (((|List| (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{primaryDecomp(I)} returns a list of primary ideals such that their intersection is the ideal \\spad{I}.")) (|radical| (((|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{radical(I)} returns the radical of the ideal \\spad{I}.")) (|prime?| (((|Boolean|) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{prime?(I)} tests if the ideal \\spad{I} is prime.")) (|zeroDimPrimary?| (((|Boolean|) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{zeroDimPrimary?(I)} tests if the ideal \\spad{I} is 0-dimensional primary.")) (|zeroDimPrime?| (((|Boolean|) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{zeroDimPrime?(I)} tests if the ideal \\spad{I} is a 0-dimensional prime."))) 
 NIL 
 NIL 
-(-510) 
+(-511) 
 ((|constructor| (NIL "This domain represents identifer AST. This domain differs from Symbol in that it does not support any form of scripting. A value of this domain is a plain old identifier. \\blankline")) (|gensym| (($) "\\spad{gensym()} returns a new identifier,{} different from any other identifier in the running system"))) 
 NIL 
 NIL 
-(-511 A S) 
+(-512 A S) 
 ((|constructor| (NIL "\\indented{1}{Indexed direct products of abelian groups over an abelian group \\spad{A} of} generators indexed by the ordered set \\spad{S}. All items have finite support: only non-zero terms are stored."))) 
 NIL 
 NIL 
-(-512 A S) 
+(-513 A S) 
 ((|constructor| (NIL "\\indented{1}{Indexed direct products of abelian monoids over an abelian monoid \\spad{A} of} generators indexed by the ordered set \\spad{S}. All items have finite support. Only non-zero terms are stored."))) 
 NIL 
 NIL 
-(-513 A S) 
+(-514 A S) 
 ((|constructor| (NIL "This category represents the direct product of some set with respect to an ordered indexing set.")) (|reductum| (($ $) "\\spad{reductum(z)} returns a new element created by removing the leading coefficient/support pair from the element \\spad{z}. Error: if \\spad{z} has no support.")) (|leadingSupport| ((|#2| $) "\\spad{leadingSupport(z)} returns the index of leading (with respect to the ordering on the indexing set) monomial of \\spad{z}. Error: if \\spad{z} has no support.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(z)} returns the coefficient of the leading (with respect to the ordering on the indexing set) monomial of \\spad{z}. Error: if \\spad{z} has no support.")) (|monomial| (($ |#1| |#2|) "\\spad{monomial(a,s)} constructs a direct product element with the \\spad{s} component set to \\spad{a}")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,z)} returns the new element created by applying the function \\spad{f} to each component of the direct product element \\spad{z}."))) 
 NIL 
 NIL 
-(-514 A S) 
+(-515 A S) 
 ((|constructor| (NIL "\\indented{1}{Indexed direct products of objects over a set \\spad{A}} of generators indexed by an ordered set \\spad{S}. All items have finite support."))) 
 NIL 
 NIL 
-(-515 A S) 
+(-516 A S) 
 ((|constructor| (NIL "\\indented{1}{Indexed direct products of ordered abelian monoids \\spad{A} of} generators indexed by the ordered set \\spad{S}. The inherited order is lexicographical. All items have finite support: only non-zero terms are stored."))) 
 NIL 
 NIL 
-(-516 A S) 
+(-517 A S) 
 ((|constructor| (NIL "\\indented{1}{Indexed direct products of ordered abelian monoid sups \\spad{A},{}} generators indexed by the ordered set \\spad{S}. All items have finite support: only non-zero terms are stored."))) 
 NIL 
 NIL 
-(-517 S A B) 
+(-518 S A B) 
 ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions. The difference between this and \\spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.")) (|eval| (($ $ (|List| |#2|) (|List| |#3|)) "\\spad{eval(f, [x1,...,xn], [v1,...,vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ |#2| |#3|) "\\spad{eval(f, x, v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) 
 NIL 
 NIL 
-(-518 A B) 
+(-519 A B) 
 ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions. The difference between this and \\spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.")) (|eval| (($ $ (|List| |#1|) (|List| |#2|)) "\\spad{eval(f, [x1,...,xn], [v1,...,vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ |#1| |#2|) "\\spad{eval(f, x, v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) 
 NIL 
 NIL 
-(-519 S E |un|) 
+(-520 S E |un|) 
 ((|constructor| (NIL "Internal implementation of a free abelian monoid."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-795)))) 
-(-520 S |mn|) 
+((|HasCategory| |#2| (QUOTE (-797)))) 
+(-521 S |mn|) 
 ((|constructor| (NIL "\\indented{1}{Author: Michael Monagan July/87,{} modified \\spad{SMW} June/91} A FlexibleArray is the notion of an array intended to allow for growth at the end only. Hence the following efficient operations \\indented{2}{\\spad{append(x,a)} meaning append item \\spad{x} at the end of the array \\spad{a}} \\indented{2}{\\spad{delete(a,n)} meaning delete the last item from the array \\spad{a}} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets.")) (|shrinkable| (((|Boolean|) (|Boolean|)) "\\spad{shrinkable(b)} sets the shrinkable attribute of flexible arrays to \\spad{b} and returns the previous value")) (|physicalLength!| (($ $ (|Integer|)) "\\spad{physicalLength!(x,n)} changes the physical length of \\spad{x} to be \\spad{n} and returns the new array.")) (|physicalLength| (((|NonNegativeInteger|) $) "\\spad{physicalLength(x)} returns the number of elements \\spad{x} can accomodate before growing")) (|flexibleArray| (($ (|List| |#1|)) "\\spad{flexibleArray(l)} creates a flexible array from the list of elements \\spad{l}"))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-521) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-522) 
 ((|constructor| (NIL "This domain represents AST for conditional expressions.")) (|elseBranch| (((|SpadAst|) $) "thenBranch(\\spad{e}) returns the `else-branch' of `e'.")) (|thenBranch| (((|SpadAst|) $) "\\spad{thenBranch(e)} returns the `then-branch' of `e'.")) (|condition| (((|SpadAst|) $) "\\spad{condition(e)} returns the condition of the if-expression `e'."))) 
 NIL 
 NIL 
-(-522 |p| |n|) 
+(-523 |p| |n|) 
 ((|constructor| (NIL "InnerFiniteField(\\spad{p},{}\\spad{n}) implements finite fields with \\spad{p**n} elements where \\spad{p} is assumed prime but does not check. For a version which checks that \\spad{p} is prime,{} see \\spadtype{FiniteField}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (|HasCategory| (-585 |#1|) (QUOTE (-145))) (|HasCategory| (-585 |#1|) (QUOTE (-371)))) (|HasCategory| (-585 |#1|) (QUOTE (-147))) (|HasCategory| (-585 |#1|) (QUOTE (-371))) (|HasCategory| (-585 |#1|) (QUOTE (-145)))) 
-(-523 R |mnRow| |mnCol| |Row| |Col|) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((-3969 (|HasCategory| (-586 |#1|) (QUOTE (-145))) (|HasCategory| (-586 |#1|) (QUOTE (-372)))) (|HasCategory| (-586 |#1|) (QUOTE (-147))) (|HasCategory| (-586 |#1|) (QUOTE (-372))) (|HasCategory| (-586 |#1|) (QUOTE (-145)))) 
+(-524 R |mnRow| |mnCol| |Row| |Col|) 
 ((|constructor| (NIL "\\indented{1}{This is an internal type which provides an implementation of} 2-dimensional arrays as PrimitiveArray\\spad{'s} of PrimitiveArray\\spad{'s}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-524 S |mn|) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-525 S |mn|) 
 ((|constructor| (NIL "\\spadtype{IndexedList} is a basic implementation of the functions in \\spadtype{ListAggregate},{} often using functions in the underlying LISP system. The second parameter to the constructor (\\spad{mn}) is the beginning index of the list. That is,{} if \\spad{l} is a list,{} then \\spad{elt(l,mn)} is the first value. This constructor is probably best viewed as the implementation of singly-linked lists that are addressable by index rather than as a mere wrapper for LISP lists."))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-525 R |Row| |Col| M) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-526 R |Row| |Col| M) 
 ((|constructor| (NIL "\\spadtype{InnerMatrixLinearAlgebraFunctions} is an internal package which provides standard linear algebra functions on domains in \\spad{MatrixCategory}")) (|inverse| (((|Union| |#4| "failed") |#4|) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|generalizedInverse| ((|#4| |#4|) "\\spad{generalizedInverse(m)} returns the generalized (Moore--Penrose) inverse of the matrix \\spad{m},{} \\spadignore{i.e.} the matrix \\spad{h} such that m*h*m=h,{} h*m*h=m,{} \\spad{m*h} and \\spad{h*m} are both symmetric matrices.")) (|determinant| ((|#1| |#4|) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. an error message is returned if the matrix is not square.")) (|nullSpace| (((|List| |#3|) |#4|) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) |#4|) "\\spad{nullity(m)} returns the mullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) |#4|) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| ((|#4| |#4|) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}."))) 
 NIL 
-((|HasAttribute| |#3| (QUOTE -4428))) 
-(-526 R |Row| |Col| M QF |Row2| |Col2| M2) 
+((|HasAttribute| |#3| (QUOTE -4435))) 
+(-527 R |Row| |Col| M QF |Row2| |Col2| M2) 
 ((|constructor| (NIL "\\spadtype{InnerMatrixQuotientFieldFunctions} provides functions on matrices over an integral domain which involve the quotient field of that integral domain. The functions rowEchelon and inverse return matrices with entries in the quotient field.")) (|nullSpace| (((|List| |#3|) |#4|) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|inverse| (((|Union| |#8| "failed") |#4|) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square. Note: the result will have entries in the quotient field.")) (|rowEchelon| ((|#8| |#4|) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}. the result will have entries in the quotient field."))) 
 NIL 
-((|HasAttribute| |#7| (QUOTE -4428))) 
-(-527 R |mnRow| |mnCol|) 
+((|HasAttribute| |#7| (QUOTE -4435))) 
+(-528 R |mnRow| |mnCol|) 
 ((|constructor| (NIL "An \\spad{IndexedMatrix} is a matrix where the minimal row and column indices are parameters of the type. The domains Row and Col are both IndexedVectors. The index of the 'first' row may be obtained by calling the function \\spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \\spadfun{minColIndex}. The index of the first element of a 'Row' is the same as the index of the first column in a matrix and vice versa."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-561))) (|HasAttribute| |#1| (QUOTE (-4429 "*"))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-528) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4436 "*"))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-529) 
 ((|constructor| (NIL "This domain represents an `import' of types.")) (|imports| (((|List| (|TypeAst|)) $) "\\spad{imports(x)} returns the list of imported types.")) (|coerce| (($ (|List| (|TypeAst|))) "ts::ImportAst constructs an ImportAst for the list if types `ts'."))) 
 NIL 
 NIL 
-(-529) 
+(-530) 
 ((|constructor| (NIL "This domain represents the `in' iterator syntax.")) (|sequence| (((|SpadAst|) $) "\\spad{sequence(i)} returns the sequence expression being iterated over by `i'.")) (|iterationVar| (((|Identifier|) $) "\\spad{iterationVar(i)} returns the name of the iterating variable of the `in' iterator 'i'"))) 
 NIL 
 NIL 
-(-530 S) 
+(-531 S) 
 ((|constructor| (NIL "This category describes input byte stream conduits.")) (|readBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{readBytes!(c,b)} reads byte sequences from conduit \\spad{`c'} into the byte buffer \\spad{`b'}. The actual number of bytes written is returned,{} and the length of \\spad{`b'} is set to that amount.")) (|readUInt32!| (((|Maybe| (|UInt32|)) $) "\\spad{readUInt32!(cond)} attempts to read a UInt32 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt32!| (((|Maybe| (|Int32|)) $) "\\spad{readInt32!(cond)} attempts to read an Int32 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt16!| (((|Maybe| (|UInt16|)) $) "\\spad{readUInt16!(cond)} attempts to read a UInt16 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt16!| (((|Maybe| (|Int16|)) $) "\\spad{readInt16!(cond)} attempts to read an Int16 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt8!| (((|Maybe| (|UInt8|)) $) "\\spad{readUInt8!(cond)} attempts to read a UInt8 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt8!| (((|Maybe| (|Int8|)) $) "\\spad{readInt8!(cond)} attempts to read an Int8 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readByte!| (((|Maybe| (|Byte|)) $) "\\spad{readByte!(cond)} attempts to read a byte from the input conduit `cond'. Returns the read byte if successful,{} otherwise \\spad{nothing}."))) 
 NIL 
 NIL 
-(-531) 
+(-532) 
 ((|constructor| (NIL "This category describes input byte stream conduits.")) (|readBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{readBytes!(c,b)} reads byte sequences from conduit \\spad{`c'} into the byte buffer \\spad{`b'}. The actual number of bytes written is returned,{} and the length of \\spad{`b'} is set to that amount.")) (|readUInt32!| (((|Maybe| (|UInt32|)) $) "\\spad{readUInt32!(cond)} attempts to read a UInt32 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt32!| (((|Maybe| (|Int32|)) $) "\\spad{readInt32!(cond)} attempts to read an Int32 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt16!| (((|Maybe| (|UInt16|)) $) "\\spad{readUInt16!(cond)} attempts to read a UInt16 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt16!| (((|Maybe| (|Int16|)) $) "\\spad{readInt16!(cond)} attempts to read an Int16 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt8!| (((|Maybe| (|UInt8|)) $) "\\spad{readUInt8!(cond)} attempts to read a UInt8 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt8!| (((|Maybe| (|Int8|)) $) "\\spad{readInt8!(cond)} attempts to read an Int8 value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readByte!| (((|Maybe| (|Byte|)) $) "\\spad{readByte!(cond)} attempts to read a byte from the input conduit `cond'. Returns the read byte if successful,{} otherwise \\spad{nothing}."))) 
 NIL 
 NIL 
-(-532 GF) 
+(-533 GF) 
 ((|constructor| (NIL "InnerNormalBasisFieldFunctions(\\spad{GF}) (unexposed): This package has functions used by every normal basis finite field extension domain.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) (|Vector| |#1|)) "\\spad{minimalPolynomial(x)} \\undocumented{} See \\axiomFunFrom{minimalPolynomial}{FiniteAlgebraicExtensionField}")) (|normalElement| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{normalElement(n)} \\undocumented{} See \\axiomFunFrom{normalElement}{FiniteAlgebraicExtensionField}")) (|basis| (((|Vector| (|Vector| |#1|)) (|PositiveInteger|)) "\\spad{basis(n)} \\undocumented{} See \\axiomFunFrom{basis}{FiniteAlgebraicExtensionField}")) (|normal?| (((|Boolean|) (|Vector| |#1|)) "\\spad{normal?(x)} \\undocumented{} See \\axiomFunFrom{normal?}{FiniteAlgebraicExtensionField}")) (|lookup| (((|PositiveInteger|) (|Vector| |#1|)) "\\spad{lookup(x)} \\undocumented{} See \\axiomFunFrom{lookup}{Finite}")) (|inv| (((|Vector| |#1|) (|Vector| |#1|)) "\\spad{inv x} \\undocumented{} See \\axiomFunFrom{inv}{DivisionRing}")) (|trace| (((|Vector| |#1|) (|Vector| |#1|) (|PositiveInteger|)) "\\spad{trace(x,n)} \\undocumented{} See \\axiomFunFrom{trace}{FiniteAlgebraicExtensionField}")) (|norm| (((|Vector| |#1|) (|Vector| |#1|) (|PositiveInteger|)) "\\spad{norm(x,n)} \\undocumented{} See \\axiomFunFrom{norm}{FiniteAlgebraicExtensionField}")) (/ (((|Vector| |#1|) (|Vector| |#1|) (|Vector| |#1|)) "\\spad{x/y} \\undocumented{} See \\axiomFunFrom{/}{Field}")) (* (((|Vector| |#1|) (|Vector| |#1|) (|Vector| |#1|)) "\\spad{x*y} \\undocumented{} See \\axiomFunFrom{*}{SemiGroup}")) (** (((|Vector| |#1|) (|Vector| |#1|) (|Integer|)) "\\spad{x**n} \\undocumented{} See \\axiomFunFrom{\\spad{**}}{DivisionRing}")) (|qPot| (((|Vector| |#1|) (|Vector| |#1|) (|Integer|)) "\\spad{qPot(v,e)} computes \\spad{v**(q**e)},{} interpreting \\spad{v} as an element of normal basis field,{} \\spad{q} the size of the ground field. This is done by a cyclic \\spad{e}-shift of the vector \\spad{v}.")) (|expPot| (((|Vector| |#1|) (|Vector| |#1|) (|SingleInteger|) (|SingleInteger|)) "\\spad{expPot(v,e,d)} returns the sum from \\spad{i = 0} to \\spad{e - 1} of \\spad{v**(q**i*d)},{} interpreting \\spad{v} as an element of a normal basis field and where \\spad{q} is the size of the ground field. Note: for a description of the algorithm,{} see \\spad{T}.Itoh and \\spad{S}.Tsujii,{} \"A fast algorithm for computing multiplicative inverses in \\spad{GF}(2^m) using normal bases\",{} Information and Computation 78,{} \\spad{pp}.171-177,{} 1988.")) (|repSq| (((|Vector| |#1|) (|Vector| |#1|) (|NonNegativeInteger|)) "\\spad{repSq(v,e)} computes \\spad{v**e} by repeated squaring,{} interpreting \\spad{v} as an element of a normal basis field.")) (|dAndcExp| (((|Vector| |#1|) (|Vector| |#1|) (|NonNegativeInteger|) (|SingleInteger|)) "\\spad{dAndcExp(v,n,k)} computes \\spad{v**e} interpreting \\spad{v} as an element of normal basis field. A divide and conquer algorithm similar to the one from \\spad{D}.\\spad{R}.Stinson,{} \"Some observations on parallel Algorithms for fast exponentiation in \\spad{GF}(2^n)\",{} Siam \\spad{J}. Computation,{} Vol.19,{} No.4,{} \\spad{pp}.711-717,{} August 1990 is used. Argument \\spad{k} is a parameter of this algorithm.")) (|xn| (((|SparseUnivariatePolynomial| |#1|) (|NonNegativeInteger|)) "\\spad{xn(n)} returns the polynomial \\spad{x**n-1}.")) (|pol| (((|SparseUnivariatePolynomial| |#1|) (|Vector| |#1|)) "\\spad{pol(v)} turns the vector \\spad{[v0,...,vn]} into the polynomial \\spad{v0+v1*x+ ... + vn*x**n}.")) (|index| (((|Vector| |#1|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{index(n,m)} is a index function for vectors of length \\spad{n} over the ground field.")) (|random| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{random(n)} creates a vector over the ground field with random entries.")) (|setFieldInfo| (((|Void|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) |#1|) "\\spad{setFieldInfo(m,p)} initializes the field arithmetic,{} where \\spad{m} is the multiplication table and \\spad{p} is the respective normal element of the ground field \\spad{GF}."))) 
 NIL 
 NIL 
-(-533) 
+(-534) 
 ((|constructor| (NIL "This domain provides representation for binary files open for input operations. `Binary' here means that the conduits do not interpret their contents.")) (|position!| (((|SingleInteger|) $ (|SingleInteger|)) "position(\\spad{f},{}\\spad{p}) sets the current byte-position to `i'.")) (|position| (((|SingleInteger|) $) "\\spad{position(f)} returns the current byte-position in the file \\spad{`f'}.")) (|isOpen?| (((|Boolean|) $) "\\spad{isOpen?(ifile)} holds if `ifile' is in open state.")) (|eof?| (((|Boolean|) $) "\\spad{eof?(ifile)} holds when the last read reached end of file.")) (|inputBinaryFile| (($ (|String|)) "\\spad{inputBinaryFile(f)} returns an input conduit obtained by opening the file named by \\spad{`f'} as a binary file.") (($ (|FileName|)) "\\spad{inputBinaryFile(f)} returns an input conduit obtained by opening the file named by \\spad{`f'} as a binary file."))) 
 NIL 
 NIL 
-(-534 R) 
+(-535 R) 
 ((|constructor| (NIL "This package provides operations to create incrementing functions.")) (|incrementBy| (((|Mapping| |#1| |#1|) |#1|) "\\spad{incrementBy(n)} produces a function which adds \\spad{n} to whatever argument it is given. For example,{} if {\\spad{f} \\spad{:=} increment(\\spad{n})} then \\spad{f x} is \\spad{x+n}.")) (|increment| (((|Mapping| |#1| |#1|)) "\\spad{increment()} produces a function which adds \\spad{1} to whatever argument it is given. For example,{} if {\\spad{f} \\spad{:=} increment()} then \\spad{f x} is \\spad{x+1}."))) 
 NIL 
 NIL 
-(-535 |Varset|) 
+(-536 |Varset|) 
 ((|constructor| (NIL "\\indented{2}{IndexedExponents of an ordered set of variables gives a representation} for the degree of polynomials in commuting variables. It gives an ordered pairing of non negative integer exponents with variables"))) 
 NIL 
 NIL 
-(-536 K -3498 |Par|) 
+(-537 K -3505 |Par|) 
 ((|constructor| (NIL "This package is the inner package to be used by NumericRealEigenPackage and NumericComplexEigenPackage for the computation of numeric eigenvalues and eigenvectors.")) (|innerEigenvectors| (((|List| (|Record| (|:| |outval| |#2|) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| |#2|))))) (|Matrix| |#1|) |#3| (|Mapping| (|Factored| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|))) "\\spad{innerEigenvectors(m,eps,factor)} computes explicitly the eigenvalues and the correspondent eigenvectors of the matrix \\spad{m}. The parameter \\spad{eps} determines the type of the output,{} \\spad{factor} is the univariate factorizer to \\spad{br} used to reduce the characteristic polynomial into irreducible factors.")) (|solve1| (((|List| |#2|) (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{solve1(pol, eps)} finds the roots of the univariate polynomial polynomial \\spad{pol} to precision eps. If \\spad{K} is \\spad{Fraction Integer} then only the real roots are returned,{} if \\spad{K} is \\spad{Complex Fraction Integer} then all roots are found.")) (|charpol| (((|SparseUnivariatePolynomial| |#1|) (|Matrix| |#1|)) "\\spad{charpol(m)} computes the characteristic polynomial of a matrix \\spad{m} with entries in \\spad{K}. This function returns a polynomial over \\spad{K},{} while the general one (that is in EiegenPackage) returns Fraction \\spad{P} \\spad{K}"))) 
 NIL 
 NIL 
-(-537) 
+(-538) 
 NIL 
 NIL 
 NIL 
-(-538) 
+(-539) 
 ((|constructor| (NIL "Default infinity signatures for the interpreter; Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989")) (|minusInfinity| (((|OrderedCompletion| (|Integer|))) "\\spad{minusInfinity()} returns minusInfinity.")) (|plusInfinity| (((|OrderedCompletion| (|Integer|))) "\\spad{plusInfinity()} returns plusIinfinity.")) (|infinity| (((|OnePointCompletion| (|Integer|))) "\\spad{infinity()} returns infinity."))) 
 NIL 
 NIL 
-(-539) 
+(-540) 
 ((|constructor| (NIL "Domain of parsed forms which can be passed to the interpreter. This is also the interface between algebra code and facilities in the interpreter.")) (|compile| (((|Symbol|) (|Symbol|) (|List| $)) "\\spad{compile(f, [t1,...,tn])} forces the interpreter to compile the function \\spad{f} with signature \\spad{(t1,...,tn) -> ?}. returns the symbol \\spad{f} if successful. Error: if \\spad{f} was not defined beforehand in the interpreter,{} or if the \\spad{ti}\\spad{'s} are not valid types,{} or if the compiler fails.")) (|declare| (((|Symbol|) (|List| $)) "\\spad{declare(t)} returns a name \\spad{f} such that \\spad{f} has been declared to the interpreter to be of type \\spad{t},{} but has not been assigned a value yet. Note: \\spad{t} should be created as \\spad{devaluate(T)\\$Lisp} where \\spad{T} is the actual type of \\spad{f} (this hack is required for the case where \\spad{T} is a mapping type).")) (|parseString| (($ (|String|)) "parseString is the inverse of unparse. It parses a string to InputForm.")) (|unparse| (((|String|) $) "\\spad{unparse(f)} returns a string \\spad{s} such that the parser would transform \\spad{s} to \\spad{f}. Error: if \\spad{f} is not the parsed form of a string.")) (|flatten| (($ $) "\\spad{flatten(s)} returns an input form corresponding to \\spad{s} with all the nested operations flattened to triples using new local variables. If \\spad{s} is a piece of code,{} this speeds up the compilation tremendously later on.")) ((|One|) (($) "\\spad{1} returns the input form corresponding to 1.")) ((|Zero|) (($) "\\spad{0} returns the input form corresponding to 0.")) (** (($ $ (|Integer|)) "\\spad{a ** b} returns the input form corresponding to \\spad{a ** b}.") (($ $ (|NonNegativeInteger|)) "\\spad{a ** b} returns the input form corresponding to \\spad{a ** b}.")) (/ (($ $ $) "\\spad{a / b} returns the input form corresponding to \\spad{a / b}.")) (* (($ $ $) "\\spad{a * b} returns the input form corresponding to \\spad{a * b}.")) (+ (($ $ $) "\\spad{a + b} returns the input form corresponding to \\spad{a + b}.")) (|lambda| (($ $ (|List| (|Symbol|))) "\\spad{lambda(code, [x1,...,xn])} returns the input form corresponding to \\spad{(x1,...,xn) +-> code} if \\spad{n > 1},{} or to \\spad{x1 +-> code} if \\spad{n = 1}.")) (|function| (($ $ (|List| (|Symbol|)) (|Symbol|)) "\\spad{function(code, [x1,...,xn], f)} returns the input form corresponding to \\spad{f(x1,...,xn) == code}.")) (|binary| (($ $ (|List| $)) "\\spad{binary(op, [a1,...,an])} returns the input form corresponding to \\spad{a1 op a2 op ... op an}.")) (|convert| (($ (|SExpression|)) "\\spad{convert(s)} makes \\spad{s} into an input form.")) (|interpret| (((|Any|) $) "\\spad{interpret(f)} passes \\spad{f} to the interpreter."))) 
 NIL 
 NIL 
-(-540 R) 
+(-541 R) 
 ((|constructor| (NIL "Tools for manipulating input forms.")) (|interpret| ((|#1| (|InputForm|)) "\\spad{interpret(f)} passes \\spad{f} to the interpreter,{} and transforms the result into an object of type \\spad{R}.")) (|packageCall| (((|InputForm|) (|Symbol|)) "\\spad{packageCall(f)} returns the input form corresponding to \\spad{f}\\$\\spad{R}."))) 
 NIL 
 NIL 
-(-541 |Coef| UTS) 
+(-542 |Coef| UTS) 
 ((|constructor| (NIL "This package computes infinite products of univariate Taylor series over an integral domain of characteristic 0.")) (|generalInfiniteProduct| ((|#2| |#2| (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| ((|#2| |#2|) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| ((|#2| |#2|) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| ((|#2| |#2|) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) 
 NIL 
 NIL 
-(-542 K -3498 |Par|) 
+(-543 K -3505 |Par|) 
 ((|constructor| (NIL "This is an internal package for computing approximate solutions to systems of polynomial equations. The parameter \\spad{K} specifies the coefficient field of the input polynomials and must be either \\spad{Fraction(Integer)} or \\spad{Complex(Fraction Integer)}. The parameter \\spad{F} specifies where the solutions must lie and can be one of the following: \\spad{Float},{} \\spad{Fraction(Integer)},{} \\spad{Complex(Float)},{} \\spad{Complex(Fraction Integer)}. The last parameter specifies the type of the precision operand and must be either \\spad{Fraction(Integer)} or \\spad{Float}.")) (|makeEq| (((|List| (|Equation| (|Polynomial| |#2|))) (|List| |#2|) (|List| (|Symbol|))) "\\spad{makeEq(lsol,lvar)} returns a list of equations formed by corresponding members of \\spad{lvar} and \\spad{lsol}.")) (|innerSolve| (((|List| (|List| |#2|)) (|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|)) |#3|) "\\spad{innerSolve(lnum,lden,lvar,eps)} returns a list of solutions of the system of polynomials \\spad{lnum},{} with the side condition that none of the members of \\spad{lden} vanish identically on any solution. Each solution is expressed as a list corresponding to the list of variables in \\spad{lvar} and with precision specified by \\spad{eps}.")) (|innerSolve1| (((|List| |#2|) (|Polynomial| |#1|) |#3|) "\\spad{innerSolve1(p,eps)} returns the list of the zeros of the polynomial \\spad{p} with precision \\spad{eps}.") (((|List| |#2|) (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{innerSolve1(up,eps)} returns the list of the zeros of the univariate polynomial \\spad{up} with precision \\spad{eps}."))) 
 NIL 
 NIL 
-(-543 R BP |pMod| |nextMod|) 
+(-544 R BP |pMod| |nextMod|) 
 ((|reduction| ((|#2| |#2| |#1|) "\\spad{reduction(f,p)} reduces the coefficients of the polynomial \\spad{f} modulo the prime \\spad{p}.")) (|modularGcd| ((|#2| (|List| |#2|)) "\\spad{modularGcd(listf)} computes the \\spad{gcd} of the list of polynomials \\spad{listf} by modular methods.")) (|modularGcdPrimitive| ((|#2| (|List| |#2|)) "\\spad{modularGcdPrimitive(f1,f2)} computes the \\spad{gcd} of the two polynomials \\spad{f1} and \\spad{f2} by modular methods."))) 
 NIL 
 NIL 
-(-544 OV E R P) 
+(-545 OV E R P) 
 ((|constructor| (NIL "\\indented{2}{This is an inner package for factoring multivariate polynomials} over various coefficient domains in characteristic 0. The univariate factor operation is passed as a parameter. Multivariate hensel lifting is used to lift the univariate factorization")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|) (|Mapping| (|Factored| (|SparseUnivariatePolynomial| |#3|)) (|SparseUnivariatePolynomial| |#3|))) "\\spad{factor(p,ufact)} factors the multivariate polynomial \\spad{p} by specializing variables and calling the univariate factorizer \\spad{ufact}. \\spad{p} is represented as a univariate polynomial with multivariate coefficients.") (((|Factored| |#4|) |#4| (|Mapping| (|Factored| (|SparseUnivariatePolynomial| |#3|)) (|SparseUnivariatePolynomial| |#3|))) "\\spad{factor(p,ufact)} factors the multivariate polynomial \\spad{p} by specializing variables and calling the univariate factorizer \\spad{ufact}."))) 
 NIL 
 NIL 
-(-545 K UP |Coef| UTS) 
+(-546 K UP |Coef| UTS) 
 ((|constructor| (NIL "This package computes infinite products of univariate Taylor series over an arbitrary finite field.")) (|generalInfiniteProduct| ((|#4| |#4| (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| ((|#4| |#4|) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| ((|#4| |#4|) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| ((|#4| |#4|) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) 
 NIL 
 NIL 
-(-546 |Coef| UTS) 
+(-547 |Coef| UTS) 
 ((|constructor| (NIL "This package computes infinite products of univariate Taylor series over a field of prime order.")) (|generalInfiniteProduct| ((|#2| |#2| (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| ((|#2| |#2|) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| ((|#2| |#2|) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| ((|#2| |#2|) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) 
 NIL 
 NIL 
-(-547 R UP) 
+(-548 R UP) 
 ((|constructor| (NIL "Find the sign of a polynomial around a point or infinity.")) (|signAround| (((|Union| (|Integer|) #1="failed") |#2| |#1| (|Mapping| (|Union| (|Integer|) #1#) |#1|)) "\\spad{signAround(u,r,f)} \\undocumented") (((|Union| (|Integer|) #1#) |#2| |#1| (|Integer|) (|Mapping| (|Union| (|Integer|) #1#) |#1|)) "\\spad{signAround(u,r,i,f)} \\undocumented") (((|Union| (|Integer|) #1#) |#2| (|Integer|) (|Mapping| (|Union| (|Integer|) #1#) |#1|)) "\\spad{signAround(u,i,f)} \\undocumented"))) 
 NIL 
 NIL 
-(-548 S) 
+(-549 S) 
 ((|constructor| (NIL "An \\spad{IntegerNumberSystem} is a model for the integers.")) (|invmod| (($ $ $) "\\spad{invmod(a,b)},{} \\spad{0<=a<b>1},{} \\spad{(a,b)=1} means \\spad{1/a mod b}.")) (|powmod| (($ $ $ $) "\\spad{powmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a**b mod p}.")) (|mulmod| (($ $ $ $) "\\spad{mulmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a*b mod p}.")) (|submod| (($ $ $ $) "\\spad{submod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a-b mod p}.")) (|addmod| (($ $ $ $) "\\spad{addmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a+b mod p}.")) (|mask| (($ $) "\\spad{mask(n)} returns \\spad{2**n-1} (an \\spad{n} bit mask).")) (|dec| (($ $) "\\spad{dec(x)} returns \\spad{x - 1}.")) (|inc| (($ $) "\\spad{inc(x)} returns \\spad{x + 1}.")) (|copy| (($ $) "\\spad{copy(n)} gives a copy of \\spad{n}.")) (|random| (($ $) "\\spad{random(a)} creates a random element from 0 to \\spad{a-1}.") (($) "\\spad{random()} creates a random element.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(n)} creates a rational number,{} or returns \"failed\" if this is not possible.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(n)} creates a rational number (see \\spadtype{Fraction Integer})..")) (|rational?| (((|Boolean|) $) "\\spad{rational?(n)} tests if \\spad{n} is a rational number (see \\spadtype{Fraction Integer}).")) (|symmetricRemainder| (($ $ $) "\\spad{symmetricRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{ -b/2 <= r < b/2 }.")) (|positiveRemainder| (($ $ $) "\\spad{positiveRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{0 <= r < b} and \\spad{r == a rem b}.")) (|bit?| (((|Boolean|) $ $) "\\spad{bit?(n,i)} returns \\spad{true} if and only if \\spad{i}-th bit of \\spad{n} is a 1.")) (|shift| (($ $ $) "\\spad{shift(a,i)} shift \\spad{a} by \\spad{i} digits.")) (|length| (($ $) "\\spad{length(a)} length of \\spad{a} in digits.")) (|base| (($) "\\spad{base()} returns the base for the operations of \\spad{IntegerNumberSystem}.")) (|multiplicativeValuation| ((|attribute|) "euclideanSize(a*b) returns \\spad{euclideanSize(a)*euclideanSize(b)}.")) (|even?| (((|Boolean|) $) "\\spad{even?(n)} returns \\spad{true} if and only if \\spad{n} is even.")) (|odd?| (((|Boolean|) $) "\\spad{odd?(n)} returns \\spad{true} if and only if \\spad{n} is odd."))) 
 NIL 
 NIL 
-(-549) 
+(-550) 
 ((|constructor| (NIL "An \\spad{IntegerNumberSystem} is a model for the integers.")) (|invmod| (($ $ $) "\\spad{invmod(a,b)},{} \\spad{0<=a<b>1},{} \\spad{(a,b)=1} means \\spad{1/a mod b}.")) (|powmod| (($ $ $ $) "\\spad{powmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a**b mod p}.")) (|mulmod| (($ $ $ $) "\\spad{mulmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a*b mod p}.")) (|submod| (($ $ $ $) "\\spad{submod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a-b mod p}.")) (|addmod| (($ $ $ $) "\\spad{addmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a+b mod p}.")) (|mask| (($ $) "\\spad{mask(n)} returns \\spad{2**n-1} (an \\spad{n} bit mask).")) (|dec| (($ $) "\\spad{dec(x)} returns \\spad{x - 1}.")) (|inc| (($ $) "\\spad{inc(x)} returns \\spad{x + 1}.")) (|copy| (($ $) "\\spad{copy(n)} gives a copy of \\spad{n}.")) (|random| (($ $) "\\spad{random(a)} creates a random element from 0 to \\spad{a-1}.") (($) "\\spad{random()} creates a random element.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(n)} creates a rational number,{} or returns \"failed\" if this is not possible.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(n)} creates a rational number (see \\spadtype{Fraction Integer})..")) (|rational?| (((|Boolean|) $) "\\spad{rational?(n)} tests if \\spad{n} is a rational number (see \\spadtype{Fraction Integer}).")) (|symmetricRemainder| (($ $ $) "\\spad{symmetricRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{ -b/2 <= r < b/2 }.")) (|positiveRemainder| (($ $ $) "\\spad{positiveRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{0 <= r < b} and \\spad{r == a rem b}.")) (|bit?| (((|Boolean|) $ $) "\\spad{bit?(n,i)} returns \\spad{true} if and only if \\spad{i}-th bit of \\spad{n} is a 1.")) (|shift| (($ $ $) "\\spad{shift(a,i)} shift \\spad{a} by \\spad{i} digits.")) (|length| (($ $) "\\spad{length(a)} length of \\spad{a} in digits.")) (|base| (($) "\\spad{base()} returns the base for the operations of \\spad{IntegerNumberSystem}.")) (|multiplicativeValuation| ((|attribute|) "euclideanSize(a*b) returns \\spad{euclideanSize(a)*euclideanSize(b)}.")) (|even?| (((|Boolean|) $) "\\spad{even?(n)} returns \\spad{true} if and only if \\spad{n} is even.")) (|odd?| (((|Boolean|) $) "\\spad{odd?(n)} returns \\spad{true} if and only if \\spad{n} is odd."))) 
-((-4425 . T) (-4426 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4432 . T) (-4433 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-550) 
+(-551) 
 ((|constructor| (NIL "\\spadtype{Integer} provides the domain of arbitrary precision integers.")) (|infinite| ((|attribute|) "nextItem never returns \"failed\".")) (|noetherian| ((|attribute|) "ascending chain condition on ideals.")) (|canonicalsClosed| ((|attribute|) "two positives multiply to give positive.")) (|canonical| ((|attribute|) "mathematical equality is data structure equality."))) 
-((-4409 . T) (-4415 . T) (-4419 . T) (-4414 . T) (-4425 . T) (-4426 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4416 . T) (-4422 . T) (-4426 . T) (-4421 . T) (-4432 . T) (-4433 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-551) 
+(-552) 
 ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 16 bits."))) 
 NIL 
 NIL 
-(-552) 
+(-553) 
 ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 32 bits."))) 
 NIL 
 NIL 
-(-553) 
+(-554) 
 ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 64 bits."))) 
 NIL 
 NIL 
-(-554) 
+(-555) 
 ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 8 bits."))) 
 NIL 
 NIL 
-(-555 |Key| |Entry| |addDom|) 
+(-556 |Key| |Entry| |addDom|) 
 ((|constructor| (NIL "This domain is used to provide a conditional \"add\" domain for the implementation of \\spadtype{Table}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-1105))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-556 R -3498) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-1107))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-557 R -3505) 
 ((|constructor| (NIL "This package provides functions for the integration of algebraic integrands over transcendental functions.")) (|algint| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|SparseUnivariatePolynomial| |#2|) (|SparseUnivariatePolynomial| |#2|))) "\\spad{algint(f, x, y, d)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x}; \\spad{d} is the derivation to use on \\spad{k[x]}."))) 
 NIL 
 NIL 
-(-557 R0 -3498 UP UPUP R) 
+(-558 R0 -3505 UP UPUP R) 
 ((|constructor| (NIL "This package provides functions for integrating a function on an algebraic curve.")) (|palginfieldint| (((|Union| |#5| "failed") |#5| (|Mapping| |#3| |#3|)) "\\spad{palginfieldint(f, d)} returns an algebraic function \\spad{g} such that \\spad{dg = f} if such a \\spad{g} exists,{} \"failed\" otherwise. Argument \\spad{f} must be a pure algebraic function.")) (|palgintegrate| (((|IntegrationResult| |#5|) |#5| (|Mapping| |#3| |#3|)) "\\spad{palgintegrate(f, d)} integrates \\spad{f} with respect to the derivation \\spad{d}. Argument \\spad{f} must be a pure algebraic function.")) (|algintegrate| (((|IntegrationResult| |#5|) |#5| (|Mapping| |#3| |#3|)) "\\spad{algintegrate(f, d)} integrates \\spad{f} with respect to the derivation \\spad{d}."))) 
 NIL 
 NIL 
-(-558) 
+(-559) 
 ((|constructor| (NIL "This package provides functions to lookup bits in integers")) (|bitTruth| (((|Boolean|) (|Integer|) (|Integer|)) "\\spad{bitTruth(n,m)} returns \\spad{true} if coefficient of 2**m in abs(\\spad{n}) is 1")) (|bitCoef| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{bitCoef(n,m)} returns the coefficient of 2**m in abs(\\spad{n})")) (|bitLength| (((|Integer|) (|Integer|)) "\\spad{bitLength(n)} returns the number of bits to represent abs(\\spad{n})"))) 
 NIL 
 NIL 
-(-559 R) 
+(-560 R) 
 ((|constructor| (NIL "\\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This category implements of interval arithmetic and transcendental + functions over intervals.")) (|contains?| (((|Boolean|) $ |#1|) "\\spad{contains?(i,f)} returns \\spad{true} if \\axiom{\\spad{f}} is contained within the interval \\axiom{\\spad{i}},{} \\spad{false} otherwise.")) (|negative?| (((|Boolean|) $) "\\spad{negative?(u)} returns \\axiom{\\spad{true}} if every element of \\spad{u} is negative,{} \\axiom{\\spad{false}} otherwise.")) (|positive?| (((|Boolean|) $) "\\spad{positive?(u)} returns \\axiom{\\spad{true}} if every element of \\spad{u} is positive,{} \\axiom{\\spad{false}} otherwise.")) (|width| ((|#1| $) "\\spad{width(u)} returns \\axiom{sup(\\spad{u}) - inf(\\spad{u})}.")) (|sup| ((|#1| $) "\\spad{sup(u)} returns the supremum of \\axiom{\\spad{u}}.")) (|inf| ((|#1| $) "\\spad{inf(u)} returns the infinum of \\axiom{\\spad{u}}.")) (|qinterval| (($ |#1| |#1|) "\\spad{qinterval(inf,sup)} creates a new interval \\axiom{[\\spad{inf},{}\\spad{sup}]},{} without checking the ordering on the elements.")) (|interval| (($ (|Fraction| (|Integer|))) "\\spad{interval(f)} creates a new interval around \\spad{f}.") (($ |#1|) "\\spad{interval(f)} creates a new interval around \\spad{f}.") (($ |#1| |#1|) "\\spad{interval(inf,sup)} creates a new interval,{} either \\axiom{[\\spad{inf},{}\\spad{sup}]} if \\axiom{\\spad{inf} \\spad{<=} \\spad{sup}} or \\axiom{[\\spad{sup},{}in]} otherwise."))) 
-((-4203 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4210 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-560 S) 
+(-561 S) 
 ((|constructor| (NIL "The category of commutative integral domains,{} \\spadignore{i.e.} commutative rings with no zero divisors. \\blankline Conditional attributes: \\indented{2}{canonicalUnitNormal\\tab{20}the canonical field is the same for all associates} \\indented{2}{canonicalsClosed\\tab{20}the product of two canonicals is itself canonical}")) (|unit?| (((|Boolean|) $) "\\spad{unit?(x)} tests whether \\spad{x} is a unit,{} \\spadignore{i.e.} is invertible.")) (|associates?| (((|Boolean|) $ $) "\\spad{associates?(x,y)} tests whether \\spad{x} and \\spad{y} are associates,{} \\spadignore{i.e.} differ by a unit factor.")) (|unitCanonical| (($ $) "\\spad{unitCanonical(x)} returns \\spad{unitNormal(x).canonical}.")) (|unitNormal| (((|Record| (|:| |unit| $) (|:| |canonical| $) (|:| |associate| $)) $) "\\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \\spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the \"canonical\" element is the same across all associates of \\spad{x} if \\spad{unitNormal(x) = [u,c,a]} then \\spad{u*c = x},{} \\spad{a*u = 1}.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} either returns an element \\spad{c} such that \\spad{c*b=a} or \"failed\" if no such element can be found."))) 
 NIL 
 NIL 
-(-561) 
+(-562) 
 ((|constructor| (NIL "The category of commutative integral domains,{} \\spadignore{i.e.} commutative rings with no zero divisors. \\blankline Conditional attributes: \\indented{2}{canonicalUnitNormal\\tab{20}the canonical field is the same for all associates} \\indented{2}{canonicalsClosed\\tab{20}the product of two canonicals is itself canonical}")) (|unit?| (((|Boolean|) $) "\\spad{unit?(x)} tests whether \\spad{x} is a unit,{} \\spadignore{i.e.} is invertible.")) (|associates?| (((|Boolean|) $ $) "\\spad{associates?(x,y)} tests whether \\spad{x} and \\spad{y} are associates,{} \\spadignore{i.e.} differ by a unit factor.")) (|unitCanonical| (($ $) "\\spad{unitCanonical(x)} returns \\spad{unitNormal(x).canonical}.")) (|unitNormal| (((|Record| (|:| |unit| $) (|:| |canonical| $) (|:| |associate| $)) $) "\\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \\spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the \"canonical\" element is the same across all associates of \\spad{x} if \\spad{unitNormal(x) = [u,c,a]} then \\spad{u*c = x},{} \\spad{a*u = 1}.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} either returns an element \\spad{c} such that \\spad{c*b=a} or \"failed\" if no such element can be found."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-562 R -3498) 
+(-563 R -3505) 
 ((|constructor| (NIL "This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for elemntary functions.")) (|lfextlimint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #1="failed") |#2| (|Symbol|) (|Kernel| |#2|) (|List| (|Kernel| |#2|))) "\\spad{lfextlimint(f,x,k,[k1,...,kn])} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f - c dk/dx}. Value \\spad{h} is looked for in a field containing \\spad{f} and \\spad{k1},{}...,{}\\spad{kn} (the \\spad{ki}\\spad{'s} must be logs).")) (|lfintegrate| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{lfintegrate(f, x)} = \\spad{g} such that \\spad{dg/dx = f}.")) (|lfinfieldint| (((|Union| |#2| "failed") |#2| (|Symbol|)) "\\spad{lfinfieldint(f, x)} returns a function \\spad{g} such that \\spad{dg/dx = f} if \\spad{g} exists,{} \"failed\" otherwise.")) (|lflimitedint| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Symbol|) (|List| |#2|)) "\\spad{lflimitedint(f,x,[g1,...,gn])} returns functions \\spad{[h,[[ci, gi]]]} such that the \\spad{gi}\\spad{'s} are among \\spad{[g1,...,gn]},{} and \\spad{d(h+sum(ci log(gi)))/dx = f},{} if possible,{} \"failed\" otherwise.")) (|lfextendedint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #1#) |#2| (|Symbol|) |#2|) "\\spad{lfextendedint(f, x, g)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f - cg},{} if (\\spad{h},{} \\spad{c}) exist,{} \"failed\" otherwise."))) 
 NIL 
 NIL 
-(-563 I) 
+(-564 I) 
 ((|constructor| (NIL "\\indented{1}{This Package contains basic methods for integer factorization.} The factor operation employs trial division up to 10,{}000. It then tests to see if \\spad{n} is a perfect power before using Pollards rho method. Because Pollards method may fail,{} the result of factor may contain composite factors. We should also employ Lenstra\\spad{'s} eliptic curve method.")) (|PollardSmallFactor| (((|Union| |#1| "failed") |#1|) "\\spad{PollardSmallFactor(n)} returns a factor of \\spad{n} or \"failed\" if no one is found")) (|BasicMethod| (((|Factored| |#1|) |#1|) "\\spad{BasicMethod(n)} returns the factorization of integer \\spad{n} by trial division")) (|squareFree| (((|Factored| |#1|) |#1|) "\\spad{squareFree(n)} returns the square free factorization of integer \\spad{n}")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(n)} returns the full factorization of integer \\spad{n}"))) 
 NIL 
 NIL 
-(-564) 
+(-565) 
 ((|constructor| (NIL "\\blankline")) (|entry| (((|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| #1="Continuous at the end points") (|:| |lowerSingular| #2="There is a singularity at the lower end point") (|:| |upperSingular| #3="There is a singularity at the upper end point") (|:| |bothSingular| #4="There are singularities at both end points") (|:| |notEvaluated| #5="End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| #6="Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| #7="The range is finite") (|:| |lowerInfinite| #8="The bottom of range is infinite") (|:| |upperInfinite| #9="The top of range is infinite") (|:| |bothInfinite| #10="Both top and bottom points are infinite") (|:| |notEvaluated| #11="Range not yet evaluated")))) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{entry(n)} \\undocumented{}")) (|entries| (((|List| (|Record| (|:| |key| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| #1#) (|:| |lowerSingular| #2#) (|:| |upperSingular| #3#) (|:| |bothSingular| #4#) (|:| |notEvaluated| #5#))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| #6#))) (|:| |range| (|Union| (|:| |finite| #7#) (|:| |lowerInfinite| #8#) (|:| |upperInfinite| #9#) (|:| |bothInfinite| #10#) (|:| |notEvaluated| #11#))))))) $) "\\spad{entries(x)} \\undocumented{}")) (|showAttributes| (((|Union| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| #1#) (|:| |lowerSingular| #2#) (|:| |upperSingular| #3#) (|:| |bothSingular| #4#) (|:| |notEvaluated| #5#))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| #6#))) (|:| |range| (|Union| (|:| |finite| #7#) (|:| |lowerInfinite| #8#) (|:| |upperInfinite| #9#) (|:| |bothInfinite| #10#) (|:| |notEvaluated| #11#)))) "failed") (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{showAttributes(x)} \\undocumented{}")) (|insert!| (($ (|Record| (|:| |key| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| #1#) (|:| |lowerSingular| #2#) (|:| |upperSingular| #3#) (|:| |bothSingular| #4#) (|:| |notEvaluated| #5#))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| #6#))) (|:| |range| (|Union| (|:| |finite| #7#) (|:| |lowerInfinite| #8#) (|:| |upperInfinite| #9#) (|:| |bothInfinite| #10#) (|:| |notEvaluated| #11#))))))) "\\spad{insert!(r)} inserts an entry \\spad{r} into theIFTable")) (|fTable| (($ (|List| (|Record| (|:| |key| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| #1#) (|:| |lowerSingular| #2#) (|:| |upperSingular| #3#) (|:| |bothSingular| #4#) (|:| |notEvaluated| #5#))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| #6#))) (|:| |range| (|Union| (|:| |finite| #7#) (|:| |lowerInfinite| #8#) (|:| |upperInfinite| #9#) (|:| |bothInfinite| #10#) (|:| |notEvaluated| #11#)))))))) "\\spad{fTable(l)} creates a functions table from the elements of \\spad{l}.")) (|keys| (((|List| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) $) "\\spad{keys(f)} returns the list of keys of \\spad{f}")) (|clearTheFTable| (((|Void|)) "\\spad{clearTheFTable()} clears the current table of functions.")) (|showTheFTable| (($) "\\spad{showTheFTable()} returns the current table of functions."))) 
 NIL 
 NIL 
-(-565 R -3498 L) 
+(-566 R -3505 L) 
 ((|constructor| (NIL "This internal package rationalises integrands on curves of the form: \\indented{2}{\\spad{y\\^2 = a x\\^2 + b x + c}} \\indented{2}{\\spad{y\\^2 = (a x + b) / (c x + d)}} \\indented{2}{\\spad{f(x, y) = 0} where \\spad{f} has degree 1 in \\spad{x}} The rationalization is done for integration,{} limited integration,{} extended integration and the risch differential equation.")) (|palgLODE0| (((|Record| (|:| |particular| (|Union| |#2| #1="failed")) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgLODE0(op,g,x,y,z,t,c)} returns the solution of \\spad{op f = g} Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}.") (((|Record| (|:| |particular| (|Union| |#2| #1#)) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgLODE0(op, g, x, y, d, p)} returns the solution of \\spad{op f = g}. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}.")) (|lift| (((|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) (|SparseUnivariatePolynomial| |#2|) (|Kernel| |#2|)) "\\spad{lift(u,k)} \\undocumented")) (|multivariate| ((|#2| (|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) (|Kernel| |#2|) |#2|) "\\spad{multivariate(u,k,f)} \\undocumented")) (|univariate| (((|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|SparseUnivariatePolynomial| |#2|)) "\\spad{univariate(f,k,k,p)} \\undocumented")) (|palgRDE0| (((|Union| |#2| #2="failed") |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| #2#) |#2| |#2| (|Symbol|)) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgRDE0(f, g, x, y, foo, t, c)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{foo},{} called by \\spad{foo(a, b, x)},{} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}.") (((|Union| |#2| #2#) |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| #2#) |#2| |#2| (|Symbol|)) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgRDE0(f, g, x, y, foo, d, p)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}. Argument \\spad{foo},{} called by \\spad{foo(a, b, x)},{} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}.")) (|palglimint0| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) #3="failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palglimint0(f, x, y, [u1,...,un], z, t, c)} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}\\spad{'s} are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}.") (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) #3#) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palglimint0(f, x, y, [u1,...,un], d, p)} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}\\spad{'s} are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}.")) (|palgextint0| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #4="failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgextint0(f, x, y, g, z, t, c)} returns functions \\spad{[h, d]} such that \\spad{dh/dx = f(x,y) - d g},{} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy},{} and \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{z} is a dummy variable not appearing in \\spad{f(x,y)}. The operation returns \"failed\" if no such functions exist.") (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #4#) |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgextint0(f, x, y, g, d, p)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f(x,y) - c g},{} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2 y(x)\\^2 = P(x)},{} or \"failed\" if no such functions exist.")) (|palgint0| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgint0(f, x, y, z, t, c)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{z} is a dummy variable not appearing in \\spad{f(x,y)}.") (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgint0(f, x, y, d, p)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2 y(x)\\^2 = P(x)}."))) 
 NIL 
-((|HasCategory| |#3| (LIST (QUOTE -661) (|devaluate| |#2|)))) 
-(-566) 
+((|HasCategory| |#3| (LIST (QUOTE -663) (|devaluate| |#2|)))) 
+(-567) 
 ((|constructor| (NIL "This package provides various number theoretic functions on the integers.")) (|sumOfKthPowerDivisors| (((|Integer|) (|Integer|) (|NonNegativeInteger|)) "\\spad{sumOfKthPowerDivisors(n,k)} returns the sum of the \\spad{k}th powers of the integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. the sum of the \\spad{k}th powers of the divisors of \\spad{n} is often denoted by \\spad{sigma_k(n)}.")) (|sumOfDivisors| (((|Integer|) (|Integer|)) "\\spad{sumOfDivisors(n)} returns the sum of the integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. The sum of the divisors of \\spad{n} is often denoted by \\spad{sigma(n)}.")) (|numberOfDivisors| (((|Integer|) (|Integer|)) "\\spad{numberOfDivisors(n)} returns the number of integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. The number of divisors of \\spad{n} is often denoted by \\spad{tau(n)}.")) (|moebiusMu| (((|Integer|) (|Integer|)) "\\spad{moebiusMu(n)} returns the Moebius function \\spad{mu(n)}. \\spad{mu(n)} is either \\spad{-1},{}0 or 1 as follows: \\spad{mu(n) = 0} if \\spad{n} is divisible by a square > 1,{} \\spad{mu(n) = (-1)^k} if \\spad{n} is square-free and has \\spad{k} distinct prime divisors.")) (|legendre| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{legendre(a,p)} returns the Legendre symbol \\spad{L(a/p)}. \\spad{L(a/p) = (-1)**((p-1)/2) mod p} (\\spad{p} prime),{} which is 0 if \\spad{a} is 0,{} 1 if \\spad{a} is a quadratic residue \\spad{mod p} and \\spad{-1} otherwise. Note: because the primality test is expensive,{} if it is known that \\spad{p} is prime then use \\spad{jacobi(a,p)}.")) (|jacobi| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{jacobi(a,b)} returns the Jacobi symbol \\spad{J(a/b)}. When \\spad{b} is odd,{} \\spad{J(a/b) = product(L(a/p) for p in factor b )}. Note: by convention,{} 0 is returned if \\spad{gcd(a,b) ~= 1}. Iterative \\spad{O(log(b)^2)} version coded by Michael Monagan June 1987.")) (|harmonic| (((|Fraction| (|Integer|)) (|Integer|)) "\\spad{harmonic(n)} returns the \\spad{n}th harmonic number. This is \\spad{H[n] = sum(1/k,k=1..n)}.")) (|fibonacci| (((|Integer|) (|Integer|)) "\\spad{fibonacci(n)} returns the \\spad{n}th Fibonacci number. the Fibonacci numbers \\spad{F[n]} are defined by \\spad{F[0] = F[1] = 1} and \\spad{F[n] = F[n-1] + F[n-2]}. The algorithm has running time \\spad{O(log(n)^3)}. Reference: Knuth,{} The Art of Computer Programming Vol 2,{} Semi-Numerical Algorithms.")) (|eulerPhi| (((|Integer|) (|Integer|)) "\\spad{eulerPhi(n)} returns the number of integers between 1 and \\spad{n} (including 1) which are relatively prime to \\spad{n}. This is the Euler phi function \\spad{\\phi(n)} is also called the totient function.")) (|euler| (((|Integer|) (|Integer|)) "\\spad{euler(n)} returns the \\spad{n}th Euler number. This is \\spad{2^n E(n,1/2)},{} where \\spad{E(n,x)} is the \\spad{n}th Euler polynomial.")) (|divisors| (((|List| (|Integer|)) (|Integer|)) "\\spad{divisors(n)} returns a list of the divisors of \\spad{n}.")) (|chineseRemainder| (((|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{chineseRemainder(x1,m1,x2,m2)} returns \\spad{w},{} where \\spad{w} is such that \\spad{w = x1 mod m1} and \\spad{w = x2 mod m2}. Note: \\spad{m1} and \\spad{m2} must be relatively prime.")) (|bernoulli| (((|Fraction| (|Integer|)) (|Integer|)) "\\spad{bernoulli(n)} returns the \\spad{n}th Bernoulli number. this is \\spad{B(n,0)},{} where \\spad{B(n,x)} is the \\spad{n}th Bernoulli polynomial."))) 
 NIL 
 NIL 
-(-567 -3498 UP UPUP R) 
+(-568 -3505 UP UPUP R) 
 ((|constructor| (NIL "algebraic Hermite redution.")) (|HermiteIntegrate| (((|Record| (|:| |answer| |#4|) (|:| |logpart| |#4|)) |#4| (|Mapping| |#2| |#2|)) "\\spad{HermiteIntegrate(f, ')} returns \\spad{[g,h]} such that \\spad{f = g' + h} and \\spad{h} has a only simple finite normal poles."))) 
 NIL 
 NIL 
-(-568 -3498 UP) 
+(-569 -3505 UP) 
 ((|constructor| (NIL "Hermite integration,{} transcendental case.")) (|HermiteIntegrate| (((|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |logpart| (|Fraction| |#2|)) (|:| |specpart| (|Fraction| |#2|)) (|:| |polypart| |#2|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{HermiteIntegrate(f, D)} returns \\spad{[g, h, s, p]} such that \\spad{f = Dg + h + s + p},{} \\spad{h} has a squarefree denominator normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and all the squarefree factors of the denominator of \\spad{s} are special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D}. Furthermore,{} \\spad{h} and \\spad{s} have no polynomial parts. \\spad{D} is the derivation to use on \\spadtype{UP}."))) 
 NIL 
 NIL 
-(-569) 
+(-570) 
 ((|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))) (|:| |extra| (|Result|))) (|NumericalIntegrationProblem|) (|RoutinesTable|)) "\\spad{measure(prob,R)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical integration problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} listed in \\axiom{\\spad{R}} of \\axiom{category} \\axiomType{NumericalIntegrationCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information.") (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|))) (|:| |extra| (|Result|))) (|NumericalIntegrationProblem|)) "\\spad{measure(prob)} is a top level ANNA function for identifying the most appropriate numerical routine for solving the numerical integration problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} of \\axiom{category} \\axiomType{NumericalIntegrationCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information.")) (|integrate| (((|Union| (|Result|) "failed") (|Expression| (|Float|)) (|SegmentBinding| (|OrderedCompletion| (|Float|))) (|Symbol|)) "\\spad{integrate(exp, x = a..b, numerical)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range,{} {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.\\newline \\blankline Default values for the absolute and relative error are used. \\blankline It is an error if the last argument is not {\\spad{\\tt} numerical}.") (((|Union| (|Result|) "failed") (|Expression| (|Float|)) (|SegmentBinding| (|OrderedCompletion| (|Float|))) (|String|)) "\\spad{integrate(exp, x = a..b, \"numerical\")} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range,{} {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.\\newline \\blankline Default values for the absolute and relative error are used. \\blankline It is an error of the last argument is not {\\spad{\\tt} \"numerical\"}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|)))) (|Float|) (|Float|) (|RoutinesTable|)) "\\spad{integrate(exp, [a..b,c..d,...], epsabs, epsrel, routines)} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges to the required absolute and relative accuracy,{} using the routines available in the RoutinesTable provided. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|)))) (|Float|) (|Float|)) "\\spad{integrate(exp, [a..b,c..d,...], epsabs, epsrel)} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges to the required absolute and relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|)))) (|Float|)) "\\spad{integrate(exp, [a..b,c..d,...], epsrel)} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges to the required relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline If epsrel = 0,{} a default absolute accuracy is used.") (((|Result|) (|Expression| (|Float|)) (|List| (|Segment| (|OrderedCompletion| (|Float|))))) "\\spad{integrate(exp, [a..b,c..d,...])} is a top level ANNA function to integrate a multivariate expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given set of ranges. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline Default values for the absolute and relative error are used.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|)))) "\\spad{integrate(exp, a..b)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline Default values for the absolute and relative error are used.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|))) (|Float|)) "\\spad{integrate(exp, a..b, epsrel)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}} to the required relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}. \\blankline If epsrel = 0,{} a default absolute accuracy is used.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|))) (|Float|) (|Float|)) "\\spad{integrate(exp, a..b, epsabs, epsrel)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}} to the required absolute and relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|NumericalIntegrationProblem|)) "\\spad{integrate(IntegrationProblem)} is a top level ANNA function to integrate an expression over a given range or ranges to the required absolute and relative accuracy. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|Segment| (|OrderedCompletion| (|Float|))) (|Float|) (|Float|) (|RoutinesTable|)) "\\spad{integrate(exp, a..b, epsrel, routines)} is a top level ANNA function to integrate an expression,{} {\\spad{\\tt} \\spad{exp}},{} over a given range {\\spad{\\tt} a} to {\\spad{\\tt} \\spad{b}} to the required absolute and relative accuracy using the routines available in the RoutinesTable provided. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalIntegrationCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline It then performs the integration of the given expression on that \\axiom{domain}."))) 
 NIL 
 NIL 
-(-570 R -3498 L) 
+(-571 R -3505 L) 
 ((|constructor| (NIL "This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for pure algebraic integrands.")) (|palgLODE| (((|Record| (|:| |particular| (|Union| |#2| #1="failed")) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Symbol|)) "\\spad{palgLODE(op, g, kx, y, x)} returns the solution of \\spad{op f = g}. \\spad{y} is an algebraic function of \\spad{x}.")) (|palgRDE| (((|Union| |#2| #1#) |#2| |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| #1#) |#2| |#2| (|Symbol|))) "\\spad{palgRDE(nfp, f, g, x, y, foo)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} \"failed\" otherwise; \\spad{y} is an algebraic function of \\spad{x}; \\spad{foo(a, b, x)} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}. \\spad{nfp} is \\spad{n * df/dx}.")) (|palglimint| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|)) "\\spad{palglimint(f, x, y, [u1,...,un])} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}\\spad{'s} are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} \"failed\" otherwise; \\spad{y} is an algebraic function of \\spad{x}.")) (|palgextint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2|) "\\spad{palgextint(f, x, y, g)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f(x,y) - c g},{} where \\spad{y} is an algebraic function of \\spad{x}; returns \"failed\" if no such functions exist.")) (|palgint| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|)) "\\spad{palgint(f, x, y)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x}."))) 
 NIL 
-((|HasCategory| |#3| (LIST (QUOTE -661) (|devaluate| |#2|)))) 
-(-571 R -3498) 
+((|HasCategory| |#3| (LIST (QUOTE -663) (|devaluate| |#2|)))) 
+(-572 R -3505) 
 ((|constructor| (NIL "\\spadtype{PatternMatchIntegration} provides functions that use the pattern matcher to find some indefinite and definite integrals involving special functions and found in the litterature.")) (|pmintegrate| (((|Union| |#2| "failed") |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|)) "\\spad{pmintegrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b} if it can be found by the built-in pattern matching rules.") (((|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|)) "\\spad{pmintegrate(f, x)} returns either \"failed\" or \\spad{[g,h]} such that \\spad{integrate(f,x) = g + integrate(h,x)}.")) (|pmComplexintegrate| (((|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|)) "\\spad{pmComplexintegrate(f, x)} returns either \"failed\" or \\spad{[g,h]} such that \\spad{integrate(f,x) = g + integrate(h,x)}. It only looks for special complex integrals that pmintegrate does not return.")) (|splitConstant| (((|Record| (|:| |const| |#2|) (|:| |nconst| |#2|)) |#2| (|Symbol|)) "\\spad{splitConstant(f, x)} returns \\spad{[c, g]} such that \\spad{f = c * g} and \\spad{c} does not involve \\spad{t}."))) 
 NIL 
-((-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-1143)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-633))))) 
-(-572 -3498 UP) 
+((-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-1145)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-635))))) 
+(-573 -3505 UP) 
 ((|constructor| (NIL "This package provides functions for the base case of the Risch algorithm.")) (|limitedint| (((|Union| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|)))))) "failed") (|Fraction| |#2|) (|List| (|Fraction| |#2|))) "\\spad{limitedint(f, [g1,...,gn])} returns fractions \\spad{[h,[[ci, gi]]]} such that the \\spad{gi}\\spad{'s} are among \\spad{[g1,...,gn]},{} \\spad{ci' = 0},{} and \\spad{(h+sum(ci log(gi)))' = f},{} if possible,{} \"failed\" otherwise.")) (|extendedint| (((|Union| (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{extendedint(f, g)} returns fractions \\spad{[h, c]} such that \\spad{c' = 0} and \\spad{h' = f - cg},{} if \\spad{(h, c)} exist,{} \"failed\" otherwise.")) (|infieldint| (((|Union| (|Fraction| |#2|) "failed") (|Fraction| |#2|)) "\\spad{infieldint(f)} returns \\spad{g} such that \\spad{g' = f} or \"failed\" if the integral of \\spad{f} is not a rational function.")) (|integrate| (((|IntegrationResult| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{integrate(f)} returns \\spad{g} such that \\spad{g' = f}."))) 
 NIL 
 NIL 
-(-573 S) 
+(-574 S) 
 ((|constructor| (NIL "Provides integer testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991")) (|integerIfCan| (((|Union| (|Integer|) "failed") |#1|) "\\spad{integerIfCan(x)} returns \\spad{x} as an integer,{} \"failed\" if \\spad{x} is not an integer.")) (|integer?| (((|Boolean|) |#1|) "\\spad{integer?(x)} is \\spad{true} if \\spad{x} is an integer,{} \\spad{false} otherwise.")) (|integer| (((|Integer|) |#1|) "\\spad{integer(x)} returns \\spad{x} as an integer; error if \\spad{x} is not an integer."))) 
 NIL 
 NIL 
-(-574 -3498) 
+(-575 -3505) 
 ((|constructor| (NIL "This package provides functions for the integration of rational functions.")) (|extendedIntegrate| (((|Union| (|Record| (|:| |ratpart| (|Fraction| (|Polynomial| |#1|))) (|:| |coeff| (|Fraction| (|Polynomial| |#1|)))) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|Fraction| (|Polynomial| |#1|))) "\\spad{extendedIntegrate(f, x, g)} returns fractions \\spad{[h, c]} such that \\spad{dc/dx = 0} and \\spad{dh/dx = f - cg},{} if \\spad{(h, c)} exist,{} \"failed\" otherwise.")) (|limitedIntegrate| (((|Union| (|Record| (|:| |mainpart| (|Fraction| (|Polynomial| |#1|))) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| (|Polynomial| |#1|))) (|:| |logand| (|Fraction| (|Polynomial| |#1|))))))) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{limitedIntegrate(f, x, [g1,...,gn])} returns fractions \\spad{[h, [[ci,gi]]]} such that the \\spad{gi}\\spad{'s} are among \\spad{[g1,...,gn]},{} \\spad{dci/dx = 0},{} and \\spad{d(h + sum(ci log(gi)))/dx = f} if possible,{} \"failed\" otherwise.")) (|infieldIntegrate| (((|Union| (|Fraction| (|Polynomial| |#1|)) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{infieldIntegrate(f, x)} returns a fraction \\spad{g} such that \\spad{dg/dx = f} if \\spad{g} exists,{} \"failed\" otherwise.")) (|internalIntegrate| (((|IntegrationResult| (|Fraction| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{internalIntegrate(f, x)} returns \\spad{g} such that \\spad{dg/dx = f}."))) 
 NIL 
 NIL 
-(-575 R) 
+(-576 R) 
 ((|constructor| (NIL "\\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This domain is an implementation of interval arithmetic and transcendental + functions over intervals."))) 
-((-4203 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4210 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-576) 
+(-577) 
 ((|constructor| (NIL "This package provides the implementation for the \\spadfun{solveLinearPolynomialEquation} operation over the integers. It uses a lifting technique from the package GenExEuclid")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| (|Integer|))) "failed") (|List| (|SparseUnivariatePolynomial| (|Integer|))) (|SparseUnivariatePolynomial| (|Integer|))) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}\\spad{'s} exists."))) 
 NIL 
 NIL 
-(-577 R -3498) 
+(-578 R -3505) 
 ((|constructor| (NIL "\\indented{1}{Tools for the integrator} Author: Manuel Bronstein Date Created: 25 April 1990 Date Last Updated: 9 June 1993 Keywords: elementary,{} function,{} integration.")) (|intPatternMatch| (((|IntegrationResult| |#2|) |#2| (|Symbol|) (|Mapping| (|IntegrationResult| |#2|) |#2| (|Symbol|)) (|Mapping| (|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|))) "\\spad{intPatternMatch(f, x, int, pmint)} tries to integrate \\spad{f} first by using the integration function \\spad{int},{} and then by using the pattern match intetgration function \\spad{pmint} on any remaining unintegrable part.")) (|mkPrim| ((|#2| |#2| (|Symbol|)) "\\spad{mkPrim(f, x)} makes the logs in \\spad{f} which are linear in \\spad{x} primitive with respect to \\spad{x}.")) (|removeConstantTerm| ((|#2| |#2| (|Symbol|)) "\\spad{removeConstantTerm(f, x)} returns \\spad{f} minus any additive constant with respect to \\spad{x}.")) (|vark| (((|List| (|Kernel| |#2|)) (|List| |#2|) (|Symbol|)) "\\spad{vark([f1,...,fn],x)} returns the set-theoretic union of \\spad{(varselect(f1,x),...,varselect(fn,x))}.")) (|union| (((|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|))) "\\spad{union(l1, l2)} returns set-theoretic union of \\spad{l1} and \\spad{l2}.")) (|ksec| (((|Kernel| |#2|) (|Kernel| |#2|) (|List| (|Kernel| |#2|)) (|Symbol|)) "\\spad{ksec(k, [k1,...,kn], x)} returns the second top-level \\spad{ki} after \\spad{k} involving \\spad{x}.")) (|kmax| (((|Kernel| |#2|) (|List| (|Kernel| |#2|))) "\\spad{kmax([k1,...,kn])} returns the top-level \\spad{ki} for integration.")) (|varselect| (((|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|)) (|Symbol|)) "\\spad{varselect([k1,...,kn], x)} returns the \\spad{ki} which involve \\spad{x}."))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-286))) (|HasCategory| |#2| (QUOTE (-633))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-1181))))) (-12 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-286)))) (|HasCategory| |#1| (QUOTE (-561)))) 
-(-578 -3498 UP) 
+((-12 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-287))) (|HasCategory| |#2| (QUOTE (-635))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-1183))))) (-12 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-287)))) (|HasCategory| |#1| (QUOTE (-562)))) 
+(-579 -3505 UP) 
 ((|constructor| (NIL "This package provides functions for the transcendental case of the Risch algorithm.")) (|monomialIntPoly| (((|Record| (|:| |answer| |#2|) (|:| |polypart| |#2|)) |#2| (|Mapping| |#2| |#2|)) "\\spad{monomialIntPoly(p, ')} returns [\\spad{q},{} \\spad{r}] such that \\spad{p = q' + r} and \\spad{degree(r) < degree(t')}. Error if \\spad{degree(t') < 2}.")) (|monomialIntegrate| (((|Record| (|:| |ir| (|IntegrationResult| (|Fraction| |#2|))) (|:| |specpart| (|Fraction| |#2|)) (|:| |polypart| |#2|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomialIntegrate(f, ')} returns \\spad{[ir, s, p]} such that \\spad{f = ir' + s + p} and all the squarefree factors of the denominator of \\spad{s} are special \\spad{w}.\\spad{r}.\\spad{t} the derivation '.")) (|expintfldpoly| (((|Union| (|LaurentPolynomial| |#1| |#2|) "failed") (|LaurentPolynomial| |#1| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|)) "\\spad{expintfldpoly(p, foo)} returns \\spad{q} such that \\spad{p' = q} or \"failed\" if no such \\spad{q} exists. Argument foo is a Risch differential equation function on \\spad{F}.")) (|primintfldpoly| (((|Union| |#2| "failed") |#2| (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1="failed") |#1|) |#1|) "\\spad{primintfldpoly(p, ', t')} returns \\spad{q} such that \\spad{p' = q} or \"failed\" if no such \\spad{q} exists. Argument \\spad{t'} is the derivative of the primitive generating the extension.")) (|primlimintfrac| (((|Union| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|)))))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|List| (|Fraction| |#2|))) "\\spad{primlimintfrac(f, ', [u1,...,un])} returns \\spad{[v, [c1,...,cn]]} such that \\spad{ci' = 0} and \\spad{f = v' + +/[ci * ui'/ui]}. Error: if \\spad{degree numer f >= degree denom f}.")) (|primextintfrac| (((|Union| (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Fraction| |#2|)) "\\spad{primextintfrac(f, ', g)} returns \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0}. Error: if \\spad{degree numer f >= degree denom f} or if \\spad{degree numer g >= degree denom g} or if \\spad{denom g} is not squarefree.")) (|explimitedint| (((|Union| (|Record| (|:| |answer| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|))))))) (|:| |a0| |#1|)) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|) (|List| (|Fraction| |#2|))) "\\spad{explimitedint(f, ', foo, [u1,...,un])} returns \\spad{[v, [c1,...,cn], a]} such that \\spad{ci' = 0},{} \\spad{f = v' + a + reduce(+,[ci * ui'/ui])},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}. Returns \"failed\" if no such \\spad{v},{} \\spad{ci},{} a exist. Argument \\spad{foo} is a Risch differential equation function on \\spad{F}.")) (|primlimitedint| (((|Union| (|Record| (|:| |answer| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|))))))) (|:| |a0| |#1|)) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1#) |#1|) (|List| (|Fraction| |#2|))) "\\spad{primlimitedint(f, ', foo, [u1,...,un])} returns \\spad{[v, [c1,...,cn], a]} such that \\spad{ci' = 0},{} \\spad{f = v' + a + reduce(+,[ci * ui'/ui])},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Returns \"failed\" if no such \\spad{v},{} \\spad{ci},{} a exist. Argument \\spad{foo} is an extended integration function on \\spad{F}.")) (|expextendedint| (((|Union| (|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |a0| |#1|)) (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|) (|Fraction| |#2|)) "\\spad{expextendedint(f, ', foo, g)} returns either \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0},{} or \\spad{[v, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}. Returns \"failed\" if neither case can hold. Argument \\spad{foo} is a Risch differential equation function on \\spad{F}.")) (|primextendedint| (((|Union| (|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |a0| |#1|)) (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1#) |#1|) (|Fraction| |#2|)) "\\spad{primextendedint(f, ', foo, g)} returns either \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0},{} or \\spad{[v, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Returns \"failed\" if neither case can hold. Argument \\spad{foo} is an extended integration function on \\spad{F}.")) (|tanintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|List| |#1|) "failed") (|Integer|) |#1| |#1|)) "\\spad{tanintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}; Argument foo is a Risch differential system solver on \\spad{F}.")) (|expintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|)) "\\spad{expintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}; Argument foo is a Risch differential equation solver on \\spad{F}.")) (|primintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1#) |#1|)) "\\spad{primintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Argument foo is an extended integration function on \\spad{F}."))) 
 NIL 
 NIL 
-(-579 R -3498) 
+(-580 R -3505) 
 ((|constructor| (NIL "This package computes the inverse Laplace Transform.")) (|inverseLaplace| (((|Union| |#2| "failed") |#2| (|Symbol|) (|Symbol|)) "\\spad{inverseLaplace(f, s, t)} returns the Inverse Laplace transform of \\spad{f(s)} using \\spad{t} as the new variable or \"failed\" if unable to find a closed form."))) 
 NIL 
 NIL 
-(-580) 
+(-581) 
 ((|constructor| (NIL "This category describes byte stream conduits supporting both input and output operations."))) 
 NIL 
 NIL 
-(-581) 
+(-582) 
 ((|constructor| (NIL "\\indented{2}{This domain provides representation for binary files open} \\indented{2}{for input and output operations.} See Also: InputBinaryFile,{} OutputBinaryFile")) (|isOpen?| (((|Boolean|) $) "\\spad{isOpen?(f)} holds if \\spad{`f'} is in open state.")) (|inputOutputBinaryFile| (($ (|String|)) "\\spad{inputOutputBinaryFile(f)} returns an input/output conduit obtained by opening the file named by \\spad{`f'} as a binary file.") (($ (|FileName|)) "\\spad{inputOutputBinaryFile(f)} returns an input/output conduit obtained by opening the file designated by \\spad{`f'} as a binary file."))) 
 NIL 
 NIL 
-(-582) 
+(-583) 
 ((|constructor| (NIL "This domain provides constants to describe directions of IO conduits (file,{} etc) mode of operations.")) (|bothWays| (($) "`bothWays' indicates that an IO conduit is for both input and output.")) (|output| (($) "`output' indicates that an IO conduit is for output")) (|input| (($) "`input' indicates that an IO conduit is for input."))) 
 NIL 
 NIL 
-(-583) 
+(-584) 
 ((|constructor| (NIL "This domain provides representation for ARPA Internet IP4 addresses.")) (|resolve| (((|Maybe| $) (|Hostname|)) "\\spad{resolve(h)} returns the IP4 address of host \\spad{`h'}.")) (|bytes| (((|DataArray| 4 (|Byte|)) $) "\\spad{bytes(x)} returns the bytes of the numeric address \\spad{`x'}.")) (|ip4Address| (($ (|String|)) "\\spad{ip4Address(a)} builds a numeric address out of the ASCII form `a'."))) 
 NIL 
 NIL 
-(-584 |p| |unBalanced?|) 
+(-585 |p| |unBalanced?|) 
 ((|constructor| (NIL "This domain implements \\spad{Zp},{} the \\spad{p}-adic completion of the integers. This is an internal domain."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-585 |p|) 
+(-586 |p|) 
 ((|constructor| (NIL "InnerPrimeField(\\spad{p}) implements the field with \\spad{p} elements. Note: argument \\spad{p} MUST be a prime (this domain does not check). See \\spadtype{PrimeField} for a domain that does check."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| $ (QUOTE (-147))) (|HasCategory| $ (QUOTE (-145))) (|HasCategory| $ (QUOTE (-371)))) 
-(-586) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| $ (QUOTE (-147))) (|HasCategory| $ (QUOTE (-145))) (|HasCategory| $ (QUOTE (-372)))) 
+(-587) 
 ((|constructor| (NIL "A package to print strings without line-feed nor carriage-return.")) (|iprint| (((|Void|) (|String|)) "\\axiom{iprint(\\spad{s})} prints \\axiom{\\spad{s}} at the current position of the cursor."))) 
 NIL 
 NIL 
-(-587 -3498) 
+(-588 -3505) 
 ((|constructor| (NIL "If a function \\spad{f} has an elementary integral \\spad{g},{} then \\spad{g} can be written in the form \\spad{g = h + c1 log(u1) + c2 log(u2) + ... + cn log(un)} where \\spad{h},{} which is in the same field than \\spad{f},{} is called the rational part of the integral,{} and \\spad{c1 log(u1) + ... cn log(un)} is called the logarithmic part of the integral. This domain manipulates integrals represented in that form,{} by keeping both parts separately. The logs are not explicitly computed.")) (|differentiate| ((|#1| $ (|Symbol|)) "\\spad{differentiate(ir,x)} differentiates \\spad{ir} with respect to \\spad{x}") ((|#1| $ (|Mapping| |#1| |#1|)) "\\spad{differentiate(ir,D)} differentiates \\spad{ir} with respect to the derivation \\spad{D}.")) (|integral| (($ |#1| (|Symbol|)) "\\spad{integral(f,x)} returns the formal integral of \\spad{f} with respect to \\spad{x}") (($ |#1| |#1|) "\\spad{integral(f,x)} returns the formal integral of \\spad{f} with respect to \\spad{x}")) (|elem?| (((|Boolean|) $) "\\spad{elem?(ir)} tests if an integration result is elementary over \\spad{F?}")) (|notelem| (((|List| (|Record| (|:| |integrand| |#1|) (|:| |intvar| |#1|))) $) "\\spad{notelem(ir)} returns the non-elementary part of an integration result")) (|logpart| (((|List| (|Record| (|:| |scalar| (|Fraction| (|Integer|))) (|:| |coeff| (|SparseUnivariatePolynomial| |#1|)) (|:| |logand| (|SparseUnivariatePolynomial| |#1|)))) $) "\\spad{logpart(ir)} returns the logarithmic part of an integration result")) (|ratpart| ((|#1| $) "\\spad{ratpart(ir)} returns the rational part of an integration result")) (|mkAnswer| (($ |#1| (|List| (|Record| (|:| |scalar| (|Fraction| (|Integer|))) (|:| |coeff| (|SparseUnivariatePolynomial| |#1|)) (|:| |logand| (|SparseUnivariatePolynomial| |#1|)))) (|List| (|Record| (|:| |integrand| |#1|) (|:| |intvar| |#1|)))) "\\spad{mkAnswer(r,l,ne)} creates an integration result from a rational part \\spad{r},{} a logarithmic part \\spad{l},{} and a non-elementary part \\spad{ne}."))) 
-((-4422 . T) (-4421 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-1181))))) 
-(-588 E -3498) 
+((-4429 . T) (-4428 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-1183))))) 
+(-589 E -3505) 
 ((|constructor| (NIL "\\indented{1}{Internally used by the integration packages} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 12 August 1992 Keywords: integration.")) (|map| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") (|Mapping| |#2| |#1|) (|Union| (|Record| (|:| |mainpart| |#1|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#1|) (|:| |logand| |#1|))))) "failed")) "\\spad{map(f,ufe)} \\undocumented") (((|Union| |#2| "failed") (|Mapping| |#2| |#1|) (|Union| |#1| "failed")) "\\spad{map(f,ue)} \\undocumented") (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") (|Mapping| |#2| |#1|) (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) "failed")) "\\spad{map(f,ure)} \\undocumented") (((|IntegrationResult| |#2|) (|Mapping| |#2| |#1|) (|IntegrationResult| |#1|)) "\\spad{map(f,ire)} \\undocumented"))) 
 NIL 
 NIL 
-(-589 R -3498) 
+(-590 R -3505) 
 ((|constructor| (NIL "This package allows a sum of logs over the roots of a polynomial to be expressed as explicit logarithms and arc tangents,{} provided that the indexing polynomial can be factored into quadratics.")) (|complexExpand| ((|#2| (|IntegrationResult| |#2|)) "\\spad{complexExpand(i)} returns the expanded complex function corresponding to \\spad{i}.")) (|expand| (((|List| |#2|) (|IntegrationResult| |#2|)) "\\spad{expand(i)} returns the list of possible real functions corresponding to \\spad{i}.")) (|split| (((|IntegrationResult| |#2|) (|IntegrationResult| |#2|)) "\\spad{split(u(x) + sum_{P(a)=0} Q(a,x))} returns \\spad{u(x) + sum_{P1(a)=0} Q(a,x) + ... + sum_{Pn(a)=0} Q(a,x)} where \\spad{P1},{}...,{}\\spad{Pn} are the factors of \\spad{P}."))) 
 NIL 
 NIL 
-(-590) 
+(-591) 
 ((|constructor| (NIL "This domain provides representations for the intermediate form data structure used by the Spad elaborator."))) 
 NIL 
 NIL 
-(-591 I) 
+(-592 I) 
 ((|constructor| (NIL "The \\spadtype{IntegerRoots} package computes square roots and \\indented{2}{\\spad{n}th roots of integers efficiently.}")) (|approxSqrt| ((|#1| |#1|) "\\spad{approxSqrt(n)} returns an approximation \\spad{x} to \\spad{sqrt(n)} such that \\spad{-1 < x - sqrt(n) < 1}. Compute an approximation \\spad{s} to \\spad{sqrt(n)} such that \\indented{10}{\\spad{-1 < s - sqrt(n) < 1}} A variable precision Newton iteration is used. The running time is \\spad{O( log(n)**2 )}.")) (|perfectSqrt| (((|Union| |#1| "failed") |#1|) "\\spad{perfectSqrt(n)} returns the square root of \\spad{n} if \\spad{n} is a perfect square and returns \"failed\" otherwise")) (|perfectSquare?| (((|Boolean|) |#1|) "\\spad{perfectSquare?(n)} returns \\spad{true} if \\spad{n} is a perfect square and \\spad{false} otherwise")) (|approxNthRoot| ((|#1| |#1| (|NonNegativeInteger|)) "\\spad{approxRoot(n,r)} returns an approximation \\spad{x} to \\spad{n**(1/r)} such that \\spad{-1 < x - n**(1/r) < 1}")) (|perfectNthRoot| (((|Record| (|:| |base| |#1|) (|:| |exponent| (|NonNegativeInteger|))) |#1|) "\\spad{perfectNthRoot(n)} returns \\spad{[x,r]},{} where \\spad{n = x\\^r} and \\spad{r} is the largest integer such that \\spad{n} is a perfect \\spad{r}th power") (((|Union| |#1| "failed") |#1| (|NonNegativeInteger|)) "\\spad{perfectNthRoot(n,r)} returns the \\spad{r}th root of \\spad{n} if \\spad{n} is an \\spad{r}th power and returns \"failed\" otherwise")) (|perfectNthPower?| (((|Boolean|) |#1| (|NonNegativeInteger|)) "\\spad{perfectNthPower?(n,r)} returns \\spad{true} if \\spad{n} is an \\spad{r}th power and \\spad{false} otherwise"))) 
 NIL 
 NIL 
-(-592 GF) 
+(-593 GF) 
 ((|constructor| (NIL "This package exports the function generateIrredPoly that computes a monic irreducible polynomial of degree \\spad{n} over a finite field.")) (|generateIrredPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{generateIrredPoly(n)} generates an irreducible univariate polynomial of the given degree \\spad{n} over the finite field."))) 
 NIL 
 NIL 
-(-593 R) 
+(-594 R) 
 ((|constructor| (NIL "\\indented{2}{This package allows a sum of logs over the roots of a polynomial} \\indented{2}{to be expressed as explicit logarithms and arc tangents,{} provided} \\indented{2}{that the indexing polynomial can be factored into quadratics.} Date Created: 21 August 1988 Date Last Updated: 4 October 1993")) (|complexIntegrate| (((|Expression| |#1|) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{complexIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable.")) (|integrate| (((|Union| (|Expression| |#1|) (|List| (|Expression| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{integrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a real variable..")) (|complexExpand| (((|Expression| |#1|) (|IntegrationResult| (|Fraction| (|Polynomial| |#1|)))) "\\spad{complexExpand(i)} returns the expanded complex function corresponding to \\spad{i}.")) (|expand| (((|List| (|Expression| |#1|)) (|IntegrationResult| (|Fraction| (|Polynomial| |#1|)))) "\\spad{expand(i)} returns the list of possible real functions corresponding to \\spad{i}.")) (|split| (((|IntegrationResult| (|Fraction| (|Polynomial| |#1|))) (|IntegrationResult| (|Fraction| (|Polynomial| |#1|)))) "\\spad{split(u(x) + sum_{P(a)=0} Q(a,x))} returns \\spad{u(x) + sum_{P1(a)=0} Q(a,x) + ... + sum_{Pn(a)=0} Q(a,x)} where \\spad{P1},{}...,{}\\spad{Pn} are the factors of \\spad{P}."))) 
 NIL 
 ((|HasCategory| |#1| (QUOTE (-147)))) 
-(-594) 
+(-595) 
 ((|constructor| (NIL "IrrRepSymNatPackage contains functions for computing the ordinary irreducible representations of symmetric groups on \\spad{n} letters {\\em {1,2,...,n}} in Young\\spad{'s} natural form and their dimensions. These representations can be labelled by number partitions of \\spad{n},{} \\spadignore{i.e.} a weakly decreasing sequence of integers summing up to \\spad{n},{} \\spadignore{e.g.} {\\em [3,3,3,1]} labels an irreducible representation for \\spad{n} equals 10. Note: whenever a \\spadtype{List Integer} appears in a signature,{} a partition required.")) (|irreducibleRepresentation| (((|List| (|Matrix| (|Integer|))) (|List| (|Integer|)) (|List| (|Permutation| (|Integer|)))) "\\spad{irreducibleRepresentation(lambda,listOfPerm)} is the list of the irreducible representations corresponding to {\\em lambda} in Young\\spad{'s} natural form for the list of permutations given by {\\em listOfPerm}.") (((|List| (|Matrix| (|Integer|))) (|List| (|Integer|))) "\\spad{irreducibleRepresentation(lambda)} is the list of the two irreducible representations corresponding to the partition {\\em lambda} in Young\\spad{'s} natural form for the following two generators of the symmetric group,{} whose elements permute {\\em {1,2,...,n}},{} namely {\\em (1 2)} (2-cycle) and {\\em (1 2 ... n)} (\\spad{n}-cycle).") (((|Matrix| (|Integer|)) (|List| (|Integer|)) (|Permutation| (|Integer|))) "\\spad{irreducibleRepresentation(lambda,pi)} is the irreducible representation corresponding to partition {\\em lambda} in Young\\spad{'s} natural form of the permutation {\\em pi} in the symmetric group,{} whose elements permute {\\em {1,2,...,n}}.")) (|dimensionOfIrreducibleRepresentation| (((|NonNegativeInteger|) (|List| (|Integer|))) "\\spad{dimensionOfIrreducibleRepresentation(lambda)} is the dimension of the ordinary irreducible representation of the symmetric group corresponding to {\\em lambda}. Note: the Robinson-Thrall hook formula is implemented."))) 
 NIL 
 NIL 
-(-595 R E V P TS) 
+(-596 R E V P TS) 
 ((|constructor| (NIL "\\indented{1}{An internal package for computing the rational univariate representation} \\indented{1}{of a zero-dimensional algebraic variety given by a square-free} \\indented{1}{triangular set.} \\indented{1}{The main operation is \\axiomOpFrom{rur}{InternalRationalUnivariateRepresentationPackage}.} \\indented{1}{It is based on the {\\em generic} algorithm description in [1]. \\newline References:} [1] \\spad{D}. LAZARD \"Solving Zero-dimensional Algebraic Systems\" \\indented{4}{Journal of Symbolic Computation,{} 1992,{} 13,{} 117-131}")) (|checkRur| (((|Boolean|) |#5| (|List| |#5|)) "\\spad{checkRur(ts,lus)} returns \\spad{true} if \\spad{lus} is a rational univariate representation of \\spad{ts}.")) (|rur| (((|List| |#5|) |#5| (|Boolean|)) "\\spad{rur(ts,univ?)} returns a rational univariate representation of \\spad{ts}. This assumes that the lowest polynomial in \\spad{ts} is a variable \\spad{v} which does not occur in the other polynomials of \\spad{ts}. This variable will be used to define the simple algebraic extension over which these other polynomials will be rewritten as univariate polynomials with degree one. If \\spad{univ?} is \\spad{true} then these polynomials will have a constant initial."))) 
 NIL 
 NIL 
-(-596) 
+(-597) 
 ((|constructor| (NIL "This domain represents a `has' expression.")) (|rhs| (((|SpadAst|) $) "\\spad{rhs(e)} returns the right hand side of the is expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the is expression `e'."))) 
 NIL 
 NIL 
-(-597 |mn|) 
+(-598 |mn|) 
 ((|constructor| (NIL "This domain implements low-level strings"))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| (-144) (QUOTE (-853))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144))))) (-12 (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144)))))) (-3962 (-12 (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144))))) (|HasCategory| (-144) (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| (-144) (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| (-144) (QUOTE (-853))) (|HasCategory| (-144) (QUOTE (-1105)))) (|HasCategory| (-144) (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144)))))) 
-(-598 E V R P) 
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+(-599 E V R P) 
 ((|constructor| (NIL "tools for the summation packages.")) (|sum| (((|Record| (|:| |num| |#4|) (|:| |den| (|Integer|))) |#4| |#2|) "\\spad{sum(p(n), n)} returns \\spad{P(n)},{} the indefinite sum of \\spad{p(n)} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{P(n+1) - P(n) = a(n)}.") (((|Record| (|:| |num| |#4|) (|:| |den| (|Integer|))) |#4| |#2| (|Segment| |#4|)) "\\spad{sum(p(n), n = a..b)} returns \\spad{p(a) + p(a+1) + ... + p(b)}."))) 
 NIL 
 NIL 
-(-599 |Coef|) 
-((|constructor| (NIL "InnerSparseUnivariatePowerSeries is an internal domain \\indented{2}{used for creating sparse Taylor and Laurent series.}")) (|cAcsch| (($ $) "\\spad{cAcsch(f)} computes the inverse hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsech| (($ $) "\\spad{cAsech(f)} computes the inverse hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcoth| (($ $) "\\spad{cAcoth(f)} computes the inverse hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtanh| (($ $) "\\spad{cAtanh(f)} computes the inverse hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcosh| (($ $) "\\spad{cAcosh(f)} computes the inverse hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsinh| (($ $) "\\spad{cAsinh(f)} computes the inverse hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsch| (($ $) "\\spad{cCsch(f)} computes the hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSech| (($ $) "\\spad{cSech(f)} computes the hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCoth| (($ $) "\\spad{cCoth(f)} computes the hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTanh| (($ $) "\\spad{cTanh(f)} computes the hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCosh| (($ $) "\\spad{cCosh(f)} computes the hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSinh| (($ $) "\\spad{cSinh(f)} computes the hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcsc| (($ $) "\\spad{cAcsc(f)} computes the arccosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsec| (($ $) "\\spad{cAsec(f)} computes the arcsecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcot| (($ $) "\\spad{cAcot(f)} computes the arccotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtan| (($ $) "\\spad{cAtan(f)} computes the arctangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcos| (($ $) "\\spad{cAcos(f)} computes the arccosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsin| (($ $) "\\spad{cAsin(f)} computes the arcsine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsc| (($ $) "\\spad{cCsc(f)} computes the cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSec| (($ $) "\\spad{cSec(f)} computes the secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCot| (($ $) "\\spad{cCot(f)} computes the cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTan| (($ $) "\\spad{cTan(f)} computes the tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCos| (($ $) "\\spad{cCos(f)} computes the cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSin| (($ $) "\\spad{cSin(f)} computes the sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cLog| (($ $) "\\spad{cLog(f)} computes the logarithm of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cExp| (($ $) "\\spad{cExp(f)} computes the exponential of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cRationalPower| (($ $ (|Fraction| (|Integer|))) "\\spad{cRationalPower(f,r)} computes \\spad{f^r}. For use when the coefficient ring is commutative.")) (|cPower| (($ $ |#1|) "\\spad{cPower(f,r)} computes \\spad{f^r},{} where \\spad{f} has constant coefficient 1. For use when the coefficient ring is commutative.")) (|integrate| (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. Warning: function does not check for a term of degree \\spad{-1}.")) (|seriesToOutputForm| (((|OutputForm|) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) (|Reference| (|OrderedCompletion| (|Integer|))) (|Symbol|) |#1| (|Fraction| (|Integer|))) "\\spad{seriesToOutputForm(st,refer,var,cen,r)} prints the series \\spad{f((var - cen)^r)}.")) (|iCompose| (($ $ $) "\\spad{iCompose(f,g)} returns \\spad{f(g(x))}. This is an internal function which should only be called for Taylor series \\spad{f(x)} and \\spad{g(x)} such that the constant coefficient of \\spad{g(x)} is zero.")) (|taylorQuoByVar| (($ $) "\\spad{taylorQuoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...}")) (|iExquo| (((|Union| $ "failed") $ $ (|Boolean|)) "\\spad{iExquo(f,g,taylor?)} is the quotient of the power series \\spad{f} and \\spad{g}. If \\spad{taylor?} is \\spad{true},{} then we must have \\spad{order(f) >= order(g)}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(fn,f)} returns the series \\spad{sum(fn(n) * an * x^n,n = n0..)},{} where \\spad{f} is the series \\spad{sum(an * x^n,n = n0..)}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(f)} tests if \\spad{f} is a single monomial.")) (|series| (($ (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.")) (|getStream| (((|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) $) "\\spad{getStream(f)} returns the stream of terms representing the series \\spad{f}.")) (|getRef| (((|Reference| (|OrderedCompletion| (|Integer|))) $) "\\spad{getRef(f)} returns a reference containing the order to which the terms of \\spad{f} have been computed.")) (|makeSeries| (($ (|Reference| (|OrderedCompletion| (|Integer|))) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{makeSeries(refer,str)} creates a power series from the reference \\spad{refer} and the stream \\spad{str}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-550)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-550)) (|devaluate| |#1|)))) (|HasCategory| (-550) (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-366))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-550)))))) 
 (-600 |Coef|) 
+((|constructor| (NIL "InnerSparseUnivariatePowerSeries is an internal domain \\indented{2}{used for creating sparse Taylor and Laurent series.}")) (|cAcsch| (($ $) "\\spad{cAcsch(f)} computes the inverse hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsech| (($ $) "\\spad{cAsech(f)} computes the inverse hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcoth| (($ $) "\\spad{cAcoth(f)} computes the inverse hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtanh| (($ $) "\\spad{cAtanh(f)} computes the inverse hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcosh| (($ $) "\\spad{cAcosh(f)} computes the inverse hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsinh| (($ $) "\\spad{cAsinh(f)} computes the inverse hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsch| (($ $) "\\spad{cCsch(f)} computes the hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSech| (($ $) "\\spad{cSech(f)} computes the hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCoth| (($ $) "\\spad{cCoth(f)} computes the hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTanh| (($ $) "\\spad{cTanh(f)} computes the hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCosh| (($ $) "\\spad{cCosh(f)} computes the hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSinh| (($ $) "\\spad{cSinh(f)} computes the hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcsc| (($ $) "\\spad{cAcsc(f)} computes the arccosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsec| (($ $) "\\spad{cAsec(f)} computes the arcsecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcot| (($ $) "\\spad{cAcot(f)} computes the arccotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtan| (($ $) "\\spad{cAtan(f)} computes the arctangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcos| (($ $) "\\spad{cAcos(f)} computes the arccosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsin| (($ $) "\\spad{cAsin(f)} computes the arcsine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsc| (($ $) "\\spad{cCsc(f)} computes the cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSec| (($ $) "\\spad{cSec(f)} computes the secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCot| (($ $) "\\spad{cCot(f)} computes the cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTan| (($ $) "\\spad{cTan(f)} computes the tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCos| (($ $) "\\spad{cCos(f)} computes the cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSin| (($ $) "\\spad{cSin(f)} computes the sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cLog| (($ $) "\\spad{cLog(f)} computes the logarithm of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cExp| (($ $) "\\spad{cExp(f)} computes the exponential of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cRationalPower| (($ $ (|Fraction| (|Integer|))) "\\spad{cRationalPower(f,r)} computes \\spad{f^r}. For use when the coefficient ring is commutative.")) (|cPower| (($ $ |#1|) "\\spad{cPower(f,r)} computes \\spad{f^r},{} where \\spad{f} has constant coefficient 1. For use when the coefficient ring is commutative.")) (|integrate| (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. Warning: function does not check for a term of degree \\spad{-1}.")) (|seriesToOutputForm| (((|OutputForm|) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) (|Reference| (|OrderedCompletion| (|Integer|))) (|Symbol|) |#1| (|Fraction| (|Integer|))) "\\spad{seriesToOutputForm(st,refer,var,cen,r)} prints the series \\spad{f((var - cen)^r)}.")) (|iCompose| (($ $ $) "\\spad{iCompose(f,g)} returns \\spad{f(g(x))}. This is an internal function which should only be called for Taylor series \\spad{f(x)} and \\spad{g(x)} such that the constant coefficient of \\spad{g(x)} is zero.")) (|taylorQuoByVar| (($ $) "\\spad{taylorQuoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...}")) (|iExquo| (((|Union| $ "failed") $ $ (|Boolean|)) "\\spad{iExquo(f,g,taylor?)} is the quotient of the power series \\spad{f} and \\spad{g}. If \\spad{taylor?} is \\spad{true},{} then we must have \\spad{order(f) >= order(g)}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(fn,f)} returns the series \\spad{sum(fn(n) * an * x^n,n = n0..)},{} where \\spad{f} is the series \\spad{sum(an * x^n,n = n0..)}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(f)} tests if \\spad{f} is a single monomial.")) (|series| (($ (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.")) (|getStream| (((|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) $) "\\spad{getStream(f)} returns the stream of terms representing the series \\spad{f}.")) (|getRef| (((|Reference| (|OrderedCompletion| (|Integer|))) $) "\\spad{getRef(f)} returns a reference containing the order to which the terms of \\spad{f} have been computed.")) (|makeSeries| (($ (|Reference| (|OrderedCompletion| (|Integer|))) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{makeSeries(refer,str)} creates a power series from the reference \\spad{refer} and the stream \\spad{str}."))) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-551)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-551)) (|devaluate| |#1|)))) (|HasCategory| (-551) (QUOTE (-1118))) (|HasCategory| |#1| (QUOTE (-367))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-551))))) (|HasSignature| |#1| (LIST (QUOTE -4387) (LIST (|devaluate| |#1|) (QUOTE (-1183)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-551)))))) 
+(-601 |Coef|) 
 ((|constructor| (NIL "Internal package for dense Taylor series. This is an internal Taylor series type in which Taylor series are represented by a \\spadtype{Stream} of \\spadtype{Ring} elements. For univariate series,{} the \\spad{Stream} elements are the Taylor coefficients. For multivariate series,{} the \\spad{n}th Stream element is a form of degree \\spad{n} in the power series variables.")) (* (($ $ (|Integer|)) "\\spad{x*i} returns the product of integer \\spad{i} and the series \\spad{x}.")) (|order| (((|NonNegativeInteger|) $ (|NonNegativeInteger|)) "\\spad{order(x,n)} returns the minimum of \\spad{n} and the order of \\spad{x}.") (((|NonNegativeInteger|) $) "\\spad{order(x)} returns the order of a power series \\spad{x},{} \\indented{1}{\\spadignore{i.e.} the degree of the first non-zero term of the series.}")) (|pole?| (((|Boolean|) $) "\\spad{pole?(x)} tests if the series \\spad{x} has a pole. \\indented{1}{Note: this is \\spad{false} when \\spad{x} is a Taylor series.}")) (|series| (($ (|Stream| |#1|)) "\\spad{series(s)} creates a power series from a stream of \\indented{1}{ring elements.} \\indented{1}{For univariate series types,{} the stream \\spad{s} should be a stream} \\indented{1}{of Taylor coefficients. For multivariate series types,{} the} \\indented{1}{stream \\spad{s} should be a stream of forms the \\spad{n}th element} \\indented{1}{of which is a} \\indented{1}{form of degree \\spad{n} in the power series variables.}")) (|coefficients| (((|Stream| |#1|) $) "\\spad{coefficients(x)} returns a stream of ring elements. \\indented{1}{When \\spad{x} is a univariate series,{} this is a stream of Taylor} \\indented{1}{coefficients. When \\spad{x} is a multivariate series,{} the} \\indented{1}{\\spad{n}th element of the stream is a form of} \\indented{1}{degree \\spad{n} in the power series variables.}"))) 
-(((-4429 "*") |has| |#1| (-561)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-561)))) 
-(-601 A B) 
+(((-4436 "*") |has| |#1| (-562)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-562)))) 
+(-602) 
+((|constructor| (NIL "This domain provides representations for internal type form."))) 
+NIL 
+NIL 
+(-603 A B) 
 ((|constructor| (NIL "Functions defined on streams with entries in two sets.")) (|map| (((|InfiniteTuple| |#2|) (|Mapping| |#2| |#1|) (|InfiniteTuple| |#1|)) "\\spad{map(f,[x0,x1,x2,...])} returns \\spad{[f(x0),f(x1),f(x2),..]}."))) 
 NIL 
 NIL 
-(-602 A B C) 
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 ((|constructor| (NIL "Functions defined on streams with entries in two sets.")) (|map| (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|InfiniteTuple| |#1|) (|Stream| |#2|)) "\\spad{map(f,a,b)} \\undocumented") (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|Stream| |#1|) (|InfiniteTuple| |#2|)) "\\spad{map(f,a,b)} \\undocumented") (((|InfiniteTuple| |#3|) (|Mapping| |#3| |#1| |#2|) (|InfiniteTuple| |#1|) (|InfiniteTuple| |#2|)) "\\spad{map(f,a,b)} \\undocumented"))) 
 NIL 
 NIL 
-(-603 R -3498 FG) 
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 ((|constructor| (NIL "This package provides transformations from trigonometric functions to exponentials and logarithms,{} and back. \\spad{F} and \\spad{FG} should be the same type of function space.")) (|trigs2explogs| ((|#3| |#3| (|List| (|Kernel| |#3|)) (|List| (|Symbol|))) "\\spad{trigs2explogs(f, [k1,...,kn], [x1,...,xm])} rewrites all the trigonometric functions appearing in \\spad{f} and involving one of the \\spad{xi's} in terms of complex logarithms and exponentials. A kernel of the form \\spad{tan(u)} is expressed using \\spad{exp(u)**2} if it is one of the \\spad{ki's},{} in terms of \\spad{exp(2*u)} otherwise.")) (|explogs2trigs| (((|Complex| |#2|) |#3|) "\\spad{explogs2trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (F2FG ((|#3| |#2|) "\\spad{F2FG(a + sqrt(-1) b)} returns \\spad{a + i b}.")) (FG2F ((|#2| |#3|) "\\spad{FG2F(a + i b)} returns \\spad{a + sqrt(-1) b}.")) (GF2FG ((|#3| (|Complex| |#2|)) "\\spad{GF2FG(a + i b)} returns \\spad{a + i b} viewed as a function with the \\spad{i} pushed down into the coefficient domain."))) 
 NIL 
 NIL 
-(-604 S) 
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 ((|constructor| (NIL "\\indented{1}{This package implements 'infinite tuples' for the interpreter.} The representation is a stream.")) (|construct| (((|Stream| |#1|) $) "\\spad{construct(t)} converts an infinite tuple to a stream.")) (|generate| (($ (|Mapping| |#1| |#1|) |#1|) "\\spad{generate(f,s)} returns \\spad{[s,f(s),f(f(s)),...]}.")) (|select| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select(p,t)} returns \\spad{[x for x in t | p(x)]}.")) (|filterUntil| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterUntil(p,t)} returns \\spad{[x for x in t while not p(x)]}.")) (|filterWhile| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterWhile(p,t)} returns \\spad{[x for x in t while p(x)]}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,t)} replaces the tuple \\spad{t} by \\spad{[f(x) for x in t]}."))) 
 NIL 
 NIL 
-(-605 R |mn|) 
+(-607 R |mn|) 
 ((|constructor| (NIL "\\indented{2}{This type represents vector like objects with varying lengths} and a user-specified initial index."))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#1| (QUOTE (-1053))) (-12 (|HasCategory| |#1| (QUOTE (-1006))) (|HasCategory| |#1| (QUOTE (-1053)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-606 S |Index| |Entry|) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-731))) (|HasCategory| |#1| (QUOTE (-1055))) (-12 (|HasCategory| |#1| (QUOTE (-1008))) (|HasCategory| |#1| (QUOTE (-1055)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-608 S |Index| |Entry|) 
 ((|constructor| (NIL "An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.")) (|swap!| (((|Void|) $ |#2| |#2|) "\\spad{swap!(u,i,j)} interchanges elements \\spad{i} and \\spad{j} of aggregate \\spad{u}. No meaningful value is returned.")) (|fill!| (($ $ |#3|) "\\spad{fill!(u,x)} replaces each entry in aggregate \\spad{u} by \\spad{x}. The modified \\spad{u} is returned as value.")) (|first| ((|#3| $) "\\spad{first(u)} returns the first element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{first([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = \\spad{x}}. Error: if \\spad{u} is empty.")) (|minIndex| ((|#2| $) "\\spad{minIndex(u)} returns the minimum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{minIndex(a) = reduce(min,{}[\\spad{i} for \\spad{i} in indices a])}; for lists,{} \\axiom{minIndex(a) = 1}.")) (|maxIndex| ((|#2| $) "\\spad{maxIndex(u)} returns the maximum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{maxIndex(\\spad{u}) = reduce(max,{}[\\spad{i} for \\spad{i} in indices \\spad{u}])}; if \\spad{u} is a list,{} \\axiom{maxIndex(\\spad{u}) = \\#u}.")) (|entry?| (((|Boolean|) |#3| $) "\\spad{entry?(x,u)} tests if \\spad{x} equals \\axiom{\\spad{u} . \\spad{i}} for some index \\spad{i}.")) (|indices| (((|List| |#2|) $) "\\spad{indices(u)} returns a list of indices of aggregate \\spad{u} in no particular order.")) (|index?| (((|Boolean|) |#2| $) "\\spad{index?(i,u)} tests if \\spad{i} is an index of aggregate \\spad{u}.")) (|entries| (((|List| |#3|) $) "\\spad{entries(u)} returns a list of all the entries of aggregate \\spad{u} in no assumed order."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4428)) (|HasCategory| |#2| (QUOTE (-853))) (|HasAttribute| |#1| (QUOTE -4427)) (|HasCategory| |#3| (QUOTE (-1105)))) 
-(-607 |Index| |Entry|) 
+((|HasAttribute| |#1| (QUOTE -4435)) (|HasCategory| |#2| (QUOTE (-855))) (|HasAttribute| |#1| (QUOTE -4434)) (|HasCategory| |#3| (QUOTE (-1107)))) 
+(-609 |Index| |Entry|) 
 ((|constructor| (NIL "An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.")) (|swap!| (((|Void|) $ |#1| |#1|) "\\spad{swap!(u,i,j)} interchanges elements \\spad{i} and \\spad{j} of aggregate \\spad{u}. No meaningful value is returned.")) (|fill!| (($ $ |#2|) "\\spad{fill!(u,x)} replaces each entry in aggregate \\spad{u} by \\spad{x}. The modified \\spad{u} is returned as value.")) (|first| ((|#2| $) "\\spad{first(u)} returns the first element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{first([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = \\spad{x}}. Error: if \\spad{u} is empty.")) (|minIndex| ((|#1| $) "\\spad{minIndex(u)} returns the minimum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{minIndex(a) = reduce(min,{}[\\spad{i} for \\spad{i} in indices a])}; for lists,{} \\axiom{minIndex(a) = 1}.")) (|maxIndex| ((|#1| $) "\\spad{maxIndex(u)} returns the maximum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{maxIndex(\\spad{u}) = reduce(max,{}[\\spad{i} for \\spad{i} in indices \\spad{u}])}; if \\spad{u} is a list,{} \\axiom{maxIndex(\\spad{u}) = \\#u}.")) (|entry?| (((|Boolean|) |#2| $) "\\spad{entry?(x,u)} tests if \\spad{x} equals \\axiom{\\spad{u} . \\spad{i}} for some index \\spad{i}.")) (|indices| (((|List| |#1|) $) "\\spad{indices(u)} returns a list of indices of aggregate \\spad{u} in no particular order.")) (|index?| (((|Boolean|) |#1| $) "\\spad{index?(i,u)} tests if \\spad{i} is an index of aggregate \\spad{u}.")) (|entries| (((|List| |#2|) $) "\\spad{entries(u)} returns a list of all the entries of aggregate \\spad{u} in no assumed order."))) 
 NIL 
 NIL 
-(-608) 
+(-610) 
 ((|constructor| (NIL "\\indented{1}{This domain defines the datatype for the Java} Virtual Machine byte codes."))) 
 NIL 
 NIL 
-(-609) 
+(-611) 
 ((|constructor| (NIL "This domain represents the join of categories ASTs.")) (|categories| (((|List| (|TypeAst|)) $) "catehories(\\spad{x}) returns the types in the join \\spad{`x'}.")) (|coerce| (($ (|List| (|TypeAst|))) "ts::JoinAst construct the AST for a join of the types `ts'."))) 
 NIL 
 NIL 
-(-610 R A) 
+(-612 R A) 
 ((|constructor| (NIL "\\indented{1}{AssociatedJordanAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$A}} \\indented{1}{to define the new multiplications \\spad{a*b := (a *\\$A b + b *\\$A a)/2}} \\indented{1}{(anticommutator).} \\indented{1}{The usual notation \\spad{{a,b}_+} cannot be used due to} \\indented{1}{restrictions in the current language.} \\indented{1}{This domain only gives a Jordan algebra if the} \\indented{1}{Jordan-identity \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds} \\indented{1}{for all \\spad{a},{}\\spad{b},{}\\spad{c} in \\spad{A}.} \\indented{1}{This relation can be checked by} \\indented{1}{\\spadfun{jordanAdmissible?()\\$A}.} \\blankline If the underlying algebra is of type \\spadtype{FramedNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank,{} together with a fixed \\spad{R}-module basis),{} then the same is \\spad{true} for the associated Jordan algebra. Moreover,{} if the underlying algebra is of type \\spadtype{FiniteRankNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank),{} then the same \\spad{true} for the associated Jordan algebra.")) (|coerce| (($ |#2|) "\\spad{coerce(a)} coerces the element \\spad{a} of the algebra \\spad{A} to an element of the Jordan algebra \\spadtype{AssociatedJordanAlgebra}(\\spad{R},{}A)."))) 
-((-4424 -3962 (-3258 (|has| |#2| (-370 |#1|)) (|has| |#1| (-561))) (-12 (|has| |#2| (-422 |#1|)) (|has| |#1| (-561)))) (-4422 . T) (-4421 . T)) 
-((-3962 (|HasCategory| |#2| (LIST (QUOTE -370) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -422) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -422) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#2| (LIST (QUOTE -422) (|devaluate| |#1|)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#2| (LIST (QUOTE -370) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#2| (LIST (QUOTE -422) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -370) (|devaluate| |#1|)))) 
-(-611 |Entry|) 
+((-4431 -3969 (-3265 (|has| |#2| (-371 |#1|)) (|has| |#1| (-562))) (-12 (|has| |#2| (-423 |#1|)) (|has| |#1| (-562)))) (-4429 . T) (-4428 . T)) 
+((-3969 (|HasCategory| |#2| (LIST (QUOTE -371) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -371) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -371) (|devaluate| |#1|)))) 
+(-613 |Entry|) 
 ((|constructor| (NIL "This domain allows a random access file to be viewed both as a table and as a file object.")) (|pack!| (($ $) "\\spad{pack!(f)} reorganizes the file \\spad{f} on disk to recover unused space."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (QUOTE (-1163))) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| (-1163) (QUOTE (-853))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-612 S |Key| |Entry|) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (QUOTE (-1165))) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| (-1165) (QUOTE (-855))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-614 S |Key| |Entry|) 
 ((|constructor| (NIL "A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.")) (|search| (((|Union| |#3| "failed") |#2| $) "\\spad{search(k,t)} searches the table \\spad{t} for the key \\spad{k},{} returning the entry stored in \\spad{t} for key \\spad{k}. If \\spad{t} has no such key,{} \\axiom{search(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|remove!| (((|Union| |#3| "failed") |#2| $) "\\spad{remove!(k,t)} searches the table \\spad{t} for the key \\spad{k} removing (and return) the entry if there. If \\spad{t} has no such key,{} \\axiom{remove!(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|keys| (((|List| |#2|) $) "\\spad{keys(t)} returns the list the keys in table \\spad{t}.")) (|key?| (((|Boolean|) |#2| $) "\\spad{key?(k,t)} tests if \\spad{k} is a key in table \\spad{t}."))) 
 NIL 
 NIL 
-(-613 |Key| |Entry|) 
+(-615 |Key| |Entry|) 
 ((|constructor| (NIL "A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.")) (|search| (((|Union| |#2| "failed") |#1| $) "\\spad{search(k,t)} searches the table \\spad{t} for the key \\spad{k},{} returning the entry stored in \\spad{t} for key \\spad{k}. If \\spad{t} has no such key,{} \\axiom{search(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|remove!| (((|Union| |#2| "failed") |#1| $) "\\spad{remove!(k,t)} searches the table \\spad{t} for the key \\spad{k} removing (and return) the entry if there. If \\spad{t} has no such key,{} \\axiom{remove!(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|keys| (((|List| |#1|) $) "\\spad{keys(t)} returns the list the keys in table \\spad{t}.")) (|key?| (((|Boolean|) |#1| $) "\\spad{key?(k,t)} tests if \\spad{k} is a key in table \\spad{t}."))) 
-((-4428 . T)) 
+((-4435 . T)) 
 NIL 
-(-614 S) 
+(-616 S) 
 ((|constructor| (NIL "A kernel over a set \\spad{S} is an operator applied to a given list of arguments from \\spad{S}.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(op(a1,...,an), s)} tests if the name of op is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(op(a1,...,an), f)} tests if op = \\spad{f}.")) (|symbolIfCan| (((|Union| (|Symbol|) "failed") $) "\\spad{symbolIfCan(k)} returns \\spad{k} viewed as a symbol if \\spad{k} is a symbol,{} and \"failed\" otherwise.")) (|kernel| (($ (|Symbol|)) "\\spad{kernel(x)} returns \\spad{x} viewed as a kernel.") (($ (|BasicOperator|) (|List| |#1|) (|NonNegativeInteger|)) "\\spad{kernel(op, [a1,...,an], m)} returns the kernel \\spad{op(a1,...,an)} of nesting level \\spad{m}. Error: if \\spad{op} is \\spad{k}-ary for some \\spad{k} not equal to \\spad{m}.")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(k)} returns the nesting level of \\spad{k}.")) (|argument| (((|List| |#1|) $) "\\spad{argument(op(a1,...,an))} returns \\spad{[a1,...,an]}.")) (|operator| (((|BasicOperator|) $) "\\spad{operator(op(a1,...,an))} returns the operator op."))) 
 NIL 
-((|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) 
-(-615 R S) 
+((|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) 
+(-617 R S) 
 ((|constructor| (NIL "This package exports some auxiliary functions on kernels")) (|constantIfCan| (((|Union| |#1| "failed") (|Kernel| |#2|)) "\\spad{constantIfCan(k)} \\undocumented")) (|constantKernel| (((|Kernel| |#2|) |#1|) "\\spad{constantKernel(r)} \\undocumented"))) 
 NIL 
 NIL 
-(-616 S) 
+(-618 S) 
 ((|constructor| (NIL "A is coercible to \\spad{B} means any element of A can automatically be converted into an element of \\spad{B} by the interpreter.")) (|coerce| ((|#1| $) "\\spad{coerce(a)} transforms a into an element of \\spad{S}."))) 
 NIL 
 NIL 
-(-617 S) 
+(-619 S) 
 ((|constructor| (NIL "A is convertible to \\spad{B} means any element of A can be converted into an element of \\spad{B},{} but not automatically by the interpreter.")) (|convert| ((|#1| $) "\\spad{convert(a)} transforms a into an element of \\spad{S}."))) 
 NIL 
 NIL 
-(-618 -3498 UP) 
+(-620 -3505 UP) 
 ((|constructor| (NIL "\\spadtype{Kovacic} provides a modified Kovacic\\spad{'s} algorithm for solving explicitely irreducible 2nd order linear ordinary differential equations.")) (|kovacic| (((|Union| (|SparseUnivariatePolynomial| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{kovacic(a_0,a_1,a_2,ezfactor)} returns either \"failed\" or \\spad{P}(\\spad{u}) such that \\spad{\\$e^{\\int(-a_1/2a_2)} e^{\\int u}\\$} is a solution of \\indented{5}{\\spad{\\$a_2 y'' + a_1 y' + a0 y = 0\\$}} whenever \\spad{u} is a solution of \\spad{P u = 0}. The equation must be already irreducible over the rational functions. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|Union| (|SparseUnivariatePolynomial| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{kovacic(a_0,a_1,a_2)} returns either \"failed\" or \\spad{P}(\\spad{u}) such that \\spad{\\$e^{\\int(-a_1/2a_2)} e^{\\int u}\\$} is a solution of \\indented{5}{\\spad{a_2 y'' + a_1 y' + a0 y = 0}} whenever \\spad{u} is a solution of \\spad{P u = 0}. The equation must be already irreducible over the rational functions."))) 
 NIL 
 NIL 
-(-619 S) 
+(-621 S) 
 ((|constructor| (NIL "A is coercible from \\spad{B} iff any element of domain \\spad{B} can be automically converted into an element of domain A.")) (|coerce| (($ |#1|) "\\spad{coerce(s)} transforms \\spad{`s'} into an element of `\\%'."))) 
 NIL 
 NIL 
-(-620) 
+(-622) 
 ((|constructor| (NIL "This domain implements Kleene\\spad{'s} 3-valued propositional logic.")) (|case| (((|Boolean|) $ (|[\|\|]| |true|)) "\\spad{s case true} holds if the value of \\spad{`x'} is `true'.") (((|Boolean|) $ (|[\|\|]| |unknown|)) "\\spad{x case unknown} holds if the value of \\spad{`x'} is `unknown'") (((|Boolean|) $ (|[\|\|]| |false|)) "\\spad{x case false} holds if the value of \\spad{`x'} is `false'")) (|unknown| (($) "the indefinite `unknown'"))) 
 NIL 
 NIL 
-(-621 S) 
+(-623 S) 
 ((|constructor| (NIL "A is convertible from \\spad{B} iff any element of domain \\spad{B} can be explicitly converted into an element of domain A.")) (|convert| (($ |#1|) "\\spad{convert(s)} transforms \\spad{`s'} into an element of `\\%'."))) 
 NIL 
 NIL 
-(-622 A R S) 
+(-624 A R S) 
 ((|constructor| (NIL "LocalAlgebra produces the localization of an algebra,{} \\spadignore{i.e.} fractions whose numerators come from some \\spad{R} algebra.")) (|denom| ((|#3| $) "\\spad{denom x} returns the denominator of \\spad{x}.")) (|numer| ((|#1| $) "\\spad{numer x} returns the numerator of \\spad{x}.")) (/ (($ |#1| |#3|) "\\spad{a / d} divides the element \\spad{a} by \\spad{d}.") (($ $ |#3|) "\\spad{x / d} divides the element \\spad{x} by \\spad{d}."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-851)))) 
-(-623 S R) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-853)))) 
+(-625 S R) 
 ((|constructor| (NIL "The category of all left algebras over an arbitrary ring.")) (|coerce| (($ |#2|) "\\spad{coerce(r)} returns \\spad{r} * 1 where 1 is the identity of the left algebra."))) 
 NIL 
 NIL 
-(-624 R) 
+(-626 R) 
 ((|constructor| (NIL "The category of all left algebras over an arbitrary ring.")) (|coerce| (($ |#1|) "\\spad{coerce(r)} returns \\spad{r} * 1 where 1 is the identity of the left algebra."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-625 R -3498) 
+(-627 R -3505) 
 ((|constructor| (NIL "This package computes the forward Laplace Transform.")) (|laplace| ((|#2| |#2| (|Symbol|) (|Symbol|)) "\\spad{laplace(f, t, s)} returns the Laplace transform of \\spad{f(t)} using \\spad{s} as the new variable. This is \\spad{integral(exp(-s*t)*f(t), t = 0..\\%plusInfinity)}. Returns the formal object \\spad{laplace(f, t, s)} if it cannot compute the transform."))) 
 NIL 
 NIL 
-(-626 R UP) 
+(-628 R UP) 
 ((|constructor| (NIL "\\indented{1}{Univariate polynomials with negative and positive exponents.} Author: Manuel Bronstein Date Created: May 1988 Date Last Updated: 26 Apr 1990")) (|separate| (((|Record| (|:| |polyPart| $) (|:| |fracPart| (|Fraction| |#2|))) (|Fraction| |#2|)) "\\spad{separate(x)} \\undocumented")) (|monomial| (($ |#1| (|Integer|)) "\\spad{monomial(x,n)} \\undocumented")) (|coefficient| ((|#1| $ (|Integer|)) "\\spad{coefficient(x,n)} \\undocumented")) (|trailingCoefficient| ((|#1| $) "\\spad{trailingCoefficient }\\undocumented")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient }\\undocumented")) (|reductum| (($ $) "\\spad{reductum(x)} \\undocumented")) (|order| (((|Integer|) $) "\\spad{order(x)} \\undocumented")) (|degree| (((|Integer|) $) "\\spad{degree(x)} \\undocumented")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} \\undocumented"))) 
-((-4422 . T) (-4421 . T) ((-4429 "*") . T) (-4420 . T) (-4424 . T)) 
-((|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550))))) 
-(-627 R E V P TS ST) 
+((-4429 . T) (-4428 . T) ((-4436 "*") . T) (-4427 . T) (-4431 . T)) 
+((|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551))))) 
+(-629 R E V P TS ST) 
 ((|constructor| (NIL "A package for solving polynomial systems by means of Lazard triangular sets [1]. This package provides two operations. One for solving in the sense of the regular zeros,{} and the other for solving in the sense of the Zariski closure. Both produce square-free regular sets. Moreover,{} the decompositions do not contain any redundant component. However,{} only zero-dimensional regular sets are normalized,{} since normalization may be time consumming in positive dimension. The decomposition process is that of [2].\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|zeroSetSplit| (((|List| |#6|) (|List| |#4|) (|Boolean|)) "\\axiom{zeroSetSplit(\\spad{lp},{}clos?)} has the same specifications as \\axiomOpFrom{zeroSetSplit(\\spad{lp},{}clos?)}{RegularTriangularSetCategory}.")) (|normalizeIfCan| ((|#6| |#6|) "\\axiom{normalizeIfCan(\\spad{ts})} returns \\axiom{\\spad{ts}} in an normalized shape if \\axiom{\\spad{ts}} is zero-dimensional."))) 
 NIL 
 NIL 
-(-628 OV E Z P) 
+(-630 OV E Z P) 
 ((|constructor| (NIL "Package for leading coefficient determination in the lifting step. Package working for every \\spad{R} euclidean with property \\spad{\"F\"}.")) (|distFact| (((|Union| (|Record| (|:| |polfac| (|List| |#4|)) (|:| |correct| |#3|) (|:| |corrfact| (|List| (|SparseUnivariatePolynomial| |#3|)))) "failed") |#3| (|List| (|SparseUnivariatePolynomial| |#3|)) (|Record| (|:| |contp| |#3|) (|:| |factors| (|List| (|Record| (|:| |irr| |#4|) (|:| |pow| (|Integer|)))))) (|List| |#3|) (|List| |#1|) (|List| |#3|)) "\\spad{distFact(contm,unilist,plead,vl,lvar,lval)},{} where \\spad{contm} is the content of the evaluated polynomial,{} \\spad{unilist} is the list of factors of the evaluated polynomial,{} \\spad{plead} is the complete factorization of the leading coefficient,{} \\spad{vl} is the list of factors of the leading coefficient evaluated,{} \\spad{lvar} is the list of variables,{} \\spad{lval} is the list of values,{} returns a record giving the list of leading coefficients to impose on the univariate factors,{}")) (|polCase| (((|Boolean|) |#3| (|NonNegativeInteger|) (|List| |#3|)) "\\spad{polCase(contprod, numFacts, evallcs)},{} where \\spad{contprod} is the product of the content of the leading coefficient of the polynomial to be factored with the content of the evaluated polynomial,{} \\spad{numFacts} is the number of factors of the leadingCoefficient,{} and evallcs is the list of the evaluated factors of the leadingCoefficient,{} returns \\spad{true} if the factors of the leading Coefficient can be distributed with this valuation."))) 
 NIL 
 NIL 
-(-629) 
+(-631) 
 ((|constructor| (NIL "This domain represents assignment expressions.")) (|rhs| (((|SpadAst|) $) "\\spad{rhs(e)} returns the right hand side of the assignment expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the assignment expression `e'."))) 
 NIL 
 NIL 
-(-630 |VarSet| R |Order|) 
+(-632 |VarSet| R |Order|) 
 ((|constructor| (NIL "Management of the Lie Group associated with a free nilpotent Lie algebra. Every Lie bracket with length greater than \\axiom{Order} are assumed to be null. The implementation inherits from the \\spadtype{XPBWPolynomial} domain constructor: Lyndon coordinates are exponential coordinates of the second kind. \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr}).")) (|identification| (((|List| (|Equation| |#2|)) $ $) "\\axiom{identification(\\spad{g},{}\\spad{h})} returns the list of equations \\axiom{g_i = h_i},{} where \\axiom{g_i} (resp. \\axiom{h_i}) are exponential coordinates of \\axiom{\\spad{g}} (resp. \\axiom{\\spad{h}}).")) (|LyndonCoordinates| (((|List| (|Record| (|:| |k| (|LyndonWord| |#1|)) (|:| |c| |#2|))) $) "\\axiom{LyndonCoordinates(\\spad{g})} returns the exponential coordinates of \\axiom{\\spad{g}}.")) (|LyndonBasis| (((|List| (|LiePolynomial| |#1| |#2|)) (|List| |#1|)) "\\axiom{LyndonBasis(\\spad{lv})} returns the Lyndon basis of the nilpotent free Lie algebra.")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{g})} returns the list of variables of \\axiom{\\spad{g}}.")) (|mirror| (($ $) "\\axiom{mirror(\\spad{g})} is the mirror of the internal representation of \\axiom{\\spad{g}}.")) (|coerce| (((|XPBWPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{g})} returns the internal representation of \\axiom{\\spad{g}}.") (((|XDistributedPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{g})} returns the internal representation of \\axiom{\\spad{g}}.")) (|ListOfTerms| (((|List| (|Record| (|:| |k| (|PoincareBirkhoffWittLyndonBasis| |#1|)) (|:| |c| |#2|))) $) "\\axiom{ListOfTerms(\\spad{p})} returns the internal representation of \\axiom{\\spad{p}}.")) (|log| (((|LiePolynomial| |#1| |#2|) $) "\\axiom{log(\\spad{p})} returns the logarithm of \\axiom{\\spad{p}}.")) (|exp| (($ (|LiePolynomial| |#1| |#2|)) "\\axiom{exp(\\spad{p})} returns the exponential of \\axiom{\\spad{p}}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-631 R |ls|) 
+(-633 R |ls|) 
 ((|constructor| (NIL "A package for solving polynomial systems with finitely many solutions. The decompositions are given by means of regular triangular sets. The computations use lexicographical Groebner bases. The main operations are \\axiomOpFrom{lexTriangular}{LexTriangularPackage} and \\axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}. The second one provide decompositions by means of square-free regular triangular sets. Both are based on the {\\em lexTriangular} method described in [1]. They differ from the algorithm described in [2] by the fact that multiciplities of the roots are not kept. With the \\axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} operation all multiciplities are removed. With the other operation some multiciplities may remain. Both operations admit an optional argument to produce normalized triangular sets. \\newline")) (|zeroSetSplit| (((|List| (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#2|)) (|OrderedVariableList| |#2|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{zeroSetSplit(\\spad{lp},{} norm?)} decomposes the variety associated with \\axiom{\\spad{lp}} into square-free regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{\\spad{lp}} needs to generate a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.") (((|List| (|RegularChain| |#1| |#2|)) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{zeroSetSplit(\\spad{lp},{} norm?)} decomposes the variety associated with \\axiom{\\spad{lp}} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{\\spad{lp}} needs to generate a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.")) (|squareFreeLexTriangular| (((|List| (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#2|)) (|OrderedVariableList| |#2|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{squareFreeLexTriangular(base,{} norm?)} decomposes the variety associated with \\axiom{base} into square-free regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{base} needs to be a lexicographical Groebner basis of a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.")) (|lexTriangular| (((|List| (|RegularChain| |#1| |#2|)) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{lexTriangular(base,{} norm?)} decomposes the variety associated with \\axiom{base} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{base} needs to be a lexicographical Groebner basis of a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.")) (|groebner| (((|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) "\\axiom{groebner(\\spad{lp})} returns the lexicographical Groebner basis of \\axiom{\\spad{lp}}. If \\axiom{\\spad{lp}} generates a zero-dimensional ideal then the {\\em FGLM} strategy is used,{} otherwise the {\\em Sugar} strategy is used.")) (|fglmIfCan| (((|Union| (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) "failed") (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) "\\axiom{fglmIfCan(\\spad{lp})} returns the lexicographical Groebner basis of \\axiom{\\spad{lp}} by using the {\\em FGLM} strategy,{} if \\axiom{zeroDimensional?(\\spad{lp})} holds .")) (|zeroDimensional?| (((|Boolean|) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) "\\axiom{zeroDimensional?(\\spad{lp})} returns \\spad{true} iff \\axiom{\\spad{lp}} generates a zero-dimensional ideal \\spad{w}.\\spad{r}.\\spad{t}. the variables involved in \\axiom{\\spad{lp}}."))) 
 NIL 
 NIL 
-(-632 R -3498) 
+(-634 R -3505) 
 ((|constructor| (NIL "This package provides liouvillian functions over an integral domain.")) (|integral| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{integral(f,x = a..b)} denotes the definite integral of \\spad{f} with respect to \\spad{x} from \\spad{a} to \\spad{b}.") ((|#2| |#2| (|Symbol|)) "\\spad{integral(f,x)} indefinite integral of \\spad{f} with respect to \\spad{x}.")) (|dilog| ((|#2| |#2|) "\\spad{dilog(f)} denotes the dilogarithm")) (|erf| ((|#2| |#2|) "\\spad{erf(f)} denotes the error function")) (|li| ((|#2| |#2|) "\\spad{li(f)} denotes the logarithmic integral")) (|Ci| ((|#2| |#2|) "\\spad{Ci(f)} denotes the cosine integral")) (|Si| ((|#2| |#2|) "\\spad{Si(f)} denotes the sine integral")) (|Ei| ((|#2| |#2|) "\\spad{Ei(f)} denotes the exponential integral")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns the Liouvillian operator based on \\spad{op}")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} checks if \\spad{op} is Liouvillian"))) 
 NIL 
 NIL 
-(-633) 
+(-635) 
 ((|constructor| (NIL "Category for the transcendental Liouvillian functions.")) (|erf| (($ $) "\\spad{erf(x)} returns the error function of \\spad{x},{} \\spadignore{i.e.} \\spad{2 / sqrt(\\%pi)} times the integral of \\spad{exp(-x**2) dx}.")) (|dilog| (($ $) "\\spad{dilog(x)} returns the dilogarithm of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{log(x) / (1 - x) dx}.")) (|li| (($ $) "\\spad{li(x)} returns the logarithmic integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{dx / log(x)}.")) (|Ci| (($ $) "\\spad{Ci(x)} returns the cosine integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{cos(x) / x dx}.")) (|Si| (($ $) "\\spad{Si(x)} returns the sine integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{sin(x) / x dx}.")) (|Ei| (($ $) "\\spad{Ei(x)} returns the exponential integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{exp(x)/x dx}."))) 
 NIL 
 NIL 
-(-634 |lv| -3498) 
+(-636 |lv| -3505) 
 ((|constructor| (NIL "\\indented{1}{Given a Groebner basis \\spad{B} with respect to the total degree ordering for} a zero-dimensional ideal \\spad{I},{} compute a Groebner basis with respect to the lexicographical ordering by using linear algebra.")) (|transform| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{transform }\\undocumented")) (|choosemon| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{choosemon }\\undocumented")) (|intcompBasis| (((|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{intcompBasis }\\undocumented")) (|anticoord| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|List| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{anticoord }\\undocumented")) (|coord| (((|Vector| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{coord }\\undocumented")) (|computeBasis| (((|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{computeBasis }\\undocumented")) (|minPol| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|)) "\\spad{minPol }\\undocumented") (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|)) "\\spad{minPol }\\undocumented")) (|totolex| (((|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{totolex }\\undocumented")) (|groebgen| (((|Record| (|:| |glbase| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |glval| (|List| (|Integer|)))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{groebgen }\\undocumented")) (|linGenPos| (((|Record| (|:| |gblist| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |gvlist| (|List| (|Integer|)))) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{linGenPos }\\undocumented"))) 
 NIL 
 NIL 
-(-635) 
+(-637) 
 ((|constructor| (NIL "This domain provides a simple way to save values in files.")) (|setelt| (((|Any|) $ (|Symbol|) (|Any|)) "\\spad{lib.k := v} saves the value \\spad{v} in the library \\spad{lib}. It can later be extracted using the key \\spad{k}.")) (|elt| (((|Any|) $ (|Symbol|)) "\\spad{elt(lib,k)} or \\spad{lib}.\\spad{k} extracts the value corresponding to the key \\spad{k} from the library \\spad{lib}.")) (|pack!| (($ $) "\\spad{pack!(f)} reorganizes the file \\spad{f} on disk to recover unused space.")) (|library| (($ (|FileName|)) "\\spad{library(ln)} creates a new library file."))) 
-((-4428 . T)) 
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-(-636 R A) 
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+(-638 R A) 
 ((|constructor| (NIL "AssociatedLieAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$A} to define the Lie bracket \\spad{a*b := (a *\\$A b - b *\\$A a)} (commutator). Note that the notation \\spad{[a,b]} cannot be used due to restrictions of the current compiler. This domain only gives a Lie algebra if the Jacobi-identity \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds for all \\spad{a},{}\\spad{b},{}\\spad{c} in \\spad{A}. This relation can be checked by \\spad{lieAdmissible?()\\$A}. \\blankline If the underlying algebra is of type \\spadtype{FramedNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank,{} together with a fixed \\spad{R}-module basis),{} then the same is \\spad{true} for the associated Lie algebra. Also,{} if the underlying algebra is of type \\spadtype{FiniteRankNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank),{} then the same is \\spad{true} for the associated Lie algebra.")) (|coerce| (($ |#2|) "\\spad{coerce(a)} coerces the element \\spad{a} of the algebra \\spad{A} to an element of the Lie algebra \\spadtype{AssociatedLieAlgebra}(\\spad{R},{}A)."))) 
-((-4424 -3962 (-3258 (|has| |#2| (-370 |#1|)) (|has| |#1| (-561))) (-12 (|has| |#2| (-422 |#1|)) (|has| |#1| (-561)))) (-4422 . T) (-4421 . T)) 
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-(-637 S R) 
+((-4431 -3969 (-3265 (|has| |#2| (-371 |#1|)) (|has| |#1| (-562))) (-12 (|has| |#2| (-423 |#1|)) (|has| |#1| (-562)))) (-4429 . T) (-4428 . T)) 
+((-3969 (|HasCategory| |#2| (LIST (QUOTE -371) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -371) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -371) (|devaluate| |#1|)))) 
+(-639 S R) 
 ((|constructor| (NIL "\\axiom{JacobiIdentity} means that \\axiom{[\\spad{x},{}[\\spad{y},{}\\spad{z}]]+[\\spad{y},{}[\\spad{z},{}\\spad{x}]]+[\\spad{z},{}[\\spad{x},{}\\spad{y}]] = 0} holds.")) (/ (($ $ |#2|) "\\axiom{\\spad{x/r}} returns the division of \\axiom{\\spad{x}} by \\axiom{\\spad{r}}.")) (|construct| (($ $ $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket of \\axiom{\\spad{x}} and \\axiom{\\spad{y}}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-366)))) 
-(-638 R) 
+((|HasCategory| |#2| (QUOTE (-367)))) 
+(-640 R) 
 ((|constructor| (NIL "\\axiom{JacobiIdentity} means that \\axiom{[\\spad{x},{}[\\spad{y},{}\\spad{z}]]+[\\spad{y},{}[\\spad{z},{}\\spad{x}]]+[\\spad{z},{}[\\spad{x},{}\\spad{y}]] = 0} holds.")) (/ (($ $ |#1|) "\\axiom{\\spad{x/r}} returns the division of \\axiom{\\spad{x}} by \\axiom{\\spad{r}}.")) (|construct| (($ $ $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket of \\axiom{\\spad{x}} and \\axiom{\\spad{y}}."))) 
-((|JacobiIdentity| . T) (|NullSquare| . T) (-4422 . T) (-4421 . T)) 
+((|JacobiIdentity| . T) (|NullSquare| . T) (-4429 . T) (-4428 . T)) 
 NIL 
-(-639 R FE) 
+(-641 R FE) 
 ((|constructor| (NIL "PowerSeriesLimitPackage implements limits of expressions in one or more variables as one of the variables approaches a limiting value. Included are two-sided limits,{} left- and right- hand limits,{} and limits at plus or minus infinity.")) (|complexLimit| (((|Union| (|OnePointCompletion| |#2|) "failed") |#2| (|Equation| (|OnePointCompletion| |#2|))) "\\spad{complexLimit(f(x),x = a)} computes the complex limit \\spad{lim(x -> a,f(x))}.")) (|limit| (((|Union| (|OrderedCompletion| |#2|) #1="failed") |#2| (|Equation| |#2|) (|String|)) "\\spad{limit(f(x),x=a,\"left\")} computes the left hand real limit \\spad{lim(x -> a-,f(x))}; \\spad{limit(f(x),x=a,\"right\")} computes the right hand real limit \\spad{lim(x -> a+,f(x))}.") (((|Union| (|OrderedCompletion| |#2|) (|Record| (|:| |leftHandLimit| (|Union| (|OrderedCompletion| |#2|) #1#)) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| |#2|) #1#))) "failed") |#2| (|Equation| (|OrderedCompletion| |#2|))) "\\spad{limit(f(x),x = a)} computes the real limit \\spad{lim(x -> a,f(x))}."))) 
 NIL 
 NIL 
-(-640 R) 
+(-642 R) 
 ((|constructor| (NIL "Computation of limits for rational functions.")) (|complexLimit| (((|OnePointCompletion| (|Fraction| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{complexLimit(f(x),x = a)} computes the complex limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}.") (((|OnePointCompletion| (|Fraction| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|OnePointCompletion| (|Polynomial| |#1|)))) "\\spad{complexLimit(f(x),x = a)} computes the complex limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}.")) (|limit| (((|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1="failed") (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|String|)) "\\spad{limit(f(x),x,a,\"left\")} computes the real limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a} from the left; limit(\\spad{f}(\\spad{x}),{}\\spad{x},{}a,{}\"right\") computes the corresponding limit as \\spad{x} approaches \\spad{a} from the right.") (((|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) (|Record| (|:| |leftHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#)) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#))) #2="failed") (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{limit(f(x),x = a)} computes the real two-sided limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}.") (((|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) (|Record| (|:| |leftHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#)) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#))) #2#) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|OrderedCompletion| (|Polynomial| |#1|)))) "\\spad{limit(f(x),x = a)} computes the real two-sided limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}."))) 
 NIL 
 NIL 
-(-641 S R) 
+(-643 S R) 
 ((|constructor| (NIL "Test for linear dependence.")) (|solveLinear| (((|Union| (|Vector| (|Fraction| |#1|)) "failed") (|Vector| |#2|) |#2|) "\\spad{solveLinear([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such \\spad{ci}\\spad{'s} exist in the quotient field of \\spad{S}.") (((|Union| (|Vector| |#1|) "failed") (|Vector| |#2|) |#2|) "\\spad{solveLinear([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such \\spad{ci}\\spad{'s} exist in \\spad{S}.")) (|linearDependence| (((|Union| (|Vector| |#1|) "failed") (|Vector| |#2|)) "\\spad{linearDependence([v1,...,vn])} returns \\spad{[c1,...,cn]} if \\spad{c1*v1 + ... + cn*vn = 0} and not all the \\spad{ci}\\spad{'s} are 0,{} \"failed\" if the \\spad{vi}\\spad{'s} are linearly independent over \\spad{S}.")) (|linearlyDependent?| (((|Boolean|) (|Vector| |#2|)) "\\spad{linearlyDependent?([v1,...,vn])} returns \\spad{true} if the \\spad{vi}\\spad{'s} are linearly dependent over \\spad{S},{} \\spad{false} otherwise."))) 
 NIL 
-((-3748 (|HasCategory| |#1| (QUOTE (-366)))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-642 R) 
+((-3755 (|HasCategory| |#1| (QUOTE (-367)))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-644 R) 
 ((|constructor| (NIL "An extension ring with an explicit linear dependence test.")) (|reducedSystem| (((|Record| (|:| |mat| (|Matrix| |#1|)) (|:| |vec| (|Vector| |#1|))) (|Matrix| $) (|Vector| $)) "\\spad{reducedSystem(A, v)} returns a matrix \\spad{B} and a vector \\spad{w} such that \\spad{A x = v} and \\spad{B x = w} have the same solutions in \\spad{R}.") (((|Matrix| |#1|) (|Matrix| $)) "\\spad{reducedSystem(A)} returns a matrix \\spad{B} such that \\spad{A x = 0} and \\spad{B x = 0} have the same solutions in \\spad{R}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-643 R) 
+(-645 R) 
 ((|constructor| (NIL "\\indented{2}{A set is an \\spad{R}-linear set if it is stable by dilation} \\indented{2}{by elements in the ring \\spad{R}.\\space{2}This category differs from} \\indented{2}{\\spad{Module} in that no other assumption (such as addition)} \\indented{2}{is made about the underlying set.} See Also: LeftLinearSet,{} RightLinearSet."))) 
 NIL 
 NIL 
-(-644 S) 
+(-646 S) 
 ((|constructor| (NIL "\\spadtype{List} implements singly-linked lists that are addressable by indices; the index of the first element is 1. In addition to the operations provided by \\spadtype{IndexedList},{} this constructor provides some LISP-like functions such as \\spadfun{null} and \\spadfun{cons}.")) (|setDifference| (($ $ $) "\\spad{setDifference(u1,u2)} returns a list of the elements of \\spad{u1} that are not also in \\spad{u2}. The order of elements in the resulting list is unspecified.")) (|setIntersection| (($ $ $) "\\spad{setIntersection(u1,u2)} returns a list of the elements that lists \\spad{u1} and \\spad{u2} have in common. The order of elements in the resulting list is unspecified.")) (|setUnion| (($ $ $) "\\spad{setUnion(u1,u2)} appends the two lists \\spad{u1} and \\spad{u2},{} then removes all duplicates. The order of elements in the resulting list is unspecified.")) (|append| (($ $ $) "\\spad{append(u1,u2)} appends the elements of list \\spad{u1} onto the front of list \\spad{u2}. This new list and \\spad{u2} will share some structure.")) (|cons| (($ |#1| $) "\\spad{cons(element,u)} appends \\spad{element} onto the front of list \\spad{u} and returns the new list. This new list and the old one will share some structure.")) (|null| (((|Boolean|) $) "\\spad{null(u)} tests if list \\spad{u} is the empty list.")) (|nil| (($) "\\spad{nil} is the empty list."))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-824))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-645 A B) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-826))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-647 A B) 
 ((|constructor| (NIL "\\spadtype{ListFunctions2} implements utility functions that operate on two kinds of lists,{} each with a possibly different type of element.")) (|map| (((|List| |#2|) (|Mapping| |#2| |#1|) (|List| |#1|)) "\\spad{map(fn,u)} applies \\spad{fn} to each element of list \\spad{u} and returns a new list with the results. For example \\spad{map(square,[1,2,3]) = [1,4,9]}.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|List| |#1|) |#2|) "\\spad{reduce(fn,u,ident)} successively uses the binary function \\spad{fn} on the elements of list \\spad{u} and the result of previous applications. \\spad{ident} is returned if the \\spad{u} is empty. Note the order of application in the following examples: \\spad{reduce(fn,[1,2,3],0) = fn(3,fn(2,fn(1,0)))} and \\spad{reduce(*,[2,3],1) = 3 * (2 * 1)}.")) (|scan| (((|List| |#2|) (|Mapping| |#2| |#1| |#2|) (|List| |#1|) |#2|) "\\spad{scan(fn,u,ident)} successively uses the binary function \\spad{fn} to reduce more and more of list \\spad{u}. \\spad{ident} is returned if the \\spad{u} is empty. The result is a list of the reductions at each step. See \\spadfun{reduce} for more information. Examples: \\spad{scan(fn,[1,2],0) = [fn(2,fn(1,0)),fn(1,0)]} and \\spad{scan(*,[2,3],1) = [2 * 1, 3 * (2 * 1)]}."))) 
 NIL 
 NIL 
-(-646 A B) 
+(-648 A B) 
 ((|constructor| (NIL "\\spadtype{ListToMap} allows mappings to be described by a pair of lists of equal lengths. The image of an element \\spad{x},{} which appears in position \\spad{n} in the first list,{} is then the \\spad{n}th element of the second list. A default value or default function can be specified to be used when \\spad{x} does not appear in the first list. In the absence of defaults,{} an error will occur in that case.")) (|match| ((|#2| (|List| |#1|) (|List| |#2|) |#1| (|Mapping| |#2| |#1|)) "\\spad{match(la, lb, a, f)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length. and applies this map to a. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Argument \\spad{f} is a default function to call if a is not in \\spad{la}. The value returned is then obtained by applying \\spad{f} to argument a.") (((|Mapping| |#2| |#1|) (|List| |#1|) (|List| |#2|) (|Mapping| |#2| |#1|)) "\\spad{match(la, lb, f)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Argument \\spad{f} is used as the function to call when the given function argument is not in \\spad{la}. The value returned is \\spad{f} applied to that argument.") ((|#2| (|List| |#1|) (|List| |#2|) |#1| |#2|) "\\spad{match(la, lb, a, b)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length. and applies this map to a. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Argument \\spad{b} is the default target value if a is not in \\spad{la}. Error: if \\spad{la} and \\spad{lb} are not of equal length.") (((|Mapping| |#2| |#1|) (|List| |#1|) (|List| |#2|) |#2|) "\\spad{match(la, lb, b)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length,{} where \\spad{b} is used as the default target value if the given function argument is not in \\spad{la}. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Error: if \\spad{la} and \\spad{lb} are not of equal length.") ((|#2| (|List| |#1|) (|List| |#2|) |#1|) "\\spad{match(la, lb, a)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length,{} where \\spad{a} is used as the default source value if the given one is not in \\spad{la}. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Error: if \\spad{la} and \\spad{lb} are not of equal length.") (((|Mapping| |#2| |#1|) (|List| |#1|) (|List| |#2|)) "\\spad{match(la, lb)} creates a map with no default source or target values defined by lists \\spad{la} and \\spad{lb} of equal length. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Error: if \\spad{la} and \\spad{lb} are not of equal length. Note: when this map is applied,{} an error occurs when applied to a value missing from \\spad{la}."))) 
 NIL 
 NIL 
-(-647 A B C) 
+(-649 A B C) 
 ((|constructor| (NIL "\\spadtype{ListFunctions3} implements utility functions that operate on three kinds of lists,{} each with a possibly different type of element.")) (|map| (((|List| |#3|) (|Mapping| |#3| |#1| |#2|) (|List| |#1|) (|List| |#2|)) "\\spad{map(fn,list1, u2)} applies the binary function \\spad{fn} to corresponding elements of lists \\spad{u1} and \\spad{u2} and returns a list of the results (in the same order). Thus \\spad{map(/,[1,2,3],[4,5,6]) = [1/4,2/4,1/2]}. The computation terminates when the end of either list is reached. That is,{} the length of the result list is equal to the minimum of the lengths of \\spad{u1} and \\spad{u2}."))) 
 NIL 
 NIL 
-(-648 T$) 
+(-650 T$) 
 ((|constructor| (NIL "This domain represents AST for Spad literals."))) 
 NIL 
 NIL 
-(-649 R) 
+(-651 R) 
 ((|constructor| (NIL "\\indented{2}{A set is an \\spad{R}-left linear set if it is stable by left-dilation} \\indented{2}{by elements in the ring \\spad{R}.\\space{2}This category differs from} \\indented{2}{\\spad{LeftModule} in that no other assumption (such as addition)} \\indented{2}{is made about the underlying set.} See Also: RightLinearSet.")) (* (($ |#1| $) "\\spad{r*x} is the left-dilation of \\spad{x} by \\spad{r}.")) (|zero?| (((|Boolean|) $) "\\spad{zero? x} holds is \\spad{x} is the origin.")) ((|Zero|) (($) "\\spad{0} represents the origin of the linear set"))) 
 NIL 
 NIL 
-(-650 S) 
+(-652 S) 
 ((|substitute| (($ |#1| |#1| $) "\\spad{substitute(x,y,d)} replace \\spad{x}\\spad{'s} with \\spad{y}\\spad{'s} in dictionary \\spad{d}.")) (|duplicates?| (((|Boolean|) $) "\\spad{duplicates?(d)} tests if dictionary \\spad{d} has duplicate entries."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-651 R) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-653 R) 
 ((|constructor| (NIL "The category of left modules over an \\spad{rng} (ring not necessarily with unit). This is an abelian group which supports left multiplation by elements of the \\spad{rng}. \\blankline"))) 
 NIL 
 NIL 
-(-652 S E |un|) 
+(-654 S E |un|) 
 ((|constructor| (NIL "This internal package represents monoid (abelian or not,{} with or without inverses) as lists and provides some common operations to the various flavors of monoids.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapExpon(f, a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|commutativeEquality| (((|Boolean|) $ $) "\\spad{commutativeEquality(x,y)} returns \\spad{true} if \\spad{x} and \\spad{y} are equal assuming commutativity")) (|plus| (($ $ $) "\\spad{plus(x, y)} returns \\spad{x + y} where \\spad{+} is the monoid operation,{} which is assumed commutative.") (($ |#1| |#2| $) "\\spad{plus(s, e, x)} returns \\spad{e * s + x} where \\spad{+} is the monoid operation,{} which is assumed commutative.")) (|leftMult| (($ |#1| $) "\\spad{leftMult(s, a)} returns \\spad{s * a} where \\spad{*} is the monoid operation,{} which is assumed non-commutative.")) (|rightMult| (($ $ |#1|) "\\spad{rightMult(a, s)} returns \\spad{a * s} where \\spad{*} is the monoid operation,{} which is assumed non-commutative.")) (|makeUnit| (($) "\\spad{makeUnit()} returns the unit element of the monomial.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(l)} returns the number of monomials forming \\spad{l}.")) (|reverse!| (($ $) "\\spad{reverse!(l)} reverses the list of monomials forming \\spad{l},{} destroying the element \\spad{l}.")) (|reverse| (($ $) "\\spad{reverse(l)} reverses the list of monomials forming \\spad{l}. This has some effect if the monoid is non-abelian,{} \\spadignore{i.e.} \\spad{reverse(a1\\^e1 ... an\\^en) = an\\^en ... a1\\^e1} which is different.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(l, n)} returns the factor of the n^th monomial of \\spad{l}.")) (|nthExpon| ((|#2| $ (|Integer|)) "\\spad{nthExpon(l, n)} returns the exponent of the n^th monomial of \\spad{l}.")) (|makeMulti| (($ (|List| (|Record| (|:| |gen| |#1|) (|:| |exp| |#2|)))) "\\spad{makeMulti(l)} returns the element whose list of monomials is \\spad{l}.")) (|makeTerm| (($ |#1| |#2|) "\\spad{makeTerm(s, e)} returns the monomial \\spad{s} exponentiated by \\spad{e} (\\spadignore{e.g.} s^e or \\spad{e} * \\spad{s}).")) (|listOfMonoms| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| |#2|))) $) "\\spad{listOfMonoms(l)} returns the list of the monomials forming \\spad{l}.")) (|outputForm| (((|OutputForm|) $ (|Mapping| (|OutputForm|) (|OutputForm|) (|OutputForm|)) (|Mapping| (|OutputForm|) (|OutputForm|) (|OutputForm|)) (|Integer|)) "\\spad{outputForm(l, fop, fexp, unit)} converts the monoid element represented by \\spad{l} to an \\spadtype{OutputForm}. Argument unit is the output form for the \\spadignore{unit} of the monoid (\\spadignore{e.g.} 0 or 1),{} \\spad{fop(a, b)} is the output form for the monoid operation applied to \\spad{a} and \\spad{b} (\\spadignore{e.g.} \\spad{a + b},{} \\spad{a * b},{} \\spad{ab}),{} and \\spad{fexp(a, n)} is the output form for the exponentiation operation applied to \\spad{a} and \\spad{n} (\\spadignore{e.g.} \\spad{n a},{} \\spad{n * a},{} \\spad{a ** n},{} \\spad{a\\^n})."))) 
 NIL 
 NIL 
-(-653 A S) 
+(-655 A S) 
 ((|constructor| (NIL "A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \\spadfun{concat} of two lists needs only to copy its first argument,{} whereas \\spadfun{concat} of two arrays needs to copy both arguments. Most of the operations exported here apply to infinite objects (\\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \\spadtype{FiniteLinearAggregate}.")) (|setelt| ((|#2| $ (|UniversalSegment| (|Integer|)) |#2|) "\\spad{setelt(u,i..j,x)} (also written: \\axiom{\\spad{u}(\\spad{i}..\\spad{j}) \\spad{:=} \\spad{x}}) destructively replaces each element in the segment \\axiom{\\spad{u}(\\spad{i}..\\spad{j})} by \\spad{x}. The value \\spad{x} is returned. Note: \\spad{u} is destructively change so that \\axiom{\\spad{u}.\\spad{k} \\spad{:=} \\spad{x} for \\spad{k} in \\spad{i}..\\spad{j}}; its length remains unchanged.")) (|insert| (($ $ $ (|Integer|)) "\\spad{insert(v,u,k)} returns a copy of \\spad{u} having \\spad{v} inserted beginning at the \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{v},{}\\spad{u},{}\\spad{k}) = concat( \\spad{u}(0..\\spad{k}-1),{} \\spad{v},{} \\spad{u}(\\spad{k}..) )}.") (($ |#2| $ (|Integer|)) "\\spad{insert(x,u,i)} returns a copy of \\spad{u} having \\spad{x} as its \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{x},{}a,{}\\spad{k}) = concat(concat(a(0..\\spad{k}-1),{}\\spad{x}),{}a(\\spad{k}..))}.")) (|delete| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete(u,i..j)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th through \\axiom{\\spad{j}}th element deleted. Note: \\axiom{delete(a,{}\\spad{i}..\\spad{j}) = concat(a(0..\\spad{i}-1),{}a(\\spad{j+1}..))}.") (($ $ (|Integer|)) "\\spad{delete(u,i)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th element deleted. Note: for lists,{} \\axiom{delete(a,{}\\spad{i}) \\spad{==} concat(a(0..\\spad{i} - 1),{}a(\\spad{i} + 1,{}..))}.")) (|elt| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{elt(u,i..j)} (also written: \\axiom{a(\\spad{i}..\\spad{j})}) returns the aggregate of elements \\axiom{\\spad{u}} for \\spad{k} from \\spad{i} to \\spad{j} in that order. Note: in general,{} \\axiom{a.\\spad{s} = [a.\\spad{k} for \\spad{i} in \\spad{s}]}.")) (|map| (($ (|Mapping| |#2| |#2| |#2|) $ $) "\\spad{map(f,u,v)} returns a new collection \\spad{w} with elements \\axiom{\\spad{z} = \\spad{f}(\\spad{x},{}\\spad{y})} for corresponding elements \\spad{x} and \\spad{y} from \\spad{u} and \\spad{v}. Note: for linear aggregates,{} \\axiom{\\spad{w}.\\spad{i} = \\spad{f}(\\spad{u}.\\spad{i},{}\\spad{v}.\\spad{i})}.")) (|concat| (($ (|List| $)) "\\spad{concat(u)},{} where \\spad{u} is a lists of aggregates \\axiom{[a,{}\\spad{b},{}...,{}\\spad{c}]},{} returns a single aggregate consisting of the elements of \\axiom{a} followed by those of \\spad{b} followed ... by the elements of \\spad{c}. Note: \\axiom{concat(a,{}\\spad{b},{}...,{}\\spad{c}) = concat(a,{}concat(\\spad{b},{}...,{}\\spad{c}))}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} then \\axiom{\\spad{w}.\\spad{i} = \\spad{u}.\\spad{i} for \\spad{i} in indices \\spad{u}} and \\axiom{\\spad{w}.(\\spad{j} + maxIndex \\spad{u}) = \\spad{v}.\\spad{j} for \\spad{j} in indices \\spad{v}}.") (($ |#2| $) "\\spad{concat(x,u)} returns aggregate \\spad{u} with additional element at the front. Note: for lists: \\axiom{concat(\\spad{x},{}\\spad{u}) \\spad{==} concat([\\spad{x}],{}\\spad{u})}.") (($ $ |#2|) "\\spad{concat(u,x)} returns aggregate \\spad{u} with additional element \\spad{x} at the end. Note: for lists,{} \\axiom{concat(\\spad{u},{}\\spad{x}) \\spad{==} concat(\\spad{u},{}[\\spad{x}])}")) (|new| (($ (|NonNegativeInteger|) |#2|) "\\spad{new(n,x)} returns \\axiom{fill!(new \\spad{n},{}\\spad{x})}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4428))) 
-(-654 S) 
+((|HasAttribute| |#1| (QUOTE -4435))) 
+(-656 S) 
 ((|constructor| (NIL "A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \\spadfun{concat} of two lists needs only to copy its first argument,{} whereas \\spadfun{concat} of two arrays needs to copy both arguments. Most of the operations exported here apply to infinite objects (\\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \\spadtype{FiniteLinearAggregate}.")) (|setelt| ((|#1| $ (|UniversalSegment| (|Integer|)) |#1|) "\\spad{setelt(u,i..j,x)} (also written: \\axiom{\\spad{u}(\\spad{i}..\\spad{j}) \\spad{:=} \\spad{x}}) destructively replaces each element in the segment \\axiom{\\spad{u}(\\spad{i}..\\spad{j})} by \\spad{x}. The value \\spad{x} is returned. Note: \\spad{u} is destructively change so that \\axiom{\\spad{u}.\\spad{k} \\spad{:=} \\spad{x} for \\spad{k} in \\spad{i}..\\spad{j}}; its length remains unchanged.")) (|insert| (($ $ $ (|Integer|)) "\\spad{insert(v,u,k)} returns a copy of \\spad{u} having \\spad{v} inserted beginning at the \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{v},{}\\spad{u},{}\\spad{k}) = concat( \\spad{u}(0..\\spad{k}-1),{} \\spad{v},{} \\spad{u}(\\spad{k}..) )}.") (($ |#1| $ (|Integer|)) "\\spad{insert(x,u,i)} returns a copy of \\spad{u} having \\spad{x} as its \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{x},{}a,{}\\spad{k}) = concat(concat(a(0..\\spad{k}-1),{}\\spad{x}),{}a(\\spad{k}..))}.")) (|delete| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete(u,i..j)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th through \\axiom{\\spad{j}}th element deleted. Note: \\axiom{delete(a,{}\\spad{i}..\\spad{j}) = concat(a(0..\\spad{i}-1),{}a(\\spad{j+1}..))}.") (($ $ (|Integer|)) "\\spad{delete(u,i)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th element deleted. Note: for lists,{} \\axiom{delete(a,{}\\spad{i}) \\spad{==} concat(a(0..\\spad{i} - 1),{}a(\\spad{i} + 1,{}..))}.")) (|elt| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{elt(u,i..j)} (also written: \\axiom{a(\\spad{i}..\\spad{j})}) returns the aggregate of elements \\axiom{\\spad{u}} for \\spad{k} from \\spad{i} to \\spad{j} in that order. Note: in general,{} \\axiom{a.\\spad{s} = [a.\\spad{k} for \\spad{i} in \\spad{s}]}.")) (|map| (($ (|Mapping| |#1| |#1| |#1|) $ $) "\\spad{map(f,u,v)} returns a new collection \\spad{w} with elements \\axiom{\\spad{z} = \\spad{f}(\\spad{x},{}\\spad{y})} for corresponding elements \\spad{x} and \\spad{y} from \\spad{u} and \\spad{v}. Note: for linear aggregates,{} \\axiom{\\spad{w}.\\spad{i} = \\spad{f}(\\spad{u}.\\spad{i},{}\\spad{v}.\\spad{i})}.")) (|concat| (($ (|List| $)) "\\spad{concat(u)},{} where \\spad{u} is a lists of aggregates \\axiom{[a,{}\\spad{b},{}...,{}\\spad{c}]},{} returns a single aggregate consisting of the elements of \\axiom{a} followed by those of \\spad{b} followed ... by the elements of \\spad{c}. Note: \\axiom{concat(a,{}\\spad{b},{}...,{}\\spad{c}) = concat(a,{}concat(\\spad{b},{}...,{}\\spad{c}))}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} then \\axiom{\\spad{w}.\\spad{i} = \\spad{u}.\\spad{i} for \\spad{i} in indices \\spad{u}} and \\axiom{\\spad{w}.(\\spad{j} + maxIndex \\spad{u}) = \\spad{v}.\\spad{j} for \\spad{j} in indices \\spad{v}}.") (($ |#1| $) "\\spad{concat(x,u)} returns aggregate \\spad{u} with additional element at the front. Note: for lists: \\axiom{concat(\\spad{x},{}\\spad{u}) \\spad{==} concat([\\spad{x}],{}\\spad{u})}.") (($ $ |#1|) "\\spad{concat(u,x)} returns aggregate \\spad{u} with additional element \\spad{x} at the end. Note: for lists,{} \\axiom{concat(\\spad{u},{}\\spad{x}) \\spad{==} concat(\\spad{u},{}[\\spad{x}])}")) (|new| (($ (|NonNegativeInteger|) |#1|) "\\spad{new(n,x)} returns \\axiom{fill!(new \\spad{n},{}\\spad{x})}."))) 
 NIL 
 NIL 
-(-655 M R S) 
+(-657 M R S) 
 ((|constructor| (NIL "Localize(\\spad{M},{}\\spad{R},{}\\spad{S}) produces fractions with numerators from an \\spad{R} module \\spad{M} and denominators from some multiplicative subset \\spad{D} of \\spad{R}.")) (|denom| ((|#3| $) "\\spad{denom x} returns the denominator of \\spad{x}.")) (|numer| ((|#1| $) "\\spad{numer x} returns the numerator of \\spad{x}.")) (/ (($ |#1| |#3|) "\\spad{m / d} divides the element \\spad{m} by \\spad{d}.") (($ $ |#3|) "\\spad{x / d} divides the element \\spad{x} by \\spad{d}."))) 
-((-4422 . T) (-4421 . T)) 
-((|HasCategory| |#1| (QUOTE (-794)))) 
-(-656 R -3498 L) 
+((-4429 . T) (-4428 . T)) 
+((|HasCategory| |#1| (QUOTE (-796)))) 
+(-658 R -3505 L) 
 ((|constructor| (NIL "\\spad{ElementaryFunctionLODESolver} provides the top-level functions for finding closed form solutions of linear ordinary differential equations and initial value problems.")) (|solve| (((|Union| |#2| "failed") |#3| |#2| (|Symbol|) |#2| (|List| |#2|)) "\\spad{solve(op, g, x, a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{op y = g, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; \\spad{x} is the dependent variable.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) "failed") |#3| |#2| (|Symbol|)) "\\spad{solve(op, g, x)} returns either a solution of the ordinary differential equation \\spad{op y = g} or \"failed\" if no non-trivial solution can be found; When found,{} the solution is returned in the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{op y = 0}. A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; \\spad{x} is the dependent variable."))) 
 NIL 
 NIL 
-(-657 A -2822) 
+(-659 A -2829) 
 ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperator} defines a ring of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-658 A) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-660 A) 
 ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperator1} defines a ring of differential operators with coefficients in a differential ring A. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-659 A M) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-661 A M) 
 ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperator2} defines a ring of differential operators with coefficients in a differential ring A and acting on an A-module \\spad{M}. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|differentiate| (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-660 S A) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-662 S A) 
 ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|directSum| (($ $ $) "\\spad{directSum(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}.")) (|symmetricSquare| (($ $) "\\spad{symmetricSquare(a)} computes \\spad{symmetricProduct(a,a)} using a more efficient method.")) (|symmetricPower| (($ $ (|NonNegativeInteger|)) "\\spad{symmetricPower(a,n)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of \\spad{n} solutions of \\spad{a}.")) (|symmetricProduct| (($ $ $) "\\spad{symmetricProduct(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of a solution of \\spad{a} by a solution of \\spad{b}.")) (|adjoint| (($ $) "\\spad{adjoint(a)} returns the adjoint operator of a.")) (D (($) "\\spad{D()} provides the operator corresponding to a derivation in the ring \\spad{A}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-366)))) 
-(-661 A) 
+((|HasCategory| |#2| (QUOTE (-367)))) 
+(-663 A) 
 ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|directSum| (($ $ $) "\\spad{directSum(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}.")) (|symmetricSquare| (($ $) "\\spad{symmetricSquare(a)} computes \\spad{symmetricProduct(a,a)} using a more efficient method.")) (|symmetricPower| (($ $ (|NonNegativeInteger|)) "\\spad{symmetricPower(a,n)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of \\spad{n} solutions of \\spad{a}.")) (|symmetricProduct| (($ $ $) "\\spad{symmetricProduct(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of a solution of \\spad{a} by a solution of \\spad{b}.")) (|adjoint| (($ $) "\\spad{adjoint(a)} returns the adjoint operator of a.")) (D (($) "\\spad{D()} provides the operator corresponding to a derivation in the ring \\spad{A}."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-662 -3498 UP) 
+(-664 -3505 UP) 
 ((|constructor| (NIL "\\spadtype{LinearOrdinaryDifferentialOperatorFactorizer} provides a factorizer for linear ordinary differential operators whose coefficients are rational functions.")) (|factor1| (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{factor1(a)} returns the factorisation of a,{} assuming that a has no first-order right factor.")) (|factor| (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{factor(a)} returns the factorisation of a.") (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{factor(a, zeros)} returns the factorisation of a. \\spad{zeros} is a zero finder in \\spad{UP}."))) 
 NIL 
 ((|HasCategory| |#1| (QUOTE (-27)))) 
-(-663 A L) 
+(-665 A L) 
 ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorsOps} provides symmetric products and sums for linear ordinary differential operators.")) (|directSum| ((|#2| |#2| |#2| (|Mapping| |#1| |#1|)) "\\spad{directSum(a,b,D)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}. \\spad{D} is the derivation to use.")) (|symmetricPower| ((|#2| |#2| (|NonNegativeInteger|) (|Mapping| |#1| |#1|)) "\\spad{symmetricPower(a,n,D)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of \\spad{n} solutions of \\spad{a}. \\spad{D} is the derivation to use.")) (|symmetricProduct| ((|#2| |#2| |#2| (|Mapping| |#1| |#1|)) "\\spad{symmetricProduct(a,b,D)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of a solution of \\spad{a} by a solution of \\spad{b}. \\spad{D} is the derivation to use."))) 
 NIL 
 NIL 
-(-664 S) 
+(-666 S) 
 ((|constructor| (NIL "`Logic' provides the basic operations for lattices,{} \\spadignore{e.g.} boolean algebra.")) (|\\/| (($ $ $) "\\spadignore{ \\/ } returns the logical `join',{} \\spadignore{e.g.} `or'.")) (|/\\| (($ $ $) "\\spadignore { /\\ }returns the logical `meet',{} \\spadignore{e.g.} `and'.")) (~ (($ $) "\\spad{~(x)} returns the logical complement of \\spad{x}."))) 
 NIL 
 NIL 
-(-665) 
+(-667) 
 ((|constructor| (NIL "`Logic' provides the basic operations for lattices,{} \\spadignore{e.g.} boolean algebra.")) (|\\/| (($ $ $) "\\spadignore{ \\/ } returns the logical `join',{} \\spadignore{e.g.} `or'.")) (|/\\| (($ $ $) "\\spadignore { /\\ }returns the logical `meet',{} \\spadignore{e.g.} `and'.")) (~ (($ $) "\\spad{~(x)} returns the logical complement of \\spad{x}."))) 
 NIL 
 NIL 
-(-666 R) 
+(-668 R) 
 ((|constructor| (NIL "Given a PolynomialFactorizationExplicit ring,{} this package provides a defaulting rule for the \\spad{solveLinearPolynomialEquation} operation,{} by moving into the field of fractions,{} and solving it there via the \\spad{multiEuclidean} operation.")) (|solveLinearPolynomialEquationByFractions| (((|Union| (|List| (|SparseUnivariatePolynomial| |#1|)) "failed") (|List| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{solveLinearPolynomialEquationByFractions([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such exists."))) 
 NIL 
 NIL 
-(-667 |VarSet| R) 
+(-669 |VarSet| R) 
 ((|constructor| (NIL "This type supports Lie polynomials in Lyndon basis see Free Lie Algebras by \\spad{C}. Reutenauer (Oxford science publications). \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr}).")) (|construct| (($ $ (|LyndonWord| |#1|)) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket \\axiom{[\\spad{x},{}\\spad{y}]}.") (($ (|LyndonWord| |#1|) $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket \\axiom{[\\spad{x},{}\\spad{y}]}.") (($ (|LyndonWord| |#1|) (|LyndonWord| |#1|)) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket \\axiom{[\\spad{x},{}\\spad{y}]}.")) (|LiePolyIfCan| (((|Union| $ "failed") (|XDistributedPolynomial| |#1| |#2|)) "\\axiom{LiePolyIfCan(\\spad{p})} returns \\axiom{\\spad{p}} in Lyndon basis if \\axiom{\\spad{p}} is a Lie polynomial,{} otherwise \\axiom{\"failed\"} is returned."))) 
-((|JacobiIdentity| . T) (|NullSquare| . T) (-4422 . T) (-4421 . T)) 
-((|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-173)))) 
-(-668 A S) 
+((|JacobiIdentity| . T) (|NullSquare| . T) (-4429 . T) (-4428 . T)) 
+((|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-173)))) 
+(-670 A S) 
 ((|constructor| (NIL "A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.")) (|list| (($ |#2|) "\\spad{list(x)} returns the list of one element \\spad{x}."))) 
 NIL 
 NIL 
-(-669 S) 
+(-671 S) 
 ((|constructor| (NIL "A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.")) (|list| (($ |#1|) "\\spad{list(x)} returns the list of one element \\spad{x}."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-670 -3498 |Row| |Col| M) 
+(-672 -3505 |Row| |Col| M) 
 ((|constructor| (NIL "This package solves linear system in the matrix form \\spad{AX = B}.")) (|rank| (((|NonNegativeInteger|) |#4| |#3|) "\\spad{rank(A,B)} computes the rank of the complete matrix \\spad{(A|B)} of the linear system \\spad{AX = B}.")) (|hasSolution?| (((|Boolean|) |#4| |#3|) "\\spad{hasSolution?(A,B)} tests if the linear system \\spad{AX = B} has a solution.")) (|particularSolution| (((|Union| |#3| #1="failed") |#4| |#3|) "\\spad{particularSolution(A,B)} finds a particular solution of the linear system \\spad{AX = B}.")) (|solve| (((|List| (|Record| (|:| |particular| (|Union| |#3| #1#)) (|:| |basis| (|List| |#3|)))) |#4| (|List| |#3|)) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|Record| (|:| |particular| (|Union| |#3| #1#)) (|:| |basis| (|List| |#3|))) |#4| |#3|) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}."))) 
 NIL 
 NIL 
-(-671 -3498) 
+(-673 -3505) 
 ((|constructor| (NIL "This package solves linear system in the matrix form \\spad{AX = B}. It is essentially a particular instantiation of the package \\spadtype{LinearSystemMatrixPackage} for Matrix and Vector. This package\\spad{'s} existence makes it easier to use \\spadfun{solve} in the AXIOM interpreter.")) (|rank| (((|NonNegativeInteger|) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{rank(A,B)} computes the rank of the complete matrix \\spad{(A|B)} of the linear system \\spad{AX = B}.")) (|hasSolution?| (((|Boolean|) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{hasSolution?(A,B)} tests if the linear system \\spad{AX = B} has a solution.")) (|particularSolution| (((|Union| (|Vector| |#1|) #1="failed") (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{particularSolution(A,B)} finds a particular solution of the linear system \\spad{AX = B}.")) (|solve| (((|List| (|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|))))) (|List| (|List| |#1|)) (|List| (|Vector| |#1|))) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|List| (|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|))))) (|Matrix| |#1|) (|List| (|Vector| |#1|))) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|)))) (|List| (|List| |#1|)) (|Vector| |#1|)) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}.") (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|)))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}."))) 
 NIL 
 NIL 
-(-672 R E OV P) 
+(-674 R E OV P) 
 ((|constructor| (NIL "this package finds the solutions of linear systems presented as a list of polynomials.")) (|linSolve| (((|Record| (|:| |particular| (|Union| (|Vector| (|Fraction| |#4|)) "failed")) (|:| |basis| (|List| (|Vector| (|Fraction| |#4|))))) (|List| |#4|) (|List| |#3|)) "\\spad{linSolve(lp,lvar)} finds the solutions of the linear system of polynomials \\spad{lp} = 0 with respect to the list of symbols \\spad{lvar}."))) 
 NIL 
 NIL 
-(-673 |n| R) 
+(-675 |n| R) 
 ((|constructor| (NIL "LieSquareMatrix(\\spad{n},{}\\spad{R}) implements the Lie algebra of the \\spad{n} by \\spad{n} matrices over the commutative ring \\spad{R}. The Lie bracket (commutator) of the algebra is given by \\spad{a*b := (a *\\$SQMATRIX(n,R) b - b *\\$SQMATRIX(n,R) a)},{} where \\spadfun{*\\$SQMATRIX(\\spad{n},{}\\spad{R})} is the usual matrix multiplication."))) 
-((-4424 . T) (-4427 . T) (-4421 . T) (-4422 . T)) 
-((|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasAttribute| |#2| (QUOTE (-4429 #1="*"))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-561))) (-3962 (|HasAttribute| |#2| (QUOTE (-4429 #1#))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-173)))) 
-(-674) 
+((-4431 . T) (-4434 . T) (-4428 . T) (-4429 . T)) 
+((|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasAttribute| |#2| (QUOTE (-4436 #1="*"))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))))) (|HasCategory| |#2| (QUOTE (-310))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-562))) (-3969 (|HasAttribute| |#2| (QUOTE (-4436 #1#))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183))))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-173)))) 
+(-676) 
 ((|constructor| (NIL "This domain represents `literal sequence' syntax.")) (|elements| (((|List| (|SpadAst|)) $) "\\spad{elements(e)} returns the list of expressions in the `literal' list `e'."))) 
 NIL 
 NIL 
-(-675 |VarSet|) 
+(-677 |VarSet|) 
 ((|constructor| (NIL "Lyndon words over arbitrary (ordered) symbols: see Free Lie Algebras by \\spad{C}. Reutenauer (Oxford science publications). A Lyndon word is a word which is smaller than any of its right factors \\spad{w}.\\spad{r}.\\spad{t}. the pure lexicographical ordering. If \\axiom{a} and \\axiom{\\spad{b}} are two Lyndon words such that \\axiom{a < \\spad{b}} holds \\spad{w}.\\spad{r}.\\spad{t} lexicographical ordering then \\axiom{a*b} is a Lyndon word. Parenthesized Lyndon words can be generated from symbols by using the following rule: \\axiom{[[a,{}\\spad{b}],{}\\spad{c}]} is a Lyndon word iff \\axiom{a*b < \\spad{c} \\spad{<=} \\spad{b}} holds. Lyndon words are internally represented by binary trees using the \\spadtype{Magma} domain constructor. Two ordering are provided: lexicographic and length-lexicographic. \\newline Author : Michel Petitot (petitot@lifl.\\spad{fr}).")) (|LyndonWordsList| (((|List| $) (|List| |#1|) (|PositiveInteger|)) "\\axiom{LyndonWordsList(\\spad{vl},{} \\spad{n})} returns the list of Lyndon words over the alphabet \\axiom{\\spad{vl}},{} up to order \\axiom{\\spad{n}}.")) (|LyndonWordsList1| (((|OneDimensionalArray| (|List| $)) (|List| |#1|) (|PositiveInteger|)) "\\axiom{LyndonWordsList1(\\spad{vl},{} \\spad{n})} returns an array of lists of Lyndon words over the alphabet \\axiom{\\spad{vl}},{} up to order \\axiom{\\spad{n}}.")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{x})} returns the list of distinct entries of \\axiom{\\spad{x}}.")) (|lyndonIfCan| (((|Union| $ "failed") (|OrderedFreeMonoid| |#1|)) "\\axiom{lyndonIfCan(\\spad{w})} convert \\axiom{\\spad{w}} into a Lyndon word.")) (|lyndon| (($ (|OrderedFreeMonoid| |#1|)) "\\axiom{lyndon(\\spad{w})} convert \\axiom{\\spad{w}} into a Lyndon word,{} error if \\axiom{\\spad{w}} is not a Lyndon word.")) (|lyndon?| (((|Boolean|) (|OrderedFreeMonoid| |#1|)) "\\axiom{lyndon?(\\spad{w})} test if \\axiom{\\spad{w}} is a Lyndon word.")) (|factor| (((|List| $) (|OrderedFreeMonoid| |#1|)) "\\axiom{factor(\\spad{x})} returns the decreasing factorization into Lyndon words.")) (|coerce| (((|Magma| |#1|) $) "\\axiom{coerce(\\spad{x})} returns the element of \\axiomType{Magma}(VarSet) corresponding to \\axiom{\\spad{x}}.") (((|OrderedFreeMonoid| |#1|) $) "\\axiom{coerce(\\spad{x})} returns the element of \\axiomType{OrderedFreeMonoid}(VarSet) corresponding to \\axiom{\\spad{x}}.")) (|lexico| (((|Boolean|) $ $) "\\axiom{lexico(\\spad{x},{}\\spad{y})} returns \\axiom{\\spad{true}} iff \\axiom{\\spad{x}} is smaller than \\axiom{\\spad{y}} \\spad{w}.\\spad{r}.\\spad{t}. the lexicographical ordering induced by \\axiom{VarSet}.")) (|length| (((|PositiveInteger|) $) "\\axiom{length(\\spad{x})} returns the number of entries in \\axiom{\\spad{x}}.")) (|right| (($ $) "\\axiom{right(\\spad{x})} returns right subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{LyndonWord}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|left| (($ $) "\\axiom{left(\\spad{x})} returns left subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{LyndonWord}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|retractable?| (((|Boolean|) $) "\\axiom{retractable?(\\spad{x})} tests if \\axiom{\\spad{x}} is a tree with only one entry."))) 
 NIL 
 NIL 
-(-676 A S) 
+(-678 A S) 
 ((|constructor| (NIL "LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \\spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.")) (|complete| (($ $) "\\spad{complete(st)} causes all entries of 'st' to be computed. this function should only be called on streams which are known to be finite.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(st,n)} causes entries to be computed,{} if necessary,{} so that 'st' will have at least \\spad{'n'} explicit entries or so that all entries of 'st' will be computed if 'st' is finite with length \\spad{<=} \\spad{n}.")) (|numberOfComputedEntries| (((|NonNegativeInteger|) $) "\\spad{numberOfComputedEntries(st)} returns the number of explicitly computed entries of stream \\spad{st} which exist immediately prior to the time this function is called.")) (|rst| (($ $) "\\spad{rst(s)} returns a pointer to the next node of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|frst| ((|#2| $) "\\spad{frst(s)} returns the first element of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|lazyEvaluate| (($ $) "\\spad{lazyEvaluate(s)} causes one lazy evaluation of stream \\spad{s}. Caution: the first node must be a lazy evaluation mechanism (satisfies \\spad{lazy?(s) = true}) as there is no error check. Note: a call to this function may or may not produce an explicit first entry")) (|lazy?| (((|Boolean|) $) "\\spad{lazy?(s)} returns \\spad{true} if the first node of the stream \\spad{s} is a lazy evaluation mechanism which could produce an additional entry to \\spad{s}.")) (|explicitlyEmpty?| (((|Boolean|) $) "\\spad{explicitlyEmpty?(s)} returns \\spad{true} if the stream is an (explicitly) empty stream. Note: this is a null test which will not cause lazy evaluation.")) (|explicitEntries?| (((|Boolean|) $) "\\spad{explicitEntries?(s)} returns \\spad{true} if the stream \\spad{s} has explicitly computed entries,{} and \\spad{false} otherwise.")) (|select| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select(f,st)} returns a stream consisting of those elements of stream \\spad{st} satisfying the predicate \\spad{f}. Note: \\spad{select(f,st) = [x for x in st | f(x)]}.")) (|remove| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove(f,st)} returns a stream consisting of those elements of stream \\spad{st} which do not satisfy the predicate \\spad{f}. Note: \\spad{remove(f,st) = [x for x in st | not f(x)]}."))) 
 NIL 
 NIL 
-(-677 S) 
+(-679 S) 
 ((|constructor| (NIL "LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \\spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.")) (|complete| (($ $) "\\spad{complete(st)} causes all entries of 'st' to be computed. this function should only be called on streams which are known to be finite.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(st,n)} causes entries to be computed,{} if necessary,{} so that 'st' will have at least \\spad{'n'} explicit entries or so that all entries of 'st' will be computed if 'st' is finite with length \\spad{<=} \\spad{n}.")) (|numberOfComputedEntries| (((|NonNegativeInteger|) $) "\\spad{numberOfComputedEntries(st)} returns the number of explicitly computed entries of stream \\spad{st} which exist immediately prior to the time this function is called.")) (|rst| (($ $) "\\spad{rst(s)} returns a pointer to the next node of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|frst| ((|#1| $) "\\spad{frst(s)} returns the first element of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|lazyEvaluate| (($ $) "\\spad{lazyEvaluate(s)} causes one lazy evaluation of stream \\spad{s}. Caution: the first node must be a lazy evaluation mechanism (satisfies \\spad{lazy?(s) = true}) as there is no error check. Note: a call to this function may or may not produce an explicit first entry")) (|lazy?| (((|Boolean|) $) "\\spad{lazy?(s)} returns \\spad{true} if the first node of the stream \\spad{s} is a lazy evaluation mechanism which could produce an additional entry to \\spad{s}.")) (|explicitlyEmpty?| (((|Boolean|) $) "\\spad{explicitlyEmpty?(s)} returns \\spad{true} if the stream is an (explicitly) empty stream. Note: this is a null test which will not cause lazy evaluation.")) (|explicitEntries?| (((|Boolean|) $) "\\spad{explicitEntries?(s)} returns \\spad{true} if the stream \\spad{s} has explicitly computed entries,{} and \\spad{false} otherwise.")) (|select| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select(f,st)} returns a stream consisting of those elements of stream \\spad{st} satisfying the predicate \\spad{f}. Note: \\spad{select(f,st) = [x for x in st | f(x)]}.")) (|remove| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove(f,st)} returns a stream consisting of those elements of stream \\spad{st} which do not satisfy the predicate \\spad{f}. Note: \\spad{remove(f,st) = [x for x in st | not f(x)]}."))) 
 NIL 
 NIL 
-(-678 R) 
+(-680 R) 
 ((|constructor| (NIL "This domain represents three dimensional matrices over a general object type")) (|matrixDimensions| (((|Vector| (|NonNegativeInteger|)) $) "\\spad{matrixDimensions(x)} returns the dimensions of a matrix")) (|matrixConcat3D| (($ (|Symbol|) $ $) "\\spad{matrixConcat3D(s,x,y)} concatenates two 3-\\spad{D} matrices along a specified axis")) (|coerce| (((|PrimitiveArray| (|PrimitiveArray| (|PrimitiveArray| |#1|))) $) "\\spad{coerce(x)} moves from the domain to the representation type") (($ (|PrimitiveArray| (|PrimitiveArray| (|PrimitiveArray| |#1|)))) "\\spad{coerce(p)} moves from the representation type (PrimitiveArray PrimitiveArray PrimitiveArray \\spad{R}) to the domain")) (|setelt!| ((|#1| $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|) |#1|) "\\spad{setelt!(x,i,j,k,s)} (or \\spad{x}.\\spad{i}.\\spad{j}.k:=s) sets a specific element of the array to some value of type \\spad{R}")) (|elt| ((|#1| $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{elt(x,i,j,k)} extract an element from the matrix \\spad{x}")) (|construct| (($ (|List| (|List| (|List| |#1|)))) "\\spad{construct(lll)} creates a 3-\\spad{D} matrix from a List List List \\spad{R} \\spad{lll}")) (|plus| (($ $ $) "\\spad{plus(x,y)} adds two matrices,{} term by term we note that they must be the same size")) (|identityMatrix| (($ (|NonNegativeInteger|)) "\\spad{identityMatrix(n)} create an identity matrix we note that this must be square")) (|zeroMatrix| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{zeroMatrix(i,j,k)} create a matrix with all zero terms"))) 
 NIL 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-679) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-681) 
 ((|constructor| (NIL "This domain represents the syntax of a macro definition.")) (|body| (((|SpadAst|) $) "\\spad{body(m)} returns the right hand side of the definition \\spad{`m'}.")) (|head| (((|HeadAst|) $) "\\spad{head(m)} returns the head of the macro definition \\spad{`m'}. This is a list of identifiers starting with the name of the macro followed by the name of the parameters,{} if any."))) 
 NIL 
 NIL 
-(-680 |VarSet|) 
+(-682 |VarSet|) 
 ((|constructor| (NIL "This type is the basic representation of parenthesized words (binary trees over arbitrary symbols) useful in \\spadtype{LiePolynomial}. \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr}).")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{x})} returns the list of distinct entries of \\axiom{\\spad{x}}.")) (|right| (($ $) "\\axiom{right(\\spad{x})} returns right subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|retractable?| (((|Boolean|) $) "\\axiom{retractable?(\\spad{x})} tests if \\axiom{\\spad{x}} is a tree with only one entry.")) (|rest| (($ $) "\\axiom{rest(\\spad{x})} return \\axiom{\\spad{x}} without the first entry or error if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|mirror| (($ $) "\\axiom{mirror(\\spad{x})} returns the reversed word of \\axiom{\\spad{x}}. That is \\axiom{\\spad{x}} itself if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true} and \\axiom{mirror(\\spad{z}) * mirror(\\spad{y})} if \\axiom{\\spad{x}} is \\axiom{\\spad{y*z}}.")) (|lexico| (((|Boolean|) $ $) "\\axiom{lexico(\\spad{x},{}\\spad{y})} returns \\axiom{\\spad{true}} iff \\axiom{\\spad{x}} is smaller than \\axiom{\\spad{y}} \\spad{w}.\\spad{r}.\\spad{t}. the lexicographical ordering induced by \\axiom{VarSet}. \\spad{N}.\\spad{B}. This operation does not take into account the tree structure of its arguments. Thus this is not a total ordering.")) (|length| (((|PositiveInteger|) $) "\\axiom{length(\\spad{x})} returns the number of entries in \\axiom{\\spad{x}}.")) (|left| (($ $) "\\axiom{left(\\spad{x})} returns left subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|first| ((|#1| $) "\\axiom{first(\\spad{x})} returns the first entry of the tree \\axiom{\\spad{x}}.")) (|coerce| (((|OrderedFreeMonoid| |#1|) $) "\\axiom{coerce(\\spad{x})} returns the element of \\axiomType{OrderedFreeMonoid}(VarSet) corresponding to \\axiom{\\spad{x}} by removing parentheses.")) (* (($ $ $) "\\axiom{x*y} returns the tree \\axiom{[\\spad{x},{}\\spad{y}]}."))) 
 NIL 
 NIL 
-(-681 A) 
+(-683 A) 
 ((|constructor| (NIL "various Currying operations.")) (|recur| ((|#1| (|Mapping| |#1| (|NonNegativeInteger|) |#1|) (|NonNegativeInteger|) |#1|) "\\spad{recur(n,g,x)} is \\spad{g(n,g(n-1,..g(1,x)..))}.")) (|iter| ((|#1| (|Mapping| |#1| |#1|) (|NonNegativeInteger|) |#1|) "\\spad{iter(f,n,x)} applies \\spad{f n} times to \\spad{x}."))) 
 NIL 
 NIL 
-(-682 A C) 
+(-684 A C) 
 ((|constructor| (NIL "various Currying operations.")) (|arg2| ((|#2| |#1| |#2|) "\\spad{arg2(a,c)} selects its second argument.")) (|arg1| ((|#1| |#1| |#2|) "\\spad{arg1(a,c)} selects its first argument."))) 
 NIL 
 NIL 
-(-683 A B C) 
+(-685 A B C) 
 ((|constructor| (NIL "various Currying operations.")) (|comp| ((|#3| (|Mapping| |#3| |#2|) (|Mapping| |#2| |#1|) |#1|) "\\spad{comp(f,g,x)} is \\spad{f(g x)}."))) 
 NIL 
 NIL 
-(-684) 
+(-686) 
 ((|constructor| (NIL "This domain represents a mapping type AST. A mapping AST \\indented{2}{is a syntactic description of a function type,{} \\spadignore{e.g.} its result} \\indented{2}{type and the list of its argument types.}")) (|target| (((|TypeAst|) $) "\\spad{target(s)} returns the result type AST for \\spad{`s'}.")) (|source| (((|List| (|TypeAst|)) $) "\\spad{source(s)} returns the parameter type AST list of \\spad{`s'}.")) (|mappingAst| (($ (|List| (|TypeAst|)) (|TypeAst|)) "\\spad{mappingAst(s,t)} builds the mapping AST \\spad{s} \\spad{->} \\spad{t}")) (|coerce| (($ (|Signature|)) "sig::MappingAst builds a MappingAst from the Signature `sig'."))) 
 NIL 
 NIL 
-(-685 A) 
+(-687 A) 
 ((|constructor| (NIL "various Currying operations.")) (|recur| (((|Mapping| |#1| (|NonNegativeInteger|) |#1|) (|Mapping| |#1| (|NonNegativeInteger|) |#1|)) "\\spad{recur(g)} is the function \\spad{h} such that \\indented{1}{\\spad{h(n,x)= g(n,g(n-1,..g(1,x)..))}.}")) (** (((|Mapping| |#1| |#1|) (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{f**n} is the function which is the \\spad{n}-fold application \\indented{1}{of \\spad{f}.}")) (|id| ((|#1| |#1|) "\\spad{id x} is \\spad{x}.")) (|fixedPoint| (((|List| |#1|) (|Mapping| (|List| |#1|) (|List| |#1|)) (|Integer|)) "\\spad{fixedPoint(f,n)} is the fixed point of function \\indented{1}{\\spad{f} which is assumed to transform a list of length} \\indented{1}{\\spad{n}.}") ((|#1| (|Mapping| |#1| |#1|)) "\\spad{fixedPoint f} is the fixed point of function \\spad{f}. \\indented{1}{\\spadignore{i.e.} such that \\spad{fixedPoint f = f(fixedPoint f)}.}")) (|coerce| (((|Mapping| |#1|) |#1|) "\\spad{coerce A} changes its argument into a \\indented{1}{nullary function.}")) (|nullary| (((|Mapping| |#1|) |#1|) "\\spad{nullary A} changes its argument into a \\indented{1}{nullary function.}"))) 
 NIL 
 NIL 
-(-686 A C) 
+(-688 A C) 
 ((|constructor| (NIL "various Currying operations.")) (|diag| (((|Mapping| |#2| |#1|) (|Mapping| |#2| |#1| |#1|)) "\\spad{diag(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g a = f(a,a)}.}")) (|constant| (((|Mapping| |#2| |#1|) (|Mapping| |#2|)) "\\spad{vu(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g a= f ()}.}")) (|curry| (((|Mapping| |#2|) (|Mapping| |#2| |#1|) |#1|) "\\spad{cu(f,a)} is the function \\spad{g} \\indented{1}{such that \\spad{g ()= f a}.}")) (|const| (((|Mapping| |#2| |#1|) |#2|) "\\spad{const c} is a function which produces \\spad{c} when \\indented{1}{applied to its argument.}"))) 
 NIL 
 NIL 
-(-687 A B C) 
+(-689 A B C) 
 ((|constructor| (NIL "various Currying operations.")) (* (((|Mapping| |#3| |#1|) (|Mapping| |#3| |#2|) (|Mapping| |#2| |#1|)) "\\spad{f*g} is the function \\spad{h} \\indented{1}{such that \\spad{h x= f(g x)}.}")) (|twist| (((|Mapping| |#3| |#2| |#1|) (|Mapping| |#3| |#1| |#2|)) "\\spad{twist(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g (a,b)= f(b,a)}.}")) (|constantLeft| (((|Mapping| |#3| |#1| |#2|) (|Mapping| |#3| |#2|)) "\\spad{constantLeft(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g (a,b)= f b}.}")) (|constantRight| (((|Mapping| |#3| |#1| |#2|) (|Mapping| |#3| |#1|)) "\\spad{constantRight(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g (a,b)= f a}.}")) (|curryLeft| (((|Mapping| |#3| |#2|) (|Mapping| |#3| |#1| |#2|) |#1|) "\\spad{curryLeft(f,a)} is the function \\spad{g} \\indented{1}{such that \\spad{g b = f(a,b)}.}")) (|curryRight| (((|Mapping| |#3| |#1|) (|Mapping| |#3| |#1| |#2|) |#2|) "\\spad{curryRight(f,b)} is the function \\spad{g} such that \\indented{1}{\\spad{g a = f(a,b)}.}"))) 
 NIL 
 NIL 
-(-688 S R |Row| |Col|) 
+(-690 S R |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by calling the function \\spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \\spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|minordet| ((|#2| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors. Error: if the matrix is not square.")) (|determinant| ((|#2| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. Error: if the matrix is not square.")) (|nullSpace| (((|List| |#4|) $) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#2|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if matrix is not square or if the matrix is square but not invertible.") (($ $ (|NonNegativeInteger|)) "\\spad{x ** n} computes a non-negative integral power of the matrix \\spad{x}. Error: if the matrix is not square.")) (* ((|#3| |#3| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#4| $ |#4|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.") (($ (|Integer|) $) "\\spad{n * x} is an integer multiple.") (($ $ |#2|) "\\spad{x * r} is the right scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ |#2| $) "\\spad{r*x} is the left scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ $ $) "\\spad{x * y} is the product of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (- (($ $) "\\spad{-x} returns the negative of the matrix \\spad{x}.") (($ $ $) "\\spad{x - y} is the difference of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (+ (($ $ $) "\\spad{x + y} is the sum of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (|setsubMatrix!| (($ $ (|Integer|) (|Integer|) $) "\\spad{setsubMatrix(x,i1,j1,y)} destructively alters the matrix \\spad{x}. Here \\spad{x(i,j)} is set to \\spad{y(i-i1+1,j-j1+1)} for \\spad{i = i1,...,i1-1+nrows y} and \\spad{j = j1,...,j1-1+ncols y}.")) (|subMatrix| (($ $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{subMatrix(x,i1,i2,j1,j2)} extracts the submatrix \\spad{[x(i,j)]} where the index \\spad{i} ranges from \\spad{i1} to \\spad{i2} and the index \\spad{j} ranges from \\spad{j1} to \\spad{j2}.")) (|swapColumns!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapColumns!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th columns of \\spad{m}. This destructively alters the matrix.")) (|swapRows!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapRows!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th rows of \\spad{m}. This destructively alters the matrix.")) (|setelt| (($ $ (|List| (|Integer|)) (|List| (|Integer|)) $) "\\spad{setelt(x,rowList,colList,y)} destructively alters the matrix \\spad{x}. If \\spad{y} is \\spad{m}-by-\\spad{n},{} \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then \\spad{x(i<k>,j<l>)} is set to \\spad{y(k,l)} for \\spad{k = 1,...,m} and \\spad{l = 1,...,n}.")) (|elt| (($ $ (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{elt(x,rowList,colList)} returns an \\spad{m}-by-\\spad{n} matrix consisting of elements of \\spad{x},{} where \\spad{m = \\# rowList} and \\spad{n = \\# colList}. If \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then the \\spad{(k,l)}th entry of \\spad{elt(x,rowList,colList)} is \\spad{x(i<k>,j<l>)}.")) (|listOfLists| (((|List| (|List| |#2|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|vertConcat| (($ $ $) "\\spad{vertConcat(x,y)} vertically concatenates two matrices with an equal number of columns. The entries of \\spad{y} appear below of the entries of \\spad{x}. Error: if the matrices do not have the same number of columns.")) (|horizConcat| (($ $ $) "\\spad{horizConcat(x,y)} horizontally concatenates two matrices with an equal number of rows. The entries of \\spad{y} appear to the right of the entries of \\spad{x}. Error: if the matrices do not have the same number of rows.")) (|squareTop| (($ $) "\\spad{squareTop(m)} returns an \\spad{n}-by-\\spad{n} matrix consisting of the first \\spad{n} rows of the \\spad{m}-by-\\spad{n} matrix \\spad{m}. Error: if \\spad{m < n}.")) (|transpose| (($ $) "\\spad{transpose(m)} returns the transpose of the matrix \\spad{m}.") (($ |#3|) "\\spad{transpose(r)} converts the row \\spad{r} to a row matrix.")) (|coerce| (($ |#4|) "\\spad{coerce(col)} converts the column \\spad{col} to a column matrix.")) (|diagonalMatrix| (($ (|List| $)) "\\spad{diagonalMatrix([m1,...,mk])} creates a block diagonal matrix \\spad{M} with block matrices {\\em m1},{}...,{}{\\em mk} down the diagonal,{} with 0 block matrices elsewhere. More precisly: if \\spad{ri := nrows mi},{} \\spad{ci := ncols mi},{} then \\spad{m} is an (\\spad{r1+}..\\spad{+rk}) by (\\spad{c1+}..\\spad{+ck}) - matrix with entries \\spad{m.i.j = ml.(i-r1-..-r(l-1)).(j-n1-..-n(l-1))},{} if \\spad{(r1+..+r(l-1)) < i <= r1+..+rl} and \\spad{(c1+..+c(l-1)) < i <= c1+..+cl},{} \\spad{m.i.j} = 0 otherwise.") (($ (|List| |#2|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ (|NonNegativeInteger|) |#2|) "\\spad{scalarMatrix(n,r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}\\spad{'s} on the diagonal and zeroes elsewhere.")) (|matrix| (($ (|List| (|List| |#2|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|zero| (($ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{zero(m,n)} returns an \\spad{m}-by-\\spad{n} zero matrix.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|finiteAggregate| ((|attribute|) "matrices are finite")) (|shallowlyMutable| ((|attribute|) "One may destructively alter matrices"))) 
 NIL 
-((|HasAttribute| |#2| (QUOTE (-4429 "*"))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-561)))) 
-(-689 R |Row| |Col|) 
+((|HasAttribute| |#2| (QUOTE (-4436 "*"))) (|HasCategory| |#2| (QUOTE (-310))) (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-562)))) 
+(-691 R |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by calling the function \\spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \\spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|minordet| ((|#1| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors. Error: if the matrix is not square.")) (|determinant| ((|#1| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. Error: if the matrix is not square.")) (|nullSpace| (((|List| |#3|) $) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#1|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if matrix is not square or if the matrix is square but not invertible.") (($ $ (|NonNegativeInteger|)) "\\spad{x ** n} computes a non-negative integral power of the matrix \\spad{x}. Error: if the matrix is not square.")) (* ((|#2| |#2| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#3| $ |#3|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.") (($ (|Integer|) $) "\\spad{n * x} is an integer multiple.") (($ $ |#1|) "\\spad{x * r} is the right scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ |#1| $) "\\spad{r*x} is the left scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ $ $) "\\spad{x * y} is the product of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (- (($ $) "\\spad{-x} returns the negative of the matrix \\spad{x}.") (($ $ $) "\\spad{x - y} is the difference of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (+ (($ $ $) "\\spad{x + y} is the sum of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (|setsubMatrix!| (($ $ (|Integer|) (|Integer|) $) "\\spad{setsubMatrix(x,i1,j1,y)} destructively alters the matrix \\spad{x}. Here \\spad{x(i,j)} is set to \\spad{y(i-i1+1,j-j1+1)} for \\spad{i = i1,...,i1-1+nrows y} and \\spad{j = j1,...,j1-1+ncols y}.")) (|subMatrix| (($ $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{subMatrix(x,i1,i2,j1,j2)} extracts the submatrix \\spad{[x(i,j)]} where the index \\spad{i} ranges from \\spad{i1} to \\spad{i2} and the index \\spad{j} ranges from \\spad{j1} to \\spad{j2}.")) (|swapColumns!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapColumns!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th columns of \\spad{m}. This destructively alters the matrix.")) (|swapRows!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapRows!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th rows of \\spad{m}. This destructively alters the matrix.")) (|setelt| (($ $ (|List| (|Integer|)) (|List| (|Integer|)) $) "\\spad{setelt(x,rowList,colList,y)} destructively alters the matrix \\spad{x}. If \\spad{y} is \\spad{m}-by-\\spad{n},{} \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then \\spad{x(i<k>,j<l>)} is set to \\spad{y(k,l)} for \\spad{k = 1,...,m} and \\spad{l = 1,...,n}.")) (|elt| (($ $ (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{elt(x,rowList,colList)} returns an \\spad{m}-by-\\spad{n} matrix consisting of elements of \\spad{x},{} where \\spad{m = \\# rowList} and \\spad{n = \\# colList}. If \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then the \\spad{(k,l)}th entry of \\spad{elt(x,rowList,colList)} is \\spad{x(i<k>,j<l>)}.")) (|listOfLists| (((|List| (|List| |#1|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|vertConcat| (($ $ $) "\\spad{vertConcat(x,y)} vertically concatenates two matrices with an equal number of columns. The entries of \\spad{y} appear below of the entries of \\spad{x}. Error: if the matrices do not have the same number of columns.")) (|horizConcat| (($ $ $) "\\spad{horizConcat(x,y)} horizontally concatenates two matrices with an equal number of rows. The entries of \\spad{y} appear to the right of the entries of \\spad{x}. Error: if the matrices do not have the same number of rows.")) (|squareTop| (($ $) "\\spad{squareTop(m)} returns an \\spad{n}-by-\\spad{n} matrix consisting of the first \\spad{n} rows of the \\spad{m}-by-\\spad{n} matrix \\spad{m}. Error: if \\spad{m < n}.")) (|transpose| (($ $) "\\spad{transpose(m)} returns the transpose of the matrix \\spad{m}.") (($ |#2|) "\\spad{transpose(r)} converts the row \\spad{r} to a row matrix.")) (|coerce| (($ |#3|) "\\spad{coerce(col)} converts the column \\spad{col} to a column matrix.")) (|diagonalMatrix| (($ (|List| $)) "\\spad{diagonalMatrix([m1,...,mk])} creates a block diagonal matrix \\spad{M} with block matrices {\\em m1},{}...,{}{\\em mk} down the diagonal,{} with 0 block matrices elsewhere. More precisly: if \\spad{ri := nrows mi},{} \\spad{ci := ncols mi},{} then \\spad{m} is an (\\spad{r1+}..\\spad{+rk}) by (\\spad{c1+}..\\spad{+ck}) - matrix with entries \\spad{m.i.j = ml.(i-r1-..-r(l-1)).(j-n1-..-n(l-1))},{} if \\spad{(r1+..+r(l-1)) < i <= r1+..+rl} and \\spad{(c1+..+c(l-1)) < i <= c1+..+cl},{} \\spad{m.i.j} = 0 otherwise.") (($ (|List| |#1|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ (|NonNegativeInteger|) |#1|) "\\spad{scalarMatrix(n,r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}\\spad{'s} on the diagonal and zeroes elsewhere.")) (|matrix| (($ (|List| (|List| |#1|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|zero| (($ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{zero(m,n)} returns an \\spad{m}-by-\\spad{n} zero matrix.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|finiteAggregate| ((|attribute|) "matrices are finite")) (|shallowlyMutable| ((|attribute|) "One may destructively alter matrices"))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
-(-690 R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) 
+(-692 R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) 
 ((|constructor| (NIL "\\spadtype{MatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \\spadfun{map} and \\spadfun{reduce}.")) (|reduce| ((|#5| (|Mapping| |#5| |#1| |#5|) |#4| |#5|) "\\spad{reduce(f,m,r)} returns a matrix \\spad{n} where \\spad{n[i,j] = f(m[i,j],r)} for all indices \\spad{i} and \\spad{j}.")) (|map| (((|Union| |#8| "failed") (|Mapping| (|Union| |#5| "failed") |#1|) |#4|) "\\spad{map(f,m)} applies the function \\spad{f} to the elements of the matrix \\spad{m}.") ((|#8| (|Mapping| |#5| |#1|) |#4|) "\\spad{map(f,m)} applies the function \\spad{f} to the elements of the matrix \\spad{m}."))) 
 NIL 
 NIL 
-(-691 R |Row| |Col| M) 
+(-693 R |Row| |Col| M) 
 ((|constructor| (NIL "\\spadtype{MatrixLinearAlgebraFunctions} provides functions to compute inverses and canonical forms.")) (|inverse| (((|Union| |#4| "failed") |#4|) "\\spad{inverse(m)} returns the inverse of the matrix. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|normalizedDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{normalizedDivide(n,d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \\spadignore{e.g.} positive remainders")) (|rowEchelon| ((|#4| |#4|) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (|adjoint| (((|Record| (|:| |adjMat| |#4|) (|:| |detMat| |#1|)) |#4|) "\\spad{adjoint(m)} returns the ajoint matrix of \\spad{m} (\\spadignore{i.e.} the matrix \\spad{n} such that \\spad{m*n} = determinant(\\spad{m})*id) and the detrminant of \\spad{m}.")) (|invertIfCan| (((|Union| |#4| "failed") |#4|) "\\spad{invertIfCan(m)} returns the inverse of \\spad{m} over \\spad{R}")) (|fractionFreeGauss!| ((|#4| |#4|) "\\spad{fractionFreeGauss(m)} performs the fraction free gaussian elimination on the matrix \\spad{m}.")) (|nullSpace| (((|List| |#3|) |#4|) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) |#4|) "\\spad{nullity(m)} returns the mullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) |#4|) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|elColumn2!| ((|#4| |#4| |#1| (|Integer|) (|Integer|)) "\\spad{elColumn2!(m,a,i,j)} adds to column \\spad{i} a*column(\\spad{m},{}\\spad{j}) : elementary operation of second kind. (\\spad{i} \\spad{~=j})")) (|elRow2!| ((|#4| |#4| |#1| (|Integer|) (|Integer|)) "\\spad{elRow2!(m,a,i,j)} adds to row \\spad{i} a*row(\\spad{m},{}\\spad{j}) : elementary operation of second kind. (\\spad{i} \\spad{~=j})")) (|elRow1!| ((|#4| |#4| (|Integer|) (|Integer|)) "\\spad{elRow1!(m,i,j)} swaps rows \\spad{i} and \\spad{j} of matrix \\spad{m} : elementary operation of first kind")) (|minordet| ((|#1| |#4|) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors. Error: if the matrix is not square.")) (|determinant| ((|#1| |#4|) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. an error message is returned if the matrix is not square."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-561)))) 
-(-692 R) 
+((|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-562)))) 
+(-694 R) 
 ((|constructor| (NIL "\\spadtype{Matrix} is a matrix domain where 1-based indexing is used for both rows and columns.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|diagonalMatrix| (($ (|Vector| |#1|)) "\\spad{diagonalMatrix(v)} returns a diagonal matrix where the elements of \\spad{v} appear on the diagonal."))) 
-((-4427 . T) (-4428 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-561))) (|HasAttribute| |#1| (QUOTE (-4429 "*"))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-693 R) 
+((-4434 . T) (-4435 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (QUOTE (-310))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4436 "*"))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-695 R) 
 ((|constructor| (NIL "This package provides standard arithmetic operations on matrices. The functions in this package store the results of computations in existing matrices,{} rather than creating new matrices. This package works only for matrices of type Matrix and uses the internal representation of this type.")) (** (((|Matrix| |#1|) (|Matrix| |#1|) (|NonNegativeInteger|)) "\\spad{x ** n} computes the \\spad{n}-th power of a square matrix. The power \\spad{n} is assumed greater than 1.")) (|power!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|NonNegativeInteger|)) "\\spad{power!(a,b,c,m,n)} computes \\spad{m} \\spad{**} \\spad{n} and stores the result in \\spad{a}. The matrices \\spad{b} and \\spad{c} are used to store intermediate results. Error: if \\spad{a},{} \\spad{b},{} \\spad{c},{} and \\spad{m} are not square and of the same dimensions.")) (|times!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{times!(c,a,b)} computes the matrix product \\spad{a * b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have compatible dimensions.")) (|rightScalarTimes!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rightScalarTimes!(c,a,r)} computes the scalar product \\spad{a * r} and stores the result in the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions.")) (|leftScalarTimes!| (((|Matrix| |#1|) (|Matrix| |#1|) |#1| (|Matrix| |#1|)) "\\spad{leftScalarTimes!(c,r,a)} computes the scalar product \\spad{r * a} and stores the result in the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions.")) (|minus!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{!minus!(c,a,b)} computes the matrix difference \\spad{a - b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have the same dimensions.") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{minus!(c,a)} computes \\spad{-a} and stores the result in the matrix \\spad{c}. Error: if a and \\spad{c} do not have the same dimensions.")) (|plus!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{plus!(c,a,b)} computes the matrix sum \\spad{a + b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have the same dimensions.")) (|copy!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{copy!(c,a)} copies the matrix \\spad{a} into the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions."))) 
 NIL 
 NIL 
-(-694 T$) 
+(-696 T$) 
 ((|constructor| (NIL "This domain implements the notion of optional value,{} where a computation may fail to produce expected value.")) (|nothing| (($) "\\spad{nothing} represents failure or absence of value.")) (|autoCoerce| ((|#1| $) "\\spad{autoCoerce} is a courtesy coercion function used by the compiler in case it knows that \\spad{`x'} really is a \\spadtype{T}.")) (|case| (((|Boolean|) $ (|[\|\|]| |nothing|)) "\\spad{x case nothing} holds if the value for \\spad{x} is missing.") (((|Boolean|) $ (|[\|\|]| |#1|)) "\\spad{x case T} returns \\spad{true} if \\spad{x} is actually a data of type \\spad{T}.")) (|just| (($ |#1|) "\\spad{just x} injects the value \\spad{`x'} into \\%."))) 
 NIL 
 NIL 
-(-695 S -3498 FLAF FLAS) 
+(-697 S -3505 FLAF FLAS) 
 ((|constructor| (NIL "\\indented{1}{\\spadtype{MultiVariableCalculusFunctions} Package provides several} \\indented{1}{functions for multivariable calculus.} These include gradient,{} hessian and jacobian,{} divergence and laplacian. Various forms for banded and sparse storage of matrices are included.")) (|bandedJacobian| (((|Matrix| |#2|) |#3| |#4| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{bandedJacobian(vf,xlist,kl,ku)} computes the jacobian,{} the matrix of first partial derivatives,{} of the vector field \\spad{vf},{} \\spad{vf} a vector function of the variables listed in \\spad{xlist},{} \\spad{kl} is the number of nonzero subdiagonals,{} \\spad{ku} is the number of nonzero superdiagonals,{} kl+ku+1 being actual bandwidth. Stores the nonzero band in a matrix,{} dimensions kl+ku+1 by \\#xlist. The upper triangle is in the top \\spad{ku} rows,{} the diagonal is in row ku+1,{} the lower triangle in the last \\spad{kl} rows. Entries in a column in the band store correspond to entries in same column of full store. (The notation conforms to LAPACK/NAG-\\spad{F07} conventions.)")) (|jacobian| (((|Matrix| |#2|) |#3| |#4|) "\\spad{jacobian(vf,xlist)} computes the jacobian,{} the matrix of first partial derivatives,{} of the vector field \\spad{vf},{} \\spad{vf} a vector function of the variables listed in \\spad{xlist}.")) (|bandedHessian| (((|Matrix| |#2|) |#2| |#4| (|NonNegativeInteger|)) "\\spad{bandedHessian(v,xlist,k)} computes the hessian,{} the matrix of second partial derivatives,{} of the scalar field \\spad{v},{} \\spad{v} a function of the variables listed in \\spad{xlist},{} \\spad{k} is the semi-bandwidth,{} the number of nonzero subdiagonals,{} 2*k+1 being actual bandwidth. Stores the nonzero band in lower triangle in a matrix,{} dimensions \\spad{k+1} by \\#xlist,{} whose rows are the vectors formed by diagonal,{} subdiagonal,{} etc. of the real,{} full-matrix,{} hessian. (The notation conforms to LAPACK/NAG-\\spad{F07} conventions.)")) (|hessian| (((|Matrix| |#2|) |#2| |#4|) "\\spad{hessian(v,xlist)} computes the hessian,{} the matrix of second partial derivatives,{} of the scalar field \\spad{v},{} \\spad{v} a function of the variables listed in \\spad{xlist}.")) (|laplacian| ((|#2| |#2| |#4|) "\\spad{laplacian(v,xlist)} computes the laplacian of the scalar field \\spad{v},{} \\spad{v} a function of the variables listed in \\spad{xlist}.")) (|divergence| ((|#2| |#3| |#4|) "\\spad{divergence(vf,xlist)} computes the divergence of the vector field \\spad{vf},{} \\spad{vf} a vector function of the variables listed in \\spad{xlist}.")) (|gradient| (((|Vector| |#2|) |#2| |#4|) "\\spad{gradient(v,xlist)} computes the gradient,{} the vector of first partial derivatives,{} of the scalar field \\spad{v},{} \\spad{v} a function of the variables listed in \\spad{xlist}."))) 
 NIL 
 NIL 
-(-696 R Q) 
+(-698 R Q) 
 ((|constructor| (NIL "MatrixCommonDenominator provides functions to compute the common denominator of a matrix of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| (|Matrix| |#1|)) (|:| |den| |#1|)) (|Matrix| |#2|)) "\\spad{splitDenominator(q)} returns \\spad{[p, d]} such that \\spad{q = p/d} and \\spad{d} is a common denominator for the elements of \\spad{q}.")) (|clearDenominator| (((|Matrix| |#1|) (|Matrix| |#2|)) "\\spad{clearDenominator(q)} returns \\spad{p} such that \\spad{q = p/d} where \\spad{d} is a common denominator for the elements of \\spad{q}.")) (|commonDenominator| ((|#1| (|Matrix| |#2|)) "\\spad{commonDenominator(q)} returns a common denominator \\spad{d} for the elements of \\spad{q}."))) 
 NIL 
 NIL 
-(-697) 
+(-699) 
 ((|constructor| (NIL "A domain which models the complex number representation used by machines in the AXIOM-NAG link.")) (|coerce| (((|Complex| (|Float|)) $) "\\spad{coerce(u)} transforms \\spad{u} into a COmplex Float") (($ (|Complex| (|MachineInteger|))) "\\spad{coerce(u)} transforms \\spad{u} into a MachineComplex") (($ (|Complex| (|MachineFloat|))) "\\spad{coerce(u)} transforms \\spad{u} into a MachineComplex") (($ (|Complex| (|Integer|))) "\\spad{coerce(u)} transforms \\spad{u} into a MachineComplex") (($ (|Complex| (|Float|))) "\\spad{coerce(u)} transforms \\spad{u} into a MachineComplex"))) 
-((-4420 . T) (-4425 |has| (-702) (-366)) (-4419 |has| (-702) (-366)) (-1464 . T) (-4426 |has| (-702) (-6 -4426)) (-4423 |has| (-702) (-6 -4423)) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
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-(-698 S) 
+((-4427 . T) (-4432 |has| (-704) (-367)) (-4426 |has| (-704) (-367)) (-1466 . T) (-4433 |has| (-704) (-6 -4433)) (-4430 |has| (-704) (-6 -4430)) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
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+(-700 S) 
 ((|constructor| (NIL "A multi-dictionary is a dictionary which may contain duplicates. As for any dictionary,{} its size is assumed large so that copying (non-destructive) operations are generally to be avoided.")) (|duplicates| (((|List| (|Record| (|:| |entry| |#1|) (|:| |count| (|NonNegativeInteger|)))) $) "\\spad{duplicates(d)} returns a list of values which have duplicates in \\spad{d}")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(d)} destructively removes any duplicate values in dictionary \\spad{d}.")) (|insert!| (($ |#1| $ (|NonNegativeInteger|)) "\\spad{insert!(x,d,n)} destructively inserts \\spad{n} copies of \\spad{x} into dictionary \\spad{d}."))) 
-((-4428 . T)) 
+((-4435 . T)) 
 NIL 
-(-699 U) 
+(-701 U) 
 ((|constructor| (NIL "This package supports factorization and gcds of univariate polynomials over the integers modulo different primes. The inputs are given as polynomials over the integers with the prime passed explicitly as an extra argument.")) (|exptMod| ((|#1| |#1| (|Integer|) |#1| (|Integer|)) "\\spad{exptMod(f,n,g,p)} raises the univariate polynomial \\spad{f} to the \\spad{n}th power modulo the polynomial \\spad{g} and the prime \\spad{p}.")) (|separateFactors| (((|List| |#1|) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|)))) (|Integer|)) "\\spad{separateFactors(ddl, p)} refines the distinct degree factorization produced by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} to give a complete list of factors.")) (|ddFact| (((|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|)))) |#1| (|Integer|)) "\\spad{ddFact(f,p)} computes a distinct degree factorization of the polynomial \\spad{f} modulo the prime \\spad{p},{} \\spadignore{i.e.} such that each factor is a product of irreducibles of the same degrees. The input polynomial \\spad{f} is assumed to be square-free modulo \\spad{p}.")) (|factor| (((|List| |#1|) |#1| (|Integer|)) "\\spad{factor(f1,p)} returns the list of factors of the univariate polynomial \\spad{f1} modulo the integer prime \\spad{p}. Error: if \\spad{f1} is not square-free modulo \\spad{p}.")) (|linears| ((|#1| |#1| (|Integer|)) "\\spad{linears(f,p)} returns the product of all the linear factors of \\spad{f} modulo \\spad{p}. Potentially incorrect result if \\spad{f} is not square-free modulo \\spad{p}.")) (|gcd| ((|#1| |#1| |#1| (|Integer|)) "\\spad{gcd(f1,f2,p)} computes the \\spad{gcd} of the univariate polynomials \\spad{f1} and \\spad{f2} modulo the integer prime \\spad{p}."))) 
 NIL 
 NIL 
-(-700) 
+(-702) 
 ((|constructor| (NIL "\\indented{1}{<description of package>} Author: Jim Wen Date Created: \\spad{??} Date Last Updated: October 1991 by Jon Steinbach Keywords: Examples: References:")) (|ptFunc| (((|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{ptFunc(a,b,c,d)} is an internal function exported in order to compile packages.")) (|meshPar1Var| (((|ThreeSpace| (|DoubleFloat|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar1Var(s,t,u,f,s1,l)} \\undocumented")) (|meshFun2Var| (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Union| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) #1="undefined") (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshFun2Var(f,g,s1,s2,l)} \\undocumented")) (|meshPar2Var| (((|ThreeSpace| (|DoubleFloat|)) (|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(sp,f,s1,s2,l)} \\undocumented") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(f,s1,s2,l)} \\undocumented") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Union| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) #1#) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(f,g,h,j,s1,s2,l)} \\undocumented"))) 
 NIL 
 NIL 
-(-701 OV E -3498 PG) 
+(-703 OV E -3505 PG) 
 ((|constructor| (NIL "Package for factorization of multivariate polynomials over finite fields.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factor(p)} produces the complete factorization of the multivariate polynomial \\spad{p} over a finite field. \\spad{p} is represented as a univariate polynomial with multivariate coefficients over a finite field.") (((|Factored| |#4|) |#4|) "\\spad{factor(p)} produces the complete factorization of the multivariate polynomial \\spad{p} over a finite field."))) 
 NIL 
 NIL 
-(-702) 
+(-704) 
 ((|constructor| (NIL "A domain which models the floating point representation used by machines in the AXIOM-NAG link.")) (|changeBase| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{changeBase(exp,man,base)} \\undocumented{}")) (|exponent| (((|Integer|) $) "\\spad{exponent(u)} returns the exponent of \\spad{u}")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(u)} returns the mantissa of \\spad{u}")) (|coerce| (($ (|MachineInteger|)) "\\spad{coerce(u)} transforms a MachineInteger into a MachineFloat") (((|Float|) $) "\\spad{coerce(u)} transforms a MachineFloat to a standard Float")) (|minimumExponent| (((|Integer|)) "\\spad{minimumExponent()} returns the minimum exponent in the model") (((|Integer|) (|Integer|)) "\\spad{minimumExponent(e)} sets the minimum exponent in the model to \\spad{e}")) (|maximumExponent| (((|Integer|)) "\\spad{maximumExponent()} returns the maximum exponent in the model") (((|Integer|) (|Integer|)) "\\spad{maximumExponent(e)} sets the maximum exponent in the model to \\spad{e}")) (|base| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{base(b)} sets the base of the model to \\spad{b}")) (|precision| (((|PositiveInteger|)) "\\spad{precision()} returns the number of digits in the model") (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(p)} sets the number of digits in the model to \\spad{p}"))) 
-((-4203 . T) (-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4210 . T) (-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-703 R) 
+(-705 R) 
 ((|constructor| (NIL "\\indented{1}{Modular hermitian row reduction.} Author: Manuel Bronstein Date Created: 22 February 1989 Date Last Updated: 24 November 1993 Keywords: matrix,{} reduction.")) (|normalizedDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{normalizedDivide(n,d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \\spadignore{e.g.} positive remainders")) (|rowEchelonLocal| (((|Matrix| |#1|) (|Matrix| |#1|) |#1| |#1|) "\\spad{rowEchelonLocal(m, d, p)} computes the row-echelon form of \\spad{m} concatenated with \\spad{d} times the identity matrix over a local ring where \\spad{p} is the only prime.")) (|rowEchLocal| (((|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rowEchLocal(m,p)} computes a modular row-echelon form of \\spad{m},{} finding an appropriate modulus over a local ring where \\spad{p} is the only prime.")) (|rowEchelon| (((|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rowEchelon(m, d)} computes a modular row-echelon form mod \\spad{d} of \\indented{3}{[\\spad{d}\\space{5}]} \\indented{3}{[\\space{2}\\spad{d}\\space{3}]} \\indented{3}{[\\space{4}. ]} \\indented{3}{[\\space{5}\\spad{d}]} \\indented{3}{[\\space{3}\\spad{M}\\space{2}]} where \\spad{M = m mod d}.")) (|rowEch| (((|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{rowEch(m)} computes a modular row-echelon form of \\spad{m},{} finding an appropriate modulus."))) 
 NIL 
 NIL 
-(-704) 
+(-706) 
 ((|constructor| (NIL "A domain which models the integer representation used by machines in the AXIOM-NAG link.")) (|coerce| (((|Expression| $) (|Expression| (|Integer|))) "\\spad{coerce(x)} returns \\spad{x} with coefficients in the domain")) (|maxint| (((|PositiveInteger|)) "\\spad{maxint()} returns the maximum integer in the model") (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{maxint(u)} sets the maximum integer in the model to \\spad{u}"))) 
-((-4426 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4433 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-705 S D1 D2 I) 
+(-707 S D1 D2 I) 
 ((|constructor| (NIL "transforms top-level objects into compiled functions.")) (|compiledFunction| (((|Mapping| |#4| |#2| |#3|) |#1| (|Symbol|) (|Symbol|)) "\\spad{compiledFunction(expr,x,y)} returns a function \\spad{f: (D1, D2) -> I} defined by \\spad{f(x, y) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\spad{(D1, D2)}")) (|binaryFunction| (((|Mapping| |#4| |#2| |#3|) (|Symbol|)) "\\spad{binaryFunction(s)} is a local function"))) 
 NIL 
 NIL 
-(-706 S) 
+(-708 S) 
 ((|constructor| (NIL "MakeFloatCompiledFunction transforms top-level objects into compiled Lisp functions whose arguments are Lisp floats. This by-passes the \\Language{} compiler and interpreter,{} thereby gaining several orders of magnitude.")) (|makeFloatFunction| (((|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) |#1| (|Symbol|) (|Symbol|)) "\\spad{makeFloatFunction(expr, x, y)} returns a Lisp function \\spad{f: (\\axiomType{DoubleFloat}, \\axiomType{DoubleFloat}) -> \\axiomType{DoubleFloat}} defined by \\spad{f(x, y) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\spad{(\\axiomType{DoubleFloat}, \\axiomType{DoubleFloat})}.") (((|Mapping| (|DoubleFloat|) (|DoubleFloat|)) |#1| (|Symbol|)) "\\spad{makeFloatFunction(expr, x)} returns a Lisp function \\spad{f: \\axiomType{DoubleFloat} -> \\axiomType{DoubleFloat}} defined by \\spad{f(x) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\axiomType{DoubleFloat}."))) 
 NIL 
 NIL 
-(-707 S) 
+(-709 S) 
 ((|constructor| (NIL "transforms top-level objects into interpreter functions.")) (|function| (((|Symbol|) |#1| (|Symbol|) (|List| (|Symbol|))) "\\spad{function(e, foo, [x1,...,xn])} creates a function \\spad{foo(x1,...,xn) == e}.") (((|Symbol|) |#1| (|Symbol|) (|Symbol|) (|Symbol|)) "\\spad{function(e, foo, x, y)} creates a function \\spad{foo(x, y) = e}.") (((|Symbol|) |#1| (|Symbol|) (|Symbol|)) "\\spad{function(e, foo, x)} creates a function \\spad{foo(x) == e}.") (((|Symbol|) |#1| (|Symbol|)) "\\spad{function(e, foo)} creates a function \\spad{foo() == e}."))) 
 NIL 
 NIL 
-(-708 S T$) 
+(-710 S T$) 
 ((|constructor| (NIL "MakeRecord is used internally by the interpreter to create record types which are used for doing parallel iterations on streams.")) (|makeRecord| (((|Record| (|:| |part1| |#1|) (|:| |part2| |#2|)) |#1| |#2|) "\\spad{makeRecord(a,b)} creates a record object with type Record(part1:S,{} part2:R),{} where part1 is \\spad{a} and part2 is \\spad{b}."))) 
 NIL 
 NIL 
-(-709 S -3074 I) 
+(-711 S -3081 I) 
 ((|constructor| (NIL "transforms top-level objects into compiled functions.")) (|compiledFunction| (((|Mapping| |#3| |#2|) |#1| (|Symbol|)) "\\spad{compiledFunction(expr, x)} returns a function \\spad{f: D -> I} defined by \\spad{f(x) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\spad{D}.")) (|unaryFunction| (((|Mapping| |#3| |#2|) (|Symbol|)) "\\spad{unaryFunction(a)} is a local function"))) 
 NIL 
 NIL 
-(-710 E OV R P) 
+(-712 E OV R P) 
 ((|constructor| (NIL "This package provides the functions for the multivariate \"lifting\",{} using an algorithm of Paul Wang. This package will work for every euclidean domain \\spad{R} which has property \\spad{F},{} \\spadignore{i.e.} there exists a factor operation in \\spad{R[x]}.")) (|lifting1| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|SparseUnivariatePolynomial| |#4|)) (|List| |#3|) (|List| |#4|) (|List| (|List| (|Record| (|:| |expt| (|NonNegativeInteger|)) (|:| |pcoef| |#4|)))) (|List| (|NonNegativeInteger|)) (|Vector| (|List| (|SparseUnivariatePolynomial| |#3|))) |#3|) "\\spad{lifting1(u,lv,lu,lr,lp,lt,ln,t,r)} \\undocumented")) (|lifting| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|SparseUnivariatePolynomial| |#3|)) (|List| |#3|) (|List| |#4|) (|List| (|NonNegativeInteger|)) |#3|) "\\spad{lifting(u,lv,lu,lr,lp,ln,r)} \\undocumented")) (|corrPoly| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| |#3|) (|List| (|NonNegativeInteger|)) (|List| (|SparseUnivariatePolynomial| |#4|)) (|Vector| (|List| (|SparseUnivariatePolynomial| |#3|))) |#3|) "\\spad{corrPoly(u,lv,lr,ln,lu,t,r)} \\undocumented"))) 
 NIL 
 NIL 
-(-711 R) 
+(-713 R) 
 ((|constructor| (NIL "This is the category of linear operator rings with one generator. The generator is not named by the category but can always be constructed as \\spad{monomial(1,1)}. \\blankline For convenience,{} call the generator \\spad{G}. Then each value is equal to \\indented{4}{\\spad{sum(a(i)*G**i, i = 0..n)}} for some unique \\spad{n} and \\spad{a(i)} in \\spad{R}. \\blankline Note that multiplication is not necessarily commutative. In fact,{} if \\spad{a} is in \\spad{R},{} it is quite normal to have \\spad{a*G \\~= G*a}.")) (|monomial| (($ |#1| (|NonNegativeInteger|)) "\\spad{monomial(c,k)} produces \\spad{c} times the \\spad{k}-th power of the generating operator,{} \\spad{monomial(1,1)}.")) (|coefficient| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coefficient(l,k)} is \\spad{a(k)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|reductum| (($ $) "\\spad{reductum(l)} is \\spad{l - monomial(a(n),n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(l)} is \\spad{a(n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|minimumDegree| (((|NonNegativeInteger|) $) "\\spad{minimumDegree(l)} is the smallest \\spad{k} such that \\spad{a(k) \\~= 0} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(l)} is \\spad{n} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-712 R1 UP1 UPUP1 R2 UP2 UPUP2) 
+(-714 R1 UP1 UPUP1 R2 UP2 UPUP2) 
 ((|constructor| (NIL "Lifting of a map through 2 levels of polynomials.")) (|map| ((|#6| (|Mapping| |#4| |#1|) |#3|) "\\spad{map(f, p)} lifts \\spad{f} to the domain of \\spad{p} then applies it to \\spad{p}."))) 
 NIL 
 NIL 
-(-713) 
+(-715) 
 ((|constructor| (NIL "\\spadtype{MathMLFormat} provides a coercion from \\spadtype{OutputForm} to MathML format.")) (|display| (((|Void|) (|String|)) "prints the string returned by coerce,{} adding <math ...> tags.")) (|exprex| (((|String|) (|OutputForm|)) "coverts \\spadtype{OutputForm} to \\spadtype{String} with the structure preserved with braces. Actually this is not quite accurate. The function \\spadfun{precondition} is first applied to the \\spadtype{OutputForm} expression before \\spadfun{exprex}. The raw \\spadtype{OutputForm} and the nature of the \\spadfun{precondition} function is still obscure to me at the time of this writing (2007-02-14).")) (|coerceL| (((|String|) (|OutputForm|)) "coerceS(\\spad{o}) changes \\spad{o} in the standard output format to MathML format and displays result as one long string.")) (|coerceS| (((|String|) (|OutputForm|)) "\\spad{coerceS(o)} changes \\spad{o} in the standard output format to MathML format and displays formatted result.")) (|coerce| (((|String|) (|OutputForm|)) "coerceS(\\spad{o}) changes \\spad{o} in the standard output format to MathML format."))) 
 NIL 
 NIL 
-(-714 R |Mod| -2217 -3943 |exactQuo|) 
+(-716 R |Mod| -2224 -3950 |exactQuo|) 
 ((|constructor| (NIL "\\indented{1}{These domains are used for the factorization and gcds} of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{ModularRing},{} \\spadtype{EuclideanModularRing}")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#1| |#2|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#1| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#2| $) "\\spad{modulus(x)} \\undocumented"))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-715 R |Rep|) 
+(-717 R |Rep|) 
 ((|constructor| (NIL "This package \\undocumented")) (|frobenius| (($ $) "\\spad{frobenius(x)} \\undocumented")) (|computePowers| (((|PrimitiveArray| $)) "\\spad{computePowers()} \\undocumented")) (|pow| (((|PrimitiveArray| $)) "\\spad{pow()} \\undocumented")) (|An| (((|Vector| |#1|) $) "\\spad{An(x)} \\undocumented")) (|UnVectorise| (($ (|Vector| |#1|)) "\\spad{UnVectorise(v)} \\undocumented")) (|Vectorise| (((|Vector| |#1|) $) "\\spad{Vectorise(x)} \\undocumented")) (|lift| ((|#2| $) "\\spad{lift(x)} \\undocumented")) (|reduce| (($ |#2|) "\\spad{reduce(x)} \\undocumented")) (|modulus| ((|#2|) "\\spad{modulus()} \\undocumented")) (|setPoly| ((|#2| |#2|) "\\spad{setPoly(x)} \\undocumented"))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4423 |has| |#1| (-366)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
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-(-716 IS E |ff|) 
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+((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-1088) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-1088) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-1088) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-1088) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-1088) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-1157))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-234))) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(-718 IS E |ff|) 
 ((|constructor| (NIL "This package \\undocumented")) (|construct| (($ |#1| |#2|) "\\spad{construct(i,e)} \\undocumented")) (|index| ((|#1| $) "\\spad{index(x)} \\undocumented")) (|exponent| ((|#2| $) "\\spad{exponent(x)} \\undocumented"))) 
 NIL 
 NIL 
-(-717 R M) 
+(-719 R M) 
 ((|constructor| (NIL "Algebra of ADDITIVE operators on a module.")) (|makeop| (($ |#1| (|FreeGroup| (|BasicOperator|))) "\\spad{makeop should} be local but conditional")) (|opeval| ((|#2| (|BasicOperator|) |#2|) "\\spad{opeval should} be local but conditional")) (** (($ $ (|Integer|)) "\\spad{op**n} \\undocumented") (($ (|BasicOperator|) (|Integer|)) "\\spad{op**n} \\undocumented")) (|evaluateInverse| (($ $ (|Mapping| |#2| |#2|)) "\\spad{evaluateInverse(x,f)} \\undocumented")) (|evaluate| (($ $ (|Mapping| |#2| |#2|)) "\\spad{evaluate(f, u +-> g u)} attaches the map \\spad{g} to \\spad{f}. \\spad{f} must be a basic operator \\spad{g} MUST be additive,{} \\spadignore{i.e.} \\spad{g(a + b) = g(a) + g(b)} for any \\spad{a},{} \\spad{b} in \\spad{M}. This implies that \\spad{g(n a) = n g(a)} for any \\spad{a} in \\spad{M} and integer \\spad{n > 0}.")) (|conjug| ((|#1| |#1|) "\\spad{conjug(x)}should be local but conditional")) (|adjoint| (($ $ $) "\\spad{adjoint(op1, op2)} sets the adjoint of \\spad{op1} to be op2. \\spad{op1} must be a basic operator") (($ $) "\\spad{adjoint(op)} returns the adjoint of the operator \\spad{op}."))) 
-((-4422 |has| |#1| (-173)) (-4421 |has| |#1| (-173)) (-4424 . T)) 
+((-4429 |has| |#1| (-173)) (-4428 |has| |#1| (-173)) (-4431 . T)) 
 ((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147)))) 
-(-718 R |Mod| -2217 -3943 |exactQuo|) 
+(-720 R |Mod| -2224 -3950 |exactQuo|) 
 ((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{EuclideanModularRing} ,{}\\spadtype{ModularField}")) (|inv| (($ $) "\\spad{inv(x)} \\undocumented")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} \\undocumented")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#1| |#2|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#1| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#2| $) "\\spad{modulus(x)} \\undocumented"))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-719 S R) 
+(-721 S R) 
 ((|constructor| (NIL "The category of modules over a commutative ring. \\blankline"))) 
 NIL 
 NIL 
-(-720 R) 
+(-722 R) 
 ((|constructor| (NIL "The category of modules over a commutative ring. \\blankline"))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 NIL 
-(-721 -3498) 
+(-723 -3505) 
 ((|constructor| (NIL "\\indented{1}{MoebiusTransform(\\spad{F}) is the domain of fractional linear (Moebius)} transformations over \\spad{F}.")) (|eval| (((|OnePointCompletion| |#1|) $ (|OnePointCompletion| |#1|)) "\\spad{eval(m,x)} returns \\spad{(a*x + b)/(c*x + d)} where \\spad{m = moebius(a,b,c,d)} (see \\spadfunFrom{moebius}{MoebiusTransform}).") ((|#1| $ |#1|) "\\spad{eval(m,x)} returns \\spad{(a*x + b)/(c*x + d)} where \\spad{m = moebius(a,b,c,d)} (see \\spadfunFrom{moebius}{MoebiusTransform}).")) (|recip| (($ $) "\\spad{recip(m)} = recip() * \\spad{m}") (($) "\\spad{recip()} returns \\spad{matrix [[0,1],[1,0]]} representing the map \\spad{x -> 1 / x}.")) (|scale| (($ $ |#1|) "\\spad{scale(m,h)} returns \\spad{scale(h) * m} (see \\spadfunFrom{shift}{MoebiusTransform}).") (($ |#1|) "\\spad{scale(k)} returns \\spad{matrix [[k,0],[0,1]]} representing the map \\spad{x -> k * x}.")) (|shift| (($ $ |#1|) "\\spad{shift(m,h)} returns \\spad{shift(h) * m} (see \\spadfunFrom{shift}{MoebiusTransform}).") (($ |#1|) "\\spad{shift(k)} returns \\spad{matrix [[1,k],[0,1]]} representing the map \\spad{x -> x + k}.")) (|moebius| (($ |#1| |#1| |#1| |#1|) "\\spad{moebius(a,b,c,d)} returns \\spad{matrix [[a,b],[c,d]]}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-722 S) 
+(-724 S) 
 ((|constructor| (NIL "Monad is the class of all multiplicative monads,{} \\spadignore{i.e.} sets with a binary operation.")) (** (($ $ (|PositiveInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|PositiveInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,1) := a}.")) (|rightPower| (($ $ (|PositiveInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,1) := a}.")) (* (($ $ $) "\\spad{a*b} is the product of \\spad{a} and \\spad{b} in a set with a binary operation."))) 
 NIL 
 NIL 
-(-723) 
+(-725) 
 ((|constructor| (NIL "Monad is the class of all multiplicative monads,{} \\spadignore{i.e.} sets with a binary operation.")) (** (($ $ (|PositiveInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|PositiveInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,1) := a}.")) (|rightPower| (($ $ (|PositiveInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,1) := a}.")) (* (($ $ $) "\\spad{a*b} is the product of \\spad{a} and \\spad{b} in a set with a binary operation."))) 
 NIL 
 NIL 
-(-724 S) 
+(-726 S) 
 ((|constructor| (NIL "\\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \\indented{1}{\\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \\indented{3}{leftIdentity(\"*\":(\\%,{}\\%)\\spad{->}\\%,{}1)\\space{3}\\tab{30} 1*x=x} \\indented{3}{rightIdentity(\"*\":(\\%,{}\\%)\\spad{->}\\%,{}1)\\space{2}\\tab{30} x*1=x} Common Additional Axioms \\indented{3}{unitsKnown---if \"recip\" says \"failed\",{} that PROVES input wasn\\spad{'t} a unit}")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|NonNegativeInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,0) := 1}.")) (|rightPower| (($ $ (|NonNegativeInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,0) := 1}.")) (|one?| (((|Boolean|) $) "\\spad{one?(a)} tests whether \\spad{a} is the unit 1.")) ((|One|) (($) "1 returns the unit element,{} denoted by 1."))) 
 NIL 
 NIL 
-(-725) 
+(-727) 
 ((|constructor| (NIL "\\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \\indented{1}{\\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \\indented{3}{leftIdentity(\"*\":(\\%,{}\\%)\\spad{->}\\%,{}1)\\space{3}\\tab{30} 1*x=x} \\indented{3}{rightIdentity(\"*\":(\\%,{}\\%)\\spad{->}\\%,{}1)\\space{2}\\tab{30} x*1=x} Common Additional Axioms \\indented{3}{unitsKnown---if \"recip\" says \"failed\",{} that PROVES input wasn\\spad{'t} a unit}")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn\\spad{'t} exist or cannot be determined (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|NonNegativeInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,0) := 1}.")) (|rightPower| (($ $ (|NonNegativeInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,0) := 1}.")) (|one?| (((|Boolean|) $) "\\spad{one?(a)} tests whether \\spad{a} is the unit 1.")) ((|One|) (($) "1 returns the unit element,{} denoted by 1."))) 
 NIL 
 NIL 
-(-726 S R UP) 
+(-728 S R UP) 
 ((|constructor| (NIL "A \\spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.")) (|derivationCoordinates| (((|Matrix| |#2|) (|Vector| $) (|Mapping| |#2| |#2|)) "\\spad{derivationCoordinates(b, ')} returns \\spad{M} such that \\spad{b' = M b}.")) (|lift| ((|#3| $) "\\spad{lift(z)} returns a minimal degree univariate polynomial up such that \\spad{z=reduce up}.")) (|convert| (($ |#3|) "\\spad{convert(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|reduce| (((|Union| $ "failed") (|Fraction| |#3|)) "\\spad{reduce(frac)} converts the fraction \\spad{frac} to an algebra element.") (($ |#3|) "\\spad{reduce(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|definingPolynomial| ((|#3|) "\\spad{definingPolynomial()} returns the minimal polynomial which \\spad{generator()} satisfies.")) (|generator| (($) "\\spad{generator()} returns the generator for this domain."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-353))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-371)))) 
-(-727 R UP) 
+((|HasCategory| |#2| (QUOTE (-354))) (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-372)))) 
+(-729 R UP) 
 ((|constructor| (NIL "A \\spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.")) (|derivationCoordinates| (((|Matrix| |#1|) (|Vector| $) (|Mapping| |#1| |#1|)) "\\spad{derivationCoordinates(b, ')} returns \\spad{M} such that \\spad{b' = M b}.")) (|lift| ((|#2| $) "\\spad{lift(z)} returns a minimal degree univariate polynomial up such that \\spad{z=reduce up}.")) (|convert| (($ |#2|) "\\spad{convert(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|reduce| (((|Union| $ "failed") (|Fraction| |#2|)) "\\spad{reduce(frac)} converts the fraction \\spad{frac} to an algebra element.") (($ |#2|) "\\spad{reduce(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|definingPolynomial| ((|#2|) "\\spad{definingPolynomial()} returns the minimal polynomial which \\spad{generator()} satisfies.")) (|generator| (($) "\\spad{generator()} returns the generator for this domain."))) 
-((-4420 |has| |#1| (-366)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 |has| |#1| (-367)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-728 S) 
+(-730 S) 
 ((|constructor| (NIL "The class of multiplicative monoids,{} \\spadignore{i.e.} semigroups with a multiplicative identity element. \\blankline")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} tries to compute the multiplicative inverse for \\spad{x} or \"failed\" if it cannot find the inverse (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (|one?| (((|Boolean|) $) "\\spad{one?(x)} tests if \\spad{x} is equal to 1.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) ((|One|) (($) "1 is the multiplicative identity."))) 
 NIL 
 NIL 
-(-729) 
+(-731) 
 ((|constructor| (NIL "The class of multiplicative monoids,{} \\spadignore{i.e.} semigroups with a multiplicative identity element. \\blankline")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} tries to compute the multiplicative inverse for \\spad{x} or \"failed\" if it cannot find the inverse (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (|one?| (((|Boolean|) $) "\\spad{one?(x)} tests if \\spad{x} is equal to 1.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) ((|One|) (($) "1 is the multiplicative identity."))) 
 NIL 
 NIL 
-(-730 -3498 UP) 
+(-732 -3505 UP) 
 ((|constructor| (NIL "Tools for handling monomial extensions.")) (|decompose| (((|Record| (|:| |poly| |#2|) (|:| |normal| (|Fraction| |#2|)) (|:| |special| (|Fraction| |#2|))) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{decompose(f, D)} returns \\spad{[p,n,s]} such that \\spad{f = p+n+s},{} all the squarefree factors of \\spad{denom(n)} are normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{denom(s)} is special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and \\spad{n} and \\spad{s} are proper fractions (no pole at infinity). \\spad{D} is the derivation to use.")) (|normalDenom| ((|#2| (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{normalDenom(f, D)} returns the product of all the normal factors of \\spad{denom(f)}. \\spad{D} is the derivation to use.")) (|splitSquarefree| (((|Record| (|:| |normal| (|Factored| |#2|)) (|:| |special| (|Factored| |#2|))) |#2| (|Mapping| |#2| |#2|)) "\\spad{splitSquarefree(p, D)} returns \\spad{[n_1 n_2\\^2 ... n_m\\^m, s_1 s_2\\^2 ... s_q\\^q]} such that \\spad{p = n_1 n_2\\^2 ... n_m\\^m s_1 s_2\\^2 ... s_q\\^q},{} each \\spad{n_i} is normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D} and each \\spad{s_i} is special \\spad{w}.\\spad{r}.\\spad{t} \\spad{D}. \\spad{D} is the derivation to use.")) (|split| (((|Record| (|:| |normal| |#2|) (|:| |special| |#2|)) |#2| (|Mapping| |#2| |#2|)) "\\spad{split(p, D)} returns \\spad{[n,s]} such that \\spad{p = n s},{} all the squarefree factors of \\spad{n} are normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and \\spad{s} is special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D}. \\spad{D} is the derivation to use."))) 
 NIL 
 NIL 
-(-731 |VarSet| E1 E2 R S PR PS) 
+(-733 |VarSet| E1 E2 R S PR PS) 
 ((|constructor| (NIL "\\indented{1}{Utilities for MPolyCat} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 28 March 1990 (\\spad{PG})")) (|reshape| ((|#7| (|List| |#5|) |#6|) "\\spad{reshape(l,p)} \\undocumented")) (|map| ((|#7| (|Mapping| |#5| |#4|) |#6|) "\\spad{map(f,p)} \\undocumented"))) 
 NIL 
 NIL 
-(-732 |Vars1| |Vars2| E1 E2 R PR1 PR2) 
+(-734 |Vars1| |Vars2| E1 E2 R PR1 PR2) 
 ((|constructor| (NIL "This package \\undocumented")) (|map| ((|#7| (|Mapping| |#2| |#1|) |#6|) "\\spad{map(f,x)} \\undocumented"))) 
 NIL 
 NIL 
-(-733 E OV R PPR) 
+(-735 E OV R PPR) 
 ((|constructor| (NIL "\\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are polynomials over some ring \\spad{R} over which we can factor. It is used internally by packages such as the solve package which need to work with polynomials in a specific set of variables with coefficients which are polynomials in all the other variables.")) (|factor| (((|Factored| |#4|) |#4|) "\\spad{factor(p)} factors a polynomial with polynomial coefficients.")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} makes an element from symbol \\spad{s} or fails.")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) 
 NIL 
 NIL 
-(-734 |vl| R) 
+(-736 |vl| R) 
 ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are from a user specified list of symbols. The ordering is specified by the position of the variable in the list. The coefficient ring may be non commutative,{} but the variables are assumed to commute."))) 
-(((-4429 "*") |has| |#2| (-173)) (-4420 |has| |#2| (-561)) (-4425 |has| |#2| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#2| (QUOTE (-914))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-914)))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-561)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-867 |#1|) (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-366))) (|HasAttribute| |#2| (QUOTE -4425)) (|HasCategory| |#2| (QUOTE (-456))) (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
-(-735 E OV R PRF) 
+(((-4436 "*") |has| |#2| (-173)) (-4427 |has| |#2| (-562)) (-4432 |has| |#2| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (-3969 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-3969 (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-3969 (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-173))) (-3969 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-869 |#1|) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (QUOTE (-367))) (|HasAttribute| |#2| (QUOTE -4432)) (|HasCategory| |#2| (QUOTE (-457))) (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
+(-737 E OV R PRF) 
 ((|constructor| (NIL "\\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are rational functions over some ring \\spad{R} over which we can factor. It is used internally by packages such as primary decomposition which need to work with polynomials with rational function coefficients,{} \\spadignore{i.e.} themselves fractions of polynomials.")) (|factor| (((|Factored| |#4|) |#4|) "\\spad{factor(prf)} factors a polynomial with rational function coefficients.")) (|pushuconst| ((|#4| (|Fraction| (|Polynomial| |#3|)) |#2|) "\\spad{pushuconst(r,var)} takes a rational function and raises all occurances of the variable \\spad{var} to the polynomial level.")) (|pushucoef| ((|#4| (|SparseUnivariatePolynomial| (|Polynomial| |#3|)) |#2|) "\\spad{pushucoef(upoly,var)} converts the anonymous univariate polynomial \\spad{upoly} to a polynomial in \\spad{var} over rational functions.")) (|pushup| ((|#4| |#4| |#2|) "\\spad{pushup(prf,var)} raises all occurences of the variable \\spad{var} in the coefficients of the polynomial \\spad{prf} back to the polynomial level.")) (|pushdterm| ((|#4| (|SparseUnivariatePolynomial| |#4|) |#2|) "\\spad{pushdterm(monom,var)} pushes all top level occurences of the variable \\spad{var} into the coefficient domain for the monomial \\spad{monom}.")) (|pushdown| ((|#4| |#4| |#2|) "\\spad{pushdown(prf,var)} pushes all top level occurences of the variable \\spad{var} into the coefficient domain for the polynomial \\spad{prf}.")) (|totalfract| (((|Record| (|:| |sup| (|Polynomial| |#3|)) (|:| |inf| (|Polynomial| |#3|))) |#4|) "\\spad{totalfract(prf)} takes a polynomial whose coefficients are themselves fractions of polynomials and returns a record containing the numerator and denominator resulting from putting \\spad{prf} over a common denominator.")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) 
 NIL 
 NIL 
-(-736 E OV R P) 
+(-738 E OV R P) 
 ((|constructor| (NIL "\\indented{1}{MRationalFactorize contains the factor function for multivariate} polynomials over the quotient field of a ring \\spad{R} such that the package MultivariateFactorize can factor multivariate polynomials over \\spad{R}.")) (|factor| (((|Factored| |#4|) |#4|) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} with coefficients which are fractions of elements of \\spad{R}."))) 
 NIL 
 NIL 
-(-737 R S M) 
+(-739 R S M) 
 ((|constructor| (NIL "MonoidRingFunctions2 implements functions between two monoid rings defined with the same monoid over different rings.")) (|map| (((|MonoidRing| |#2| |#3|) (|Mapping| |#2| |#1|) (|MonoidRing| |#1| |#3|)) "\\spad{map(f,u)} maps \\spad{f} onto the coefficients \\spad{f} the element \\spad{u} of the monoid ring to create an element of a monoid ring with the same monoid \\spad{b}."))) 
 NIL 
 NIL 
-(-738 R M) 
+(-740 R M) 
 ((|constructor| (NIL "\\spadtype{MonoidRing}(\\spad{R},{}\\spad{M}),{} implements the algebra of all maps from the monoid \\spad{M} to the commutative ring \\spad{R} with finite support. Multiplication of two maps \\spad{f} and \\spad{g} is defined to map an element \\spad{c} of \\spad{M} to the (convolution) sum over {\\em f(a)g(b)} such that {\\em ab = c}. Thus \\spad{M} can be identified with a canonical basis and the maps can also be considered as formal linear combinations of the elements in \\spad{M}. Scalar multiples of a basis element are called monomials. A prominent example is the class of polynomials where the monoid is a direct product of the natural numbers with pointwise addition. When \\spad{M} is \\spadtype{FreeMonoid Symbol},{} one gets polynomials in infinitely many non-commuting variables. Another application area is representation theory of finite groups \\spad{G},{} where modules over \\spadtype{MonoidRing}(\\spad{R},{}\\spad{G}) are studied.")) (|reductum| (($ $) "\\spad{reductum(f)} is \\spad{f} minus its leading monomial.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(f)} gives the coefficient of \\spad{f},{} whose corresponding monoid element is the greatest among all those with non-zero coefficients.")) (|leadingMonomial| ((|#2| $) "\\spad{leadingMonomial(f)} gives the monomial of \\spad{f} whose corresponding monoid element is the greatest among all those with non-zero coefficients.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(f)} is the number of non-zero coefficients with respect to the canonical basis.")) (|monomials| (((|List| $) $) "\\spad{monomials(f)} gives the list of all monomials whose sum is \\spad{f}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(f)} lists all non-zero coefficients.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(f)} tests if \\spad{f} is a single monomial.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|terms| (((|List| (|Record| (|:| |coef| |#1|) (|:| |monom| |#2|))) $) "\\spad{terms(f)} gives the list of non-zero coefficients combined with their corresponding basis element as records. This is the internal representation.")) (|coerce| (($ (|List| (|Record| (|:| |coef| |#1|) (|:| |monom| |#2|)))) "\\spad{coerce(lt)} converts a list of terms and coefficients to a member of the domain.")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(f,m)} extracts the coefficient of \\spad{m} in \\spad{f} with respect to the canonical basis \\spad{M}.")) (|monomial| (($ |#1| |#2|) "\\spad{monomial(r,m)} creates a scalar multiple of the basis element \\spad{m}."))) 
-((-4422 |has| |#1| (-173)) (-4421 |has| |#1| (-173)) (-4424 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#2| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-853)))) 
-(-739 S) 
+((-4429 |has| |#1| (-173)) (-4428 |has| |#1| (-173)) (-4431 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#2| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-855)))) 
+(-741 S) 
 ((|constructor| (NIL "A multiset is a set with multiplicities.")) (|remove!| (($ (|Mapping| (|Boolean|) |#1|) $ (|Integer|)) "\\spad{remove!(p,ms,number)} removes destructively at most \\spad{number} copies of elements \\spad{x} such that \\spad{p(x)} is \\spadfun{\\spad{true}} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.") (($ |#1| $ (|Integer|)) "\\spad{remove!(x,ms,number)} removes destructively at most \\spad{number} copies of element \\spad{x} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.")) (|remove| (($ (|Mapping| (|Boolean|) |#1|) $ (|Integer|)) "\\spad{remove(p,ms,number)} removes at most \\spad{number} copies of elements \\spad{x} such that \\spad{p(x)} is \\spadfun{\\spad{true}} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.") (($ |#1| $ (|Integer|)) "\\spad{remove(x,ms,number)} removes at most \\spad{number} copies of element \\spad{x} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.")) (|members| (((|List| |#1|) $) "\\spad{members(ms)} returns a list of the elements of \\spad{ms} {\\em without} their multiplicity. See also \\spadfun{parts}.")) (|multiset| (($ (|List| |#1|)) "\\spad{multiset(ls)} creates a multiset with elements from \\spad{ls}.") (($ |#1|) "\\spad{multiset(s)} creates a multiset with singleton \\spad{s}.") (($) "\\spad{multiset()}\\$\\spad{D} creates an empty multiset of domain \\spad{D}."))) 
-((-4427 . T) (-4417 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-740 S) 
+((-4434 . T) (-4424 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-742 S) 
 ((|constructor| (NIL "A multi-set aggregate is a set which keeps track of the multiplicity of its elements."))) 
-((-4417 . T) (-4428 . T)) 
+((-4424 . T) (-4435 . T)) 
 NIL 
-(-741) 
+(-743) 
 ((|constructor| (NIL "\\spadtype{MoreSystemCommands} implements an interface with the system command facility. These are the commands that are issued from source files or the system interpreter and they start with a close parenthesis,{} \\spadignore{e.g.} \\spadsyscom{what} commands.")) (|systemCommand| (((|Void|) (|String|)) "\\spad{systemCommand(cmd)} takes the string \\spadvar{\\spad{cmd}} and passes it to the runtime environment for execution as a system command. Although various things may be printed,{} no usable value is returned."))) 
 NIL 
 NIL 
-(-742 S) 
+(-744 S) 
 ((|constructor| (NIL "This package exports tools for merging lists")) (|mergeDifference| (((|List| |#1|) (|List| |#1|) (|List| |#1|)) "\\spad{mergeDifference(l1,l2)} returns a list of elements in \\spad{l1} not present in \\spad{l2}. Assumes lists are ordered and all \\spad{x} in \\spad{l2} are also in \\spad{l1}."))) 
 NIL 
 NIL 
-(-743 |Coef| |Var|) 
+(-745 |Coef| |Var|) 
 ((|constructor| (NIL "\\spadtype{MultivariateTaylorSeriesCategory} is the most general multivariate Taylor series category.")) (|integrate| (($ $ |#2|) "\\spad{integrate(f,x)} returns the anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{x} with constant coefficient 1. We may integrate a series when we can divide coefficients by integers.")) (|polynomial| (((|Polynomial| |#1|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| (((|NonNegativeInteger|) $ |#2| (|NonNegativeInteger|)) "\\spad{order(f,x,n)} returns \\spad{min(n,order(f,x))}.") (((|NonNegativeInteger|) $ |#2|) "\\spad{order(f,x)} returns the order of \\spad{f} viewed as a series in \\spad{x} may result in an infinite loop if \\spad{f} has no non-zero terms.")) (|monomial| (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{monomial(a,[x1,x2,...,xk],[n1,n2,...,nk])} returns \\spad{a * x1^n1 * ... * xk^nk}.") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{monomial(a,x,n)} returns \\spad{a*x^n}.")) (|extend| (($ $ (|NonNegativeInteger|)) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree \\spad{<= n} to be computed.")) (|coefficient| (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{coefficient(f,[x1,x2,...,xk],[n1,n2,...,nk])} returns the coefficient of \\spad{x1^n1 * ... * xk^nk} in \\spad{f}.") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{coefficient(f,x,n)} returns the coefficient of \\spad{x^n} in \\spad{f}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4422 . T) (-4421 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
-(-744 OV E R P) 
+(-746 OV E R P) 
 ((|constructor| (NIL "\\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \\spadtype{Integer} or \\spadtype{Fraction Integer}.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} over its coefficient domain where \\spad{p} is represented as a univariate polynomial with multivariate coefficients") (((|Factored| |#4|) |#4|) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} over its coefficient domain"))) 
 NIL 
 NIL 
-(-745 E OV R P) 
+(-747 E OV R P) 
 ((|constructor| (NIL "Author : \\spad{P}.Gianni This package provides the functions for the computation of the square free decomposition of a multivariate polynomial. It uses the package GenExEuclid for the resolution of the equation \\spad{Af + Bg = h} and its generalization to \\spad{n} polynomials over an integral domain and the package \\spad{MultivariateLifting} for the \"multivariate\" lifting.")) (|normDeriv2| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#3|) (|Integer|)) "\\spad{normDeriv2 should} be local")) (|myDegree| (((|List| (|NonNegativeInteger|)) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|NonNegativeInteger|)) "\\spad{myDegree should} be local")) (|lift| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#3|) |#4| (|List| |#2|) (|List| (|NonNegativeInteger|)) (|List| |#3|)) "\\spad{lift should} be local")) (|check| (((|Boolean|) (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|)))) (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|))))) "\\spad{check should} be local")) (|coefChoose| ((|#4| (|Integer|) (|Factored| |#4|)) "\\spad{coefChoose should} be local")) (|intChoose| (((|Record| (|:| |upol| (|SparseUnivariatePolynomial| |#3|)) (|:| |Lval| (|List| |#3|)) (|:| |Lfact| (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|))))) (|:| |ctpol| |#3|)) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|List| |#3|))) "\\spad{intChoose should} be local")) (|nsqfree| (((|Record| (|:| |unitPart| |#4|) (|:| |suPart| (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#4|)) (|:| |exponent| (|Integer|)))))) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|List| |#3|))) "\\spad{nsqfree should} be local")) (|consnewpol| (((|Record| (|:| |pol| (|SparseUnivariatePolynomial| |#4|)) (|:| |polval| (|SparseUnivariatePolynomial| |#3|))) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#3|) (|Integer|)) "\\spad{consnewpol should} be local")) (|univcase| (((|Factored| |#4|) |#4| |#2|) "\\spad{univcase should} be local")) (|compdegd| (((|Integer|) (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|))))) "\\spad{compdegd should} be local")) (|squareFreePrim| (((|Factored| |#4|) |#4|) "\\spad{squareFreePrim(p)} compute the square free decomposition of a primitive multivariate polynomial \\spad{p}.")) (|squareFree| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{squareFree(p)} computes the square free decomposition of a multivariate polynomial \\spad{p} presented as a univariate polynomial with multivariate coefficients.") (((|Factored| |#4|) |#4|) "\\spad{squareFree(p)} computes the square free decomposition of a multivariate polynomial \\spad{p}."))) 
 NIL 
 NIL 
-(-746 S R) 
+(-748 S R) 
 ((|constructor| (NIL "NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \\indented{3}{\\spad{r*}(a*b) = (r*a)\\spad{*b} = a*(\\spad{r*b})}")) (|plenaryPower| (($ $ (|PositiveInteger|)) "\\spad{plenaryPower(a,n)} is recursively defined to be \\spad{plenaryPower(a,n-1)*plenaryPower(a,n-1)} for \\spad{n>1} and \\spad{a} for \\spad{n=1}."))) 
 NIL 
 NIL 
-(-747 R) 
+(-749 R) 
 ((|constructor| (NIL "NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \\indented{3}{\\spad{r*}(a*b) = (r*a)\\spad{*b} = a*(\\spad{r*b})}")) (|plenaryPower| (($ $ (|PositiveInteger|)) "\\spad{plenaryPower(a,n)} is recursively defined to be \\spad{plenaryPower(a,n-1)*plenaryPower(a,n-1)} for \\spad{n>1} and \\spad{a} for \\spad{n=1}."))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 NIL 
-(-748) 
+(-750) 
 ((|constructor| (NIL "This package uses the NAG Library to compute the zeros of a polynomial with real or complex coefficients. See \\downlink{Manual Page}{manpageXXc02}.")) (|c02agf| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Boolean|) (|Integer|)) "\\spad{c02agf(a,n,scale,ifail)} finds all the roots of a real polynomial equation,{} using a variant of Laguerre\\spad{'s} Method. See \\downlink{Manual Page}{manpageXXc02agf}.")) (|c02aff| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Boolean|) (|Integer|)) "\\spad{c02aff(a,n,scale,ifail)} finds all the roots of a complex polynomial equation,{} using a variant of Laguerre\\spad{'s} Method. See \\downlink{Manual Page}{manpageXXc02aff}."))) 
 NIL 
 NIL 
-(-749) 
+(-751) 
 ((|constructor| (NIL "This package uses the NAG Library to calculate real zeros of continuous real functions of one or more variables. (Complex equations must be expressed in terms of the equivalent larger system of real equations.) See \\downlink{Manual Page}{manpageXXc05}.")) (|c05pbf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp35| FCN)))) "\\spad{c05pbf(n,ldfjac,lwa,x,xtol,ifail,fcn)} is an easy-to-use routine to find a solution of a system of nonlinear equations by a modification of the Powell hybrid method. The user must provide the Jacobian. See \\downlink{Manual Page}{manpageXXc05pbf}.")) (|c05nbf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp6| FCN)))) "\\spad{c05nbf(n,lwa,x,xtol,ifail,fcn)} is an easy-to-use routine to find a solution of a system of nonlinear equations by a modification of the Powell hybrid method. See \\downlink{Manual Page}{manpageXXc05nbf}.")) (|c05adf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| F)))) "\\spad{c05adf(a,b,eps,eta,ifail,f)} locates a zero of a continuous function in a given interval by a combination of the methods of linear interpolation,{} extrapolation and bisection. See \\downlink{Manual Page}{manpageXXc05adf}."))) 
 NIL 
 NIL 
-(-750) 
+(-752) 
 ((|constructor| (NIL "This package uses the NAG Library to calculate the discrete Fourier transform of a sequence of real or complex data values,{} and applies it to calculate convolutions and correlations. See \\downlink{Manual Page}{manpageXXc06}.")) (|c06gsf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06gsf(m,n,x,ifail)} takes \\spad{m} Hermitian sequences,{} each containing \\spad{n} data values,{} and forms the real and imaginary parts of the \\spad{m} corresponding complex sequences. See \\downlink{Manual Page}{manpageXXc06gsf}.")) (|c06gqf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06gqf(m,n,x,ifail)} forms the complex conjugates,{} each containing \\spad{n} data values. See \\downlink{Manual Page}{manpageXXc06gqf}.")) (|c06gcf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06gcf(n,y,ifail)} forms the complex conjugate of a sequence of \\spad{n} data values. See \\downlink{Manual Page}{manpageXXc06gcf}.")) (|c06gbf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06gbf(n,x,ifail)} forms the complex conjugate of \\spad{n} data values. See \\downlink{Manual Page}{manpageXXc06gbf}.")) (|c06fuf| (((|Result|) (|Integer|) (|Integer|) (|String|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06fuf(m,n,init,x,y,trigm,trign,ifail)} computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values. This routine is designed to be particularly efficient on vector processors. See \\downlink{Manual Page}{manpageXXc06fuf}.")) (|c06frf| (((|Result|) (|Integer|) (|Integer|) (|String|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06frf(m,n,init,x,y,trig,ifail)} computes the discrete Fourier transforms of \\spad{m} sequences,{} each containing \\spad{n} complex data values. This routine is designed to be particularly efficient on vector processors. See \\downlink{Manual Page}{manpageXXc06frf}.")) (|c06fqf| (((|Result|) (|Integer|) (|Integer|) (|String|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06fqf(m,n,init,x,trig,ifail)} computes the discrete Fourier transforms of \\spad{m} Hermitian sequences,{} each containing \\spad{n} complex data values. This routine is designed to be particularly efficient on vector processors. See \\downlink{Manual Page}{manpageXXc06fqf}.")) (|c06fpf| (((|Result|) (|Integer|) (|Integer|) (|String|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06fpf(m,n,init,x,trig,ifail)} computes the discrete Fourier transforms of \\spad{m} sequences,{} each containing \\spad{n} real data values. This routine is designed to be particularly efficient on vector processors. See \\downlink{Manual Page}{manpageXXc06fpf}.")) (|c06ekf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06ekf(job,n,x,y,ifail)} calculates the circular convolution of two real vectors of period \\spad{n}. No extra workspace is required. See \\downlink{Manual Page}{manpageXXc06ekf}.")) (|c06ecf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06ecf(n,x,y,ifail)} calculates the discrete Fourier transform of a sequence of \\spad{n} complex data values. (No extra workspace required.) See \\downlink{Manual Page}{manpageXXc06ecf}.")) (|c06ebf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06ebf(n,x,ifail)} calculates the discrete Fourier transform of a Hermitian sequence of \\spad{n} complex data values. (No extra workspace required.) See \\downlink{Manual Page}{manpageXXc06ebf}.")) (|c06eaf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{c06eaf(n,x,ifail)} calculates the discrete Fourier transform of a sequence of \\spad{n} real data values. (No extra workspace required.) See \\downlink{Manual Page}{manpageXXc06eaf}."))) 
 NIL 
 NIL 
-(-751) 
+(-753) 
 ((|constructor| (NIL "This package uses the NAG Library to calculate the numerical value of definite integrals in one or more dimensions and to evaluate weights and abscissae of integration rules. See \\downlink{Manual Page}{manpageXXd01}.")) (|d01gbf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp4| FUNCTN)))) "\\spad{d01gbf(ndim,a,b,maxcls,eps,lenwrk,mincls,wrkstr,ifail,functn)} returns an approximation to the integral of a function over a hyper-rectangular region,{} using a Monte Carlo method. An approximate relative error estimate is also returned. This routine is suitable for low accuracy work. See \\downlink{Manual Page}{manpageXXd01gbf}.")) (|d01gaf| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|)) "\\spad{d01gaf(x,y,n,ifail)} integrates a function which is specified numerically at four or more points,{} over the whole of its specified range,{} using third-order finite-difference formulae with error estimates,{} according to a method due to Gill and Miller. See \\downlink{Manual Page}{manpageXXd01gaf}.")) (|d01fcf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp4| FUNCTN)))) "\\spad{d01fcf(ndim,a,b,maxpts,eps,lenwrk,minpts,ifail,functn)} attempts to evaluate a multi-dimensional integral (up to 15 dimensions),{} with constant and finite limits,{} to a specified relative accuracy,{} using an adaptive subdivision strategy. See \\downlink{Manual Page}{manpageXXd01fcf}.")) (|d01bbf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{d01bbf(a,b,itype,n,gtype,ifail)} returns the weight appropriate to a Gaussian quadrature. The formulae provided are Gauss-Legendre,{} Gauss-Rational,{} Gauss- Laguerre and Gauss-Hermite. See \\downlink{Manual Page}{manpageXXd01bbf}.")) (|d01asf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| G)))) "\\spad{d01asf(a,omega,key,epsabs,limlst,lw,liw,ifail,g)} calculates an approximation to the sine or the cosine transform of a function \\spad{g} over [a,{}infty): See \\downlink{Manual Page}{manpageXXd01asf}.")) (|d01aqf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| G)))) "\\spad{d01aqf(a,b,c,epsabs,epsrel,lw,liw,ifail,g)} calculates an approximation to the Hilbert transform of a function \\spad{g}(\\spad{x}) over [a,{}\\spad{b}]: See \\downlink{Manual Page}{manpageXXd01aqf}.")) (|d01apf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| G)))) "\\spad{d01apf(a,b,alfa,beta,key,epsabs,epsrel,lw,liw,ifail,g)} is an adaptive integrator which calculates an approximation to the integral of a function \\spad{g}(\\spad{x})\\spad{w}(\\spad{x}) over a finite interval [a,{}\\spad{b}]: See \\downlink{Manual Page}{manpageXXd01apf}.")) (|d01anf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| G)))) "\\spad{d01anf(a,b,omega,key,epsabs,epsrel,lw,liw,ifail,g)} calculates an approximation to the sine or the cosine transform of a function \\spad{g} over [a,{}\\spad{b}]: See \\downlink{Manual Page}{manpageXXd01anf}.")) (|d01amf| (((|Result|) (|DoubleFloat|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| F)))) "\\spad{d01amf(bound,inf,epsabs,epsrel,lw,liw,ifail,f)} calculates an approximation to the integral of a function \\spad{f}(\\spad{x}) over an infinite or semi-infinite interval [a,{}\\spad{b}]: See \\downlink{Manual Page}{manpageXXd01amf}.")) (|d01alf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| F)))) "\\spad{d01alf(a,b,npts,points,epsabs,epsrel,lw,liw,ifail,f)} is a general purpose integrator which calculates an approximation to the integral of a function \\spad{f}(\\spad{x}) over a finite interval [a,{}\\spad{b}]: See \\downlink{Manual Page}{manpageXXd01alf}.")) (|d01akf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| F)))) "\\spad{d01akf(a,b,epsabs,epsrel,lw,liw,ifail,f)} is an adaptive integrator,{} especially suited to oscillating,{} non-singular integrands,{} which calculates an approximation to the integral of a function \\spad{f}(\\spad{x}) over a finite interval [a,{}\\spad{b}]: See \\downlink{Manual Page}{manpageXXd01akf}.")) (|d01ajf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp1| F)))) "\\spad{d01ajf(a,b,epsabs,epsrel,lw,liw,ifail,f)} is a general-purpose integrator which calculates an approximation to the integral of a function \\spad{f}(\\spad{x}) over a finite interval [a,{}\\spad{b}]: See \\downlink{Manual Page}{manpageXXd01ajf}."))) 
 NIL 
 NIL 
-(-752) 
+(-754) 
 ((|constructor| (NIL "This package uses the NAG Library to calculate the numerical solution of ordinary differential equations. There are two main types of problem,{} those in which all boundary conditions are specified at one point (initial-value problems),{} and those in which the boundary conditions are distributed between two or more points (boundary- value problems and eigenvalue problems). Routines are available for initial-value problems,{} two-point boundary-value problems and Sturm-Liouville eigenvalue problems. See \\downlink{Manual Page}{manpageXXd02}.")) (|d02raf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp41| FCN JACOBF JACEPS))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp42| G JACOBG JACGEP)))) "\\spad{d02raf(n,mnp,numbeg,nummix,tol,init,iy,ijac,lwork,liwork,np,x,y,deleps,ifail,fcn,g)} solves the two-point boundary-value problem with general boundary conditions for a system of ordinary differential equations,{} using a deferred correction technique and Newton iteration. See \\downlink{Manual Page}{manpageXXd02raf}.")) (|d02kef| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp10| COEFFN))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp80| BDYVAL))) (|FileName|) (|FileName|)) "\\spad{d02kef(xpoint,m,k,tol,maxfun,match,elam,delam,hmax,maxit,ifail,coeffn,bdyval,monit,report)} finds a specified eigenvalue of a regular singular second- order Sturm-Liouville system on a finite or infinite range,{} using a Pruefer transformation and a shooting method. It also reports values of the eigenfunction and its derivatives. Provision is made for discontinuities in the coefficient functions or their derivatives. See \\downlink{Manual Page}{manpageXXd02kef}. Files \\spad{monit} and \\spad{report} will be used to define the subroutines for the MONIT and REPORT arguments. See \\downlink{Manual Page}{manpageXXd02gbf}.") (((|Result|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp10| COEFFN))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp80| BDYVAL)))) "\\spad{d02kef(xpoint,m,k,tol,maxfun,match,elam,delam,hmax,maxit,ifail,coeffn,bdyval)} finds a specified eigenvalue of a regular singular second- order Sturm-Liouville system on a finite or infinite range,{} using a Pruefer transformation and a shooting method. It also reports values of the eigenfunction and its derivatives. Provision is made for discontinuities in the coefficient functions or their derivatives. See \\downlink{Manual Page}{manpageXXd02kef}. ASP domains Asp12 and Asp33 are used to supply default subroutines for the MONIT and REPORT arguments via their \\axiomOp{outputAsFortran} operation.")) (|d02gbf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp77| FCNF))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp78| FCNG)))) "\\spad{d02gbf(a,b,n,tol,mnp,lw,liw,c,d,gam,x,np,ifail,fcnf,fcng)} solves a general linear two-point boundary value problem for a system of ordinary differential equations using a deferred correction technique. See \\downlink{Manual Page}{manpageXXd02gbf}.")) (|d02gaf| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp7| FCN)))) "\\spad{d02gaf(u,v,n,a,b,tol,mnp,lw,liw,x,np,ifail,fcn)} solves the two-point boundary-value problem with assigned boundary values for a system of ordinary differential equations,{} using a deferred correction technique and a Newton iteration. See \\downlink{Manual Page}{manpageXXd02gaf}.")) (|d02ejf| (((|Result|) (|DoubleFloat|) (|Integer|) (|Integer|) (|String|) (|Integer|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp9| G))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp7| FCN))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp31| PEDERV))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp8| OUTPUT)))) "\\spad{d02ejf(xend,m,n,relabs,iw,x,y,tol,ifail,g,fcn,pederv,output)} integrates a stiff system of first-order ordinary differential equations over an interval with suitable initial conditions,{} using a variable-order,{} variable-step method implementing the Backward Differentiation Formulae (\\spad{BDF}),{} until a user-specified function,{} if supplied,{} of the solution is zero,{} and returns the solution at points specified by the user,{} if desired. See \\downlink{Manual Page}{manpageXXd02ejf}.")) (|d02cjf| (((|Result|) (|DoubleFloat|) (|Integer|) (|Integer|) (|DoubleFloat|) (|String|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp9| G))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp7| FCN))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp8| OUTPUT)))) "\\spad{d02cjf(xend,m,n,tol,relabs,x,y,ifail,g,fcn,output)} integrates a system of first-order ordinary differential equations over a range with suitable initial conditions,{} using a variable-order,{} variable-step Adams method until a user-specified function,{} if supplied,{} of the solution is zero,{} and returns the solution at points specified by the user,{} if desired. See \\downlink{Manual Page}{manpageXXd02cjf}.")) (|d02bhf| (((|Result|) (|DoubleFloat|) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp9| G))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp7| FCN)))) "\\spad{d02bhf(xend,n,irelab,hmax,x,y,tol,ifail,g,fcn)} integrates a system of first-order ordinary differential equations over an interval with suitable initial conditions,{} using a Runge-Kutta-Merson method,{} until a user-specified function of the solution is zero. See \\downlink{Manual Page}{manpageXXd02bhf}.")) (|d02bbf| (((|Result|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp7| FCN))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp8| OUTPUT)))) "\\spad{d02bbf(xend,m,n,irelab,x,y,tol,ifail,fcn,output)} integrates a system of first-order ordinary differential equations over an interval with suitable initial conditions,{} using a Runge-Kutta-Merson method,{} and returns the solution at points specified by the user. See \\downlink{Manual Page}{manpageXXd02bbf}."))) 
 NIL 
 NIL 
-(-753) 
+(-755) 
 ((|constructor| (NIL "This package uses the NAG Library to solve partial differential equations. See \\downlink{Manual Page}{manpageXXd03}.")) (|d03faf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|ThreeDimensionalMatrix| (|DoubleFloat|)) (|Integer|)) "\\spad{d03faf(xs,xf,l,lbdcnd,bdxs,bdxf,ys,yf,m,mbdcnd,bdys,bdyf,zs,zf,n,nbdcnd,bdzs,bdzf,lambda,ldimf,mdimf,lwrk,f,ifail)} solves the Helmholtz equation in Cartesian co-ordinates in three dimensions using the standard seven-point finite difference approximation. This routine is designed to be particularly efficient on vector processors. See \\downlink{Manual Page}{manpageXXd03faf}.")) (|d03eef| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|String|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp73| PDEF))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp74| BNDY)))) "\\spad{d03eef(xmin,xmax,ymin,ymax,ngx,ngy,lda,scheme,ifail,pdef,bndy)} discretizes a second order elliptic partial differential equation (PDE) on a rectangular region. See \\downlink{Manual Page}{manpageXXd03eef}.")) (|d03edf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{d03edf(ngx,ngy,lda,maxit,acc,iout,a,rhs,ub,ifail)} solves seven-diagonal systems of linear equations which arise from the discretization of an elliptic partial differential equation on a rectangular region. This routine uses a multigrid technique. See \\downlink{Manual Page}{manpageXXd03edf}."))) 
 NIL 
 NIL 
-(-754) 
+(-756) 
 ((|constructor| (NIL "This package uses the NAG Library to calculate the interpolation of a function of one or two variables. When provided with the value of the function (and possibly one or more of its lowest-order derivatives) at each of a number of values of the variable(\\spad{s}),{} the routines provide either an interpolating function or an interpolated value. For some of the interpolating functions,{} there are supporting routines to evaluate,{} differentiate or integrate them. See \\downlink{Manual Page}{manpageXXe01}.")) (|e01sff| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{e01sff(m,x,y,f,rnw,fnodes,px,py,ifail)} evaluates at a given point the two-dimensional interpolating function computed by E01SEF. See \\downlink{Manual Page}{manpageXXe01sff}.")) (|e01sef| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{e01sef(m,x,y,f,nw,nq,rnw,rnq,ifail)} generates a two-dimensional surface interpolating a set of scattered data points,{} using a modified Shepard method. See \\downlink{Manual Page}{manpageXXe01sef}.")) (|e01sbf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{e01sbf(m,x,y,f,triang,grads,px,py,ifail)} evaluates at a given point the two-dimensional interpolant function computed by E01SAF. See \\downlink{Manual Page}{manpageXXe01sbf}.")) (|e01saf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e01saf(m,x,y,f,ifail)} generates a two-dimensional surface interpolating a set of scattered data points,{} using the method of Renka and Cline. See \\downlink{Manual Page}{manpageXXe01saf}.")) (|e01daf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e01daf(mx,my,x,y,f,ifail)} computes a bicubic spline interpolating surface through a set of data values,{} given on a rectangular grid in the \\spad{x}-\\spad{y} plane. See \\downlink{Manual Page}{manpageXXe01daf}.")) (|e01bhf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{e01bhf(n,x,f,d,a,b,ifail)} evaluates the definite integral of a piecewise cubic Hermite interpolant over the interval [a,{}\\spad{b}]. See \\downlink{Manual Page}{manpageXXe01bhf}.")) (|e01bgf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e01bgf(n,x,f,d,m,px,ifail)} evaluates a piecewise cubic Hermite interpolant and its first derivative at a set of points. See \\downlink{Manual Page}{manpageXXe01bgf}.")) (|e01bff| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e01bff(n,x,f,d,m,px,ifail)} evaluates a piecewise cubic Hermite interpolant at a set of points. See \\downlink{Manual Page}{manpageXXe01bff}.")) (|e01bef| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e01bef(n,x,f,ifail)} computes a monotonicity-preserving piecewise cubic Hermite interpolant to a set of data points. See \\downlink{Manual Page}{manpageXXe01bef}.")) (|e01baf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{e01baf(m,x,y,lck,lwrk,ifail)} determines a cubic spline to a given set of data. See \\downlink{Manual Page}{manpageXXe01baf}."))) 
 NIL 
 NIL 
-(-755) 
+(-757) 
 ((|constructor| (NIL "This package uses the NAG Library to find a function which approximates a set of data points. Typically the data contain random errors,{} as of experimental measurement,{} which need to be smoothed out. To seek an approximation to the data,{} it is first necessary to specify for the approximating function a mathematical form (a polynomial,{} for example) which contains a number of unspecified coefficients: the appropriate fitting routine then derives for the coefficients the values which provide the best fit of that particular form. The package deals mainly with curve and surface fitting (\\spadignore{i.e.} fitting with functions of one and of two variables) when a polynomial or a cubic spline is used as the fitting function,{} since these cover the most common needs. However,{} fitting with other functions and/or more variables can be undertaken by means of general linear or nonlinear routines (some of which are contained in other packages) depending on whether the coefficients in the function occur linearly or nonlinearly. Cases where a graph rather than a set of data points is given can be treated simply by first reading a suitable set of points from the graph. The package also contains routines for evaluating,{} differentiating and integrating polynomial and spline curves and surfaces,{} once the numerical values of their coefficients have been determined. See \\downlink{Manual Page}{manpageXXe02}.")) (|e02zaf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{e02zaf(px,py,lamda,mu,m,x,y,npoint,nadres,ifail)} sorts two-dimensional data into rectangular panels. See \\downlink{Manual Page}{manpageXXe02zaf}.")) (|e02gaf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e02gaf(m,la,nplus2,toler,a,b,ifail)} calculates an \\spad{l} solution to an over-determined system of \\indented{22}{1} linear equations. See \\downlink{Manual Page}{manpageXXe02gaf}.")) (|e02dff| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{e02dff(mx,my,px,py,x,y,lamda,mu,c,lwrk,liwrk,ifail)} calculates values of a bicubic spline representation. The spline is evaluated at all points on a rectangular grid. See \\downlink{Manual Page}{manpageXXe02dff}.")) (|e02def| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e02def(m,px,py,x,y,lamda,mu,c,ifail)} calculates values of a bicubic spline representation. See \\downlink{Manual Page}{manpageXXe02def}.")) (|e02ddf| (((|Result|) (|String|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e02ddf(start,m,x,y,f,w,s,nxest,nyest,lwrk,liwrk,nx,lamda,ny,mu,wrk,ifail)} computes a bicubic spline approximation to a set of scattered data are located automatically,{} but a single parameter must be specified to control the trade-off between closeness of fit and smoothness of fit. See \\downlink{Manual Page}{manpageXXe02ddf}.")) (|e02dcf| (((|Result|) (|String|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Integer|)) "\\spad{e02dcf(start,mx,x,my,y,f,s,nxest,nyest,lwrk,liwrk,nx,lamda,ny,mu,wrk,iwrk,ifail)} computes a bicubic spline approximation to a set of data values,{} given on a rectangular grid in the \\spad{x}-\\spad{y} plane. The knots of the spline are located automatically,{} but a single parameter must be specified to control the trade-off between closeness of fit and smoothness of fit. See \\downlink{Manual Page}{manpageXXe02dcf}.")) (|e02daf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e02daf(m,px,py,x,y,f,w,mu,point,npoint,nc,nws,eps,lamda,ifail)} forms a minimal,{} weighted least-squares bicubic spline surface fit with prescribed knots to a given set of data points. See \\downlink{Manual Page}{manpageXXe02daf}.")) (|e02bef| (((|Result|) (|String|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|))) "\\spad{e02bef(start,m,x,y,w,s,nest,lwrk,n,lamda,ifail,wrk,iwrk)} computes a cubic spline approximation to an arbitrary set of data points. The knot are located automatically,{} but a single parameter must be specified to control the trade-off between closeness of fit and smoothness of fit. See \\downlink{Manual Page}{manpageXXe02bef}.")) (|e02bdf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e02bdf(ncap7,lamda,c,ifail)} computes the definite integral from its \\spad{B}-spline representation. See \\downlink{Manual Page}{manpageXXe02bdf}.")) (|e02bcf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|Integer|)) "\\spad{e02bcf(ncap7,lamda,c,x,left,ifail)} evaluates a cubic spline and its first three derivatives from its \\spad{B}-spline representation. See \\downlink{Manual Page}{manpageXXe02bcf}.")) (|e02bbf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|)) "\\spad{e02bbf(ncap7,lamda,c,x,ifail)} evaluates a cubic spline representation. See \\downlink{Manual Page}{manpageXXe02bbf}.")) (|e02baf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e02baf(m,ncap7,x,y,w,lamda,ifail)} computes a weighted least-squares approximation to an arbitrary set of data points by a cubic splines prescribed by the user. Cubic spline can also be carried out. See \\downlink{Manual Page}{manpageXXe02baf}.")) (|e02akf| (((|Result|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|)) "\\spad{e02akf(np1,xmin,xmax,a,ia1,la,x,ifail)} evaluates a polynomial from its Chebyshev-series representation,{} allowing an arbitrary index increment for accessing the array of coefficients. See \\downlink{Manual Page}{manpageXXe02akf}.")) (|e02ajf| (((|Result|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{e02ajf(np1,xmin,xmax,a,ia1,la,qatm1,iaint1,laint,ifail)} determines the coefficients in the Chebyshev-series representation of the indefinite integral of a polynomial given in Chebyshev-series form. See \\downlink{Manual Page}{manpageXXe02ajf}.")) (|e02ahf| (((|Result|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{e02ahf(np1,xmin,xmax,a,ia1,la,iadif1,ladif,ifail)} determines the coefficients in the Chebyshev-series representation of the derivative of a polynomial given in Chebyshev-series form. See \\downlink{Manual Page}{manpageXXe02ahf}.")) (|e02agf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|Integer|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{e02agf(m,kplus1,nrows,xmin,xmax,x,y,w,mf,xf,yf,lyf,ip,lwrk,liwrk,ifail)} computes constrained weighted least-squares polynomial approximations in Chebyshev-series form to an arbitrary set of data points. The values of the approximations and any number of their derivatives can be specified at selected points. See \\downlink{Manual Page}{manpageXXe02agf}.")) (|e02aef| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|)) "\\spad{e02aef(nplus1,a,xcap,ifail)} evaluates a polynomial from its Chebyshev-series representation. See \\downlink{Manual Page}{manpageXXe02aef}.")) (|e02adf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e02adf(m,kplus1,nrows,x,y,w,ifail)} computes weighted least-squares polynomial approximations to an arbitrary set of data points. See \\downlink{Manual Page}{manpageXXe02adf}."))) 
 NIL 
 NIL 
-(-756) 
+(-758) 
 ((|constructor| (NIL "This package uses the NAG Library to perform optimization. An optimization problem involves minimizing a function (called the objective function) of several variables,{} possibly subject to restrictions on the values of the variables defined by a set of constraint functions. The routines in the NAG Foundation Library are concerned with function minimization only,{} since the problem of maximizing a given function can be transformed into a minimization problem simply by multiplying the function by \\spad{-1}. See \\downlink{Manual Page}{manpageXXe04}.")) (|e04ycf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e04ycf(job,m,n,fsumsq,s,lv,v,ifail)} returns estimates of elements of the variance matrix of the estimated regression coefficients for a nonlinear least squares problem. The estimates are derived from the Jacobian of the function \\spad{f}(\\spad{x}) at the solution. See \\downlink{Manual Page}{manpageXXe04ycf}.")) (|e04ucf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Boolean|) (|DoubleFloat|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Boolean|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Boolean|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|Integer|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp55| CONFUN))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp49| OBJFUN)))) "\\spad{e04ucf(n,nclin,ncnln,nrowa,nrowj,nrowr,a,bl,bu,liwork,lwork,sta,cra,der,fea,fun,hes,infb,infs,linf,lint,list,maji,majp,mini,minp,mon,nonf,opt,ste,stao,stac,stoo,stoc,ve,istate,cjac,clamda,r,x,ifail,confun,objfun)} is designed to minimize an arbitrary smooth function subject to constraints on the variables,{} linear constraints. (E04UCF may be used for unconstrained,{} bound-constrained and linearly constrained optimization.) The user must provide subroutines that define the objective and constraint functions and as many of their first partial derivatives as possible. Unspecified derivatives are approximated by finite differences. All matrices are treated as dense,{} and hence E04UCF is not intended for large sparse problems. See \\downlink{Manual Page}{manpageXXe04ucf}.")) (|e04naf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Boolean|) (|Boolean|) (|Boolean|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp20| QPHESS)))) "\\spad{e04naf(itmax,msglvl,n,nclin,nctotl,nrowa,nrowh,ncolh,bigbnd,a,bl,bu,cvec,featol,hess,cold,lpp,orthog,liwork,lwork,x,istate,ifail,qphess)} is a comprehensive programming (\\spad{QP}) or linear programming (\\spad{LP}) problems. It is not intended for large sparse problems. See \\downlink{Manual Page}{manpageXXe04naf}.")) (|e04mbf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Boolean|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{e04mbf(itmax,msglvl,n,nclin,nctotl,nrowa,a,bl,bu,cvec,linobj,liwork,lwork,x,ifail)} is an easy-to-use routine for solving linear programming problems,{} or for finding a feasible point for such problems. It is not intended for large sparse problems. See \\downlink{Manual Page}{manpageXXe04mbf}.")) (|e04jaf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp24| FUNCT1)))) "\\spad{e04jaf(n,ibound,liw,lw,bl,bu,x,ifail,funct1)} is an easy-to-use quasi-Newton algorithm for finding a minimum of a function \\spad{F}(\\spad{x} ,{}\\spad{x} ,{}...,{}\\spad{x} ),{} subject to fixed upper and \\indented{25}{1\\space{2}2\\space{6}\\spad{n}} lower bounds of the independent variables \\spad{x} ,{}\\spad{x} ,{}...,{}\\spad{x} ,{} using \\indented{43}{1\\space{2}2\\space{6}\\spad{n}} function values only. See \\downlink{Manual Page}{manpageXXe04jaf}.")) (|e04gcf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp19| LSFUN2)))) "\\spad{e04gcf(m,n,liw,lw,x,ifail,lsfun2)} is an easy-to-use quasi-Newton algorithm for finding an unconstrained minimum of \\spad{m} nonlinear functions in \\spad{n} variables (m>=n). First derivatives are required. See \\downlink{Manual Page}{manpageXXe04gcf}.")) (|e04fdf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp50| LSFUN1)))) "\\spad{e04fdf(m,n,liw,lw,x,ifail,lsfun1)} is an easy-to-use algorithm for finding an unconstrained minimum of a sum of squares of \\spad{m} nonlinear functions in \\spad{n} variables (m>=n). No derivatives are required. See \\downlink{Manual Page}{manpageXXe04fdf}.")) (|e04dgf| (((|Result|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|DoubleFloat|) (|Boolean|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp49| OBJFUN)))) "\\spad{e04dgf(n,es,fu,it,lin,list,ma,op,pr,sta,sto,ve,x,ifail,objfun)} minimizes an unconstrained nonlinear function of several variables using a pre-conditioned,{} limited memory quasi-Newton conjugate gradient method. First derivatives are required. The routine is intended for use on large scale problems. See \\downlink{Manual Page}{manpageXXe04dgf}."))) 
 NIL 
 NIL 
-(-757) 
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 ((|constructor| (NIL "This package uses the NAG Library to provide facilities for matrix factorizations and associated transformations. See \\downlink{Manual Page}{manpageXXf01}.")) (|f01ref| (((|Result|) (|String|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|)) "\\spad{f01ref(wheret,m,n,ncolq,lda,theta,a,ifail)} returns the first \\spad{ncolq} columns of the complex \\spad{m} by \\spad{m} unitary matrix \\spad{Q},{} where \\spad{Q} is given as the product of Householder transformation matrices. See \\downlink{Manual Page}{manpageXXf01ref}.")) (|f01rdf| (((|Result|) (|String|) (|String|) (|Integer|) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|)) "\\spad{f01rdf(trans,wheret,m,n,a,lda,theta,ncolb,ldb,b,ifail)} performs one of the transformations See \\downlink{Manual Page}{manpageXXf01rdf}.")) (|f01rcf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|)) "\\spad{f01rcf(m,n,lda,a,ifail)} finds the \\spad{QR} factorization of the complex \\spad{m} by \\spad{n} matrix A,{} where m>=n. See \\downlink{Manual Page}{manpageXXf01rcf}.")) (|f01qef| (((|Result|) (|String|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f01qef(wheret,m,n,ncolq,lda,zeta,a,ifail)} returns the first \\spad{ncolq} columns of the real \\spad{m} by \\spad{m} orthogonal matrix \\spad{Q},{} where \\spad{Q} is given as the product of Householder transformation matrices. See \\downlink{Manual Page}{manpageXXf01qef}.")) (|f01qdf| (((|Result|) (|String|) (|String|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f01qdf(trans,wheret,m,n,a,lda,zeta,ncolb,ldb,b,ifail)} performs one of the transformations See \\downlink{Manual Page}{manpageXXf01qdf}.")) (|f01qcf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f01qcf(m,n,lda,a,ifail)} finds the \\spad{QR} factorization of the real \\spad{m} by \\spad{n} matrix A,{} where m>=n. See \\downlink{Manual Page}{manpageXXf01qcf}.")) (|f01mcf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|Integer|)) (|Integer|)) "\\spad{f01mcf(n,avals,lal,nrow,ifail)} computes the Cholesky factorization of a real symmetric positive-definite variable-bandwidth matrix. See \\downlink{Manual Page}{manpageXXf01mcf}.")) (|f01maf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|List| (|Boolean|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Matrix| (|Integer|)) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{f01maf(n,nz,licn,lirn,abort,avals,irn,icn,droptl,densw,ifail)} computes an incomplete Cholesky factorization of a real sparse symmetric positive-definite matrix A. See \\downlink{Manual Page}{manpageXXf01maf}.")) (|f01bsf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|Integer|)) (|Matrix| (|Integer|)) (|Matrix| (|Integer|)) (|Matrix| (|Integer|)) (|Boolean|) (|DoubleFloat|) (|Boolean|) (|Matrix| (|Integer|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f01bsf(n,nz,licn,ivect,jvect,icn,ikeep,grow,eta,abort,idisp,avals,ifail)} factorizes a real sparse matrix using the pivotal sequence previously obtained by F01BRF when a matrix of the same sparsity pattern was factorized. See \\downlink{Manual Page}{manpageXXf01bsf}.")) (|f01brf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Boolean|) (|Boolean|) (|List| (|Boolean|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Matrix| (|Integer|)) (|Integer|)) "\\spad{f01brf(n,nz,licn,lirn,pivot,lblock,grow,abort,a,irn,icn,ifail)} factorizes a real sparse matrix. The routine either forms the LU factorization of a permutation of the entire matrix,{} or,{} optionally,{} first permutes the matrix to block lower triangular form and then only factorizes the diagonal blocks. See \\downlink{Manual Page}{manpageXXf01brf}."))) 
 NIL 
 NIL 
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 ((|constructor| (NIL "This package uses the NAG Library to compute \\begin{items} \\item eigenvalues and eigenvectors of a matrix \\item eigenvalues and eigenvectors of generalized matrix eigenvalue problems \\item singular values and singular vectors of a matrix. \\end{items} See \\downlink{Manual Page}{manpageXXf02}.")) (|f02xef| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Boolean|) (|Integer|) (|Boolean|) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|)) "\\spad{f02xef(m,n,lda,ncolb,ldb,wantq,ldq,wantp,ldph,a,b,ifail)} returns all,{} or part,{} of the singular value decomposition of a general complex matrix. See \\downlink{Manual Page}{manpageXXf02xef}.")) (|f02wef| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Boolean|) (|Integer|) (|Boolean|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02wef(m,n,lda,ncolb,ldb,wantq,ldq,wantp,ldpt,a,b,ifail)} returns all,{} or part,{} of the singular value decomposition of a general real matrix. See \\downlink{Manual Page}{manpageXXf02wef}.")) (|f02fjf| (((|Result|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp27| DOT))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp28| IMAGE))) (|FileName|)) "\\spad{f02fjf(n,k,tol,novecs,nrx,lwork,lrwork,liwork,m,noits,x,ifail,dot,image,monit)} finds eigenvalues of a real sparse symmetric or generalized symmetric eigenvalue problem. See \\downlink{Manual Page}{manpageXXf02fjf}.") (((|Result|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp27| DOT))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp28| IMAGE)))) "\\spad{f02fjf(n,k,tol,novecs,nrx,lwork,lrwork,liwork,m,noits,x,ifail,dot,image)} finds eigenvalues of a real sparse symmetric or generalized symmetric eigenvalue problem. See \\downlink{Manual Page}{manpageXXf02fjf}.")) (|f02bjf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Boolean|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02bjf(n,ia,ib,eps1,matv,iv,a,b,ifail)} calculates all the eigenvalues and,{} if required,{} all the eigenvectors of the generalized eigenproblem Ax=(lambda)\\spad{Bx} where A and \\spad{B} are real,{} square matrices,{} using the \\spad{QZ} algorithm. See \\downlink{Manual Page}{manpageXXf02bjf}.")) (|f02bbf| (((|Result|) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02bbf(ia,n,alb,ub,m,iv,a,ifail)} calculates selected eigenvalues of a real symmetric matrix by reduction to tridiagonal form,{} bisection and inverse iteration,{} where the selected eigenvalues lie within a given interval. See \\downlink{Manual Page}{manpageXXf02bbf}.")) (|f02axf| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{f02axf(ar,iar,ai,iai,n,ivr,ivi,ifail)} calculates all the eigenvalues of a complex Hermitian matrix. See \\downlink{Manual Page}{manpageXXf02axf}.")) (|f02awf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02awf(iar,iai,n,ar,ai,ifail)} calculates all the eigenvalues of a complex Hermitian matrix. See \\downlink{Manual Page}{manpageXXf02awf}.")) (|f02akf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02akf(iar,iai,n,ivr,ivi,ar,ai,ifail)} calculates all the eigenvalues of a complex matrix. See \\downlink{Manual Page}{manpageXXf02akf}.")) (|f02ajf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02ajf(iar,iai,n,ar,ai,ifail)} calculates all the eigenvalue. See \\downlink{Manual Page}{manpageXXf02ajf}.")) (|f02agf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02agf(ia,n,ivr,ivi,a,ifail)} calculates all the eigenvalues of a real unsymmetric matrix. See \\downlink{Manual Page}{manpageXXf02agf}.")) (|f02aff| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02aff(ia,n,a,ifail)} calculates all the eigenvalues of a real unsymmetric matrix. See \\downlink{Manual Page}{manpageXXf02aff}.")) (|f02aef| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02aef(ia,ib,n,iv,a,b,ifail)} calculates all the eigenvalues of Ax=(lambda)\\spad{Bx},{} where A is a real symmetric matrix and \\spad{B} is a real symmetric positive-definite matrix. See \\downlink{Manual Page}{manpageXXf02aef}.")) (|f02adf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02adf(ia,ib,n,a,b,ifail)} calculates all the eigenvalues of Ax=(lambda)\\spad{Bx},{} where A is a real symmetric matrix and \\spad{B} is a real symmetric positive- definite matrix. See \\downlink{Manual Page}{manpageXXf02adf}.")) (|f02abf| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{f02abf(a,ia,n,iv,ifail)} calculates all the eigenvalues of a real symmetric matrix. See \\downlink{Manual Page}{manpageXXf02abf}.")) (|f02aaf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f02aaf(ia,n,a,ifail)} calculates all the eigenvalue. See \\downlink{Manual Page}{manpageXXf02aaf}."))) 
 NIL 
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 ((|constructor| (NIL "This package uses the NAG Library to solve the matrix equation \\axiom{AX=B},{} where \\axiom{\\spad{B}} may be a single vector or a matrix of multiple right-hand sides. The matrix \\axiom{A} may be real,{} complex,{} symmetric,{} Hermitian positive- definite,{} or sparse. It may also be rectangular,{} in which case a least-squares solution is obtained. See \\downlink{Manual Page}{manpageXXf04}.")) (|f04qaf| (((|Result|) (|Integer|) (|Integer|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp30| APROD)))) "\\spad{f04qaf(m,n,damp,atol,btol,conlim,itnlim,msglvl,lrwork,liwork,b,ifail,aprod)} solves sparse unsymmetric equations,{} sparse linear least- squares problems and sparse damped linear least-squares problems,{} using a Lanczos algorithm. See \\downlink{Manual Page}{manpageXXf04qaf}.")) (|f04mcf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{f04mcf(n,al,lal,d,nrow,ir,b,nrb,iselct,nrx,ifail)} computes the approximate solution of a system of real linear equations with multiple right-hand sides,{} AX=B,{} where A is a symmetric positive-definite variable-bandwidth matrix,{} which has previously been factorized by F01MCF. Related systems may also be solved. See \\downlink{Manual Page}{manpageXXf04mcf}.")) (|f04mbf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Boolean|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp28| APROD))) (|Union| (|:| |fn| (|FileName|)) (|:| |fp| (|Asp34| MSOLVE)))) "\\spad{f04mbf(n,b,precon,shift,itnlim,msglvl,lrwork,liwork,rtol,ifail,aprod,msolve)} solves a system of real sparse symmetric linear equations using a Lanczos algorithm. See \\downlink{Manual Page}{manpageXXf04mbf}.")) (|f04maf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|Integer|)) (|Integer|) (|Matrix| (|Integer|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Matrix| (|Integer|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|Integer|)) (|Integer|)) "\\spad{f04maf(n,nz,avals,licn,irn,lirn,icn,wkeep,ikeep,inform,b,acc,noits,ifail)} \\spad{e} a sparse symmetric positive-definite system of linear equations,{} Ax=b,{} using a pre-conditioned conjugate gradient method,{} where A has been factorized by F01MAF. See \\downlink{Manual Page}{manpageXXf04maf}.")) (|f04jgf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|DoubleFloat|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f04jgf(m,n,nra,tol,lwork,a,b,ifail)} finds the solution of a linear least-squares problem,{} Ax=b ,{} where A is a real \\spad{m} by \\spad{n} (m>=n) matrix and \\spad{b} is an \\spad{m} element vector. If the matrix of observations is not of full rank,{} then the minimal least-squares solution is returned. See \\downlink{Manual Page}{manpageXXf04jgf}.")) (|f04faf| (((|Result|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f04faf(job,n,d,e,b,ifail)} calculates the approximate solution of a set of real symmetric positive-definite tridiagonal linear equations. See \\downlink{Manual Page}{manpageXXf04faf}.")) (|f04axf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|Integer|)) (|Matrix| (|Integer|)) (|Integer|) (|Matrix| (|Integer|)) (|Matrix| (|DoubleFloat|))) "\\spad{f04axf(n,a,licn,icn,ikeep,mtype,idisp,rhs)} calculates the approximate solution of a set of real sparse linear equations with a single right-hand side,{} Ax=b or \\indented{1}{\\spad{T}} A \\spad{x=b},{} where A has been factorized by F01BRF or F01BSF. See \\downlink{Manual Page}{manpageXXf04axf}.")) (|f04atf| (((|Result|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{f04atf(a,ia,b,n,iaa,ifail)} calculates the accurate solution of a set of real linear equations with a single right-hand side,{} using an LU factorization with partial pivoting,{} and iterative refinement. See \\downlink{Manual Page}{manpageXXf04atf}.")) (|f04asf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f04asf(ia,b,n,a,ifail)} calculates the accurate solution of a set of real symmetric positive-definite linear equations with a single right- hand side,{} Ax=b,{} using a Cholesky factorization and iterative refinement. See \\downlink{Manual Page}{manpageXXf04asf}.")) (|f04arf| (((|Result|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|)) "\\spad{f04arf(ia,b,n,a,ifail)} calculates the approximate solution of a set of real linear equations with a single right-hand side,{} using an LU factorization with partial pivoting. See \\downlink{Manual Page}{manpageXXf04arf}.")) (|f04adf| (((|Result|) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|Complex| (|DoubleFloat|))) (|Integer|)) "\\spad{f04adf(ia,b,ib,n,m,ic,a,ifail)} calculates the approximate solution of a set of complex linear equations with multiple right-hand sides,{} using an LU factorization with partial pivoting. See \\downlink{Manual Page}{manpageXXf04adf}."))) 
 NIL 
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-(-760) 
+(-762) 
 ((|constructor| (NIL "This package uses the NAG Library to compute matrix factorizations,{} and to solve systems of linear equations following the matrix factorizations. See \\downlink{Manual Page}{manpageXXf07}.")) (|f07fef| (((|Result|) (|String|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|))) "\\spad{f07fef(uplo,n,nrhs,a,lda,ldb,b)} (DPOTRS) solves a real symmetric positive-definite system of linear equations with multiple right-hand sides,{} AX=B,{} where A has been factorized by F07FDF (DPOTRF). See \\downlink{Manual Page}{manpageXXf07fef}.")) (|f07fdf| (((|Result|) (|String|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|))) "\\spad{f07fdf(uplo,n,lda,a)} (DPOTRF) computes the Cholesky factorization of a real symmetric positive-definite matrix. See \\downlink{Manual Page}{manpageXXf07fdf}.")) (|f07aef| (((|Result|) (|String|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|)) (|Integer|) (|Matrix| (|Integer|)) (|Integer|) (|Matrix| (|DoubleFloat|))) "\\spad{f07aef(trans,n,nrhs,a,lda,ipiv,ldb,b)} (DGETRS) solves a real system of linear equations with \\indented{36}{\\spad{T}} multiple right-hand sides,{} AX=B or A \\spad{X=B},{} where A has been factorized by F07ADF (DGETRF). See \\downlink{Manual Page}{manpageXXf07aef}.")) (|f07adf| (((|Result|) (|Integer|) (|Integer|) (|Integer|) (|Matrix| (|DoubleFloat|))) "\\spad{f07adf(m,n,lda,a)} (DGETRF) computes the LU factorization of a real \\spad{m} by \\spad{n} matrix. See \\downlink{Manual Page}{manpageXXf07adf}."))) 
 NIL 
 NIL 
-(-761) 
+(-763) 
 ((|constructor| (NIL "This package uses the NAG Library to compute some commonly occurring physical and mathematical functions. See \\downlink{Manual Page}{manpageXXs}.")) (|s21bdf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{s21bdf(x,y,z,r,ifail)} returns a value of the symmetrised elliptic integral of the third kind,{} via the routine name. See \\downlink{Manual Page}{manpageXXs21bdf}.")) (|s21bcf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{s21bcf(x,y,z,ifail)} returns a value of the symmetrised elliptic integral of the second kind,{} via the routine name. See \\downlink{Manual Page}{manpageXXs21bcf}.")) (|s21bbf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{s21bbf(x,y,z,ifail)} returns a value of the symmetrised elliptic integral of the first kind,{} via the routine name. See \\downlink{Manual Page}{manpageXXs21bbf}.")) (|s21baf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{s21baf(x,y,ifail)} returns a value of an elementary integral,{} which occurs as a degenerate case of an elliptic integral of the first kind,{} via the routine name. See \\downlink{Manual Page}{manpageXXs21baf}.")) (|s20adf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s20adf(x,ifail)} returns a value for the Fresnel Integral \\spad{C}(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs20adf}.")) (|s20acf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s20acf(x,ifail)} returns a value for the Fresnel Integral \\spad{S}(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs20acf}.")) (|s19adf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s19adf(x,ifail)} returns a value for the Kelvin function kei(\\spad{x}) via the routine name. See \\downlink{Manual Page}{manpageXXs19adf}.")) (|s19acf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s19acf(x,ifail)} returns a value for the Kelvin function ker(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs19acf}.")) (|s19abf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s19abf(x,ifail)} returns a value for the Kelvin function bei(\\spad{x}) via the routine name. See \\downlink{Manual Page}{manpageXXs19abf}.")) (|s19aaf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s19aaf(x,ifail)} returns a value for the Kelvin function ber(\\spad{x}) via the routine name. See \\downlink{Manual Page}{manpageXXs19aaf}.")) (|s18def| (((|Result|) (|DoubleFloat|) (|Complex| (|DoubleFloat|)) (|Integer|) (|String|) (|Integer|)) "\\spad{s18def(fnu,z,n,scale,ifail)} returns a sequence of values for the modified Bessel functions \\indented{1}{\\spad{I}\\space{6}(\\spad{z}) for complex \\spad{z},{} non-negative (nu) and} \\indented{2}{(nu)\\spad{+n}} \\spad{n=0},{}1,{}...,{}\\spad{N}-1,{} with an option for exponential scaling. See \\downlink{Manual Page}{manpageXXs18def}.")) (|s18dcf| (((|Result|) (|DoubleFloat|) (|Complex| (|DoubleFloat|)) (|Integer|) (|String|) (|Integer|)) "\\spad{s18dcf(fnu,z,n,scale,ifail)} returns a sequence of values for the modified Bessel functions \\indented{1}{\\spad{K}\\space{6}(\\spad{z}) for complex \\spad{z},{} non-negative (nu) and} \\indented{2}{(nu)\\spad{+n}} \\spad{n=0},{}1,{}...,{}\\spad{N}-1,{} with an option for exponential scaling. See \\downlink{Manual Page}{manpageXXs18dcf}.")) (|s18aff| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s18aff(x,ifail)} returns a value for the modified Bessel Function \\indented{1}{\\spad{I} (\\spad{x}),{} via the routine name.} \\indented{2}{1} See \\downlink{Manual Page}{manpageXXs18aff}.")) (|s18aef| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s18aef(x,ifail)} returns the value of the modified Bessel Function \\indented{1}{\\spad{I} (\\spad{x}),{} via the routine name.} \\indented{2}{0} See \\downlink{Manual Page}{manpageXXs18aef}.")) (|s18adf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s18adf(x,ifail)} returns the value of the modified Bessel Function \\indented{1}{\\spad{K} (\\spad{x}),{} via the routine name.} \\indented{2}{1} See \\downlink{Manual Page}{manpageXXs18adf}.")) (|s18acf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s18acf(x,ifail)} returns the value of the modified Bessel Function \\indented{1}{\\spad{K} (\\spad{x}),{} via the routine name.} \\indented{2}{0} See \\downlink{Manual Page}{manpageXXs18acf}.")) (|s17dlf| (((|Result|) (|Integer|) (|DoubleFloat|) (|Complex| (|DoubleFloat|)) (|Integer|) (|String|) (|Integer|)) "\\spad{s17dlf(m,fnu,z,n,scale,ifail)} returns a sequence of values for the Hankel functions \\indented{2}{(1)\\space{11}(2)} \\indented{1}{\\spad{H}\\space{6}(\\spad{z}) or \\spad{H}\\space{6}(\\spad{z}) for complex \\spad{z},{} non-negative (nu) and} \\indented{2}{(nu)\\spad{+n}\\space{8}(nu)\\spad{+n}} \\spad{n=0},{}1,{}...,{}\\spad{N}-1,{} with an option for exponential scaling. See \\downlink{Manual Page}{manpageXXs17dlf}.")) (|s17dhf| (((|Result|) (|String|) (|Complex| (|DoubleFloat|)) (|String|) (|Integer|)) "\\spad{s17dhf(deriv,z,scale,ifail)} returns the value of the Airy function \\spad{Bi}(\\spad{z}) or its derivative Bi'(\\spad{z}) for complex \\spad{z},{} with an option for exponential scaling. See \\downlink{Manual Page}{manpageXXs17dhf}.")) (|s17dgf| (((|Result|) (|String|) (|Complex| (|DoubleFloat|)) (|String|) (|Integer|)) "\\spad{s17dgf(deriv,z,scale,ifail)} returns the value of the Airy function \\spad{Ai}(\\spad{z}) or its derivative Ai'(\\spad{z}) for complex \\spad{z},{} with an option for exponential scaling. See \\downlink{Manual Page}{manpageXXs17dgf}.")) (|s17def| (((|Result|) (|DoubleFloat|) (|Complex| (|DoubleFloat|)) (|Integer|) (|String|) (|Integer|)) "\\spad{s17def(fnu,z,n,scale,ifail)} returns a sequence of values for the Bessel functions \\indented{1}{\\spad{J}\\space{6}(\\spad{z}) for complex \\spad{z},{} non-negative (nu) and \\spad{n=0},{}1,{}...,{}\\spad{N}-1,{}} \\indented{2}{(nu)\\spad{+n}} with an option for exponential scaling. See \\downlink{Manual Page}{manpageXXs17def}.")) (|s17dcf| (((|Result|) (|DoubleFloat|) (|Complex| (|DoubleFloat|)) (|Integer|) (|String|) (|Integer|)) "\\spad{s17dcf(fnu,z,n,scale,ifail)} returns a sequence of values for the Bessel functions \\indented{1}{\\spad{Y}\\space{6}(\\spad{z}) for complex \\spad{z},{} non-negative (nu) and \\spad{n=0},{}1,{}...,{}\\spad{N}-1,{}} \\indented{2}{(nu)\\spad{+n}} with an option for exponential scaling. See \\downlink{Manual Page}{manpageXXs17dcf}.")) (|s17akf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17akf(x,ifail)} returns a value for the derivative of the Airy function \\spad{Bi}(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs17akf}.")) (|s17ajf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17ajf(x,ifail)} returns a value of the derivative of the Airy function \\spad{Ai}(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs17ajf}.")) (|s17ahf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17ahf(x,ifail)} returns a value of the Airy function,{} \\spad{Bi}(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs17ahf}.")) (|s17agf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17agf(x,ifail)} returns a value for the Airy function,{} \\spad{Ai}(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs17agf}.")) (|s17aff| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17aff(x,ifail)} returns the value of the Bessel Function \\indented{1}{\\spad{J} (\\spad{x}),{} via the routine name.} \\indented{2}{1} See \\downlink{Manual Page}{manpageXXs17aff}.")) (|s17aef| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17aef(x,ifail)} returns the value of the Bessel Function \\indented{1}{\\spad{J} (\\spad{x}),{} via the routine name.} \\indented{2}{0} See \\downlink{Manual Page}{manpageXXs17aef}.")) (|s17adf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17adf(x,ifail)} returns the value of the Bessel Function \\indented{1}{\\spad{Y} (\\spad{x}),{} via the routine name.} \\indented{2}{1} See \\downlink{Manual Page}{manpageXXs17adf}.")) (|s17acf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s17acf(x,ifail)} returns the value of the Bessel Function \\indented{1}{\\spad{Y} (\\spad{x}),{} via the routine name.} \\indented{2}{0} See \\downlink{Manual Page}{manpageXXs17acf}.")) (|s15aef| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s15aef(x,ifail)} returns the value of the error function erf(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs15aef}.")) (|s15adf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s15adf(x,ifail)} returns the value of the complementary error function,{} erfc(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs15adf}.")) (|s14baf| (((|Result|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|)) "\\spad{s14baf(a,x,tol,ifail)} computes values for the incomplete gamma functions \\spad{P}(a,{}\\spad{x}) and \\spad{Q}(a,{}\\spad{x}). See \\downlink{Manual Page}{manpageXXs14baf}.")) (|s14abf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s14abf(x,ifail)} returns a value for the log,{} \\spad{ln}(Gamma(\\spad{x})),{} via the routine name. See \\downlink{Manual Page}{manpageXXs14abf}.")) (|s14aaf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s14aaf(x,ifail)} returns the value of the Gamma function (Gamma)(\\spad{x}),{} via the routine name. See \\downlink{Manual Page}{manpageXXs14aaf}.")) (|s13adf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s13adf(x,ifail)} returns the value of the sine integral See \\downlink{Manual Page}{manpageXXs13adf}.")) (|s13acf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s13acf(x,ifail)} returns the value of the cosine integral See \\downlink{Manual Page}{manpageXXs13acf}.")) (|s13aaf| (((|Result|) (|DoubleFloat|) (|Integer|)) "\\spad{s13aaf(x,ifail)} returns the value of the exponential integral \\indented{1}{\\spad{E} (\\spad{x}),{} via the routine name.} \\indented{2}{1} See \\downlink{Manual Page}{manpageXXs13aaf}.")) (|s01eaf| (((|Result|) (|Complex| (|DoubleFloat|)) (|Integer|)) "\\spad{s01eaf(z,ifail)} S01EAF evaluates the exponential function exp(\\spad{z}) ,{} for complex \\spad{z}. See \\downlink{Manual Page}{manpageXXs01eaf}."))) 
 NIL 
 NIL 
-(-762) 
+(-764) 
 ((|constructor| (NIL "Support functions for the NAG Library Link functions")) (|restorePrecision| (((|Void|)) "\\spad{restorePrecision()} \\undocumented{}")) (|checkPrecision| (((|Boolean|)) "\\spad{checkPrecision()} \\undocumented{}")) (|dimensionsOf| (((|SExpression|) (|Symbol|) (|Matrix| (|Integer|))) "\\spad{dimensionsOf(s,m)} \\undocumented{}") (((|SExpression|) (|Symbol|) (|Matrix| (|DoubleFloat|))) "\\spad{dimensionsOf(s,m)} \\undocumented{}")) (|aspFilename| (((|String|) (|String|)) "\\spad{aspFilename(\"f\")} returns a String consisting of \\spad{\"f\"} suffixed with \\indented{1}{an extension identifying the current AXIOM session.}")) (|fortranLinkerArgs| (((|String|)) "\\spad{fortranLinkerArgs()} returns the current linker arguments")) (|fortranCompilerName| (((|String|)) "\\spad{fortranCompilerName()} returns the name of the currently selected \\indented{1}{Fortran compiler}"))) 
 NIL 
 NIL 
-(-763 S) 
+(-765 S) 
 ((|constructor| (NIL "NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \\indented{2}{\\spad{x*}(\\spad{y+z}) = x*y + \\spad{x*z}} \\indented{2}{(x+y)\\spad{*z} = \\spad{x*z} + \\spad{y*z}} Common Additional Axioms \\indented{2}{noZeroDivisors\\space{2}ab = 0 \\spad{=>} a=0 or \\spad{b=0}}")) (|antiCommutator| (($ $ $) "\\spad{antiCommutator(a,b)} returns \\spad{a*b+b*a}.")) (|commutator| (($ $ $) "\\spad{commutator(a,b)} returns \\spad{a*b-b*a}.")) (|associator| (($ $ $ $) "\\spad{associator(a,b,c)} returns \\spad{(a*b)*c-a*(b*c)}."))) 
 NIL 
 NIL 
-(-764) 
+(-766) 
 ((|constructor| (NIL "NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \\indented{2}{\\spad{x*}(\\spad{y+z}) = x*y + \\spad{x*z}} \\indented{2}{(x+y)\\spad{*z} = \\spad{x*z} + \\spad{y*z}} Common Additional Axioms \\indented{2}{noZeroDivisors\\space{2}ab = 0 \\spad{=>} a=0 or \\spad{b=0}}")) (|antiCommutator| (($ $ $) "\\spad{antiCommutator(a,b)} returns \\spad{a*b+b*a}.")) (|commutator| (($ $ $) "\\spad{commutator(a,b)} returns \\spad{a*b-b*a}.")) (|associator| (($ $ $ $) "\\spad{associator(a,b,c)} returns \\spad{(a*b)*c-a*(b*c)}."))) 
 NIL 
 NIL 
-(-765 S) 
+(-767 S) 
 ((|constructor| (NIL "A NonAssociativeRing is a non associative \\spad{rng} which has a unit,{} the multiplication is not necessarily commutative or associative.")) (|coerce| (($ (|Integer|)) "\\spad{coerce(n)} coerces the integer \\spad{n} to an element of the ring.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring."))) 
 NIL 
 NIL 
-(-766) 
+(-768) 
 ((|constructor| (NIL "A NonAssociativeRing is a non associative \\spad{rng} which has a unit,{} the multiplication is not necessarily commutative or associative.")) (|coerce| (($ (|Integer|)) "\\spad{coerce(n)} coerces the integer \\spad{n} to an element of the ring.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring."))) 
 NIL 
 NIL 
-(-767 |Par|) 
+(-769 |Par|) 
 ((|constructor| (NIL "This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the complex rational numbers. The results are expressed either as complex floating numbers or as complex rational numbers depending on the type of the precision parameter.")) (|complexEigenvectors| (((|List| (|Record| (|:| |outval| (|Complex| |#1|)) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| (|Complex| |#1|)))))) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) |#1|) "\\spad{complexEigenvectors(m,eps)} returns a list of records each one containing a complex eigenvalue,{} its algebraic multiplicity,{} and a list of associated eigenvectors. All these results are computed to precision \\spad{eps} and are expressed as complex floats or complex rational numbers depending on the type of \\spad{eps} (float or rational).")) (|complexEigenvalues| (((|List| (|Complex| |#1|)) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) |#1|) "\\spad{complexEigenvalues(m,eps)} computes the eigenvalues of the matrix \\spad{m} to precision \\spad{eps}. The eigenvalues are expressed as complex floats or complex rational numbers depending on the type of \\spad{eps} (float or rational).")) (|characteristicPolynomial| (((|Polynomial| (|Complex| (|Fraction| (|Integer|)))) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) (|Symbol|)) "\\spad{characteristicPolynomial(m,x)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over Complex Rationals with variable \\spad{x}.") (((|Polynomial| (|Complex| (|Fraction| (|Integer|)))) (|Matrix| (|Complex| (|Fraction| (|Integer|))))) "\\spad{characteristicPolynomial(m)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over complex rationals with a new symbol as variable."))) 
 NIL 
 NIL 
-(-768 -3498) 
+(-770 -3505) 
 ((|constructor| (NIL "\\spadtype{NumericContinuedFraction} provides functions \\indented{2}{for converting floating point numbers to continued fractions.}")) (|continuedFraction| (((|ContinuedFraction| (|Integer|)) |#1|) "\\spad{continuedFraction(f)} converts the floating point number \\spad{f} to a reduced continued fraction."))) 
 NIL 
 NIL 
-(-769 P -3498) 
+(-771 P -3505) 
 ((|constructor| (NIL "This package provides a division and related operations for \\spadtype{MonogenicLinearOperator}\\spad{s} over a \\spadtype{Field}. Since the multiplication is in general non-commutative,{} these operations all have left- and right-hand versions. This package provides the operations based on left-division.")) (|leftLcm| ((|#1| |#1| |#1|) "\\spad{leftLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = a*aa = b*bb} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using left-division.")) (|leftGcd| ((|#1| |#1| |#1|) "\\spad{leftGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = aa*g}} \\indented{3}{\\spad{b = bb*g}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using left-division.")) (|leftExactQuotient| (((|Union| |#1| "failed") |#1| |#1|) "\\spad{leftExactQuotient(a,b)} computes the value \\spad{q},{} if it exists,{} \\indented{1}{such that \\spad{a = b*q}.}")) (|leftRemainder| ((|#1| |#1| |#1|) "\\spad{leftRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|leftQuotient| ((|#1| |#1| |#1|) "\\spad{leftQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|leftDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{leftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division\\spad{''}."))) 
 NIL 
 NIL 
-(-770 T$) 
+(-772 T$) 
 NIL 
 NIL 
 NIL 
-(-771 UP -3498) 
+(-773 UP -3505) 
 ((|constructor| (NIL "In this package \\spad{F} is a framed algebra over the integers (typically \\spad{F = Z[a]} for some algebraic integer a). The package provides functions to compute the integral closure of \\spad{Z} in the quotient quotient field of \\spad{F}.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| (|Integer|))) (|:| |basisDen| (|Integer|)) (|:| |basisInv| (|Matrix| (|Integer|)))) (|Integer|)) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{Z} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{Z}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| (|Integer|))) (|:| |basisDen| (|Integer|)) (|:| |basisInv| (|Matrix| (|Integer|))))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{Z} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{Z}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|discriminant| (((|Integer|)) "\\spad{discriminant()} returns the discriminant of the integral closure of \\spad{Z} in the quotient field of the framed algebra \\spad{F}."))) 
 NIL 
 NIL 
-(-772) 
+(-774) 
 ((|retract| (((|Union| (|:| |nia| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |mdnia| (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|List| (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))))) $) "\\spad{retract(x)} \\undocumented{}")) (|coerce| (($ (|Union| (|:| |nia| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |mdnia| (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|List| (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))))) "\\spad{coerce(x)} \\undocumented{}") (($ (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|List| (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{coerce(x)} \\undocumented{}") (($ (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{coerce(x)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-773 R) 
+(-775 R) 
 ((|constructor| (NIL "NonLinearSolvePackage is an interface to \\spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving. The solutions are given in the algebraic closure of \\spad{R} whenever possible.")) (|solve| (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{solve(lp)} finds the solution in the algebraic closure of \\spad{R} of the list \\spad{lp} of rational functions with respect to all the symbols appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{solve(lp,lv)} finds the solutions in the algebraic closure of \\spad{R} of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}.")) (|solveInField| (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{solveInField(lp)} finds the solution of the list \\spad{lp} of rational functions with respect to all the symbols appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{solveInField(lp,lv)} finds the solutions of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}."))) 
 NIL 
 NIL 
-(-774) 
+(-776) 
 ((|constructor| (NIL "\\spadtype{NonNegativeInteger} provides functions for non \\indented{2}{negative integers.}")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} means multiplication is commutative : \\spad{x*y = y*x}.")) (|random| (($ $) "\\spad{random(n)} returns a random integer from 0 to \\spad{n-1}.")) (|shift| (($ $ (|Integer|)) "\\spad{shift(a,i)} shift \\spad{a} by \\spad{i} bits.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} returns the quotient of \\spad{a} and \\spad{b},{} or \"failed\" if \\spad{b} is zero or \\spad{a} rem \\spad{b} is zero.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(a,b)} returns a record containing both remainder and quotient.")) (|gcd| (($ $ $) "\\spad{gcd(a,b)} computes the greatest common divisor of two non negative integers \\spad{a} and \\spad{b}.")) (|rem| (($ $ $) "\\spad{a rem b} returns the remainder of \\spad{a} and \\spad{b}.")) (|quo| (($ $ $) "\\spad{a quo b} returns the quotient of \\spad{a} and \\spad{b},{} forgetting the remainder."))) 
-(((-4429 "*") . T)) 
+(((-4436 "*") . T)) 
 NIL 
-(-775 R -3498) 
+(-777 R -3505) 
 ((|constructor| (NIL "NonLinearFirstOrderODESolver provides a function for finding closed form first integrals of nonlinear ordinary differential equations of order 1.")) (|solve| (((|Union| |#2| "failed") |#2| |#2| (|BasicOperator|) (|Symbol|)) "\\spad{solve(M(x,y), N(x,y), y, x)} returns \\spad{F(x,y)} such that \\spad{F(x,y) = c} for a constant \\spad{c} is a first integral of the equation \\spad{M(x,y) dx + N(x,y) dy = 0},{} or \"failed\" if no first-integral can be found."))) 
 NIL 
 NIL 
-(-776) 
+(-778) 
 ((|constructor| (NIL "\\spadtype{None} implements a type with no objects. It is mainly used in technical situations where such a thing is needed (\\spadignore{e.g.} the interpreter and some of the internal \\spadtype{Expression} code)."))) 
 NIL 
 NIL 
-(-777 S) 
+(-779 S) 
 ((|constructor| (NIL "\\spadtype{NoneFunctions1} implements functions on \\spadtype{None}. It particular it includes a particulary dangerous coercion from any other type to \\spadtype{None}.")) (|coerce| (((|None|) |#1|) "\\spad{coerce(x)} changes \\spad{x} into an object of type \\spadtype{None}."))) 
 NIL 
 NIL 
-(-778 R |PolR| E |PolE|) 
+(-780 R |PolR| E |PolE|) 
 ((|constructor| (NIL "This package implements the norm of a polynomial with coefficients in a monogenic algebra (using resultants)")) (|norm| ((|#2| |#4|) "\\spad{norm q} returns the norm of \\spad{q},{} \\spadignore{i.e.} the product of all the conjugates of \\spad{q}."))) 
 NIL 
 NIL 
-(-779 R E V P TS) 
+(-781 R E V P TS) 
 ((|constructor| (NIL "A package for computing normalized assocites of univariate polynomials with coefficients in a tower of simple extensions of a field.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of \\spad{gcd} over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of AAECC11} \\indented{5}{Paris,{} 1995.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.}")) (|normInvertible?| (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{normInvertible?(\\spad{p},{}\\spad{ts})} is an internal subroutine,{} exported only for developement.")) (|outputArgs| (((|Void|) (|String|) (|String|) |#4| |#5|) "\\axiom{outputArgs(\\spad{s1},{}\\spad{s2},{}\\spad{p},{}\\spad{ts})} is an internal subroutine,{} exported only for developement.")) (|normalize| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{normalize(\\spad{p},{}\\spad{ts})} normalizes \\axiom{\\spad{p}} \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}.")) (|normalizedAssociate| ((|#4| |#4| |#5|) "\\axiom{normalizedAssociate(\\spad{p},{}\\spad{ts})} returns a normalized polynomial \\axiom{\\spad{n}} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts} such that \\axiom{\\spad{n}} and \\axiom{\\spad{p}} are associates \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts} and assuming that \\axiom{\\spad{p}} is invertible \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}.")) (|recip| (((|Record| (|:| |num| |#4|) (|:| |den| |#4|)) |#4| |#5|) "\\axiom{recip(\\spad{p},{}\\spad{ts})} returns the inverse of \\axiom{\\spad{p}} \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts} assuming that \\axiom{\\spad{p}} is invertible \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}."))) 
 NIL 
 NIL 
-(-780 -3498 |ExtF| |SUEx| |ExtP| |n|) 
+(-782 -3505 |ExtF| |SUEx| |ExtP| |n|) 
 ((|constructor| (NIL "This package \\undocumented")) (|Frobenius| ((|#4| |#4|) "\\spad{Frobenius(x)} \\undocumented")) (|retractIfCan| (((|Union| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|)) "failed") |#4|) "\\spad{retractIfCan(x)} \\undocumented")) (|normFactors| (((|List| |#4|) |#4|) "\\spad{normFactors(x)} \\undocumented"))) 
 NIL 
 NIL 
-(-781 BP E OV R P) 
+(-783 BP E OV R P) 
 ((|constructor| (NIL "Package for the determination of the coefficients in the lifting process. Used by \\spadtype{MultivariateLifting}. This package will work for every euclidean domain \\spad{R} which has property \\spad{F},{} \\spadignore{i.e.} there exists a factor operation in \\spad{R[x]}.")) (|listexp| (((|List| (|NonNegativeInteger|)) |#1|) "\\spad{listexp }\\undocumented")) (|npcoef| (((|Record| (|:| |deter| (|List| (|SparseUnivariatePolynomial| |#5|))) (|:| |dterm| (|List| (|List| (|Record| (|:| |expt| (|NonNegativeInteger|)) (|:| |pcoef| |#5|))))) (|:| |nfacts| (|List| |#1|)) (|:| |nlead| (|List| |#5|))) (|SparseUnivariatePolynomial| |#5|) (|List| |#1|) (|List| |#5|)) "\\spad{npcoef }\\undocumented"))) 
 NIL 
 NIL 
-(-782 |Par|) 
+(-784 |Par|) 
 ((|constructor| (NIL "This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the Rational Numbers. The results are expressed as floating numbers or as rational numbers depending on the type of the parameter Par.")) (|realEigenvectors| (((|List| (|Record| (|:| |outval| |#1|) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| |#1|))))) (|Matrix| (|Fraction| (|Integer|))) |#1|) "\\spad{realEigenvectors(m,eps)} returns a list of records each one containing a real eigenvalue,{} its algebraic multiplicity,{} and a list of associated eigenvectors. All these results are computed to precision \\spad{eps} as floats or rational numbers depending on the type of \\spad{eps} .")) (|realEigenvalues| (((|List| |#1|) (|Matrix| (|Fraction| (|Integer|))) |#1|) "\\spad{realEigenvalues(m,eps)} computes the eigenvalues of the matrix \\spad{m} to precision \\spad{eps}. The eigenvalues are expressed as floats or rational numbers depending on the type of \\spad{eps} (float or rational).")) (|characteristicPolynomial| (((|Polynomial| (|Fraction| (|Integer|))) (|Matrix| (|Fraction| (|Integer|))) (|Symbol|)) "\\spad{characteristicPolynomial(m,x)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over \\spad{RN} with variable \\spad{x}. Fraction \\spad{P} \\spad{RN}.") (((|Polynomial| (|Fraction| (|Integer|))) (|Matrix| (|Fraction| (|Integer|)))) "\\spad{characteristicPolynomial(m)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over \\spad{RN} with a new symbol as variable."))) 
 NIL 
 NIL 
-(-783 R |VarSet|) 
+(-785 R |VarSet|) 
 ((|constructor| (NIL "A post-facto extension for \\axiomType{\\spad{SMP}} in order to speed up operations related to pseudo-division and \\spad{gcd}. This domain is based on the \\axiomType{NSUP} constructor which is itself a post-facto extension of the \\axiomType{SUP} constructor."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
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-(-784 R) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
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+(-786 R) 
 ((|constructor| (NIL "A post-facto extension for \\axiomType{SUP} in order to speed up operations related to pseudo-division and \\spad{gcd} for both \\axiomType{SUP} and,{} consequently,{} \\axiomType{NSMP}.")) (|halfExtendedResultant2| (((|Record| (|:| |resultant| |#1|) (|:| |coef2| $)) $ $) "\\axiom{halfExtendedResultant2(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca]} such that \\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{} \\spad{cb}]}")) (|halfExtendedResultant1| (((|Record| (|:| |resultant| |#1|) (|:| |coef1| $)) $ $) "\\axiom{halfExtendedResultant1(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca]} such that \\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{} \\spad{cb}]}")) (|extendedResultant| (((|Record| (|:| |resultant| |#1|) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{}\\spad{cb}]} such that \\axiom{\\spad{r}} is the resultant of \\axiom{a} and \\axiom{\\spad{b}} and \\axiom{\\spad{r} = ca * a + \\spad{cb} * \\spad{b}}")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{halfExtendedSubResultantGcd2(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}\\spad{cb}]} such that \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} \\spad{cb}]}")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{halfExtendedSubResultantGcd1(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} such that \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} \\spad{cb}]}")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} \\spad{cb}]} such that \\axiom{\\spad{g}} is a \\spad{gcd} of \\axiom{a} and \\axiom{\\spad{b}} in \\axiom{\\spad{R^}(\\spad{-1}) \\spad{P}} and \\axiom{\\spad{g} = ca * a + \\spad{cb} * \\spad{b}}")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns \\axiom{resultant(a,{}\\spad{b})} if \\axiom{a} and \\axiom{\\spad{b}} has no non-trivial \\spad{gcd} in \\axiom{\\spad{R^}(\\spad{-1}) \\spad{P}} otherwise the non-zero sub-resultant with smallest index.")) (|subResultantsChain| (((|List| $) $ $) "\\axiom{subResultantsChain(a,{}\\spad{b})} returns the list of the non-zero sub-resultants of \\axiom{a} and \\axiom{\\spad{b}} sorted by increasing degree.")) (|lazyPseudoQuotient| (($ $ $) "\\axiom{lazyPseudoQuotient(a,{}\\spad{b})} returns \\axiom{\\spad{q}} if \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}")) (|lazyPseudoDivide| (((|Record| (|:| |coef| |#1|) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{\\spad{c^n} * a = \\spad{q*b} \\spad{+r}} and \\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]} where \\axiom{\\spad{n} + \\spad{g} = max(0,{} degree(\\spad{b}) - degree(a) + 1)}.")) (|lazyPseudoRemainder| (($ $ $) "\\axiom{lazyPseudoRemainder(a,{}\\spad{b})} returns \\axiom{\\spad{r}} if \\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]}. This lazy pseudo-remainder is computed by means of the \\axiomOpFrom{fmecg}{NewSparseUnivariatePolynomial} operation.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| |#1|) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]} such that \\axiom{\\spad{r}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{b}} divides \\axiom{\\spad{c^n} * a - \\spad{r}} where \\axiom{\\spad{c}} is \\axiom{leadingCoefficient(\\spad{b})} and \\axiom{\\spad{n}} is as small as possible with the previous properties.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} returns \\axiom{\\spad{r}} such that \\axiom{\\spad{r}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{b}} divides \\axiom{a \\spad{-r}} where \\axiom{\\spad{b}} is monic.")) (|fmecg| (($ $ (|NonNegativeInteger|) |#1| $) "\\axiom{fmecg(\\spad{p1},{}\\spad{e},{}\\spad{r},{}\\spad{p2})} returns \\axiom{\\spad{p1} - \\spad{r} * X**e * \\spad{p2}} where \\axiom{\\spad{X}} is \\axiom{monomial(1,{}1)}"))) 
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-(-785 R S) 
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+(-787 R S) 
 ((|constructor| (NIL "This package lifts a mapping from coefficient rings \\spad{R} to \\spad{S} to a mapping from sparse univariate polynomial over \\spad{R} to a sparse univariate polynomial over \\spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.")) (|map| (((|NewSparseUnivariatePolynomial| |#2|) (|Mapping| |#2| |#1|) (|NewSparseUnivariatePolynomial| |#1|)) "\\axiom{map(func,{} poly)} creates a new polynomial by applying func to every non-zero coefficient of the polynomial poly."))) 
 NIL 
 NIL 
-(-786 R) 
+(-788 R) 
 ((|constructor| (NIL "This package provides polynomials as functions on a ring.")) (|eulerE| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{eulerE(n,r)} \\undocumented")) (|bernoulliB| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{bernoulliB(n,r)} \\undocumented")) (|cyclotomic| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{cyclotomic(n,r)} \\undocumented"))) 
 NIL 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550)))))) 
-(-787 R E V P) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551)))))) 
+(-789 R E V P) 
 ((|constructor| (NIL "The category of normalized triangular sets. A triangular set \\spad{ts} is said normalized if for every algebraic variable \\spad{v} of \\spad{ts} the polynomial \\spad{select(ts,v)} is normalized \\spad{w}.\\spad{r}.\\spad{t}. every polynomial in \\spad{collectUnder(ts,v)}. A polynomial \\spad{p} is said normalized \\spad{w}.\\spad{r}.\\spad{t}. a non-constant polynomial \\spad{q} if \\spad{p} is constant or \\spad{degree(p,mdeg(q)) = 0} and \\spad{init(p)} is normalized \\spad{w}.\\spad{r}.\\spad{t}. \\spad{q}. One of the important features of normalized triangular sets is that they are regular sets.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)} \\indented{1}{[3] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of \\spad{gcd} over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of AAECC11} \\indented{5}{Paris,{} 1995.} \\indented{1}{[4] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.}"))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-788 S) 
+(-790 S) 
 ((|constructor| (NIL "Numeric provides real and complex numerical evaluation functions for various symbolic types.")) (|numericIfCan| (((|Union| (|Float|) "failed") (|Expression| |#1|) (|PositiveInteger|)) "\\spad{numericIfCan(x, n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Expression| |#1|)) "\\spad{numericIfCan(x)} returns a real approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{numericIfCan(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Fraction| (|Polynomial| |#1|))) "\\spad{numericIfCan(x)} returns a real approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{numericIfCan(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Polynomial| |#1|)) "\\spad{numericIfCan(x)} returns a real approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.")) (|complexNumericIfCan| (((|Union| (|Complex| (|Float|)) "failed") (|Expression| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Expression| (|Complex| |#1|))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Expression| |#1|) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Expression| |#1|)) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| (|Complex| |#1|))) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| (|Complex| |#1|)))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| |#1|))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| |#1|)) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| (|Complex| |#1|))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not constant.")) (|complexNumeric| (((|Complex| (|Float|)) (|Expression| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Expression| (|Complex| |#1|))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Expression| |#1|) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Expression| |#1|)) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| (|Complex| |#1|))) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| (|Complex| |#1|)))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x}") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| |#1|))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Polynomial| |#1|)) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Polynomial| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Polynomial| (|Complex| |#1|))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Complex| |#1|) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Complex| |#1|)) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) |#1| (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) |#1|) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.")) (|numeric| (((|Float|) (|Expression| |#1|) (|PositiveInteger|)) "\\spad{numeric(x, n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) (|Expression| |#1|)) "\\spad{numeric(x)} returns a real approximation of \\spad{x}.") (((|Float|) (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{numeric(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) (|Fraction| (|Polynomial| |#1|))) "\\spad{numeric(x)} returns a real approximation of \\spad{x}.") (((|Float|) (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{numeric(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) (|Polynomial| |#1|)) "\\spad{numeric(x)} returns a real approximation of \\spad{x}.") (((|Float|) |#1| (|PositiveInteger|)) "\\spad{numeric(x, n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) |#1|) "\\spad{numeric(x)} returns a real approximation of \\spad{x}."))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-853)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-1053))) (|HasCategory| |#1| (QUOTE (-173)))) 
-(-789) 
+((-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-855)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-1055))) (|HasCategory| |#1| (QUOTE (-173)))) 
+(-791) 
 ((|constructor| (NIL "NumberFormats provides function to format and read arabic and roman numbers,{} to convert numbers to strings and to read floating-point numbers.")) (|ScanFloatIgnoreSpacesIfCan| (((|Union| (|Float|) "failed") (|String|)) "\\spad{ScanFloatIgnoreSpacesIfCan(s)} tries to form a floating point number from the string \\spad{s} ignoring any spaces.")) (|ScanFloatIgnoreSpaces| (((|Float|) (|String|)) "\\spad{ScanFloatIgnoreSpaces(s)} forms a floating point number from the string \\spad{s} ignoring any spaces. Error is generated if the string is not recognised as a floating point number.")) (|ScanRoman| (((|PositiveInteger|) (|String|)) "\\spad{ScanRoman(s)} forms an integer from a Roman numeral string \\spad{s}.")) (|FormatRoman| (((|String|) (|PositiveInteger|)) "\\spad{FormatRoman(n)} forms a Roman numeral string from an integer \\spad{n}.")) (|ScanArabic| (((|PositiveInteger|) (|String|)) "\\spad{ScanArabic(s)} forms an integer from an Arabic numeral string \\spad{s}.")) (|FormatArabic| (((|String|) (|PositiveInteger|)) "\\spad{FormatArabic(n)} forms an Arabic numeral string from an integer \\spad{n}."))) 
 NIL 
 NIL 
-(-790) 
+(-792) 
 ((|numericalIntegration| (((|Result|) (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|List| (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))) (|Result|)) "\\spad{numericalIntegration(args,hints)} performs the integration of the function given the strategy or method returned by \\axiomFun{measure}.") (((|Result|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))) (|Result|)) "\\spad{numericalIntegration(args,hints)} performs the integration of the function given the strategy or method returned by \\axiomFun{measure}.")) (|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|)) (|:| |extra| (|Result|))) (|RoutinesTable|) (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|List| (|Segment| (|OrderedCompletion| (|DoubleFloat|))))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve a problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far.") (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|)) (|:| |extra| (|Result|))) (|RoutinesTable|) (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve a problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far."))) 
 NIL 
 NIL 
-(-791) 
+(-793) 
 ((|constructor| (NIL "This package is a suite of functions for the numerical integration of an ordinary differential equation of \\spad{n} variables: \\blankline \\indented{8}{\\center{dy/dx = \\spad{f}(\\spad{y},{}\\spad{x})\\space{5}\\spad{y} is an \\spad{n}-vector}} \\blankline \\par All the routines are based on a 4-th order Runge-Kutta kernel. These routines generally have as arguments: \\spad{n},{} the number of dependent variables; \\spad{x1},{} the initial point; \\spad{h},{} the step size; \\spad{y},{} a vector of initial conditions of length \\spad{n} which upon exit contains the solution at \\spad{x1 + h}; \\spad{derivs},{} a function which computes the right hand side of the ordinary differential equation: \\spad{derivs(dydx,y,x)} computes \\spad{dydx},{} a vector which contains the derivative information. \\blankline \\par In order of increasing complexity:\\begin{items} \\blankline \\item \\spad{rk4(y,n,x1,h,derivs)} advances the solution vector to \\spad{x1 + h} and return the values in \\spad{y}. \\blankline \\item \\spad{rk4(y,n,x1,h,derivs,t1,t2,t3,t4)} is the same as \\spad{rk4(y,n,x1,h,derivs)} except that you must provide 4 scratch arrays \\spad{t1}-\\spad{t4} of size \\spad{n}. \\blankline \\item Starting with \\spad{y} at \\spad{x1},{} \\spad{rk4f(y,n,x1,x2,ns,derivs)} uses \\spad{ns} fixed steps of a 4-th order Runge-Kutta integrator to advance the solution vector to \\spad{x2} and return the values in \\spad{y}. Argument \\spad{x2},{} is the final point,{} and \\spad{ns},{} the number of steps to take. \\blankline \\item \\spad{rk4qc(y,n,x1,step,eps,yscal,derivs)} takes a 5-th order Runge-Kutta step with monitoring of local truncation to ensure accuracy and adjust stepsize. The function takes two half steps and one full step and scales the difference in solutions at the final point. If the error is within \\spad{eps},{} the step is taken and the result is returned. If the error is not within \\spad{eps},{} the stepsize if decreased and the procedure is tried again until the desired accuracy is reached. Upon input,{} an trial step size must be given and upon return,{} an estimate of the next step size to use is returned as well as the step size which produced the desired accuracy. The scaled error is computed as \\center{\\spad{error = MAX(ABS((y2steps(i) - y1step(i))/yscal(i)))}} and this is compared against \\spad{eps}. If this is greater than \\spad{eps},{} the step size is reduced accordingly to \\center{\\spad{hnew = 0.9 * hdid * (error/eps)**(-1/4)}} If the error criterion is satisfied,{} then we check if the step size was too fine and return a more efficient one. If \\spad{error > \\spad{eps} * (6.0E-04)} then the next step size should be \\center{\\spad{hnext = 0.9 * hdid * (error/\\spad{eps})\\spad{**}(-1/5)}} Otherwise \\spad{hnext = 4.0 * hdid} is returned. A more detailed discussion of this and related topics can be found in the book \"Numerical Recipies\" by \\spad{W}.Press,{} \\spad{B}.\\spad{P}. Flannery,{} \\spad{S}.A. Teukolsky,{} \\spad{W}.\\spad{T}. Vetterling published by Cambridge University Press. Argument \\spad{step} is a record of 3 floating point numbers \\spad{(try , did , next)},{} \\spad{eps} is the required accuracy,{} \\spad{yscal} is the scaling vector for the difference in solutions. On input,{} \\spad{step.try} should be the guess at a step size to achieve the accuracy. On output,{} \\spad{step.did} contains the step size which achieved the accuracy and \\spad{step.next} is the next step size to use. \\blankline \\item \\spad{rk4qc(y,n,x1,step,eps,yscal,derivs,t1,t2,t3,t4,t5,t6,t7)} is the same as \\spad{rk4qc(y,n,x1,step,eps,yscal,derivs)} except that the user must provide the 7 scratch arrays \\spad{t1-t7} of size \\spad{n}. \\blankline \\item \\spad{rk4a(y,n,x1,x2,eps,h,ns,derivs)} is a driver program which uses \\spad{rk4qc} to integrate \\spad{n} ordinary differential equations starting at \\spad{x1} to \\spad{x2},{} keeping the local truncation error to within \\spad{eps} by changing the local step size. The scaling vector is defined as \\center{\\spad{yscal(i) = abs(y(i)) + abs(h*dydx(i)) + tiny}} where \\spad{y(i)} is the solution at location \\spad{x},{} \\spad{dydx} is the ordinary differential equation\\spad{'s} right hand side,{} \\spad{h} is the current step size and \\spad{tiny} is 10 times the smallest positive number representable. The user must supply an estimate for a trial step size and the maximum number of calls to \\spad{rk4qc} to use. Argument \\spad{x2} is the final point,{} \\spad{eps} is local truncation,{} \\spad{ns} is the maximum number of call to \\spad{rk4qc} to use. \\end{items}")) (|rk4f| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Integer|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4f(y,n,x1,x2,ns,derivs)} uses a 4-th order Runge-Kutta method to numerically integrate the ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector. Starting with \\spad{y} at \\spad{x1},{} this function uses \\spad{ns} fixed steps of a 4-th order Runge-Kutta integrator to advance the solution vector to \\spad{x2} and return the values in \\spad{y}. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.")) (|rk4qc| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Record| (|:| |try| (|Float|)) (|:| |did| (|Float|)) (|:| |next| (|Float|))) (|Float|) (|Vector| (|Float|)) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|))) "\\spad{rk4qc(y,n,x1,step,eps,yscal,derivs,t1,t2,t3,t4,t5,t6,t7)} is a subfunction for the numerical integration of an ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector using a 4-th order Runge-Kutta method. This function takes a 5-th order Runge-Kutta \\spad{step} with monitoring of local truncation to ensure accuracy and adjust stepsize. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.") (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Record| (|:| |try| (|Float|)) (|:| |did| (|Float|)) (|:| |next| (|Float|))) (|Float|) (|Vector| (|Float|)) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4qc(y,n,x1,step,eps,yscal,derivs)} is a subfunction for the numerical integration of an ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector using a 4-th order Runge-Kutta method. This function takes a 5-th order Runge-Kutta \\spad{step} with monitoring of local truncation to ensure accuracy and adjust stepsize. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.")) (|rk4a| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4a(y,n,x1,x2,eps,h,ns,derivs)} is a driver function for the numerical integration of an ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector using a 4-th order Runge-Kutta method. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.")) (|rk4| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|))) "\\spad{rk4(y,n,x1,h,derivs,t1,t2,t3,t4)} is the same as \\spad{rk4(y,n,x1,h,derivs)} except that you must provide 4 scratch arrays \\spad{t1}-\\spad{t4} of size \\spad{n}. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.") (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4(y,n,x1,h,derivs)} uses a 4-th order Runge-Kutta method to numerically integrate the ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector. Argument \\spad{y} is a vector of initial conditions of length \\spad{n} which upon exit contains the solution at \\spad{x1 + h},{} \\spad{n} is the number of dependent variables,{} \\spad{x1} is the initial point,{} \\spad{h} is the step size,{} and \\spad{derivs} is a function which computes the right hand side of the ordinary differential equation. For details,{} see \\spadtype{NumericalOrdinaryDifferentialEquations}."))) 
 NIL 
 NIL 
-(-792) 
+(-794) 
 ((|constructor| (NIL "This suite of routines performs numerical quadrature using algorithms derived from the basic trapezoidal rule. Because the error term of this rule contains only even powers of the step size (for open and closed versions),{} fast convergence can be obtained if the integrand is sufficiently smooth. \\blankline Each routine returns a Record of type TrapAns,{} which contains\\indent{3} \\newline value (\\spadtype{Float}):\\tab{20} estimate of the integral \\newline error (\\spadtype{Float}):\\tab{20} estimate of the error in the computation \\newline totalpts (\\spadtype{Integer}):\\tab{20} total number of function evaluations \\newline success (\\spadtype{Boolean}):\\tab{20} if the integral was computed within the user specified error criterion \\indent{0}\\indent{0} To produce this estimate,{} each routine generates an internal sequence of sub-estimates,{} denoted by {\\em S(i)},{} depending on the routine,{} to which the various convergence criteria are applied. The user must supply a relative accuracy,{} \\spad{eps_r},{} and an absolute accuracy,{} \\spad{eps_a}. Convergence is obtained when either \\center{\\spad{ABS(S(i) - S(i-1)) < eps_r * ABS(S(i-1))}} \\center{or \\spad{ABS(S(i) - S(i-1)) < eps_a}} are \\spad{true} statements. \\blankline The routines come in three families and three flavors: \\newline\\tab{3} closed:\\tab{20}romberg,{}\\tab{30}simpson,{}\\tab{42}trapezoidal \\newline\\tab{3} open: \\tab{20}rombergo,{}\\tab{30}simpsono,{}\\tab{42}trapezoidalo \\newline\\tab{3} adaptive closed:\\tab{20}aromberg,{}\\tab{30}asimpson,{}\\tab{42}atrapezoidal \\par The {\\em S(i)} for the trapezoidal family is the value of the integral using an equally spaced absicca trapezoidal rule for that level of refinement. \\par The {\\em S(i)} for the simpson family is the value of the integral using an equally spaced absicca simpson rule for that level of refinement. \\par The {\\em S(i)} for the romberg family is the estimate of the integral using an equally spaced absicca romberg method. For the \\spad{i}\\spad{-}th level,{} this is an appropriate combination of all the previous trapezodial estimates so that the error term starts with the \\spad{2*(i+1)} power only. \\par The three families come in a closed version,{} where the formulas include the endpoints,{} an open version where the formulas do not include the endpoints and an adaptive version,{} where the user is required to input the number of subintervals over which the appropriate closed family integrator will apply with the usual convergence parmeters for each subinterval. This is useful where a large number of points are needed only in a small fraction of the entire domain. \\par Each routine takes as arguments: \\newline \\spad{f}\\tab{10} integrand \\newline a\\tab{10} starting point \\newline \\spad{b}\\tab{10} ending point \\newline \\spad{eps_r}\\tab{10} relative error \\newline \\spad{eps_a}\\tab{10} absolute error \\newline \\spad{nmin} \\tab{10} refinement level when to start checking for convergence (> 1) \\newline \\spad{nmax} \\tab{10} maximum level of refinement \\par The adaptive routines take as an additional parameter \\newline \\spad{nint}\\tab{10} the number of independent intervals to apply a closed \\indented{1}{family integrator of the same name.} \\par Notes: \\newline Closed family level \\spad{i} uses \\spad{1 + 2**i} points. \\newline Open family level \\spad{i} uses \\spad{1 + 3**i} points.")) (|trapezoidalo| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{trapezoidalo(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the trapezoidal method to numerically integrate function \\spad{fn} over the open interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|simpsono| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{simpsono(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the simpson method to numerically integrate function \\spad{fn} over the open interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|rombergo| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{rombergo(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the romberg method to numerically integrate function \\spad{fn} over the open interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|trapezoidal| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{trapezoidal(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the trapezoidal method to numerically integrate function \\spadvar{\\spad{fn}} over the closed interval \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|simpson| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{simpson(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the simpson method to numerically integrate function \\spad{fn} over the closed interval \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|romberg| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{romberg(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the romberg method to numerically integrate function \\spadvar{\\spad{fn}} over the closed interval \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|atrapezoidal| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{atrapezoidal(fn,a,b,epsrel,epsabs,nmin,nmax,nint)} uses the adaptive trapezoidal method to numerically integrate function \\spad{fn} over the closed interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax},{} and where \\spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|asimpson| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{asimpson(fn,a,b,epsrel,epsabs,nmin,nmax,nint)} uses the adaptive simpson method to numerically integrate function \\spad{fn} over the closed interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax},{} and where \\spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|aromberg| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{aromberg(fn,a,b,epsrel,epsabs,nmin,nmax,nint)} uses the adaptive romberg method to numerically integrate function \\spad{fn} over the closed interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax},{} and where \\spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details."))) 
 NIL 
 NIL 
-(-793 |Curve|) 
+(-795 |Curve|) 
 ((|constructor| (NIL "\\indented{1}{Author: Clifton \\spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.")) (|tube| (((|TubePlot| |#1|) |#1| (|DoubleFloat|) (|Integer|)) "\\spad{tube(c,r,n)} creates a tube of radius \\spad{r} around the curve \\spad{c}."))) 
 NIL 
 NIL 
-(-794) 
+(-796) 
 ((|constructor| (NIL "Ordered sets which are also abelian groups,{} such that the addition preserves the ordering."))) 
 NIL 
 NIL 
-(-795) 
+(-797) 
 ((|constructor| (NIL "Ordered sets which are also abelian monoids,{} such that the addition preserves the ordering."))) 
 NIL 
 NIL 
-(-796) 
+(-798) 
 ((|constructor| (NIL "This domain is an OrderedAbelianMonoid with a \\spadfun{sup} operation added. The purpose of the \\spadfun{sup} operator in this domain is to act as a supremum with respect to the partial order imposed by \\spadop{-},{} rather than with respect to the total \\spad{>} order (since that is \"max\"). \\blankline")) (|sup| (($ $ $) "\\spad{sup(x,y)} returns the least element from which both \\spad{x} and \\spad{y} can be subtracted."))) 
 NIL 
 NIL 
-(-797) 
+(-799) 
 ((|constructor| (NIL "Ordered sets which are also abelian semigroups,{} such that the addition preserves the ordering. \\indented{2}{\\spad{ x < y => x+z < y+z}}"))) 
 NIL 
 NIL 
-(-798 S R) 
+(-800 S R) 
 ((|constructor| (NIL "OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.")) (|inv| (($ $) "\\spad{inv(o)} returns the inverse of \\spad{o} if it exists.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(o)} returns the real part if all seven imaginary parts are 0,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(o)} returns the real part if all seven imaginary parts are 0. Error: if \\spad{o} is not rational.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(o)} tests if \\spad{o} is rational,{} \\spadignore{i.e.} that all seven imaginary parts are 0.")) (|abs| ((|#2| $) "\\spad{abs(o)} computes the absolute value of an octonion,{} equal to the square root of the \\spadfunFrom{norm}{Octonion}.")) (|octon| (($ |#2| |#2| |#2| |#2| |#2| |#2| |#2| |#2|) "\\spad{octon(re,ri,rj,rk,rE,rI,rJ,rK)} constructs an octonion from scalars.")) (|norm| ((|#2| $) "\\spad{norm(o)} returns the norm of an octonion,{} equal to the sum of the squares of its coefficients.")) (|imagK| ((|#2| $) "\\spad{imagK(o)} extracts the imaginary \\spad{K} part of octonion \\spad{o}.")) (|imagJ| ((|#2| $) "\\spad{imagJ(o)} extracts the imaginary \\spad{J} part of octonion \\spad{o}.")) (|imagI| ((|#2| $) "\\spad{imagI(o)} extracts the imaginary \\spad{I} part of octonion \\spad{o}.")) (|imagE| ((|#2| $) "\\spad{imagE(o)} extracts the imaginary \\spad{E} part of octonion \\spad{o}.")) (|imagk| ((|#2| $) "\\spad{imagk(o)} extracts the \\spad{k} part of octonion \\spad{o}.")) (|imagj| ((|#2| $) "\\spad{imagj(o)} extracts the \\spad{j} part of octonion \\spad{o}.")) (|imagi| ((|#2| $) "\\spad{imagi(o)} extracts the \\spad{i} part of octonion \\spad{o}.")) (|real| ((|#2| $) "\\spad{real(o)} extracts real part of octonion \\spad{o}.")) (|conjugate| (($ $) "\\spad{conjugate(o)} negates the imaginary parts \\spad{i},{}\\spad{j},{}\\spad{k},{}\\spad{E},{}\\spad{I},{}\\spad{J},{}\\spad{K} of octonian \\spad{o}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-549))) (|HasCategory| |#2| (QUOTE (-1064))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-371)))) 
-(-799 R) 
+((|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-372)))) 
+(-801 R) 
 ((|constructor| (NIL "OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.")) (|inv| (($ $) "\\spad{inv(o)} returns the inverse of \\spad{o} if it exists.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(o)} returns the real part if all seven imaginary parts are 0,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(o)} returns the real part if all seven imaginary parts are 0. Error: if \\spad{o} is not rational.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(o)} tests if \\spad{o} is rational,{} \\spadignore{i.e.} that all seven imaginary parts are 0.")) (|abs| ((|#1| $) "\\spad{abs(o)} computes the absolute value of an octonion,{} equal to the square root of the \\spadfunFrom{norm}{Octonion}.")) (|octon| (($ |#1| |#1| |#1| |#1| |#1| |#1| |#1| |#1|) "\\spad{octon(re,ri,rj,rk,rE,rI,rJ,rK)} constructs an octonion from scalars.")) (|norm| ((|#1| $) "\\spad{norm(o)} returns the norm of an octonion,{} equal to the sum of the squares of its coefficients.")) (|imagK| ((|#1| $) "\\spad{imagK(o)} extracts the imaginary \\spad{K} part of octonion \\spad{o}.")) (|imagJ| ((|#1| $) "\\spad{imagJ(o)} extracts the imaginary \\spad{J} part of octonion \\spad{o}.")) (|imagI| ((|#1| $) "\\spad{imagI(o)} extracts the imaginary \\spad{I} part of octonion \\spad{o}.")) (|imagE| ((|#1| $) "\\spad{imagE(o)} extracts the imaginary \\spad{E} part of octonion \\spad{o}.")) (|imagk| ((|#1| $) "\\spad{imagk(o)} extracts the \\spad{k} part of octonion \\spad{o}.")) (|imagj| ((|#1| $) "\\spad{imagj(o)} extracts the \\spad{j} part of octonion \\spad{o}.")) (|imagi| ((|#1| $) "\\spad{imagi(o)} extracts the \\spad{i} part of octonion \\spad{o}.")) (|real| ((|#1| $) "\\spad{real(o)} extracts real part of octonion \\spad{o}.")) (|conjugate| (($ $) "\\spad{conjugate(o)} negates the imaginary parts \\spad{i},{}\\spad{j},{}\\spad{k},{}\\spad{E},{}\\spad{I},{}\\spad{J},{}\\spad{K} of octonian \\spad{o}."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-800) 
+(-802) 
 ((|constructor| (NIL "Ordered sets which are also abelian cancellation monoids,{} such that the addition preserves the ordering."))) 
 NIL 
 NIL 
-(-801 R) 
+(-803 R) 
 ((|constructor| (NIL "Octonion implements octonions (Cayley-Dixon algebra) over a commutative ring,{} an eight-dimensional non-associative algebra,{} doubling the quaternions in the same way as doubling the complex numbers to get the quaternions the main constructor function is {\\em octon} which takes 8 arguments: the real part,{} the \\spad{i} imaginary part,{} the \\spad{j} imaginary part,{} the \\spad{k} imaginary part,{} (as with quaternions) and in addition the imaginary parts \\spad{E},{} \\spad{I},{} \\spad{J},{} \\spad{K}.")) (|octon| (($ (|Quaternion| |#1|) (|Quaternion| |#1|)) "\\spad{octon(qe,qE)} constructs an octonion from two quaternions using the relation {\\em O = Q + QE}."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (LIST (QUOTE -518) (QUOTE (-1181)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -288) (|devaluate| |#1|) (|devaluate| |#1|))) (-3962 (|HasCategory| (-1000 |#1|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-1000 |#1|) (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-1064))) (|HasCategory| |#1| (QUOTE (-549))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1000 |#1|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-1000 |#1|) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550))))) 
-(-802 -3962 R OS S) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -289) (|devaluate| |#1|) (|devaluate| |#1|))) (-3969 (|HasCategory| (-1002 |#1|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-1002 |#1|) (LIST (QUOTE -1044) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (QUOTE (-550))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| (-1002 |#1|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-1002 |#1|) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551))))) 
+(-804 -3969 R OS S) 
 ((|constructor| (NIL "OctonionCategoryFunctions2 implements functions between two octonion domains defined over different rings. The function map is used to coerce between octonion types.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the component parts of the octonion \\spad{u}."))) 
 NIL 
 NIL 
-(-803) 
+(-805) 
 ((|ODESolve| (((|Result|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{ODESolve(args)} performs the integration of the function given the strategy or method returned by \\axiomFun{measure}.")) (|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|))) (|RoutinesTable|) (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve a problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far."))) 
 NIL 
 NIL 
-(-804 R -3498 L) 
+(-806 R -3505 L) 
 ((|constructor| (NIL "Solution of linear ordinary differential equations,{} constant coefficient case.")) (|constDsolve| (((|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Symbol|)) "\\spad{constDsolve(op, g, x)} returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular solution of the equation \\spad{op y = g},{} and the \\spad{yi}\\spad{'s} form a basis for the solutions of \\spad{op y = 0}."))) 
 NIL 
 NIL 
-(-805 R -3498) 
+(-807 R -3505) 
 ((|constructor| (NIL "\\spad{ElementaryFunctionODESolver} provides the top-level functions for finding closed form solutions of ordinary differential equations and initial value problems.")) (|solve| (((|Union| |#2| #1="failed") |#2| (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{solve(eq, y, x = a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{eq, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \\spad{dy/dx = f(x,y)}.") (((|Union| |#2| #1#) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{solve(eq, y, x = a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{eq, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \\spad{dy/dx = f(x,y)}.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#2| #2="failed") |#2| (|BasicOperator|) (|Symbol|)) "\\spad{solve(eq, y, x)} returns either a solution of the ordinary differential equation \\spad{eq} or \"failed\" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{f(x,y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \\spad{f}(\\spad{x},{}\\spad{y})},{} a solution is of the form \\spad{h(x,y)} where \\spad{h(x,y) = c} is a first integral of the equation for any constant \\spad{c}.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#2| #2#) (|Equation| |#2|) (|BasicOperator|) (|Symbol|)) "\\spad{solve(eq, y, x)} returns either a solution of the ordinary differential equation \\spad{eq} or \"failed\" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{f(x,y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \\spad{f}(\\spad{x},{}\\spad{y})},{} a solution is of the form \\spad{h(x,y)} where \\spad{h(x,y) = c} is a first integral of the equation for any constant \\spad{c}; error if the equation is not one of those 2 forms.") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|List| |#2|) (|List| (|BasicOperator|)) (|Symbol|)) "\\spad{solve([eq_1,...,eq_n], [y_1,...,y_n], x)} returns either \"failed\" or,{} if the equations form a fist order linear system,{} a solution of the form \\spad{[y_p, [b_1,...,b_n]]} where \\spad{h_p} is a particular solution and \\spad{[b_1,...b_m]} are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Symbol|)) "\\spad{solve([eq_1,...,eq_n], [y_1,...,y_n], x)} returns either \"failed\" or,{} if the equations form a fist order linear system,{} a solution of the form \\spad{[y_p, [b_1,...,b_n]]} where \\spad{h_p} is a particular solution and \\spad{[b_1,...b_m]} are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system") (((|Union| (|List| (|Vector| |#2|)) "failed") (|Matrix| |#2|) (|Symbol|)) "\\spad{solve(m, x)} returns a basis for the solutions of \\spad{D y = m y}. \\spad{x} is the dependent variable.") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|Matrix| |#2|) (|Vector| |#2|) (|Symbol|)) "\\spad{solve(m, v, x)} returns \\spad{[v_p, [v_1,...,v_m]]} such that the solutions of the system \\spad{D y = m y + v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{D y = m y}. \\spad{x} is the dependent variable."))) 
 NIL 
 NIL 
-(-806) 
+(-808) 
 ((|constructor| (NIL "\\axiom{ODEIntensityFunctionsTable()} provides a dynamic table and a set of functions to store details found out about sets of ODE\\spad{'s}.")) (|showIntensityFunctions| (((|Union| (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|))) "failed") (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{showIntensityFunctions(k)} returns the entries in the table of intensity functions \\spad{k}.")) (|insert!| (($ (|Record| (|:| |key| (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|)))))) "\\spad{insert!(r)} inserts an entry \\spad{r} into theIFTable")) (|iFTable| (($ (|List| (|Record| (|:| |key| (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |stiffness| (|Float|)) (|:| |stability| (|Float|)) (|:| |expense| (|Float|)) (|:| |accuracy| (|Float|)) (|:| |intermediateResults| (|Float|))))))) "\\spad{iFTable(l)} creates an intensity-functions table from the elements of \\spad{l}.")) (|keys| (((|List| (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) $) "\\spad{keys(tab)} returns the list of keys of \\spad{f}")) (|clearTheIFTable| (((|Void|)) "\\spad{clearTheIFTable()} clears the current table of intensity functions.")) (|showTheIFTable| (($) "\\spad{showTheIFTable()} returns the current table of intensity functions."))) 
 NIL 
 NIL 
-(-807 R -3498) 
+(-809 R -3505) 
 ((|constructor| (NIL "\\spadtype{ODEIntegration} provides an interface to the integrator. This package is intended for use by the differential equations solver but not at top-level.")) (|diff| (((|Mapping| |#2| |#2|) (|Symbol|)) "\\spad{diff(x)} returns the derivation with respect to \\spad{x}.")) (|expint| ((|#2| |#2| (|Symbol|)) "\\spad{expint(f, x)} returns e^{the integral of \\spad{f} with respect to \\spad{x}}.")) (|int| ((|#2| |#2| (|Symbol|)) "\\spad{int(f, x)} returns the integral of \\spad{f} with respect to \\spad{x}."))) 
 NIL 
 NIL 
-(-808) 
+(-810) 
 ((|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalODEProblem|) (|RoutinesTable|)) "\\spad{measure(prob,R)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical ODE problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} listed in \\axiom{\\spad{R}} of \\axiom{category} \\axiomType{OrdinaryDifferentialEquationsSolverCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information. It predicts the likely most effective NAG numerical Library routine to solve the input set of ODEs by checking various attributes of the system of ODEs and calculating a measure of compatibility of each routine to these attributes.") (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalODEProblem|)) "\\spad{measure(prob)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical ODE problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} of \\axiom{category} \\axiomType{OrdinaryDifferentialEquationsSolverCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information. It predicts the likely most effective NAG numerical Library routine to solve the input set of ODEs by checking various attributes of the system of ODEs and calculating a measure of compatibility of each routine to these attributes.")) (|solve| (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Expression| (|Float|)) (|List| (|Float|)) (|Float|) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,G,intVals,epsabs,epsrel)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to an absolute error requirement \\axiom{\\spad{epsabs}} and relative error \\axiom{\\spad{epsrel}}. The values of \\spad{Y}[1]..\\spad{Y}[\\spad{n}] will be output for the values of \\spad{X} in \\axiom{\\spad{intVals}}. The calculation will stop if the function \\spad{G}(\\spad{X},{}\\spad{Y}[1],{}..,{}\\spad{Y}[\\spad{n}]) evaluates to zero before \\spad{X} = \\spad{xEnd}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Expression| (|Float|)) (|List| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,G,intVals,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. The values of \\spad{Y}[1]..\\spad{Y}[\\spad{n}] will be output for the values of \\spad{X} in \\axiom{\\spad{intVals}}. The calculation will stop if the function \\spad{G}(\\spad{X},{}\\spad{Y}[1],{}..,{}\\spad{Y}[\\spad{n}]) evaluates to zero before \\spad{X} = \\spad{xEnd}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|List| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,intVals,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. The values of \\spad{Y}[1]..\\spad{Y}[\\spad{n}] will be output for the values of \\spad{X} in \\axiom{\\spad{intVals}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Expression| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,G,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. The calculation will stop if the function \\spad{G}(\\spad{X},{}\\spad{Y}[1],{}..,{}\\spad{Y}[\\spad{n}]) evaluates to zero before \\spad{X} = \\spad{xEnd}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|)) (|Float|)) "\\spad{solve(f,xStart,xEnd,yInitial,tol)} is a top level ANNA function to solve numerically a system of ordinary differential equations,{} \\axiom{\\spad{f}},{} \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}] from \\axiom{\\spad{xStart}} to \\axiom{\\spad{xEnd}} with the initial values for \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (\\axiom{\\spad{yInitial}}) to a tolerance \\axiom{\\spad{tol}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|Vector| (|Expression| (|Float|))) (|Float|) (|Float|) (|List| (|Float|))) "\\spad{solve(f,xStart,xEnd,yInitial)} is a top level ANNA function to solve numerically a system of ordinary differential equations \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}],{} together with a starting value for \\spad{X} and \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (called the initial conditions) and a final value of \\spad{X}. A default value is used for the accuracy requirement. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|NumericalODEProblem|) (|RoutinesTable|)) "\\spad{solve(odeProblem,R)} is a top level ANNA function to solve numerically a system of ordinary differential equations \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}],{} together with starting values for \\spad{X} and \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (called the initial conditions),{} a final value of \\spad{X},{} an accuracy requirement and any intermediate points at which the result is required. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} contained in the table of routines \\axiom{\\spad{R}} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine.") (((|Result|) (|NumericalODEProblem|)) "\\spad{solve(odeProblem)} is a top level ANNA function to solve numerically a system of ordinary differential equations \\spadignore{i.e.} equations for the derivatives \\spad{Y}[1]'..\\spad{Y}[\\spad{n}]' defined in terms of \\spad{X},{}\\spad{Y}[1]..\\spad{Y}[\\spad{n}],{} together with starting values for \\spad{X} and \\spad{Y}[1]..\\spad{Y}[\\spad{n}] (called the initial conditions),{} a final value of \\spad{X},{} an accuracy requirement and any intermediate points at which the result is required. \\blankline It iterates over the \\axiom{domains} of \\axiomType{OrdinaryDifferentialEquationsSolverCategory} to get the name and other relevant information of the the (domain of the) numerical routine likely to be the most appropriate,{} \\spadignore{i.e.} have the best \\axiom{measure}. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of ODE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine."))) 
 NIL 
 NIL 
-(-809 -3498 UP UPUP R) 
+(-811 -3505 UP UPUP R) 
 ((|constructor| (NIL "In-field solution of an linear ordinary differential equation,{} pure algebraic case.")) (|algDsolve| (((|Record| (|:| |particular| (|Union| |#4| "failed")) (|:| |basis| (|List| |#4|))) (|LinearOrdinaryDifferentialOperator1| |#4|) |#4|) "\\spad{algDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no solution in \\spad{R}. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{y_i's} form a basis for the solutions in \\spad{R} of the homogeneous equation."))) 
 NIL 
 NIL 
-(-810 -3498 UP L LQ) 
+(-812 -3505 UP L LQ) 
 ((|constructor| (NIL "\\spad{PrimitiveRatDE} provides functions for in-field solutions of linear \\indented{1}{ordinary differential equations,{} in the transcendental case.} \\indented{1}{The derivation to use is given by the parameter \\spad{L}.}")) (|splitDenominator| (((|Record| (|:| |eq| |#3|) (|:| |rh| (|List| (|Fraction| |#2|)))) |#4| (|List| (|Fraction| |#2|))) "\\spad{splitDenominator(op, [g1,...,gm])} returns \\spad{op0, [h1,...,hm]} such that the equations \\spad{op y = c1 g1 + ... + cm gm} and \\spad{op0 y = c1 h1 + ... + cm hm} have the same solutions.")) (|indicialEquation| ((|#2| |#4| |#1|) "\\spad{indicialEquation(op, a)} returns the indicial equation of \\spad{op} at \\spad{a}.") ((|#2| |#3| |#1|) "\\spad{indicialEquation(op, a)} returns the indicial equation of \\spad{op} at \\spad{a}.")) (|indicialEquations| (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#4| |#2|) "\\spad{indicialEquations(op, p)} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op} above the roots of \\spad{p},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#4|) "\\spad{indicialEquations op} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#3| |#2|) "\\spad{indicialEquations(op, p)} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op} above the roots of \\spad{p},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#3|) "\\spad{indicialEquations op} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}\\spad{'s} are the affine singularities of \\spad{op},{} and the \\spad{e_i}\\spad{'s} are the indicial equations at each \\spad{d_i}.")) (|denomLODE| ((|#2| |#3| (|List| (|Fraction| |#2|))) "\\spad{denomLODE(op, [g1,...,gm])} returns a polynomial \\spad{d} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{p/d} for some polynomial \\spad{p}.") (((|Union| |#2| "failed") |#3| (|Fraction| |#2|)) "\\spad{denomLODE(op, g)} returns a polynomial \\spad{d} such that any rational solution of \\spad{op y = g} is of the form \\spad{p/d} for some polynomial \\spad{p},{} and \"failed\",{} if the equation has no rational solution."))) 
 NIL 
 NIL 
-(-811) 
+(-813) 
 ((|retract| (((|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|))) $) "\\spad{retract(x)} \\undocumented{}")) (|coerce| (($ (|Record| (|:| |xinit| (|DoubleFloat|)) (|:| |xend| (|DoubleFloat|)) (|:| |fn| (|Vector| (|Expression| (|DoubleFloat|)))) (|:| |yinit| (|List| (|DoubleFloat|))) (|:| |intvals| (|List| (|DoubleFloat|))) (|:| |g| (|Expression| (|DoubleFloat|))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{coerce(x)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-812 -3498 UP L LQ) 
+(-814 -3505 UP L LQ) 
 ((|constructor| (NIL "In-field solution of Riccati equations,{} primitive case.")) (|changeVar| ((|#3| |#3| (|Fraction| |#2|)) "\\spad{changeVar(+/[ai D^i], a)} returns the operator \\spad{+/[ai (D+a)^i]}.") ((|#3| |#3| |#2|) "\\spad{changeVar(+/[ai D^i], a)} returns the operator \\spad{+/[ai (D+a)^i]}.")) (|singRicDE| (((|List| (|Record| (|:| |frac| (|Fraction| |#2|)) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{singRicDE(op, zeros, ezfactor)} returns \\spad{[[f1, L1], [f2, L2], ... , [fk, Lk]]} such that the singular part of any rational solution of the associated Riccati equation of \\spad{op y=0} must be one of the \\spad{fi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z=y e^{-int p}} is \\spad{Li z=0}. \\spad{zeros(C(x),H(x,y))} returns all the \\spad{P_i(x)}\\spad{'s} such that \\spad{H(x,P_i(x)) = 0 modulo C(x)}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.")) (|polyRicDE| (((|List| (|Record| (|:| |poly| |#2|) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#1|) |#2|)) "\\spad{polyRicDE(op, zeros)} returns \\spad{[[p1, L1], [p2, L2], ... , [pk, Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y=0} must be one of the \\spad{pi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z=y e^{-int p}} is \\spad{Li z =0}. \\spad{zeros} is a zero finder in \\spad{UP}.")) (|constantCoefficientRicDE| (((|List| (|Record| (|:| |constant| |#1|) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#1|) |#2|)) "\\spad{constantCoefficientRicDE(op, ric)} returns \\spad{[[a1, L1], [a2, L2], ... , [ak, Lk]]} such that any rational solution with no polynomial part of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{ai}\\spad{'s} in which case the equation for \\spad{z = y e^{-int ai}} is \\spad{Li z = 0}. \\spad{ric} is a Riccati equation solver over \\spad{F},{} whose input is the associated linear equation.")) (|leadingCoefficientRicDE| (((|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |eq| |#2|))) |#3|) "\\spad{leadingCoefficientRicDE(op)} returns \\spad{[[m1, p1], [m2, p2], ... , [mk, pk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must have degree \\spad{mj} for some \\spad{j},{} and its leading coefficient is then a zero of \\spad{pj}. In addition,{}\\spad{m1>m2> ... >mk}.")) (|denomRicDE| ((|#2| |#3|) "\\spad{denomRicDE(op)} returns a polynomial \\spad{d} such that any rational solution of the associated Riccati equation of \\spad{op y = 0} is of the form \\spad{p/d + q'/q + r} for some polynomials \\spad{p} and \\spad{q} and a reduced \\spad{r}. Also,{} \\spad{deg(p) < deg(d)} and {\\spad{gcd}(\\spad{d},{}\\spad{q}) = 1}."))) 
 NIL 
 NIL 
-(-813 -3498 UP) 
+(-815 -3505 UP) 
 ((|constructor| (NIL "\\spad{RationalLODE} provides functions for in-field solutions of linear \\indented{1}{ordinary differential equations,{} in the rational case.}")) (|indicialEquationAtInfinity| ((|#2| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))) "\\spad{indicialEquationAtInfinity op} returns the indicial equation of \\spad{op} at infinity.") ((|#2| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{indicialEquationAtInfinity op} returns the indicial equation of \\spad{op} at infinity.")) (|ratDsolve| (((|Record| (|:| |basis| (|List| (|Fraction| |#2|))) (|:| |mat| (|Matrix| |#1|))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|List| (|Fraction| |#2|))) "\\spad{ratDsolve(op, [g1,...,gm])} returns \\spad{[[h1,...,hq], M]} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{d1 h1 + ... + dq hq} where \\spad{M [d1,...,dq,c1,...,cm] = 0}.") (((|Record| (|:| |particular| (|Union| (|Fraction| |#2|) #1="failed")) (|:| |basis| (|List| (|Fraction| |#2|)))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{ratDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no rational solution. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{yi}\\spad{'s} form a basis for the rational solutions of the homogeneous equation.") (((|Record| (|:| |basis| (|List| (|Fraction| |#2|))) (|:| |mat| (|Matrix| |#1|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|List| (|Fraction| |#2|))) "\\spad{ratDsolve(op, [g1,...,gm])} returns \\spad{[[h1,...,hq], M]} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{d1 h1 + ... + dq hq} where \\spad{M [d1,...,dq,c1,...,cm] = 0}.") (((|Record| (|:| |particular| (|Union| (|Fraction| |#2|) #1#)) (|:| |basis| (|List| (|Fraction| |#2|)))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{ratDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no rational solution. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{yi}\\spad{'s} form a basis for the rational solutions of the homogeneous equation."))) 
 NIL 
 NIL 
-(-814 -3498 L UP A LO) 
+(-816 -3505 L UP A LO) 
 ((|constructor| (NIL "Elimination of an algebraic from the coefficentss of a linear ordinary differential equation.")) (|reduceLODE| (((|Record| (|:| |mat| (|Matrix| |#2|)) (|:| |vec| (|Vector| |#1|))) |#5| |#4|) "\\spad{reduceLODE(op, g)} returns \\spad{[m, v]} such that any solution in \\spad{A} of \\spad{op z = g} is of the form \\spad{z = (z_1,...,z_m) . (b_1,...,b_m)} where the \\spad{b_i's} are the basis of \\spad{A} over \\spad{F} returned by \\spadfun{basis}() from \\spad{A},{} and the \\spad{z_i's} satisfy the differential system \\spad{M.z = v}."))) 
 NIL 
 NIL 
-(-815 -3498 UP) 
+(-817 -3505 UP) 
 ((|constructor| (NIL "In-field solution of Riccati equations,{} rational case.")) (|polyRicDE| (((|List| (|Record| (|:| |poly| |#2|) (|:| |eq| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{polyRicDE(op, zeros)} returns \\spad{[[p1, L1], [p2, L2], ... , [pk,Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{pi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z = y e^{-int p}} is \\spad{Li z = 0}. \\spad{zeros} is a zero finder in \\spad{UP}.")) (|singRicDE| (((|List| (|Record| (|:| |frac| (|Fraction| |#2|)) (|:| |eq| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{singRicDE(op, ezfactor)} returns \\spad{[[f1,L1], [f2,L2],..., [fk,Lk]]} such that the singular \\spad{++} part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{fi}\\spad{'s} (up to the constant coefficient),{} in which case the equation for \\spad{z = y e^{-int ai}} is \\spad{Li z = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.")) (|ricDsolve| (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))) "\\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, zeros, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{ricDsolve(op, zeros)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, zeros, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{ricDsolve(op, zeros)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}."))) 
 NIL 
 ((|HasCategory| |#1| (QUOTE (-27)))) 
-(-816 -3498 LO) 
+(-818 -3505 LO) 
 ((|constructor| (NIL "SystemODESolver provides tools for triangulating and solving some systems of linear ordinary differential equations.")) (|solveInField| (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) "failed")) (|:| |basis| (|List| (|Vector| |#1|)))) (|Matrix| |#2|) (|Vector| |#1|) (|Mapping| (|Record| (|:| |particular| (|Union| |#1| "failed")) (|:| |basis| (|List| |#1|))) |#2| |#1|)) "\\spad{solveInField(m, v, solve)} returns \\spad{[[v_1,...,v_m], v_p]} such that the solutions in \\spad{F} of the system \\spad{m x = v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{m x = 0}. Argument \\spad{solve} is a function for solving a single linear ordinary differential equation in \\spad{F}.")) (|solve| (((|Union| (|Record| (|:| |particular| (|Vector| |#1|)) (|:| |basis| (|Matrix| |#1|))) "failed") (|Matrix| |#1|) (|Vector| |#1|) (|Mapping| (|Union| (|Record| (|:| |particular| |#1|) (|:| |basis| (|List| |#1|))) "failed") |#2| |#1|)) "\\spad{solve(m, v, solve)} returns \\spad{[[v_1,...,v_m], v_p]} such that the solutions in \\spad{F} of the system \\spad{D x = m x + v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{D x = m x}. Argument \\spad{solve} is a function for solving a single linear ordinary differential equation in \\spad{F}.")) (|triangulate| (((|Record| (|:| |mat| (|Matrix| |#2|)) (|:| |vec| (|Vector| |#1|))) (|Matrix| |#2|) (|Vector| |#1|)) "\\spad{triangulate(m, v)} returns \\spad{[m_0, v_0]} such that \\spad{m_0} is upper triangular and the system \\spad{m_0 x = v_0} is equivalent to \\spad{m x = v}.") (((|Record| (|:| A (|Matrix| |#1|)) (|:| |eqs| (|List| (|Record| (|:| C (|Matrix| |#1|)) (|:| |g| (|Vector| |#1|)) (|:| |eq| |#2|) (|:| |rh| |#1|))))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{triangulate(M,v)} returns \\spad{A,[[C_1,g_1,L_1,h_1],...,[C_k,g_k,L_k,h_k]]} such that under the change of variable \\spad{y = A z},{} the first order linear system \\spad{D y = M y + v} is uncoupled as \\spad{D z_i = C_i z_i + g_i} and each \\spad{C_i} is a companion matrix corresponding to the scalar equation \\spad{L_i z_j = h_i}."))) 
 NIL 
 NIL 
-(-817 -3498 LODO) 
+(-819 -3505 LODO) 
 ((|constructor| (NIL "\\spad{ODETools} provides tools for the linear ODE solver.")) (|particularSolution| (((|Union| |#1| "failed") |#2| |#1| (|List| |#1|) (|Mapping| |#1| |#1|)) "\\spad{particularSolution(op, g, [f1,...,fm], I)} returns a particular solution \\spad{h} of the equation \\spad{op y = g} where \\spad{[f1,...,fm]} are linearly independent and \\spad{op(fi)=0}. The value \"failed\" is returned if no particular solution is found. Note: the method of variations of parameters is used.")) (|variationOfParameters| (((|Union| (|Vector| |#1|) "failed") |#2| |#1| (|List| |#1|)) "\\spad{variationOfParameters(op, g, [f1,...,fm])} returns \\spad{[u1,...,um]} such that a particular solution of the equation \\spad{op y = g} is \\spad{f1 int(u1) + ... + fm int(um)} where \\spad{[f1,...,fm]} are linearly independent and \\spad{op(fi)=0}. The value \"failed\" is returned if \\spad{m < n} and no particular solution is found.")) (|wronskianMatrix| (((|Matrix| |#1|) (|List| |#1|) (|NonNegativeInteger|)) "\\spad{wronskianMatrix([f1,...,fn], q, D)} returns the \\spad{q x n} matrix \\spad{m} whose i^th row is \\spad{[f1^(i-1),...,fn^(i-1)]}.") (((|Matrix| |#1|) (|List| |#1|)) "\\spad{wronskianMatrix([f1,...,fn])} returns the \\spad{n x n} matrix \\spad{m} whose i^th row is \\spad{[f1^(i-1),...,fn^(i-1)]}."))) 
 NIL 
 NIL 
-(-818 -3023 S |f|) 
+(-820 -3030 S |f|) 
 ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The ordering on the type is determined by its third argument which represents the less than function on vectors. This type is a suitable third argument for \\spadtype{GeneralDistributedMultivariatePolynomial}."))) 
-((-4421 |has| |#2| (-1053)) (-4422 |has| |#2| (-1053)) (-4424 |has| |#2| (-6 -4424)) ((-4429 "*") |has| |#2| (-173)) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-1053)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#2| (QUOTE (-366))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-1053)))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-366)))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-796))) (-3962 (|HasCategory| |#2| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-851)))) (|HasCategory| |#2| (QUOTE (-851))) (|HasCategory| |#2| (QUOTE (-729))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-1053)))) (|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (-3962 (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-1053))) 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 ((|constructor| (NIL "\\spadtype{OrderlyDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is orderly. This is analogous to the domain \\spadtype{Polynomial}. \\blankline"))) 
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-(-820 |Kernels| R |var|) 
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+((|HasCategory| |#1| (QUOTE (-916))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-823 (-1183)) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-823 (-1183)) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-823 (-1183)) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-823 (-1183)) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-823 (-1183)) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(-822 |Kernels| R |var|) 
 ((|constructor| (NIL "This constructor produces an ordinary differential ring from a partial differential ring by specifying a variable."))) 
-(((-4429 "*") |has| |#2| (-366)) (-4420 |has| |#2| (-366)) (-4425 |has| |#2| (-366)) (-4419 |has| |#2| (-366)) (-4424 . T) (-4422 . T) (-4421 . T)) 
-((|HasCategory| |#2| (QUOTE (-366)))) 
-(-821 S) 
+(((-4436 "*") |has| |#2| (-367)) (-4427 |has| |#2| (-367)) (-4432 |has| |#2| (-367)) (-4426 |has| |#2| (-367)) (-4431 . T) (-4429 . T) (-4428 . T)) 
+((|HasCategory| |#2| (QUOTE (-367)))) 
+(-823 S) 
 ((|constructor| (NIL "\\spadtype{OrderlyDifferentialVariable} adds a commonly used orderly ranking to the set of derivatives of an ordered list of differential indeterminates. An orderly ranking is a ranking \\spadfun{<} of the derivatives with the property that for two derivatives \\spad{u} and \\spad{v},{} \\spad{u} \\spadfun{<} \\spad{v} if the \\spadfun{order} of \\spad{u} is less than that of \\spad{v}. This domain belongs to \\spadtype{DifferentialVariableCategory}. It defines \\spadfun{weight} to be just \\spadfun{order},{} and it defines an orderly ranking \\spadfun{<} on derivatives \\spad{u} via the lexicographic order on the pair (\\spadfun{order}(\\spad{u}),{} \\spadfun{variable}(\\spad{u}))."))) 
 NIL 
 NIL 
-(-822 S) 
+(-824 S) 
 ((|constructor| (NIL "\\indented{3}{The free monoid on a set \\spad{S} is the monoid of finite products of} the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}\\spad{'s} are in \\spad{S},{} and the \\spad{ni}\\spad{'s} are non-negative integers. The multiplication is not commutative. For two elements \\spad{x} and \\spad{y} the relation \\spad{x < y} holds if either \\spad{length(x) < length(y)} holds or if these lengths are equal and if \\spad{x} is smaller than \\spad{y} \\spad{w}.\\spad{r}.\\spad{t}. the lexicographical ordering induced by \\spad{S}. This domain inherits implementation from \\spadtype{FreeMonoid}.")) (|varList| (((|List| |#1|) $) "\\spad{varList(x)} returns the list of variables of \\spad{x}.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length(x)} returns the length of \\spad{x}.")) (|div| (((|Union| (|Record| (|:| |lm| $) (|:| |rm| $)) "failed") $ $) "\\spad{x div y} returns the left and right exact quotients of \\spad{x} by \\spad{y},{} that is \\spad{[l, r]} such that \\spad{x = l * y * r}. \"failed\" is returned iff \\spad{x} is not of the form \\spad{l * y * r}. monomial of \\spad{x}.")) (|rquo| (((|Union| $ "failed") $ |#1|) "\\spad{rquo(x, s)} returns the exact right quotient of \\spad{x} by \\spad{s}.")) (|lquo| (((|Union| $ "failed") $ |#1|) "\\spad{lquo(x, s)} returns the exact left quotient of \\spad{x} by \\spad{s}.")) (|lexico| (((|Boolean|) $ $) "\\spad{lexico(x,y)} returns \\spad{true} iff \\spad{x} is smaller than \\spad{y} \\spad{w}.\\spad{r}.\\spad{t}. the pure lexicographical ordering induced by \\spad{S}.")) (|mirror| (($ $) "\\spad{mirror(x)} returns the reversed word of \\spad{x}.")) (|rest| (($ $) "\\spad{rest(x)} returns \\spad{x} except the first letter.")) (|first| ((|#1| $) "\\spad{first(x)} returns the first letter of \\spad{x}."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-853)))) 
-(-823) 
+((|HasCategory| |#1| (QUOTE (-855)))) 
+(-825) 
 ((|constructor| (NIL "The category of ordered commutative integral domains,{} where ordering and the arithmetic operations are compatible \\blankline"))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-824) 
+(-826) 
 ((|constructor| (NIL "\\spadtype{OpenMath} provides operations for exporting an object in OpenMath format.")) (|OMwrite| (((|Void|) (|OpenMathDevice|) $ (|Boolean|)) "\\spad{OMwrite(dev, u, true)} writes the OpenMath form of \\axiom{\\spad{u}} to the OpenMath device \\axiom{\\spad{dev}} as a complete OpenMath object; OMwrite(\\spad{dev},{} \\spad{u},{} \\spad{false}) writes the object as an OpenMath fragment.") (((|Void|) (|OpenMathDevice|) $) "\\spad{OMwrite(dev, u)} writes the OpenMath form of \\axiom{\\spad{u}} to the OpenMath device \\axiom{\\spad{dev}} as a complete OpenMath object.") (((|String|) $ (|Boolean|)) "\\spad{OMwrite(u, true)} returns the OpenMath \\spad{XML} encoding of \\axiom{\\spad{u}} as a complete OpenMath object; OMwrite(\\spad{u},{} \\spad{false}) returns the OpenMath \\spad{XML} encoding of \\axiom{\\spad{u}} as an OpenMath fragment.") (((|String|) $) "\\spad{OMwrite(u)} returns the OpenMath \\spad{XML} encoding of \\axiom{\\spad{u}} as a complete OpenMath object."))) 
 NIL 
 NIL 
-(-825) 
+(-827) 
 ((|constructor| (NIL "\\spadtype{OpenMathConnection} provides low-level functions for handling connections to and from \\spadtype{OpenMathDevice}\\spad{s}.")) (|OMbindTCP| (((|Boolean|) $ (|SingleInteger|)) "\\spad{OMbindTCP}")) (|OMconnectTCP| (((|Boolean|) $ (|String|) (|SingleInteger|)) "\\spad{OMconnectTCP}")) (|OMconnOutDevice| (((|OpenMathDevice|) $) "\\spad{OMconnOutDevice:}")) (|OMconnInDevice| (((|OpenMathDevice|) $) "\\spad{OMconnInDevice:}")) (|OMcloseConn| (((|Void|) $) "\\spad{OMcloseConn}")) (|OMmakeConn| (($ (|SingleInteger|)) "\\spad{OMmakeConn}"))) 
 NIL 
 NIL 
-(-826) 
+(-828) 
 ((|constructor| (NIL "\\spadtype{OpenMathDevice} provides support for reading and writing openMath objects to files,{} strings etc. It also provides access to low-level operations from within the interpreter.")) (|OMgetType| (((|Symbol|) $) "\\spad{OMgetType(dev)} returns the type of the next object on \\axiom{\\spad{dev}}.")) (|OMgetSymbol| (((|Record| (|:| |cd| (|String|)) (|:| |name| (|String|))) $) "\\spad{OMgetSymbol(dev)} reads a symbol from \\axiom{\\spad{dev}}.")) (|OMgetString| (((|String|) $) "\\spad{OMgetString(dev)} reads a string from \\axiom{\\spad{dev}}.")) (|OMgetVariable| (((|Symbol|) $) "\\spad{OMgetVariable(dev)} reads a variable from \\axiom{\\spad{dev}}.")) (|OMgetFloat| (((|DoubleFloat|) $) "\\spad{OMgetFloat(dev)} reads a float from \\axiom{\\spad{dev}}.")) (|OMgetInteger| (((|Integer|) $) "\\spad{OMgetInteger(dev)} reads an integer from \\axiom{\\spad{dev}}.")) (|OMgetEndObject| (((|Void|) $) "\\spad{OMgetEndObject(dev)} reads an end object token from \\axiom{\\spad{dev}}.")) (|OMgetEndError| (((|Void|) $) "\\spad{OMgetEndError(dev)} reads an end error token from \\axiom{\\spad{dev}}.")) (|OMgetEndBVar| (((|Void|) $) "\\spad{OMgetEndBVar(dev)} reads an end bound variable list token from \\axiom{\\spad{dev}}.")) (|OMgetEndBind| (((|Void|) $) "\\spad{OMgetEndBind(dev)} reads an end binder token from \\axiom{\\spad{dev}}.")) (|OMgetEndAttr| (((|Void|) $) "\\spad{OMgetEndAttr(dev)} reads an end attribute token from \\axiom{\\spad{dev}}.")) (|OMgetEndAtp| (((|Void|) $) "\\spad{OMgetEndAtp(dev)} reads an end attribute pair token from \\axiom{\\spad{dev}}.")) (|OMgetEndApp| (((|Void|) $) "\\spad{OMgetEndApp(dev)} reads an end application token from \\axiom{\\spad{dev}}.")) (|OMgetObject| (((|Void|) $) "\\spad{OMgetObject(dev)} reads a begin object token from \\axiom{\\spad{dev}}.")) (|OMgetError| (((|Void|) $) "\\spad{OMgetError(dev)} reads a begin error token from \\axiom{\\spad{dev}}.")) (|OMgetBVar| (((|Void|) $) "\\spad{OMgetBVar(dev)} reads a begin bound variable list token from \\axiom{\\spad{dev}}.")) (|OMgetBind| (((|Void|) $) "\\spad{OMgetBind(dev)} reads a begin binder token from \\axiom{\\spad{dev}}.")) (|OMgetAttr| (((|Void|) $) "\\spad{OMgetAttr(dev)} reads a begin attribute token from \\axiom{\\spad{dev}}.")) (|OMgetAtp| (((|Void|) $) "\\spad{OMgetAtp(dev)} reads a begin attribute pair token from \\axiom{\\spad{dev}}.")) (|OMgetApp| (((|Void|) $) "\\spad{OMgetApp(dev)} reads a begin application token from \\axiom{\\spad{dev}}.")) (|OMputSymbol| (((|Void|) $ (|String|) (|String|)) "\\spad{OMputSymbol(dev,cd,s)} writes the symbol \\axiom{\\spad{s}} from \\spad{CD} \\axiom{\\spad{cd}} to \\axiom{\\spad{dev}}.")) (|OMputString| (((|Void|) $ (|String|)) "\\spad{OMputString(dev,i)} writes the string \\axiom{\\spad{i}} to \\axiom{\\spad{dev}}.")) (|OMputVariable| (((|Void|) $ (|Symbol|)) "\\spad{OMputVariable(dev,i)} writes the variable \\axiom{\\spad{i}} to \\axiom{\\spad{dev}}.")) (|OMputFloat| (((|Void|) $ (|DoubleFloat|)) "\\spad{OMputFloat(dev,i)} writes the float \\axiom{\\spad{i}} to \\axiom{\\spad{dev}}.")) (|OMputInteger| (((|Void|) $ (|Integer|)) "\\spad{OMputInteger(dev,i)} writes the integer \\axiom{\\spad{i}} to \\axiom{\\spad{dev}}.")) (|OMputEndObject| (((|Void|) $) "\\spad{OMputEndObject(dev)} writes an end object token to \\axiom{\\spad{dev}}.")) (|OMputEndError| (((|Void|) $) "\\spad{OMputEndError(dev)} writes an end error token to \\axiom{\\spad{dev}}.")) (|OMputEndBVar| (((|Void|) $) "\\spad{OMputEndBVar(dev)} writes an end bound variable list token to \\axiom{\\spad{dev}}.")) (|OMputEndBind| (((|Void|) $) "\\spad{OMputEndBind(dev)} writes an end binder token to \\axiom{\\spad{dev}}.")) (|OMputEndAttr| (((|Void|) $) "\\spad{OMputEndAttr(dev)} writes an end attribute token to \\axiom{\\spad{dev}}.")) (|OMputEndAtp| (((|Void|) $) "\\spad{OMputEndAtp(dev)} writes an end attribute pair token to \\axiom{\\spad{dev}}.")) (|OMputEndApp| (((|Void|) $) "\\spad{OMputEndApp(dev)} writes an end application token to \\axiom{\\spad{dev}}.")) (|OMputObject| (((|Void|) $) "\\spad{OMputObject(dev)} writes a begin object token to \\axiom{\\spad{dev}}.")) (|OMputError| (((|Void|) $) "\\spad{OMputError(dev)} writes a begin error token to \\axiom{\\spad{dev}}.")) (|OMputBVar| (((|Void|) $) "\\spad{OMputBVar(dev)} writes a begin bound variable list token to \\axiom{\\spad{dev}}.")) (|OMputBind| (((|Void|) $) "\\spad{OMputBind(dev)} writes a begin binder token to \\axiom{\\spad{dev}}.")) (|OMputAttr| (((|Void|) $) "\\spad{OMputAttr(dev)} writes a begin attribute token to \\axiom{\\spad{dev}}.")) (|OMputAtp| (((|Void|) $) "\\spad{OMputAtp(dev)} writes a begin attribute pair token to \\axiom{\\spad{dev}}.")) (|OMputApp| (((|Void|) $) "\\spad{OMputApp(dev)} writes a begin application token to \\axiom{\\spad{dev}}.")) (|OMsetEncoding| (((|Void|) $ (|OpenMathEncoding|)) "\\spad{OMsetEncoding(dev,enc)} sets the encoding used for reading or writing OpenMath objects to or from \\axiom{\\spad{dev}} to \\axiom{\\spad{enc}}.")) (|OMclose| (((|Void|) $) "\\spad{OMclose(dev)} closes \\axiom{\\spad{dev}},{} flushing output if necessary.")) (|OMopenString| (($ (|String|) (|OpenMathEncoding|)) "\\spad{OMopenString(s,mode)} opens the string \\axiom{\\spad{s}} for reading or writing OpenMath objects in encoding \\axiom{enc}.")) (|OMopenFile| (($ (|String|) (|String|) (|OpenMathEncoding|)) "\\spad{OMopenFile(f,mode,enc)} opens file \\axiom{\\spad{f}} for reading or writing OpenMath objects (depending on \\axiom{\\spad{mode}} which can be \\spad{\"r\"},{} \\spad{\"w\"} or \"a\" for read,{} write and append respectively),{} in the encoding \\axiom{\\spad{enc}}."))) 
 NIL 
 NIL 
-(-827) 
+(-829) 
 ((|constructor| (NIL "\\spadtype{OpenMathEncoding} is the set of valid OpenMath encodings.")) (|OMencodingBinary| (($) "\\spad{OMencodingBinary()} is the constant for the OpenMath binary encoding.")) (|OMencodingSGML| (($) "\\spad{OMencodingSGML()} is the constant for the deprecated OpenMath SGML encoding.")) (|OMencodingXML| (($) "\\spad{OMencodingXML()} is the constant for the OpenMath \\spad{XML} encoding.")) (|OMencodingUnknown| (($) "\\spad{OMencodingUnknown()} is the constant for unknown encoding types. If this is used on an input device,{} the encoding will be autodetected. It is invalid to use it on an output device."))) 
 NIL 
 NIL 
-(-828) 
+(-830) 
 ((|constructor| (NIL "\\spadtype{OpenMathError} is the domain of OpenMath errors.")) (|omError| (($ (|OpenMathErrorKind|) (|List| (|Symbol|))) "\\spad{omError(k,l)} creates an instance of OpenMathError.")) (|errorInfo| (((|List| (|Symbol|)) $) "\\spad{errorInfo(u)} returns information about the error \\spad{u}.")) (|errorKind| (((|OpenMathErrorKind|) $) "\\spad{errorKind(u)} returns the type of error which \\spad{u} represents."))) 
 NIL 
 NIL 
-(-829) 
+(-831) 
 ((|constructor| (NIL "\\spadtype{OpenMathErrorKind} represents different kinds of OpenMath errors: specifically parse errors,{} unknown \\spad{CD} or symbol errors,{} and read errors.")) (|OMReadError?| (((|Boolean|) $) "\\spad{OMReadError?(u)} tests whether \\spad{u} is an OpenMath read error.")) (|OMUnknownSymbol?| (((|Boolean|) $) "\\spad{OMUnknownSymbol?(u)} tests whether \\spad{u} is an OpenMath unknown symbol error.")) (|OMUnknownCD?| (((|Boolean|) $) "\\spad{OMUnknownCD?(u)} tests whether \\spad{u} is an OpenMath unknown \\spad{CD} error.")) (|OMParseError?| (((|Boolean|) $) "\\spad{OMParseError?(u)} tests whether \\spad{u} is an OpenMath parsing error.")) (|coerce| (($ (|Symbol|)) "\\spad{coerce(u)} creates an OpenMath error object of an appropriate type if \\axiom{\\spad{u}} is one of \\axiom{OMParseError},{} \\axiom{OMReadError},{} \\axiom{OMUnknownCD} or \\axiom{OMUnknownSymbol},{} otherwise it raises a runtime error."))) 
 NIL 
 NIL 
-(-830 R) 
+(-832 R) 
 ((|constructor| (NIL "\\spadtype{ExpressionToOpenMath} provides support for converting objects of type \\spadtype{Expression} into OpenMath."))) 
 NIL 
 NIL 
-(-831 P R) 
+(-833 P R) 
 ((|constructor| (NIL "This constructor creates the \\spadtype{MonogenicLinearOperator} domain which is ``opposite\\spad{''} in the ring sense to \\spad{P}. That is,{} as sets \\spad{P = \\$} but \\spad{a * b} in \\spad{\\$} is equal to \\spad{b * a} in \\spad{P}.")) (|po| ((|#1| $) "\\spad{po(q)} creates a value in \\spad{P} equal to \\spad{q} in \\$.")) (|op| (($ |#1|) "\\spad{op(p)} creates a value in \\$ equal to \\spad{p} in \\spad{P}."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 ((|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-234)))) 
-(-832) 
+(-834) 
 ((|constructor| (NIL "\\spadtype{OpenMathPackage} provides some simple utilities to make reading OpenMath objects easier.")) (|OMunhandledSymbol| (((|Exit|) (|String|) (|String|)) "\\spad{OMunhandledSymbol(s,cd)} raises an error if AXIOM reads a symbol which it is unable to handle. Note that this is different from an unexpected symbol.")) (|OMsupportsSymbol?| (((|Boolean|) (|String|) (|String|)) "\\spad{OMsupportsSymbol?(s,cd)} returns \\spad{true} if AXIOM supports symbol \\axiom{\\spad{s}} from \\spad{CD} \\axiom{\\spad{cd}},{} \\spad{false} otherwise.")) (|OMsupportsCD?| (((|Boolean|) (|String|)) "\\spad{OMsupportsCD?(cd)} returns \\spad{true} if AXIOM supports \\axiom{\\spad{cd}},{} \\spad{false} otherwise.")) (|OMlistSymbols| (((|List| (|String|)) (|String|)) "\\spad{OMlistSymbols(cd)} lists all the symbols in \\axiom{\\spad{cd}}.")) (|OMlistCDs| (((|List| (|String|))) "\\spad{OMlistCDs()} lists all the \\spad{CDs} supported by AXIOM.")) (|OMreadStr| (((|Any|) (|String|)) "\\spad{OMreadStr(f)} reads an OpenMath object from \\axiom{\\spad{f}} and passes it to AXIOM.")) (|OMreadFile| (((|Any|) (|String|)) "\\spad{OMreadFile(f)} reads an OpenMath object from \\axiom{\\spad{f}} and passes it to AXIOM.")) (|OMread| (((|Any|) (|OpenMathDevice|)) "\\spad{OMread(dev)} reads an OpenMath object from \\axiom{\\spad{dev}} and passes it to AXIOM."))) 
 NIL 
 NIL 
-(-833 S) 
+(-835 S) 
 ((|constructor| (NIL "to become an in order iterator")) (|min| ((|#1| $) "\\spad{min(u)} returns the smallest entry in the multiset aggregate \\spad{u}."))) 
-((-4427 . T) (-4417 . T) (-4428 . T)) 
+((-4434 . T) (-4424 . T) (-4435 . T)) 
 NIL 
-(-834) 
+(-836) 
 ((|constructor| (NIL "\\spadtype{OpenMathServerPackage} provides the necessary operations to run AXIOM as an OpenMath server,{} reading/writing objects to/from a port. Please note the facilities available here are very basic. The idea is that a user calls \\spadignore{e.g.} \\axiom{Omserve(4000,{}60)} and then another process sends OpenMath objects to port 4000 and reads the result.")) (|OMserve| (((|Void|) (|SingleInteger|) (|SingleInteger|)) "\\spad{OMserve(portnum,timeout)} puts AXIOM into server mode on port number \\axiom{\\spad{portnum}}. The parameter \\axiom{\\spad{timeout}} specifies the \\spad{timeout} period for the connection.")) (|OMsend| (((|Void|) (|OpenMathConnection|) (|Any|)) "\\spad{OMsend(c,u)} attempts to output \\axiom{\\spad{u}} on \\aciom{\\spad{c}} in OpenMath.")) (|OMreceive| (((|Any|) (|OpenMathConnection|)) "\\spad{OMreceive(c)} reads an OpenMath object from connection \\axiom{\\spad{c}} and returns the appropriate AXIOM object."))) 
 NIL 
 NIL 
-(-835 R) 
+(-837 R) 
 ((|constructor| (NIL "Adjunction of a complex infinity to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a finite rational number if it is one,{} \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a finite rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a finite rational number.")) (|infinite?| (((|Boolean|) $) "\\spad{infinite?(x)} tests if \\spad{x} is infinite.")) (|finite?| (((|Boolean|) $) "\\spad{finite?(x)} tests if \\spad{x} is finite.")) (|infinity| (($) "\\spad{infinity()} returns infinity."))) 
-((-4424 |has| |#1| (-851))) 
-((|HasCategory| |#1| (QUOTE (-851))) (|HasCategory| |#1| (QUOTE (-21))) (-3962 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-851)))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (-3962 (|HasCategory| |#1| (QUOTE (-851))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-549)))) 
-(-836 R S) 
+((-4431 |has| |#1| (-853))) 
+((|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-21))) (-3969 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-853)))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (-3969 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-550)))) 
+(-838 R S) 
 ((|constructor| (NIL "Lifting of maps to one-point completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989")) (|map| (((|OnePointCompletion| |#2|) (|Mapping| |#2| |#1|) (|OnePointCompletion| |#1|) (|OnePointCompletion| |#2|)) "\\spad{map(f, r, i)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(infinity) = \\spad{i}.") (((|OnePointCompletion| |#2|) (|Mapping| |#2| |#1|) (|OnePointCompletion| |#1|)) "\\spad{map(f, r)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(infinity) = infinity."))) 
 NIL 
 NIL 
-(-837 R) 
+(-839 R) 
 ((|constructor| (NIL "Algebra of ADDITIVE operators over a ring."))) 
-((-4422 |has| |#1| (-173)) (-4421 |has| |#1| (-173)) (-4424 . T)) 
+((-4429 |has| |#1| (-173)) (-4428 |has| |#1| (-173)) (-4431 . T)) 
 ((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147)))) 
-(-838 A S) 
+(-840 A S) 
 ((|constructor| (NIL "This category specifies the interface for operators used to build terms,{} in the sense of Universal Algebra. The domain parameter \\spad{S} provides representation for the `external name' of an operator.")) (|is?| (((|Boolean|) $ |#2|) "\\spad{is?(op,n)} holds if the name of the operator \\spad{op} is \\spad{n}.")) (|arity| (((|Arity|) $) "\\spad{arity(op)} returns the arity of the operator \\spad{op}.")) (|name| ((|#2| $) "\\spad{name(op)} returns the externam name of \\spad{op}."))) 
 NIL 
 NIL 
-(-839 S) 
+(-841 S) 
 ((|constructor| (NIL "This category specifies the interface for operators used to build terms,{} in the sense of Universal Algebra. The domain parameter \\spad{S} provides representation for the `external name' of an operator.")) (|is?| (((|Boolean|) $ |#1|) "\\spad{is?(op,n)} holds if the name of the operator \\spad{op} is \\spad{n}.")) (|arity| (((|Arity|) $) "\\spad{arity(op)} returns the arity of the operator \\spad{op}.")) (|name| ((|#1| $) "\\spad{name(op)} returns the externam name of \\spad{op}."))) 
 NIL 
 NIL 
-(-840) 
+(-842) 
 ((|constructor| (NIL "This package exports tools to create AXIOM Library information databases.")) (|getDatabase| (((|Database| (|IndexCard|)) (|String|)) "\\spad{getDatabase(\"char\")} returns a list of appropriate entries in the browser database. The legal values for \\spad{\"char\"} are \"o\" (operations),{} \\spad{\"k\"} (constructors),{} \\spad{\"d\"} (domains),{} \\spad{\"c\"} (categories) or \\spad{\"p\"} (packages)."))) 
 NIL 
 NIL 
-(-841) 
+(-843) 
 ((|constructor| (NIL "This the datatype for an operator-signature pair.")) (|construct| (($ (|Identifier|) (|Signature|)) "\\spad{construct(op,sig)} construct a signature-operator with operator name `op',{} and signature `sig'.")) (|signature| (((|Signature|) $) "\\spad{signature(x)} returns the signature of \\spad{`x'}."))) 
 NIL 
 NIL 
-(-842) 
+(-844) 
 ((|numericalOptimization| (((|Result|) (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|)))))) "\\spad{numericalOptimization(args)} performs the optimization of the function given the strategy or method returned by \\axiomFun{measure}.") (((|Result|) (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|))))) "\\spad{numericalOptimization(args)} performs the optimization of the function given the strategy or method returned by \\axiomFun{measure}.")) (|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|))) (|RoutinesTable|) (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|))))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve an optimization problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far.") (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|))) (|RoutinesTable|) (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|)))))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve an optimization problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far."))) 
 NIL 
 NIL 
-(-843) 
+(-845) 
 ((|goodnessOfFit| (((|Result|) (|List| (|Expression| (|Float|))) (|List| (|Float|))) "\\spad{goodnessOfFit(lf,start)} is a top level ANNA function to check to goodness of fit of a least squares model \\spadignore{i.e.} the minimization of a set of functions,{} \\axiom{\\spad{lf}},{} of one or more variables without constraints. \\blankline The parameter \\axiom{\\spad{start}} is a list of the initial guesses of the values of the variables. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}. It then calls the numerical routine \\axiomType{E04YCF} to get estimates of the variance-covariance matrix of the regression coefficients of the least-squares problem. \\blankline It thus returns both the results of the optimization and the variance-covariance calculation. goodnessOfFit(\\spad{lf},{}\\spad{start}) is a top level function to iterate over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}. It then checks the goodness of fit of the least squares model.") (((|Result|) (|NumericalOptimizationProblem|)) "\\spad{goodnessOfFit(prob)} is a top level ANNA function to check to goodness of fit of a least squares model as defined within \\axiom{\\spad{prob}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}. It then calls the numerical routine \\axiomType{E04YCF} to get estimates of the variance-covariance matrix of the regression coefficients of the least-squares problem. \\blankline It thus returns both the results of the optimization and the variance-covariance calculation.")) (|optimize| (((|Result|) (|List| (|Expression| (|Float|))) (|List| (|Float|))) "\\spad{optimize(lf,start)} is a top level ANNA function to minimize a set of functions,{} \\axiom{\\spad{lf}},{} of one or more variables without constraints \\spadignore{i.e.} a least-squares problem. \\blankline The parameter \\axiom{\\spad{start}} is a list of the initial guesses of the values of the variables. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Float|))) "\\spad{optimize(f,start)} is a top level ANNA function to minimize a function,{} \\axiom{\\spad{f}},{} of one or more variables without constraints. \\blankline The parameter \\axiom{\\spad{start}} is a list of the initial guesses of the values of the variables. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Float|)) (|List| (|OrderedCompletion| (|Float|))) (|List| (|OrderedCompletion| (|Float|)))) "\\spad{optimize(f,start,lower,upper)} is a top level ANNA function to minimize a function,{} \\axiom{\\spad{f}},{} of one or more variables with simple constraints. The bounds on the variables are defined in \\axiom{\\spad{lower}} and \\axiom{\\spad{upper}}. \\blankline The parameter \\axiom{\\spad{start}} is a list of the initial guesses of the values of the variables. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}.") (((|Result|) (|Expression| (|Float|)) (|List| (|Float|)) (|List| (|OrderedCompletion| (|Float|))) (|List| (|Expression| (|Float|))) (|List| (|OrderedCompletion| (|Float|)))) "\\spad{optimize(f,start,lower,cons,upper)} is a top level ANNA function to minimize a function,{} \\axiom{\\spad{f}},{} of one or more variables with the given constraints. \\blankline These constraints may be simple constraints on the variables in which case \\axiom{\\spad{cons}} would be an empty list and the bounds on those variables defined in \\axiom{\\spad{lower}} and \\axiom{\\spad{upper}},{} or a mixture of simple,{} linear and non-linear constraints,{} where \\axiom{\\spad{cons}} contains the linear and non-linear constraints and the bounds on these are added to \\axiom{\\spad{upper}} and \\axiom{\\spad{lower}}. \\blankline The parameter \\axiom{\\spad{start}} is a list of the initial guesses of the values of the variables. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}.") (((|Result|) (|NumericalOptimizationProblem|)) "\\spad{optimize(prob)} is a top level ANNA function to minimize a function or a set of functions with any constraints as defined within \\axiom{\\spad{prob}}. \\blankline It iterates over the \\axiom{domains} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}.") (((|Result|) (|NumericalOptimizationProblem|) (|RoutinesTable|)) "\\spad{optimize(prob,routines)} is a top level ANNA function to minimize a function or a set of functions with any constraints as defined within \\axiom{\\spad{prob}}. \\blankline It iterates over the \\axiom{domains} listed in \\axiom{\\spad{routines}} of \\axiomType{NumericalOptimizationCategory} to get the name and other relevant information of the best \\axiom{measure} and then optimize the function on that \\axiom{domain}.")) (|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalOptimizationProblem|) (|RoutinesTable|)) "\\spad{measure(prob,R)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical optimization problem defined by \\axiom{\\spad{prob}} by checking various attributes of the functions and calculating a measure of compatibility of each routine to these attributes. \\blankline It calls each \\axiom{domain} listed in \\axiom{\\spad{R}} of \\axiom{category} \\axiomType{NumericalOptimizationCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information.") (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalOptimizationProblem|)) "\\spad{measure(prob)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical optimization problem defined by \\axiom{\\spad{prob}} by checking various attributes of the functions and calculating a measure of compatibility of each routine to these attributes. \\blankline It calls each \\axiom{domain} of \\axiom{category} \\axiomType{NumericalOptimizationCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information."))) 
 NIL 
 NIL 
-(-844) 
+(-846) 
 ((|retract| (((|Union| (|:| |noa| (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|)))))) (|:| |lsa| (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|)))))) $) "\\spad{retract(x)} \\undocumented{}")) (|coerce| (($ (|Union| (|:| |noa| (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|)))))) (|:| |lsa| (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|))))))) "\\spad{coerce(x)} \\undocumented{}") (($ (|Record| (|:| |lfn| (|List| (|Expression| (|DoubleFloat|)))) (|:| |init| (|List| (|DoubleFloat|))))) "\\spad{coerce(x)} \\undocumented{}") (($ (|Record| (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |init| (|List| (|DoubleFloat|))) (|:| |lb| (|List| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |cf| (|List| (|Expression| (|DoubleFloat|)))) (|:| |ub| (|List| (|OrderedCompletion| (|DoubleFloat|)))))) "\\spad{coerce(x)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-845 R) 
+(-847 R) 
 ((|constructor| (NIL "Adjunction of two real infinites quantities to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a finite rational number if it is one and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a finite rational number. Error: if \\spad{x} cannot be so converted.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a finite rational number.")) (|whatInfinity| (((|SingleInteger|) $) "\\spad{whatInfinity(x)} returns 0 if \\spad{x} is finite,{} 1 if \\spad{x} is +infinity,{} and \\spad{-1} if \\spad{x} is -infinity.")) (|infinite?| (((|Boolean|) $) "\\spad{infinite?(x)} tests if \\spad{x} is +infinity or -infinity,{}")) (|finite?| (((|Boolean|) $) "\\spad{finite?(x)} tests if \\spad{x} is finite.")) (|minusInfinity| (($) "\\spad{minusInfinity()} returns -infinity.")) (|plusInfinity| (($) "\\spad{plusInfinity()} returns +infinity."))) 
-((-4424 |has| |#1| (-851))) 
-((|HasCategory| |#1| (QUOTE (-851))) (|HasCategory| |#1| (QUOTE (-21))) (-3962 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-851)))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (-3962 (|HasCategory| |#1| (QUOTE (-851))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-549)))) 
-(-846 R S) 
+((-4431 |has| |#1| (-853))) 
+((|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-21))) (-3969 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-853)))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (-3969 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-550)))) 
+(-848 R S) 
 ((|constructor| (NIL "Lifting of maps to ordered completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989")) (|map| (((|OrderedCompletion| |#2|) (|Mapping| |#2| |#1|) (|OrderedCompletion| |#1|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|)) "\\spad{map(f, r, p, m)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(plusInfinity) = \\spad{p} and that \\spad{f}(minusInfinity) = \\spad{m}.") (((|OrderedCompletion| |#2|) (|Mapping| |#2| |#1|) (|OrderedCompletion| |#1|)) "\\spad{map(f, r)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(plusInfinity) = plusInfinity and that \\spad{f}(minusInfinity) = minusInfinity."))) 
 NIL 
 NIL 
-(-847) 
+(-849) 
 ((|constructor| (NIL "Ordered finite sets.")) (|max| (($) "\\spad{max} is the maximum value of \\%.")) (|min| (($) "\\spad{min} is the minimum value of \\%."))) 
 NIL 
 NIL 
-(-848 -3023 S) 
+(-850 -3030 S) 
 ((|constructor| (NIL "\\indented{3}{This package provides ordering functions on vectors which} are suitable parameters for OrderedDirectProduct.")) (|reverseLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{reverseLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by the reverse lexicographic ordering.")) (|totalLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{totalLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by lexicographic ordering.")) (|pureLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{pureLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the lexicographic ordering."))) 
 NIL 
 NIL 
-(-849) 
+(-851) 
 ((|constructor| (NIL "Ordered sets which are also monoids,{} such that multiplication preserves the ordering. \\blankline"))) 
 NIL 
 NIL 
-(-850 S) 
+(-852 S) 
 ((|constructor| (NIL "Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \\blankline")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x}.")) (|sign| (((|Integer|) $) "\\spad{sign(x)} is 1 if \\spad{x} is positive,{} \\spad{-1} if \\spad{x} is negative,{} 0 if \\spad{x} equals 0.")) (|negative?| (((|Boolean|) $) "\\spad{negative?(x)} tests whether \\spad{x} is strictly less than 0.")) (|positive?| (((|Boolean|) $) "\\spad{positive?(x)} tests whether \\spad{x} is strictly greater than 0."))) 
 NIL 
 NIL 
-(-851) 
+(-853) 
 ((|constructor| (NIL "Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \\blankline")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x}.")) (|sign| (((|Integer|) $) "\\spad{sign(x)} is 1 if \\spad{x} is positive,{} \\spad{-1} if \\spad{x} is negative,{} 0 if \\spad{x} equals 0.")) (|negative?| (((|Boolean|) $) "\\spad{negative?(x)} tests whether \\spad{x} is strictly less than 0.")) (|positive?| (((|Boolean|) $) "\\spad{positive?(x)} tests whether \\spad{x} is strictly greater than 0."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-852 S) 
+(-854 S) 
 ((|constructor| (NIL "The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \\spad{(a,b)} exactly one of the following relations holds \\spad{a<b or a=b or b<a} and the relation is transitive,{} \\spadignore{i.e.} \\spad{a<b and b<c => a<c}.")) (|min| (($ $ $) "\\spad{min(x,y)} returns the minimum of \\spad{x} and \\spad{y} relative to \\spad{\"<\"}.")) (|max| (($ $ $) "\\spad{max(x,y)} returns the maximum of \\spad{x} and \\spad{y} relative to \\spad{\"<\"}.")) (<= (((|Boolean|) $ $) "\\spad{x <= y} is a less than or equal test.")) (>= (((|Boolean|) $ $) "\\spad{x >= y} is a greater than or equal test.")) (> (((|Boolean|) $ $) "\\spad{x > y} is a greater than test.")) (< (((|Boolean|) $ $) "\\spad{x < y} is a strict total ordering on the elements of the set."))) 
 NIL 
 NIL 
-(-853) 
+(-855) 
 ((|constructor| (NIL "The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \\spad{(a,b)} exactly one of the following relations holds \\spad{a<b or a=b or b<a} and the relation is transitive,{} \\spadignore{i.e.} \\spad{a<b and b<c => a<c}.")) (|min| (($ $ $) "\\spad{min(x,y)} returns the minimum of \\spad{x} and \\spad{y} relative to \\spad{\"<\"}.")) (|max| (($ $ $) "\\spad{max(x,y)} returns the maximum of \\spad{x} and \\spad{y} relative to \\spad{\"<\"}.")) (<= (((|Boolean|) $ $) "\\spad{x <= y} is a less than or equal test.")) (>= (((|Boolean|) $ $) "\\spad{x >= y} is a greater than or equal test.")) (> (((|Boolean|) $ $) "\\spad{x > y} is a greater than test.")) (< (((|Boolean|) $ $) "\\spad{x < y} is a strict total ordering on the elements of the set."))) 
 NIL 
 NIL 
-(-854 S R) 
+(-856 S R) 
 ((|constructor| (NIL "This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}. This category is an evolution of the types \\indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \\indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \\spad{M}. Watt.")) (|leftLcm| (($ $ $) "\\spad{leftLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = aa*a = bb*b} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using right-division.")) (|rightExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{rightExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = c * a + d * b = rightGcd(a, b)}.")) (|rightGcd| (($ $ $) "\\spad{rightGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = aa*g}} \\indented{3}{\\spad{b = bb*g}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using right-division.")) (|rightExactQuotient| (((|Union| $ "failed") $ $) "\\spad{rightExactQuotient(a,b)} computes the value \\spad{q},{} if it exists such that \\spad{a = q*b}.")) (|rightRemainder| (($ $ $) "\\spad{rightRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|rightQuotient| (($ $ $) "\\spad{rightQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|rightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{rightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``right division\\spad{''}.")) (|rightLcm| (($ $ $) "\\spad{rightLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = a*aa = b*bb} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using left-division.")) (|leftExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{leftExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = a * c + b * d = leftGcd(a, b)}.")) (|leftGcd| (($ $ $) "\\spad{leftGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = g*aa}} \\indented{3}{\\spad{b = g*bb}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using left-division.")) (|leftExactQuotient| (((|Union| $ "failed") $ $) "\\spad{leftExactQuotient(a,b)} computes the value \\spad{q},{} if it exists,{} \\indented{1}{such that \\spad{a = b*q}.}")) (|leftRemainder| (($ $ $) "\\spad{leftRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|leftQuotient| (($ $ $) "\\spad{leftQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|leftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{leftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division\\spad{''}.")) (|primitivePart| (($ $) "\\spad{primitivePart(l)} returns \\spad{l0} such that \\spad{l = a * l0} for some a in \\spad{R},{} and \\spad{content(l0) = 1}.")) (|content| ((|#2| $) "\\spad{content(l)} returns the \\spad{gcd} of all the coefficients of \\spad{l}.")) (|monicRightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicRightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``right division\\spad{''}.")) (|monicLeftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicLeftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``left division\\spad{''}.")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(l, a)} returns the exact quotient of \\spad{l} by a,{} returning \\axiom{\"failed\"} if this is not possible.")) (|apply| ((|#2| $ |#2| |#2|) "\\spad{apply(p, c, m)} returns \\spad{p(m)} where the action is given by \\spad{x m = c sigma(m) + delta(m)}.")) (|coefficients| (((|List| |#2|) $) "\\spad{coefficients(l)} returns the list of all the nonzero coefficients of \\spad{l}.")) (|monomial| (($ |#2| (|NonNegativeInteger|)) "\\spad{monomial(c,k)} produces \\spad{c} times the \\spad{k}-th power of the generating operator,{} \\spad{monomial(1,1)}.")) (|coefficient| ((|#2| $ (|NonNegativeInteger|)) "\\spad{coefficient(l,k)} is \\spad{a(k)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|reductum| (($ $) "\\spad{reductum(l)} is \\spad{l - monomial(a(n),n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(l)} is \\spad{a(n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|minimumDegree| (((|NonNegativeInteger|) $) "\\spad{minimumDegree(l)} is the smallest \\spad{k} such that \\spad{a(k) ~= 0} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(l)} is \\spad{n} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}"))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173)))) 
-(-855 R) 
+((|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-173)))) 
+(-857 R) 
 ((|constructor| (NIL "This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}. This category is an evolution of the types \\indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \\indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \\spad{M}. Watt.")) (|leftLcm| (($ $ $) "\\spad{leftLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = aa*a = bb*b} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using right-division.")) (|rightExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{rightExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = c * a + d * b = rightGcd(a, b)}.")) (|rightGcd| (($ $ $) "\\spad{rightGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = aa*g}} \\indented{3}{\\spad{b = bb*g}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using right-division.")) (|rightExactQuotient| (((|Union| $ "failed") $ $) "\\spad{rightExactQuotient(a,b)} computes the value \\spad{q},{} if it exists such that \\spad{a = q*b}.")) (|rightRemainder| (($ $ $) "\\spad{rightRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|rightQuotient| (($ $ $) "\\spad{rightQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|rightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{rightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``right division\\spad{''}.")) (|rightLcm| (($ $ $) "\\spad{rightLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = a*aa = b*bb} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using left-division.")) (|leftExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{leftExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = a * c + b * d = leftGcd(a, b)}.")) (|leftGcd| (($ $ $) "\\spad{leftGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = g*aa}} \\indented{3}{\\spad{b = g*bb}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using left-division.")) (|leftExactQuotient| (((|Union| $ "failed") $ $) "\\spad{leftExactQuotient(a,b)} computes the value \\spad{q},{} if it exists,{} \\indented{1}{such that \\spad{a = b*q}.}")) (|leftRemainder| (($ $ $) "\\spad{leftRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|leftQuotient| (($ $ $) "\\spad{leftQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|leftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{leftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division\\spad{''}.")) (|primitivePart| (($ $) "\\spad{primitivePart(l)} returns \\spad{l0} such that \\spad{l = a * l0} for some a in \\spad{R},{} and \\spad{content(l0) = 1}.")) (|content| ((|#1| $) "\\spad{content(l)} returns the \\spad{gcd} of all the coefficients of \\spad{l}.")) (|monicRightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicRightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``right division\\spad{''}.")) (|monicLeftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicLeftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``left division\\spad{''}.")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(l, a)} returns the exact quotient of \\spad{l} by a,{} returning \\axiom{\"failed\"} if this is not possible.")) (|apply| ((|#1| $ |#1| |#1|) "\\spad{apply(p, c, m)} returns \\spad{p(m)} where the action is given by \\spad{x m = c sigma(m) + delta(m)}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(l)} returns the list of all the nonzero coefficients of \\spad{l}.")) (|monomial| (($ |#1| (|NonNegativeInteger|)) "\\spad{monomial(c,k)} produces \\spad{c} times the \\spad{k}-th power of the generating operator,{} \\spad{monomial(1,1)}.")) (|coefficient| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coefficient(l,k)} is \\spad{a(k)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|reductum| (($ $) "\\spad{reductum(l)} is \\spad{l - monomial(a(n),n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(l)} is \\spad{a(n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|minimumDegree| (((|NonNegativeInteger|) $) "\\spad{minimumDegree(l)} is the smallest \\spad{k} such that \\spad{a(k) ~= 0} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(l)} is \\spad{n} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-856 R C) 
+(-858 R C) 
 ((|constructor| (NIL "\\spad{UnivariateSkewPolynomialCategoryOps} provides products and \\indented{1}{divisions of univariate skew polynomials.}")) (|rightDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{rightDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``right division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|leftDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{leftDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|monicRightDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{monicRightDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``right division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|monicLeftDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{monicLeftDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``left division\\spad{''}. \\spad{\\sigma} is the morphism to use.")) (|apply| ((|#1| |#2| |#1| |#1| (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{apply(p, c, m, sigma, delta)} returns \\spad{p(m)} where the action is given by \\spad{x m = c sigma(m) + delta(m)}.")) (|times| ((|#2| |#2| |#2| (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{times(p, q, sigma, delta)} returns \\spad{p * q}. \\spad{\\sigma} and \\spad{\\delta} are the maps to use."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) 
-(-857 R |sigma| -3667) 
+((|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) 
+(-859 R |sigma| -3674) 
 ((|constructor| (NIL "This is the domain of sparse univariate skew polynomials over an Ore coefficient field. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}.")) (|outputForm| (((|OutputForm|) $ (|OutputForm|)) "\\spad{outputForm(p, x)} returns the output form of \\spad{p} using \\spad{x} for the otherwise anonymous variable."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-858 |x| R |sigma| -3667) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-860 |x| R |sigma| -3674) 
 ((|constructor| (NIL "This is the domain of univariate skew polynomials over an Ore coefficient field in a named variable. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}."))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-366)))) 
-(-859 R) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-367)))) 
+(-861 R) 
 ((|constructor| (NIL "This package provides orthogonal polynomials as functions on a ring.")) (|legendreP| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{legendreP(n,x)} is the \\spad{n}-th Legendre polynomial,{} \\spad{P[n](x)}. These are defined by \\spad{1/sqrt(1-2*x*t+t**2) = sum(P[n](x)*t**n, n = 0..)}.")) (|laguerreL| ((|#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#1|) "\\spad{laguerreL(m,n,x)} is the associated Laguerre polynomial,{} \\spad{L<m>[n](x)}. This is the \\spad{m}-th derivative of \\spad{L[n](x)}.") ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{laguerreL(n,x)} is the \\spad{n}-th Laguerre polynomial,{} \\spad{L[n](x)}. These are defined by \\spad{exp(-t*x/(1-t))/(1-t) = sum(L[n](x)*t**n/n!, n = 0..)}.")) (|hermiteH| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{hermiteH(n,x)} is the \\spad{n}-th Hermite polynomial,{} \\spad{H[n](x)}. These are defined by \\spad{exp(2*t*x-t**2) = sum(H[n](x)*t**n/n!, n = 0..)}.")) (|chebyshevU| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{chebyshevU(n,x)} is the \\spad{n}-th Chebyshev polynomial of the second kind,{} \\spad{U[n](x)}. These are defined by \\spad{1/(1-2*t*x+t**2) = sum(T[n](x) *t**n, n = 0..)}.")) (|chebyshevT| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{chebyshevT(n,x)} is the \\spad{n}-th Chebyshev polynomial of the first kind,{} \\spad{T[n](x)}. These are defined by \\spad{(1-t*x)/(1-2*t*x+t**2) = sum(T[n](x) *t**n, n = 0..)}."))) 
 NIL 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550)))))) 
-(-860) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551)))))) 
+(-862) 
 ((|constructor| (NIL "Semigroups with compatible ordering."))) 
 NIL 
 NIL 
-(-861) 
+(-863) 
 ((|constructor| (NIL "\\indented{1}{Author : Larry Lambe} Date created : 14 August 1988 Date Last Updated : 11 March 1991 Description : A domain used in order to take the free \\spad{R}-module on the Integers \\spad{I}. This is actually the forgetful functor from OrderedRings to OrderedSets applied to \\spad{I}")) (|value| (((|Integer|) $) "\\spad{value(x)} returns the integer associated with \\spad{x}")) (|coerce| (($ (|Integer|)) "\\spad{coerce(i)} returns the element corresponding to \\spad{i}"))) 
 NIL 
 NIL 
-(-862) 
+(-864) 
 ((|constructor| (NIL "OutPackage allows pretty-printing from programs.")) (|outputList| (((|Void|) (|List| (|Any|))) "\\spad{outputList(l)} displays the concatenated components of the list \\spad{l} on the ``algebra output\\spad{''} stream,{} as defined by \\spadsyscom{set output algebra}; quotes are stripped from strings.")) (|output| (((|Void|) (|String|) (|OutputForm|)) "\\spad{output(s,x)} displays the string \\spad{s} followed by the form \\spad{x} on the ``algebra output\\spad{''} stream,{} as defined by \\spadsyscom{set output algebra}.") (((|Void|) (|OutputForm|)) "\\spad{output(x)} displays the output form \\spad{x} on the ``algebra output\\spad{''} stream,{} as defined by \\spadsyscom{set output algebra}.") (((|Void|) (|String|)) "\\spad{output(s)} displays the string \\spad{s} on the ``algebra output\\spad{''} stream,{} as defined by \\spadsyscom{set output algebra}."))) 
 NIL 
 NIL 
-(-863 S) 
+(-865 S) 
 ((|constructor| (NIL "This category describes output byte stream conduits.")) (|writeBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{writeBytes!(c,b)} write bytes from buffer \\spad{`b'} onto the conduit \\spad{`c'}. The actual number of written bytes is returned.")) (|writeUInt8!| (((|Maybe| (|UInt8|)) $ (|UInt8|)) "\\spad{writeUInt8!(c,b)} attempts to write the unsigned 8-bit value \\spad{`v'} on the conduit \\spad{`c'}. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeInt8!| (((|Maybe| (|Int8|)) $ (|Int8|)) "\\spad{writeInt8!(c,b)} attempts to write the 8-bit value \\spad{`v'} on the conduit \\spad{`c'}. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeByte!| (((|Maybe| (|Byte|)) $ (|Byte|)) "\\spad{writeByte!(c,b)} attempts to write the byte \\spad{`b'} on the conduit \\spad{`c'}. Returns the written byte if successful,{} otherwise,{} returns \\spad{nothing}."))) 
 NIL 
 NIL 
-(-864) 
+(-866) 
 ((|constructor| (NIL "This category describes output byte stream conduits.")) (|writeBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{writeBytes!(c,b)} write bytes from buffer \\spad{`b'} onto the conduit \\spad{`c'}. The actual number of written bytes is returned.")) (|writeUInt8!| (((|Maybe| (|UInt8|)) $ (|UInt8|)) "\\spad{writeUInt8!(c,b)} attempts to write the unsigned 8-bit value \\spad{`v'} on the conduit \\spad{`c'}. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeInt8!| (((|Maybe| (|Int8|)) $ (|Int8|)) "\\spad{writeInt8!(c,b)} attempts to write the 8-bit value \\spad{`v'} on the conduit \\spad{`c'}. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeByte!| (((|Maybe| (|Byte|)) $ (|Byte|)) "\\spad{writeByte!(c,b)} attempts to write the byte \\spad{`b'} on the conduit \\spad{`c'}. Returns the written byte if successful,{} otherwise,{} returns \\spad{nothing}."))) 
 NIL 
 NIL 
-(-865) 
+(-867) 
 ((|constructor| (NIL "This domain provides representation for binary files open for output operations. `Binary' here means that the conduits do not interpret their contents.")) (|isOpen?| (((|Boolean|) $) "open?(ifile) holds if `ifile' is in open state.")) (|outputBinaryFile| (($ (|String|)) "\\spad{outputBinaryFile(f)} returns an output conduit obtained by opening the file named by \\spad{`f'} as a binary file.") (($ (|FileName|)) "\\spad{outputBinaryFile(f)} returns an output conduit obtained by opening the file named by \\spad{`f'} as a binary file."))) 
 NIL 
 NIL 
-(-866) 
+(-868) 
 ((|constructor| (NIL "This domain is used to create and manipulate mathematical expressions for output. It is intended to provide an insulating layer between the expression rendering software (\\spadignore{e.g.} TeX,{} or Script) and the output coercions in the various domains.")) (SEGMENT (($ $) "\\spad{SEGMENT(x)} creates the prefix form: \\spad{x..}.") (($ $ $) "\\spad{SEGMENT(x,y)} creates the infix form: \\spad{x..y}.")) (|not| (($ $) "\\spad{not f} creates the equivalent prefix form.")) (|or| (($ $ $) "\\spad{f or g} creates the equivalent infix form.")) (|and| (($ $ $) "\\spad{f and g} creates the equivalent infix form.")) (|exquo| (($ $ $) "\\spad{exquo(f,g)} creates the equivalent infix form.")) (|quo| (($ $ $) "\\spad{f quo g} creates the equivalent infix form.")) (|rem| (($ $ $) "\\spad{f rem g} creates the equivalent infix form.")) (|div| (($ $ $) "\\spad{f div g} creates the equivalent infix form.")) (** (($ $ $) "\\spad{f ** g} creates the equivalent infix form.")) (/ (($ $ $) "\\spad{f / g} creates the equivalent infix form.")) (* (($ $ $) "\\spad{f * g} creates the equivalent infix form.")) (- (($ $) "\\spad{- f} creates the equivalent prefix form.") (($ $ $) "\\spad{f - g} creates the equivalent infix form.")) (+ (($ $ $) "\\spad{f + g} creates the equivalent infix form.")) (>= (($ $ $) "\\spad{f >= g} creates the equivalent infix form.")) (<= (($ $ $) "\\spad{f <= g} creates the equivalent infix form.")) (> (($ $ $) "\\spad{f > g} creates the equivalent infix form.")) (< (($ $ $) "\\spad{f < g} creates the equivalent infix form.")) (~= (($ $ $) "\\spad{f ~= g} creates the equivalent infix form.")) (= (($ $ $) "\\spad{f = g} creates the equivalent infix form.")) (|blankSeparate| (($ (|List| $)) "\\spad{blankSeparate(l)} creates the form separating the elements of \\spad{l} by blanks.")) (|semicolonSeparate| (($ (|List| $)) "\\spad{semicolonSeparate(l)} creates the form separating the elements of \\spad{l} by semicolons.")) (|commaSeparate| (($ (|List| $)) "\\spad{commaSeparate(l)} creates the form separating the elements of \\spad{l} by commas.")) (|pile| (($ (|List| $)) "\\spad{pile(l)} creates the form consisting of the elements of \\spad{l} which displays as a pile,{} \\spadignore{i.e.} the elements begin on a new line and are indented right to the same margin.")) (|paren| (($ (|List| $)) "\\spad{paren(lf)} creates the form separating the elements of \\spad{lf} by commas and encloses the result in parentheses.") (($ $) "\\spad{paren(f)} creates the form enclosing \\spad{f} in parentheses.")) (|bracket| (($ (|List| $)) "\\spad{bracket(lf)} creates the form separating the elements of \\spad{lf} by commas and encloses the result in square brackets.") (($ $) "\\spad{bracket(f)} creates the form enclosing \\spad{f} in square brackets.")) (|brace| (($ (|List| $)) "\\spad{brace(lf)} creates the form separating the elements of \\spad{lf} by commas and encloses the result in curly brackets.") (($ $) "\\spad{brace(f)} creates the form enclosing \\spad{f} in braces (curly brackets).")) (|int| (($ $ $ $) "\\spad{int(expr,lowerlimit,upperlimit)} creates the form prefixing \\spad{expr} by an integral sign with both a \\spad{lowerlimit} and \\spad{upperlimit}.") (($ $ $) "\\spad{int(expr,lowerlimit)} creates the form prefixing \\spad{expr} by an integral sign with a \\spad{lowerlimit}.") (($ $) "\\spad{int(expr)} creates the form prefixing \\spad{expr} with an integral sign.")) (|prod| (($ $ $ $) "\\spad{prod(expr,lowerlimit,upperlimit)} creates the form prefixing \\spad{expr} by a capital \\spad{pi} with both a \\spad{lowerlimit} and \\spad{upperlimit}.") (($ $ $) "\\spad{prod(expr,lowerlimit)} creates the form prefixing \\spad{expr} by a capital \\spad{pi} with a \\spad{lowerlimit}.") (($ $) "\\spad{prod(expr)} creates the form prefixing \\spad{expr} by a capital \\spad{pi}.")) (|sum| (($ $ $ $) "\\spad{sum(expr,lowerlimit,upperlimit)} creates the form prefixing \\spad{expr} by a capital sigma with both a \\spad{lowerlimit} and \\spad{upperlimit}.") (($ $ $) "\\spad{sum(expr,lowerlimit)} creates the form prefixing \\spad{expr} by a capital sigma with a \\spad{lowerlimit}.") (($ $) "\\spad{sum(expr)} creates the form prefixing \\spad{expr} by a capital sigma.")) (|overlabel| (($ $ $) "\\spad{overlabel(x,f)} creates the form \\spad{f} with \\spad{\"x} overbar\" over the top.")) (|overbar| (($ $) "\\spad{overbar(f)} creates the form \\spad{f} with an overbar.")) (|prime| (($ $ (|NonNegativeInteger|)) "\\spad{prime(f,n)} creates the form \\spad{f} followed by \\spad{n} primes.") (($ $) "\\spad{prime(f)} creates the form \\spad{f} followed by a suffix prime (single quote).")) (|dot| (($ $ (|NonNegativeInteger|)) "\\spad{dot(f,n)} creates the form \\spad{f} with \\spad{n} dots overhead.") (($ $) "\\spad{dot(f)} creates the form with a one dot overhead.")) (|quote| (($ $) "\\spad{quote(f)} creates the form \\spad{f} with a prefix quote.")) (|supersub| (($ $ (|List| $)) "\\spad{supersub(a,[sub1,super1,sub2,super2,...])} creates a form with each subscript aligned under each superscript.")) (|scripts| (($ $ (|List| $)) "\\spad{scripts(f, [sub, super, presuper, presub])} \\indented{1}{creates a form for \\spad{f} with scripts on all 4 corners.}")) (|presuper| (($ $ $) "\\spad{presuper(f,n)} creates a form for \\spad{f} presuperscripted by \\spad{n}.")) (|presub| (($ $ $) "\\spad{presub(f,n)} creates a form for \\spad{f} presubscripted by \\spad{n}.")) (|super| (($ $ $) "\\spad{super(f,n)} creates a form for \\spad{f} superscripted by \\spad{n}.")) (|sub| (($ $ $) "\\spad{sub(f,n)} creates a form for \\spad{f} subscripted by \\spad{n}.")) (|binomial| (($ $ $) "\\spad{binomial(n,m)} creates a form for the binomial coefficient of \\spad{n} and \\spad{m}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(f,n)} creates a form for the \\spad{n}th derivative of \\spad{f},{} \\spadignore{e.g.} \\spad{f'},{} \\spad{f''},{} \\spad{f'''},{} \\spad{\"f} super \\spad{iv}\".")) (|rarrow| (($ $ $) "\\spad{rarrow(f,g)} creates a form for the mapping \\spad{f -> g}.")) (|assign| (($ $ $) "\\spad{assign(f,g)} creates a form for the assignment \\spad{f := g}.")) (|slash| (($ $ $) "\\spad{slash(f,g)} creates a form for the horizontal fraction of \\spad{f} over \\spad{g}.")) (|over| (($ $ $) "\\spad{over(f,g)} creates a form for the vertical fraction of \\spad{f} over \\spad{g}.")) (|root| (($ $ $) "\\spad{root(f,n)} creates a form for the \\spad{n}th root of form \\spad{f}.") (($ $) "\\spad{root(f)} creates a form for the square root of form \\spad{f}.")) (|zag| (($ $ $) "\\spad{zag(f,g)} creates a form for the continued fraction form for \\spad{f} over \\spad{g}.")) (|matrix| (($ (|List| (|List| $))) "\\spad{matrix(llf)} makes \\spad{llf} (a list of lists of forms) into a form which displays as a matrix.")) (|box| (($ $) "\\spad{box(f)} encloses \\spad{f} in a box.")) (|label| (($ $ $) "\\spad{label(n,f)} gives form \\spad{f} an equation label \\spad{n}.")) (|string| (($ $) "\\spad{string(f)} creates \\spad{f} with string quotes.")) (|elt| (($ $ (|List| $)) "\\spad{elt(op,l)} creates a form for application of \\spad{op} to list of arguments \\spad{l}.")) (|infix?| (((|Boolean|) $) "\\spad{infix?(op)} returns \\spad{true} if \\spad{op} is an infix operator,{} and \\spad{false} otherwise.")) (|postfix| (($ $ $) "\\spad{postfix(op, a)} creates a form which prints as: a \\spad{op}.")) (|infix| (($ $ $ $) "\\spad{infix(op, a, b)} creates a form which prints as: a \\spad{op} \\spad{b}.") (($ $ (|List| $)) "\\spad{infix(f,l)} creates a form depicting the \\spad{n}-ary application of infix operation \\spad{f} to a tuple of arguments \\spad{l}.")) (|prefix| (($ $ (|List| $)) "\\spad{prefix(f,l)} creates a form depicting the \\spad{n}-ary prefix application of \\spad{f} to a tuple of arguments given by list \\spad{l}.")) (|vconcat| (($ (|List| $)) "\\spad{vconcat(u)} vertically concatenates all forms in list \\spad{u}.") (($ $ $) "\\spad{vconcat(f,g)} vertically concatenates forms \\spad{f} and \\spad{g}.")) (|hconcat| (($ (|List| $)) "\\spad{hconcat(u)} horizontally concatenates all forms in list \\spad{u}.") (($ $ $) "\\spad{hconcat(f,g)} horizontally concatenate forms \\spad{f} and \\spad{g}.")) (|center| (($ $) "\\spad{center(f)} centers form \\spad{f} in total space.") (($ $ (|Integer|)) "\\spad{center(f,n)} centers form \\spad{f} within space of width \\spad{n}.")) (|right| (($ $) "\\spad{right(f)} right-justifies form \\spad{f} in total space.") (($ $ (|Integer|)) "\\spad{right(f,n)} right-justifies form \\spad{f} within space of width \\spad{n}.")) (|left| (($ $) "\\spad{left(f)} left-justifies form \\spad{f} in total space.") (($ $ (|Integer|)) "\\spad{left(f,n)} left-justifies form \\spad{f} within space of width \\spad{n}.")) (|rspace| (($ (|Integer|) (|Integer|)) "\\spad{rspace(n,m)} creates rectangular white space,{} \\spad{n} wide by \\spad{m} high.")) (|vspace| (($ (|Integer|)) "\\spad{vspace(n)} creates white space of height \\spad{n}.")) (|hspace| (($ (|Integer|)) "\\spad{hspace(n)} creates white space of width \\spad{n}.")) (|superHeight| (((|Integer|) $) "\\spad{superHeight(f)} returns the height of form \\spad{f} above the base line.")) (|subHeight| (((|Integer|) $) "\\spad{subHeight(f)} returns the height of form \\spad{f} below the base line.")) (|height| (((|Integer|)) "\\spad{height()} returns the height of the display area (an integer).") (((|Integer|) $) "\\spad{height(f)} returns the height of form \\spad{f} (an integer).")) (|width| (((|Integer|)) "\\spad{width()} returns the width of the display area (an integer).") (((|Integer|) $) "\\spad{width(f)} returns the width of form \\spad{f} (an integer).")) (|doubleFloatFormat| (((|String|) (|String|)) "change the output format for doublefloats using lisp format strings")) (|empty| (($) "\\spad{empty()} creates an empty form.")) (|outputForm| (($ (|DoubleFloat|)) "\\spad{outputForm(sf)} creates an form for small float \\spad{sf}.") (($ (|String|)) "\\spad{outputForm(s)} creates an form for string \\spad{s}.") (($ (|Symbol|)) "\\spad{outputForm(s)} creates an form for symbol \\spad{s}.") (($ (|Integer|)) "\\spad{outputForm(n)} creates an form for integer \\spad{n}.")) (|messagePrint| (((|Void|) (|String|)) "\\spad{messagePrint(s)} prints \\spad{s} without string quotes. Note: \\spad{messagePrint(s)} is equivalent to \\spad{print message(s)}.")) (|message| (($ (|String|)) "\\spad{message(s)} creates an form with no string quotes from string \\spad{s}.")) (|print| (((|Void|) $) "\\spad{print(u)} prints the form \\spad{u}."))) 
 NIL 
 NIL 
-(-867 |VariableList|) 
+(-869 |VariableList|) 
 ((|constructor| (NIL "This domain implements ordered variables")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} returns a member of the variable set or failed"))) 
 NIL 
 NIL 
-(-868) 
+(-870) 
 ((|constructor| (NIL "This domain represents set of overloaded operators (in fact operator descriptors).")) (|members| (((|List| (|FunctionDescriptor|)) $) "\\spad{members(x)} returns the list of operator descriptors,{} \\spadignore{e.g.} signature and implementation slots,{} of the overload set \\spad{x}.")) (|name| (((|Identifier|) $) "\\spad{name(x)} returns the name of the overload set \\spad{x}."))) 
 NIL 
 NIL 
-(-869 R |vl| |wl| |wtlevel|) 
+(-871 R |vl| |wl| |wtlevel|) 
 ((|constructor| (NIL "This domain represents truncated weighted polynomials over the \"Polynomial\" type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.")) (|changeWeightLevel| (((|Void|) (|NonNegativeInteger|)) "\\spad{changeWeightLevel(n)} This changes the weight level to the new value given: \\spad{NB:} previously calculated terms are not affected")) (/ (((|Union| $ "failed") $ $) "\\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \\spad{R} is a Field)"))) 
-((-4422 |has| |#1| (-173)) (-4421 |has| |#1| (-173)) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-870 R PS UP) 
+((-4429 |has| |#1| (-173)) (-4428 |has| |#1| (-173)) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-872 R PS UP) 
 ((|constructor| (NIL "\\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \\indented{2}{\"Pade Approximants,{} Part I: Basic Theory\",{} Baker & Graves-Morris.}")) (|padecf| (((|Union| (|ContinuedFraction| |#3|) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) |#2| |#2|) "\\spad{padecf(nd,dd,ns,ds)} computes the approximant as a continued fraction of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function).")) (|pade| (((|Union| (|Fraction| |#3|) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) |#2| |#2|) "\\spad{pade(nd,dd,ns,ds)} computes the approximant as a quotient of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function)."))) 
 NIL 
 NIL 
-(-871 R |x| |pt|) 
+(-873 R |x| |pt|) 
 ((|constructor| (NIL "\\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Trager,{}Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \\indented{2}{\"Pade Approximants,{} Part I: Basic Theory\",{} Baker & Graves-Morris.}")) (|pade| (((|Union| (|Fraction| (|UnivariatePolynomial| |#2| |#1|)) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) (|UnivariateTaylorSeries| |#1| |#2| |#3|)) "\\spad{pade(nd,dd,s)} computes the quotient of polynomials (if it exists) with numerator degree at most \\spad{nd} and denominator degree at most \\spad{dd} which matches the series \\spad{s} to order \\spad{nd + dd}.") (((|Union| (|Fraction| (|UnivariatePolynomial| |#2| |#1|)) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) (|UnivariateTaylorSeries| |#1| |#2| |#3|) (|UnivariateTaylorSeries| |#1| |#2| |#3|)) "\\spad{pade(nd,dd,ns,ds)} computes the approximant as a quotient of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function)."))) 
 NIL 
 NIL 
-(-872 |p|) 
+(-874 |p|) 
 ((|constructor| (NIL "Stream-based implementation of \\spad{Zp:} \\spad{p}-adic numbers are represented as sum(\\spad{i} = 0..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in 0,{}1,{}...,{}(\\spad{p} - 1)."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-873 |p|) 
+(-875 |p|) 
 ((|constructor| (NIL "This is the catefory of stream-based representations of \\indented{2}{the \\spad{p}-adic integers.}")) (|root| (($ (|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{root(f,a)} returns a root of the polynomial \\spad{f}. Argument \\spad{a} must be a root of \\spad{f} \\spad{(mod p)}.")) (|sqrt| (($ $ (|Integer|)) "\\spad{sqrt(b,a)} returns a square root of \\spad{b}. Argument \\spad{a} is a square root of \\spad{b} \\spad{(mod p)}.")) (|approximate| (((|Integer|) $ (|Integer|)) "\\spad{approximate(x,n)} returns an integer \\spad{y} such that \\spad{y = x (mod p^n)} when \\spad{n} is positive,{} and 0 otherwise.")) (|quotientByP| (($ $) "\\spad{quotientByP(x)} returns \\spad{b},{} where \\spad{x = a + b p}.")) (|moduloP| (((|Integer|) $) "\\spad{modulo(x)} returns a,{} where \\spad{x = a + b p}.")) (|modulus| (((|Integer|)) "\\spad{modulus()} returns the value of \\spad{p}.")) (|complete| (($ $) "\\spad{complete(x)} forces the computation of all digits.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(x,n)} forces the computation of digits up to order \\spad{n}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(x)} returns the exponent of the highest power of \\spad{p} dividing \\spad{x}.")) (|digits| (((|Stream| (|Integer|)) $) "\\spad{digits(x)} returns a stream of \\spad{p}-adic digits of \\spad{x}."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-874 |p|) 
+(-876 |p|) 
 ((|constructor| (NIL "Stream-based implementation of \\spad{Qp:} numbers are represented as sum(\\spad{i} = \\spad{k}..,{} a[\\spad{i}] * p^i) where the a[\\spad{i}] lie in 0,{}1,{}...,{}(\\spad{p} - 1)."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-872 |#1|) (QUOTE (-914))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| (-872 |#1|) (QUOTE (-145))) (|HasCategory| (-872 |#1|) (QUOTE (-147))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-872 |#1|) (QUOTE (-1024))) (|HasCategory| (-872 |#1|) (QUOTE (-823))) (-3962 (|HasCategory| (-872 |#1|) (QUOTE (-823))) (|HasCategory| (-872 |#1|) (QUOTE (-853)))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-872 |#1|) (QUOTE (-1155))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| (-872 |#1|) (QUOTE (-234))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -518) (QUOTE (-1181)) (LIST (QUOTE -872) (|devaluate| |#1|)))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -311) (LIST (QUOTE -872) (|devaluate| |#1|)))) (|HasCategory| (-872 |#1|) (LIST (QUOTE -288) (LIST (QUOTE -872) (|devaluate| |#1|)) (LIST (QUOTE -872) (|devaluate| |#1|)))) (|HasCategory| (-872 |#1|) (QUOTE (-309))) (|HasCategory| (-872 |#1|) (QUOTE (-549))) (|HasCategory| (-872 |#1|) (QUOTE (-853))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-872 |#1|) (QUOTE (-914)))) (-3962 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-872 |#1|) (QUOTE (-914)))) (|HasCategory| (-872 |#1|) (QUOTE (-145))))) 
-(-875 |p| PADIC) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-874 |#1|) (QUOTE (-916))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| (-874 |#1|) (QUOTE (-145))) (|HasCategory| (-874 |#1|) (QUOTE (-147))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-874 |#1|) (QUOTE (-1026))) (|HasCategory| (-874 |#1|) (QUOTE (-825))) (-3969 (|HasCategory| (-874 |#1|) (QUOTE (-825))) (|HasCategory| (-874 |#1|) (QUOTE (-855)))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-874 |#1|) (QUOTE (-1157))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| (-874 |#1|) (QUOTE (-234))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -519) (QUOTE (-1183)) (LIST (QUOTE -874) (|devaluate| |#1|)))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -312) (LIST (QUOTE -874) (|devaluate| |#1|)))) (|HasCategory| (-874 |#1|) (LIST (QUOTE -289) (LIST (QUOTE -874) (|devaluate| |#1|)) (LIST (QUOTE -874) (|devaluate| |#1|)))) (|HasCategory| (-874 |#1|) (QUOTE (-310))) (|HasCategory| (-874 |#1|) (QUOTE (-550))) (|HasCategory| (-874 |#1|) (QUOTE (-855))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-874 |#1|) (QUOTE (-916)))) (-3969 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-874 |#1|) (QUOTE (-916)))) (|HasCategory| (-874 |#1|) (QUOTE (-145))))) 
+(-877 |p| PADIC) 
 ((|constructor| (NIL "This is the category of stream-based representations of \\spad{Qp}.")) (|removeZeroes| (($ (|Integer|) $) "\\spad{removeZeroes(n,x)} removes up to \\spad{n} leading zeroes from the \\spad{p}-adic rational \\spad{x}.") (($ $) "\\spad{removeZeroes(x)} removes leading zeroes from the representation of the \\spad{p}-adic rational \\spad{x}. A \\spad{p}-adic rational is represented by (1) an exponent and (2) a \\spad{p}-adic integer which may have leading zero digits. When the \\spad{p}-adic integer has a leading zero digit,{} a 'leading zero' is removed from the \\spad{p}-adic rational as follows: the number is rewritten by increasing the exponent by 1 and dividing the \\spad{p}-adic integer by \\spad{p}. Note: \\spad{removeZeroes(f)} removes all leading zeroes from \\spad{f}.")) (|continuedFraction| (((|ContinuedFraction| (|Fraction| (|Integer|))) $) "\\spad{continuedFraction(x)} converts the \\spad{p}-adic rational number \\spad{x} to a continued fraction.")) (|approximate| (((|Fraction| (|Integer|)) $ (|Integer|)) "\\spad{approximate(x,n)} returns a rational number \\spad{y} such that \\spad{y = x (mod p^n)}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (QUOTE (-1024))) (|HasCategory| |#2| (QUOTE (-823))) (-3962 (|HasCategory| |#2| (QUOTE (-823))) (|HasCategory| |#2| (QUOTE (-853)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-1155))) (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (LIST (QUOTE -518) (QUOTE (-1181)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -288) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-549))) (|HasCategory| |#2| (QUOTE (-853))) (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
-(-876 S T$) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (QUOTE (-1026))) (|HasCategory| |#2| (QUOTE (-825))) (-3969 (|HasCategory| |#2| (QUOTE (-825))) (|HasCategory| |#2| (QUOTE (-855)))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-1157))) (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#2| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -289) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-310))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-855))) (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
+(-878 S T$) 
 ((|constructor| (NIL "\\indented{1}{This domain provides a very simple representation} of the notion of `pair of objects'. It does not try to achieve all possible imaginable things.")) (|second| ((|#2| $) "\\spad{second(p)} extracts the second components of \\spad{`p'}.")) (|first| ((|#1| $) "\\spad{first(p)} extracts the first component of \\spad{`p'}.")) (|construct| (($ |#1| |#2|) "\\spad{construct(s,t)} is same as pair(\\spad{s},{}\\spad{t}),{} with syntactic sugar.")) (|pair| (($ |#1| |#2|) "\\spad{pair(s,t)} returns a pair object composed of \\spad{`s'} and \\spad{`t'}."))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#2| (QUOTE (-1105)))) (-3962 (-12 (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#2| (QUOTE (-1105))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))))) 
-(-877) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#2| (QUOTE (-1107)))) (-3969 (-12 (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#2| (QUOTE (-1107))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))))) 
+(-879) 
 ((|constructor| (NIL "This domain describes four groups of color shades (palettes).")) (|coerce| (($ (|Color|)) "\\spad{coerce(c)} sets the average shade for the palette to that of the indicated color \\spad{c}.")) (|shade| (((|Integer|) $) "\\spad{shade(p)} returns the shade index of the indicated palette \\spad{p}.")) (|hue| (((|Color|) $) "\\spad{hue(p)} returns the hue field of the indicated palette \\spad{p}.")) (|light| (($ (|Color|)) "\\spad{light(c)} sets the shade of a hue,{} \\spad{c},{} to it\\spad{'s} highest value.")) (|pastel| (($ (|Color|)) "\\spad{pastel(c)} sets the shade of a hue,{} \\spad{c},{} above bright,{} but below light.")) (|bright| (($ (|Color|)) "\\spad{bright(c)} sets the shade of a hue,{} \\spad{c},{} above dim,{} but below pastel.")) (|dim| (($ (|Color|)) "\\spad{dim(c)} sets the shade of a hue,{} \\spad{c},{} above dark,{} but below bright.")) (|dark| (($ (|Color|)) "\\spad{dark(c)} sets the shade of the indicated hue of \\spad{c} to it\\spad{'s} lowest value."))) 
 NIL 
 NIL 
-(-878) 
+(-880) 
 ((|constructor| (NIL "This package provides a coerce from polynomials over algebraic numbers to \\spadtype{Expression AlgebraicNumber}.")) (|coerce| (((|Expression| (|Integer|)) (|Fraction| (|Polynomial| (|AlgebraicNumber|)))) "\\spad{coerce(rf)} converts \\spad{rf},{} a fraction of polynomial \\spad{p} with algebraic number coefficients to \\spadtype{Expression Integer}.") (((|Expression| (|Integer|)) (|Polynomial| (|AlgebraicNumber|))) "\\spad{coerce(p)} converts the polynomial \\spad{p} with algebraic number coefficients to \\spadtype{Expression Integer}."))) 
 NIL 
 NIL 
-(-879) 
+(-881) 
 ((|constructor| (NIL "Representation of parameters to functions or constructors. For the most part,{} they are Identifiers. However,{} in very cases,{} they are \"flags\",{} \\spadignore{e.g.} string literals.")) (|autoCoerce| (((|String|) $) "\\spad{autoCoerce(x)@String} implicitly coerce the object \\spad{x} to \\spadtype{String}. This function is left at the discretion of the compiler.") (((|Identifier|) $) "\\spad{autoCoerce(x)@Identifier} implicitly coerce the object \\spad{x} to \\spadtype{Identifier}. This function is left at the discretion of the compiler.")) (|case| (((|Boolean|) $ (|[\|\|]| (|String|))) "\\spad{x case String} if the parameter AST object \\spad{x} designates a flag.") (((|Boolean|) $ (|[\|\|]| (|Identifier|))) "\\spad{x case Identifier} if the parameter AST object \\spad{x} designates an \\spadtype{Identifier}."))) 
 NIL 
 NIL 
-(-880 CF1 CF2) 
+(-882 CF1 CF2) 
 ((|constructor| (NIL "This package \\undocumented")) (|map| (((|ParametricPlaneCurve| |#2|) (|Mapping| |#2| |#1|) (|ParametricPlaneCurve| |#1|)) "\\spad{map(f,x)} \\undocumented"))) 
 NIL 
 NIL 
-(-881 |ComponentFunction|) 
+(-883 |ComponentFunction|) 
 ((|constructor| (NIL "ParametricPlaneCurve is used for plotting parametric plane curves in the affine plane.")) (|coordinate| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coordinate(c,i)} returns a coordinate function for \\spad{c} using 1-based indexing according to \\spad{i}. This indicates what the function for the coordinate component \\spad{i} of the plane curve is.")) (|curve| (($ |#1| |#1|) "\\spad{curve(c1,c2)} creates a plane curve from 2 component functions \\spad{c1} and \\spad{c2}."))) 
 NIL 
 NIL 
-(-882 CF1 CF2) 
+(-884 CF1 CF2) 
 ((|constructor| (NIL "This package \\undocumented")) (|map| (((|ParametricSpaceCurve| |#2|) (|Mapping| |#2| |#1|) (|ParametricSpaceCurve| |#1|)) "\\spad{map(f,x)} \\undocumented"))) 
 NIL 
 NIL 
-(-883 |ComponentFunction|) 
+(-885 |ComponentFunction|) 
 ((|constructor| (NIL "ParametricSpaceCurve is used for plotting parametric space curves in affine 3-space.")) (|coordinate| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coordinate(c,i)} returns a coordinate function of \\spad{c} using 1-based indexing according to \\spad{i}. This indicates what the function for the coordinate component,{} \\spad{i},{} of the space curve is.")) (|curve| (($ |#1| |#1| |#1|) "\\spad{curve(c1,c2,c3)} creates a space curve from 3 component functions \\spad{c1},{} \\spad{c2},{} and \\spad{c3}."))) 
 NIL 
 NIL 
-(-884) 
+(-886) 
 ((|constructor| (NIL "\\indented{1}{This package provides a simple Spad script parser.} Related Constructors: Syntax. See Also: Syntax.")) (|getSyntaxFormsFromFile| (((|List| (|Syntax|)) (|String|)) "\\spad{getSyntaxFormsFromFile(f)} parses the source file \\spad{f} (supposedly containing Spad scripts) and returns a List Syntax. The filename \\spad{f} is supposed to have the proper extension. Note that source location information is not part of result."))) 
 NIL 
 NIL 
-(-885 CF1 CF2) 
+(-887 CF1 CF2) 
 ((|constructor| (NIL "This package \\undocumented")) (|map| (((|ParametricSurface| |#2|) (|Mapping| |#2| |#1|) (|ParametricSurface| |#1|)) "\\spad{map(f,x)} \\undocumented"))) 
 NIL 
 NIL 
-(-886 |ComponentFunction|) 
+(-888 |ComponentFunction|) 
 ((|constructor| (NIL "ParametricSurface is used for plotting parametric surfaces in affine 3-space.")) (|coordinate| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coordinate(s,i)} returns a coordinate function of \\spad{s} using 1-based indexing according to \\spad{i}. This indicates what the function for the coordinate component,{} \\spad{i},{} of the surface is.")) (|surface| (($ |#1| |#1| |#1|) "\\spad{surface(c1,c2,c3)} creates a surface from 3 parametric component functions \\spad{c1},{} \\spad{c2},{} and \\spad{c3}."))) 
 NIL 
 NIL 
-(-887) 
+(-889) 
 ((|constructor| (NIL "PartitionsAndPermutations contains functions for generating streams of integer partitions,{} and streams of sequences of integers composed from a multi-set.")) (|permutations| (((|Stream| (|List| (|Integer|))) (|Integer|)) "\\spad{permutations(n)} is the stream of permutations \\indented{1}{formed from \\spad{1,2,3,...,n}.}")) (|sequences| (((|Stream| (|List| (|Integer|))) (|List| (|Integer|))) "\\spad{sequences([l0,l1,l2,..,ln])} is the set of \\indented{1}{all sequences formed from} \\spad{l0} 0\\spad{'s},{}\\spad{l1} 1\\spad{'s},{}\\spad{l2} 2\\spad{'s},{}...,{}\\spad{ln} \\spad{n}\\spad{'s}.") (((|Stream| (|List| (|Integer|))) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{sequences(l1,l2)} is the stream of all sequences that \\indented{1}{can be composed from the multiset defined from} \\indented{1}{two lists of integers \\spad{l1} and \\spad{l2}.} \\indented{1}{For example,{}the pair \\spad{([1,2,4],[2,3,5])} represents} \\indented{1}{multi-set with 1 \\spad{2},{} 2 \\spad{3}\\spad{'s},{} and 4 \\spad{5}\\spad{'s}.}")) (|shufflein| (((|Stream| (|List| (|Integer|))) (|List| (|Integer|)) (|Stream| (|List| (|Integer|)))) "\\spad{shufflein(l,st)} maps shuffle(\\spad{l},{}\\spad{u}) on to all \\indented{1}{members \\spad{u} of \\spad{st},{} concatenating the results.}")) (|shuffle| (((|Stream| (|List| (|Integer|))) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{shuffle(l1,l2)} forms the stream of all shuffles of \\spad{l1} \\indented{1}{and \\spad{l2},{} \\spadignore{i.e.} all sequences that can be formed from} \\indented{1}{merging \\spad{l1} and \\spad{l2}.}")) (|conjugates| (((|Stream| (|List| (|Integer|))) (|Stream| (|List| (|Integer|)))) "\\spad{conjugates(lp)} is the stream of conjugates of a stream \\indented{1}{of partitions \\spad{lp}.}")) (|conjugate| (((|List| (|Integer|)) (|List| (|Integer|))) "\\spad{conjugate(pt)} is the conjugate of the partition \\spad{pt}.")) (|partitions| (((|Stream| (|List| (|Integer|))) (|Integer|) (|Integer|)) "\\spad{partitions(p,l)} is the stream of all \\indented{1}{partitions whose number of} \\indented{1}{parts and largest part are no greater than \\spad{p} and \\spad{l}.}") (((|Stream| (|List| (|Integer|))) (|Integer|)) "\\spad{partitions(n)} is the stream of all partitions of \\spad{n}.") (((|Stream| (|List| (|Integer|))) (|Integer|) (|Integer|) (|Integer|)) "\\spad{partitions(p,l,n)} is the stream of partitions \\indented{1}{of \\spad{n} whose number of parts is no greater than \\spad{p}} \\indented{1}{and whose largest part is no greater than \\spad{l}.}"))) 
 NIL 
 NIL 
-(-888 R) 
+(-890 R) 
 ((|constructor| (NIL "An object \\spad{S} is Patternable over an object \\spad{R} if \\spad{S} can lift the conversions from \\spad{R} into \\spadtype{Pattern(Integer)} and \\spadtype{Pattern(Float)} to itself."))) 
 NIL 
 NIL 
-(-889 R S L) 
+(-891 R S L) 
 ((|constructor| (NIL "A PatternMatchListResult is an object internally returned by the pattern matcher when matching on lists. It is either a failed match,{} or a pair of PatternMatchResult,{} one for atoms (elements of the list),{} and one for lists.")) (|lists| (((|PatternMatchResult| |#1| |#3|) $) "\\spad{lists(r)} returns the list of matches that match lists.")) (|atoms| (((|PatternMatchResult| |#1| |#2|) $) "\\spad{atoms(r)} returns the list of matches that match atoms (elements of the lists).")) (|makeResult| (($ (|PatternMatchResult| |#1| |#2|) (|PatternMatchResult| |#1| |#3|)) "\\spad{makeResult(r1,r2)} makes the combined result [\\spad{r1},{}\\spad{r2}].")) (|new| (($) "\\spad{new()} returns a new empty match result.")) (|failed| (($) "\\spad{failed()} returns a failed match.")) (|failed?| (((|Boolean|) $) "\\spad{failed?(r)} tests if \\spad{r} is a failed match."))) 
 NIL 
 NIL 
-(-890 S) 
+(-892 S) 
 ((|constructor| (NIL "A set \\spad{R} is PatternMatchable over \\spad{S} if elements of \\spad{R} can be matched to patterns over \\spad{S}.")) (|patternMatch| (((|PatternMatchResult| |#1| $) $ (|Pattern| |#1|) (|PatternMatchResult| |#1| $)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}. res contains the variables of \\spad{pat} which are already matched and their matches (necessary for recursion). Initially,{} res is just the result of \\spadfun{new} which is an empty list of matches."))) 
 NIL 
 NIL 
-(-891 |Base| |Subject| |Pat|) 
+(-893 |Base| |Subject| |Pat|) 
 ((|constructor| (NIL "This package provides the top-level pattern macthing functions.")) (|Is| (((|PatternMatchResult| |#1| |#2|) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a match of the form \\spad{[v1 = e1,...,vn = en]}; returns an empty match if \\spad{expr} is exactly equal to pat. returns a \\spadfun{failed} match if pat does not match \\spad{expr}.") (((|List| (|Equation| (|Polynomial| |#2|))) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a list of matches \\spad{[v1 = e1,...,vn = en]}; returns an empty list if either \\spad{expr} is exactly equal to pat or if pat does not match \\spad{expr}.") (((|List| (|Equation| |#2|)) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a list of matches \\spad{[v1 = e1,...,vn = en]}; returns an empty list if either \\spad{expr} is exactly equal to pat or if pat does not match \\spad{expr}.") (((|PatternMatchListResult| |#1| |#2| (|List| |#2|)) (|List| |#2|) |#3|) "\\spad{Is([e1,...,en], pat)} matches the pattern pat on the list of expressions \\spad{[e1,...,en]} and returns the result.")) (|is?| (((|Boolean|) (|List| |#2|) |#3|) "\\spad{is?([e1,...,en], pat)} tests if the list of expressions \\spad{[e1,...,en]} matches the pattern pat.") (((|Boolean|) |#2| |#3|) "\\spad{is?(expr, pat)} tests if the expression \\spad{expr} matches the pattern pat."))) 
 NIL 
-((-12 (-3748 (|HasCategory| |#2| (QUOTE (-1053)))) (-3748 (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-1181)))))) (-12 (|HasCategory| |#2| (QUOTE (-1053))) (-3748 (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-1181)))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-1181))))) 
-(-892 R S) 
+((-12 (-3755 (|HasCategory| |#2| (QUOTE (-1055)))) (-3755 (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-1183)))))) (-12 (|HasCategory| |#2| (QUOTE (-1055))) (-3755 (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-1183)))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-1183))))) 
+(-894 R S) 
 ((|constructor| (NIL "A PatternMatchResult is an object internally returned by the pattern matcher; It is either a failed match,{} or a list of matches of the form (var,{} expr) meaning that the variable var matches the expression expr.")) (|satisfy?| (((|Union| (|Boolean|) "failed") $ (|Pattern| |#1|)) "\\spad{satisfy?(r, p)} returns \\spad{true} if the matches satisfy the top-level predicate of \\spad{p},{} \\spad{false} if they don\\spad{'t},{} and \"failed\" if not enough variables of \\spad{p} are matched in \\spad{r} to decide.")) (|construct| (($ (|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| |#2|)))) "\\spad{construct([v1,e1],...,[vn,en])} returns the match result containing the matches (\\spad{v1},{}e1),{}...,{}(\\spad{vn},{}en).")) (|destruct| (((|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| |#2|))) $) "\\spad{destruct(r)} returns the list of matches (var,{} expr) in \\spad{r}. Error: if \\spad{r} is a failed match.")) (|addMatchRestricted| (($ (|Pattern| |#1|) |#2| $ |#2|) "\\spad{addMatchRestricted(var, expr, r, val)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} provided that \\spad{expr} satisfies the predicates attached to \\spad{var},{} that \\spad{var} is not matched to another expression already,{} and that either \\spad{var} is an optional pattern variable or that \\spad{expr} is not equal to val (usually an identity).")) (|insertMatch| (($ (|Pattern| |#1|) |#2| $) "\\spad{insertMatch(var, expr, r)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} without checking predicates or previous matches for \\spad{var}.")) (|addMatch| (($ (|Pattern| |#1|) |#2| $) "\\spad{addMatch(var, expr, r)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} provided that \\spad{expr} satisfies the predicates attached to \\spad{var},{} and that \\spad{var} is not matched to another expression already.")) (|getMatch| (((|Union| |#2| "failed") (|Pattern| |#1|) $) "\\spad{getMatch(var, r)} returns the expression that \\spad{var} matches in the result \\spad{r},{} and \"failed\" if \\spad{var} is not matched in \\spad{r}.")) (|union| (($ $ $) "\\spad{union(a, b)} makes the set-union of two match results.")) (|new| (($) "\\spad{new()} returns a new empty match result.")) (|failed| (($) "\\spad{failed()} returns a failed match.")) (|failed?| (((|Boolean|) $) "\\spad{failed?(r)} tests if \\spad{r} is a failed match."))) 
 NIL 
 NIL 
-(-893 R A B) 
+(-895 R A B) 
 ((|constructor| (NIL "Lifts maps to pattern matching results.")) (|map| (((|PatternMatchResult| |#1| |#3|) (|Mapping| |#3| |#2|) (|PatternMatchResult| |#1| |#2|)) "\\spad{map(f, [(v1,a1),...,(vn,an)])} returns the matching result [(\\spad{v1},{}\\spad{f}(a1)),{}...,{}(\\spad{vn},{}\\spad{f}(an))]."))) 
 NIL 
 NIL 
-(-894 R) 
+(-896 R) 
 ((|constructor| (NIL "Patterns for use by the pattern matcher.")) (|optpair| (((|Union| (|List| $) "failed") (|List| $)) "\\spad{optpair(l)} returns \\spad{l} has the form \\spad{[a, b]} and a is optional,{} and \"failed\" otherwise.")) (|variables| (((|List| $) $) "\\spad{variables(p)} returns the list of matching variables appearing in \\spad{p}.")) (|getBadValues| (((|List| (|Any|)) $) "\\spad{getBadValues(p)} returns the list of \"bad values\" for \\spad{p}. Note: \\spad{p} is not allowed to match any of its \"bad values\".")) (|addBadValue| (($ $ (|Any|)) "\\spad{addBadValue(p, v)} adds \\spad{v} to the list of \"bad values\" for \\spad{p}. Note: \\spad{p} is not allowed to match any of its \"bad values\".")) (|resetBadValues| (($ $) "\\spad{resetBadValues(p)} initializes the list of \"bad values\" for \\spad{p} to \\spad{[]}. Note: \\spad{p} is not allowed to match any of its \"bad values\".")) (|hasTopPredicate?| (((|Boolean|) $) "\\spad{hasTopPredicate?(p)} tests if \\spad{p} has a top-level predicate.")) (|topPredicate| (((|Record| (|:| |var| (|List| (|Symbol|))) (|:| |pred| (|Any|))) $) "\\spad{topPredicate(x)} returns \\spad{[[a1,...,an], f]} where the top-level predicate of \\spad{x} is \\spad{f(a1,...,an)}. Note: \\spad{n} is 0 if \\spad{x} has no top-level predicate.")) (|setTopPredicate| (($ $ (|List| (|Symbol|)) (|Any|)) "\\spad{setTopPredicate(x, [a1,...,an], f)} returns \\spad{x} with the top-level predicate set to \\spad{f(a1,...,an)}.")) (|patternVariable| (($ (|Symbol|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\spad{patternVariable(x, c?, o?, m?)} creates a pattern variable \\spad{x},{} which is constant if \\spad{c? = true},{} optional if \\spad{o? = true},{} and multiple if \\spad{m? = true}.")) (|withPredicates| (($ $ (|List| (|Any|))) "\\spad{withPredicates(p, [p1,...,pn])} makes a copy of \\spad{p} and attaches the predicate \\spad{p1} and ... and \\spad{pn} to the copy,{} which is returned.")) (|setPredicates| (($ $ (|List| (|Any|))) "\\spad{setPredicates(p, [p1,...,pn])} attaches the predicate \\spad{p1} and ... and \\spad{pn} to \\spad{p}.")) (|predicates| (((|List| (|Any|)) $) "\\spad{predicates(p)} returns \\spad{[p1,...,pn]} such that the predicate attached to \\spad{p} is \\spad{p1} and ... and \\spad{pn}.")) (|hasPredicate?| (((|Boolean|) $) "\\spad{hasPredicate?(p)} tests if \\spad{p} has predicates attached to it.")) (|optional?| (((|Boolean|) $) "\\spad{optional?(p)} tests if \\spad{p} is a single matching variable which can match an identity.")) (|multiple?| (((|Boolean|) $) "\\spad{multiple?(p)} tests if \\spad{p} is a single matching variable allowing list matching or multiple term matching in a sum or product.")) (|generic?| (((|Boolean|) $) "\\spad{generic?(p)} tests if \\spad{p} is a single matching variable.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(p)} tests if \\spad{p} contains no matching variables.")) (|symbol?| (((|Boolean|) $) "\\spad{symbol?(p)} tests if \\spad{p} is a symbol.")) (|quoted?| (((|Boolean|) $) "\\spad{quoted?(p)} tests if \\spad{p} is of the form \\spad{'s} for a symbol \\spad{s}.")) (|inR?| (((|Boolean|) $) "\\spad{inR?(p)} tests if \\spad{p} is an atom (\\spadignore{i.e.} an element of \\spad{R}).")) (|copy| (($ $) "\\spad{copy(p)} returns a recursive copy of \\spad{p}.")) (|convert| (($ (|List| $)) "\\spad{convert([a1,...,an])} returns the pattern \\spad{[a1,...,an]}.")) (|depth| (((|NonNegativeInteger|) $) "\\spad{depth(p)} returns the nesting level of \\spad{p}.")) (/ (($ $ $) "\\spad{a / b} returns the pattern \\spad{a / b}.")) (** (($ $ $) "\\spad{a ** b} returns the pattern \\spad{a ** b}.") (($ $ (|NonNegativeInteger|)) "\\spad{a ** n} returns the pattern \\spad{a ** n}.")) (* (($ $ $) "\\spad{a * b} returns the pattern \\spad{a * b}.")) (+ (($ $ $) "\\spad{a + b} returns the pattern \\spad{a + b}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op, [a1,...,an])} returns \\spad{op(a1,...,an)}.")) (|isPower| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| $)) "failed") $) "\\spad{isPower(p)} returns \\spad{[a, b]} if \\spad{p = a ** b},{} and \"failed\" otherwise.")) (|isList| (((|Union| (|List| $) "failed") $) "\\spad{isList(p)} returns \\spad{[a1,...,an]} if \\spad{p = [a1,...,an]},{} \"failed\" otherwise.")) (|isQuotient| (((|Union| (|Record| (|:| |num| $) (|:| |den| $)) "failed") $) "\\spad{isQuotient(p)} returns \\spad{[a, b]} if \\spad{p = a / b},{} and \"failed\" otherwise.")) (|isExpt| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| (|NonNegativeInteger|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[q, n]} if \\spad{n > 0} and \\spad{p = q ** n},{} and \"failed\" otherwise.")) (|isOp| (((|Union| (|Record| (|:| |op| (|BasicOperator|)) (|:| |arg| (|List| $))) "failed") $) "\\spad{isOp(p)} returns \\spad{[op, [a1,...,an]]} if \\spad{p = op(a1,...,an)},{} and \"failed\" otherwise.") (((|Union| (|List| $) "failed") $ (|BasicOperator|)) "\\spad{isOp(p, op)} returns \\spad{[a1,...,an]} if \\spad{p = op(a1,...,an)},{} and \"failed\" otherwise.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{n > 1} and \\spad{p = a1 * ... * an},{} and \"failed\" otherwise.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[a1,...,an]} if \\spad{n > 1} \\indented{1}{and \\spad{p = a1 + ... + an},{}} and \"failed\" otherwise.")) ((|One|) (($) "1")) ((|Zero|) (($) "0"))) 
 NIL 
 NIL 
-(-895 R -3074) 
+(-897 R -3081) 
 ((|constructor| (NIL "Tools for patterns.")) (|badValues| (((|List| |#2|) (|Pattern| |#1|)) "\\spad{badValues(p)} returns the list of \"bad values\" for \\spad{p}; \\spad{p} is not allowed to match any of its \"bad values\".")) (|addBadValue| (((|Pattern| |#1|) (|Pattern| |#1|) |#2|) "\\spad{addBadValue(p, v)} adds \\spad{v} to the list of \"bad values\" for \\spad{p}; \\spad{p} is not allowed to match any of its \"bad values\".")) (|satisfy?| (((|Boolean|) (|List| |#2|) (|Pattern| |#1|)) "\\spad{satisfy?([v1,...,vn], p)} returns \\spad{f(v1,...,vn)} where \\spad{f} is the top-level predicate attached to \\spad{p}.") (((|Boolean|) |#2| (|Pattern| |#1|)) "\\spad{satisfy?(v, p)} returns \\spad{f}(\\spad{v}) where \\spad{f} is the predicate attached to \\spad{p}.")) (|predicate| (((|Mapping| (|Boolean|) |#2|) (|Pattern| |#1|)) "\\spad{predicate(p)} returns the predicate attached to \\spad{p},{} the constant function \\spad{true} if \\spad{p} has no predicates attached to it.")) (|suchThat| (((|Pattern| |#1|) (|Pattern| |#1|) (|List| (|Symbol|)) (|Mapping| (|Boolean|) (|List| |#2|))) "\\spad{suchThat(p, [a1,...,an], f)} returns a copy of \\spad{p} with the top-level predicate set to \\spad{f(a1,...,an)}.") (((|Pattern| |#1|) (|Pattern| |#1|) (|List| (|Mapping| (|Boolean|) |#2|))) "\\spad{suchThat(p, [f1,...,fn])} makes a copy of \\spad{p} and adds the predicate \\spad{f1} and ... and \\spad{fn} to the copy,{} which is returned.") (((|Pattern| |#1|) (|Pattern| |#1|) (|Mapping| (|Boolean|) |#2|)) "\\spad{suchThat(p, f)} makes a copy of \\spad{p} and adds the predicate \\spad{f} to the copy,{} which is returned."))) 
 NIL 
 NIL 
-(-896 R S) 
+(-898 R S) 
 ((|constructor| (NIL "Lifts maps to patterns.")) (|map| (((|Pattern| |#2|) (|Mapping| |#2| |#1|) (|Pattern| |#1|)) "\\spad{map(f, p)} applies \\spad{f} to all the leaves of \\spad{p} and returns the result as a pattern over \\spad{S}."))) 
 NIL 
 NIL 
-(-897 |VarSet|) 
+(-899 |VarSet|) 
 ((|constructor| (NIL "This domain provides the internal representation of polynomials in non-commutative variables written over the Poincare-Birkhoff-Witt basis. See the \\spadtype{XPBWPolynomial} domain constructor. See Free Lie Algebras by \\spad{C}. Reutenauer (Oxford science publications). \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr}).")) (|varList| (((|List| |#1|) $) "\\spad{varList([l1]*[l2]*...[ln])} returns the list of variables in the word \\spad{l1*l2*...*ln}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?([l1]*[l2]*...[ln])} returns \\spad{true} iff \\spad{n} equals \\spad{1}.")) (|rest| (($ $) "\\spad{rest([l1]*[l2]*...[ln])} returns the list \\spad{l2, .... ln}.")) (|ListOfTerms| (((|List| (|LyndonWord| |#1|)) $) "\\spad{ListOfTerms([l1]*[l2]*...[ln])} returns the list of words \\spad{l1, l2, .... ln}.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length([l1]*[l2]*...[ln])} returns the length of the word \\spad{l1*l2*...*ln}.")) (|first| (((|LyndonWord| |#1|) $) "\\spad{first([l1]*[l2]*...[ln])} returns the Lyndon word \\spad{l1}.")) (|coerce| (($ |#1|) "\\spad{coerce(v)} return \\spad{v}") (((|OrderedFreeMonoid| |#1|) $) "\\spad{coerce([l1]*[l2]*...[ln])} returns the word \\spad{l1*l2*...*ln},{} where \\spad{[l_i]} is the backeted form of the Lyndon word \\spad{l_i}.")) ((|One|) (($) "\\spad{1} returns the empty list."))) 
 NIL 
 NIL 
-(-898 UP R) 
+(-900 UP R) 
 ((|constructor| (NIL "This package \\undocumented")) (|compose| ((|#1| |#1| |#1|) "\\spad{compose(p,q)} \\undocumented"))) 
 NIL 
 NIL 
-(-899) 
+(-901) 
 ((|PDESolve| (((|Result|) (|Record| (|:| |pde| (|List| (|Expression| (|DoubleFloat|)))) (|:| |constraints| (|List| (|Record| (|:| |start| (|DoubleFloat|)) (|:| |finish| (|DoubleFloat|)) (|:| |grid| (|NonNegativeInteger|)) (|:| |boundaryType| (|Integer|)) (|:| |dStart| (|Matrix| (|DoubleFloat|))) (|:| |dFinish| (|Matrix| (|DoubleFloat|)))))) (|:| |f| (|List| (|List| (|Expression| (|DoubleFloat|))))) (|:| |st| (|String|)) (|:| |tol| (|DoubleFloat|)))) "\\spad{PDESolve(args)} performs the integration of the function given the strategy or method returned by \\axiomFun{measure}.")) (|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |explanations| (|String|))) (|RoutinesTable|) (|Record| (|:| |pde| (|List| (|Expression| (|DoubleFloat|)))) (|:| |constraints| (|List| (|Record| (|:| |start| (|DoubleFloat|)) (|:| |finish| (|DoubleFloat|)) (|:| |grid| (|NonNegativeInteger|)) (|:| |boundaryType| (|Integer|)) (|:| |dStart| (|Matrix| (|DoubleFloat|))) (|:| |dFinish| (|Matrix| (|DoubleFloat|)))))) (|:| |f| (|List| (|List| (|Expression| (|DoubleFloat|))))) (|:| |st| (|String|)) (|:| |tol| (|DoubleFloat|)))) "\\spad{measure(R,args)} calculates an estimate of the ability of a particular method to solve a problem. \\blankline This method may be either a specific NAG routine or a strategy (such as transforming the function from one which is difficult to one which is easier to solve). \\blankline It will call whichever agents are needed to perform analysis on the problem in order to calculate the measure. There is a parameter,{} labelled \\axiom{sofar},{} which would contain the best compatibility found so far."))) 
 NIL 
 NIL 
-(-900 UP -3498) 
+(-902 UP -3505) 
 ((|constructor| (NIL "This package \\undocumented")) (|rightFactorCandidate| ((|#1| |#1| (|NonNegativeInteger|)) "\\spad{rightFactorCandidate(p,n)} \\undocumented")) (|leftFactor| (((|Union| |#1| "failed") |#1| |#1|) "\\spad{leftFactor(p,q)} \\undocumented")) (|decompose| (((|Union| (|Record| (|:| |left| |#1|) (|:| |right| |#1|)) "failed") |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{decompose(up,m,n)} \\undocumented") (((|List| |#1|) |#1|) "\\spad{decompose(up)} \\undocumented"))) 
 NIL 
 NIL 
-(-901) 
+(-903) 
 ((|measure| (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalPDEProblem|) (|RoutinesTable|)) "\\spad{measure(prob,R)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical PDE problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} listed in \\axiom{\\spad{R}} of \\axiom{category} \\axiomType{PartialDifferentialEquationsSolverCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information. It predicts the likely most effective NAG numerical Library routine to solve the input set of PDEs by checking various attributes of the system of PDEs and calculating a measure of compatibility of each routine to these attributes.") (((|Record| (|:| |measure| (|Float|)) (|:| |name| (|String|)) (|:| |explanations| (|List| (|String|)))) (|NumericalPDEProblem|)) "\\spad{measure(prob)} is a top level ANNA function for identifying the most appropriate numerical routine from those in the routines table provided for solving the numerical PDE problem defined by \\axiom{\\spad{prob}}. \\blankline It calls each \\axiom{domain} of \\axiom{category} \\axiomType{PartialDifferentialEquationsSolverCategory} in turn to calculate all measures and returns the best \\spadignore{i.e.} the name of the most appropriate domain and any other relevant information. It predicts the likely most effective NAG numerical Library routine to solve the input set of PDEs by checking various attributes of the system of PDEs and calculating a measure of compatibility of each routine to these attributes.")) (|solve| (((|Result|) (|Float|) (|Float|) (|Float|) (|Float|) (|NonNegativeInteger|) (|NonNegativeInteger|) (|List| (|Expression| (|Float|))) (|List| (|List| (|Expression| (|Float|)))) (|String|)) "\\spad{solve(xmin,ymin,xmax,ymax,ngx,ngy,pde,bounds,st)} is a top level ANNA function to solve numerically a system of partial differential equations. This is defined as a list of coefficients (\\axiom{\\spad{pde}}),{} a grid (\\axiom{\\spad{xmin}},{} \\axiom{\\spad{ymin}},{} \\axiom{\\spad{xmax}},{} \\axiom{\\spad{ymax}},{} \\axiom{\\spad{ngx}},{} \\axiom{\\spad{ngy}}) and the boundary values (\\axiom{\\spad{bounds}}). A default value for tolerance is used. There is also a parameter (\\axiom{\\spad{st}}) which should contain the value \"elliptic\" if the PDE is known to be elliptic,{} or \"unknown\" if it is uncertain. This causes the routine to check whether the PDE is elliptic. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of PDE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine. \\blankline \\spad{**} At the moment,{} only Second Order Elliptic Partial Differential Equations are solved \\spad{**}") (((|Result|) (|Float|) (|Float|) (|Float|) (|Float|) (|NonNegativeInteger|) (|NonNegativeInteger|) (|List| (|Expression| (|Float|))) (|List| (|List| (|Expression| (|Float|)))) (|String|) (|DoubleFloat|)) "\\spad{solve(xmin,ymin,xmax,ymax,ngx,ngy,pde,bounds,st,tol)} is a top level ANNA function to solve numerically a system of partial differential equations. This is defined as a list of coefficients (\\axiom{\\spad{pde}}),{} a grid (\\axiom{\\spad{xmin}},{} \\axiom{\\spad{ymin}},{} \\axiom{\\spad{xmax}},{} \\axiom{\\spad{ymax}},{} \\axiom{\\spad{ngx}},{} \\axiom{\\spad{ngy}}),{} the boundary values (\\axiom{\\spad{bounds}}) and a tolerance requirement (\\axiom{\\spad{tol}}). There is also a parameter (\\axiom{\\spad{st}}) which should contain the value \"elliptic\" if the PDE is known to be elliptic,{} or \"unknown\" if it is uncertain. This causes the routine to check whether the PDE is elliptic. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of PDE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine. \\blankline \\spad{**} At the moment,{} only Second Order Elliptic Partial Differential Equations are solved \\spad{**}") (((|Result|) (|NumericalPDEProblem|) (|RoutinesTable|)) "\\spad{solve(PDEProblem,routines)} is a top level ANNA function to solve numerically a system of partial differential equations. \\blankline The method used to perform the numerical process will be one of the \\spad{routines} contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of PDE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine. \\blankline \\spad{**} At the moment,{} only Second Order Elliptic Partial Differential Equations are solved \\spad{**}") (((|Result|) (|NumericalPDEProblem|)) "\\spad{solve(PDEProblem)} is a top level ANNA function to solve numerically a system of partial differential equations. \\blankline The method used to perform the numerical process will be one of the routines contained in the NAG numerical Library. The function predicts the likely most effective routine by checking various attributes of the system of PDE\\spad{'s} and calculating a measure of compatibility of each routine to these attributes. \\blankline It then calls the resulting `best' routine. \\blankline \\spad{**} At the moment,{} only Second Order Elliptic Partial Differential Equations are solved \\spad{**}"))) 
 NIL 
 NIL 
-(-902) 
+(-904) 
 ((|retract| (((|Record| (|:| |pde| (|List| (|Expression| (|DoubleFloat|)))) (|:| |constraints| (|List| (|Record| (|:| |start| (|DoubleFloat|)) (|:| |finish| (|DoubleFloat|)) (|:| |grid| (|NonNegativeInteger|)) (|:| |boundaryType| (|Integer|)) (|:| |dStart| (|Matrix| (|DoubleFloat|))) (|:| |dFinish| (|Matrix| (|DoubleFloat|)))))) (|:| |f| (|List| (|List| (|Expression| (|DoubleFloat|))))) (|:| |st| (|String|)) (|:| |tol| (|DoubleFloat|))) $) "\\spad{retract(x)} \\undocumented{}")) (|coerce| (($ (|Record| (|:| |pde| (|List| (|Expression| (|DoubleFloat|)))) (|:| |constraints| (|List| (|Record| (|:| |start| (|DoubleFloat|)) (|:| |finish| (|DoubleFloat|)) (|:| |grid| (|NonNegativeInteger|)) (|:| |boundaryType| (|Integer|)) (|:| |dStart| (|Matrix| (|DoubleFloat|))) (|:| |dFinish| (|Matrix| (|DoubleFloat|)))))) (|:| |f| (|List| (|List| (|Expression| (|DoubleFloat|))))) (|:| |st| (|String|)) (|:| |tol| (|DoubleFloat|)))) "\\spad{coerce(x)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-903 A S) 
+(-905 A S) 
 ((|constructor| (NIL "A partial differential ring with differentiations indexed by a parameter type \\spad{S}. \\blankline")) (D (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{D(x, [s1,...,sn], [n1,...,nn])} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{D(...D(x, s1, n1)..., sn, nn)}.") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{D(x, s, n)} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{n}-th derivative of \\spad{x} with respect to \\spad{s}.") (($ $ (|List| |#2|)) "\\spad{D(x,[s1,...sn])} computes successive partial derivatives,{} \\spadignore{i.e.} \\spad{D(...D(x, s1)..., sn)}.") (($ $ |#2|) "\\spad{D(x,v)} computes the partial derivative of \\spad{x} with respect to \\spad{v}.")) (|differentiate| (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{differentiate(x, [s1,...,sn], [n1,...,nn])} computes multiple partial derivatives,{} \\spadignore{i.e.}") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{differentiate(x, s, n)} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{n}-th derivative of \\spad{x} with respect to \\spad{s}.") (($ $ (|List| |#2|)) "\\spad{differentiate(x,[s1,...sn])} computes successive partial derivatives,{} \\spadignore{i.e.} \\spad{differentiate(...differentiate(x, s1)..., sn)}.") (($ $ |#2|) "\\spad{differentiate(x,v)} computes the partial derivative of \\spad{x} with respect to \\spad{v}."))) 
 NIL 
 NIL 
-(-904 S) 
+(-906 S) 
 ((|constructor| (NIL "A partial differential ring with differentiations indexed by a parameter type \\spad{S}. \\blankline")) (D (($ $ (|List| |#1|) (|List| (|NonNegativeInteger|))) "\\spad{D(x, [s1,...,sn], [n1,...,nn])} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{D(...D(x, s1, n1)..., sn, nn)}.") (($ $ |#1| (|NonNegativeInteger|)) "\\spad{D(x, s, n)} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{n}-th derivative of \\spad{x} with respect to \\spad{s}.") (($ $ (|List| |#1|)) "\\spad{D(x,[s1,...sn])} computes successive partial derivatives,{} \\spadignore{i.e.} \\spad{D(...D(x, s1)..., sn)}.") (($ $ |#1|) "\\spad{D(x,v)} computes the partial derivative of \\spad{x} with respect to \\spad{v}.")) (|differentiate| (($ $ (|List| |#1|) (|List| (|NonNegativeInteger|))) "\\spad{differentiate(x, [s1,...,sn], [n1,...,nn])} computes multiple partial derivatives,{} \\spadignore{i.e.}") (($ $ |#1| (|NonNegativeInteger|)) "\\spad{differentiate(x, s, n)} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{n}-th derivative of \\spad{x} with respect to \\spad{s}.") (($ $ (|List| |#1|)) "\\spad{differentiate(x,[s1,...sn])} computes successive partial derivatives,{} \\spadignore{i.e.} \\spad{differentiate(...differentiate(x, s1)..., sn)}.") (($ $ |#1|) "\\spad{differentiate(x,v)} computes the partial derivative of \\spad{x} with respect to \\spad{v}."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-905 S) 
+(-907 S) 
 ((|constructor| (NIL "\\indented{1}{A PendantTree(\\spad{S})is either a leaf? and is an \\spad{S} or has} a left and a right both PendantTree(\\spad{S})\\spad{'s}")) (|ptree| (($ $ $) "\\spad{ptree(x,y)} \\undocumented") (($ |#1|) "\\spad{ptree(s)} is a leaf? pendant tree"))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-906 S) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-908 S) 
 ((|constructor| (NIL "Permutation(\\spad{S}) implements the group of all bijections \\indented{2}{on a set \\spad{S},{} which move only a finite number of points.} \\indented{2}{A permutation is considered as a map from \\spad{S} into \\spad{S}. In particular} \\indented{2}{multiplication is defined as composition of maps:} \\indented{2}{{\\em pi1 * pi2 = pi1 o pi2}.} \\indented{2}{The internal representation of permuatations are two lists} \\indented{2}{of equal length representing preimages and images.}")) (|coerceImages| (($ (|List| |#1|)) "\\spad{coerceImages(ls)} coerces the list {\\em ls} to a permutation whose image is given by {\\em ls} and the preimage is fixed to be {\\em [1,...,n]}. Note: {coerceImages(\\spad{ls})=coercePreimagesImages([1,{}...,{}\\spad{n}],{}\\spad{ls})}. We assume that both preimage and image do not contain repetitions.")) (|fixedPoints| (((|Set| |#1|) $) "\\spad{fixedPoints(p)} returns the points fixed by the permutation \\spad{p}.")) (|sort| (((|List| $) (|List| $)) "\\spad{sort(lp)} sorts a list of permutations {\\em lp} according to cycle structure first according to length of cycles,{} second,{} if \\spad{S} has \\spadtype{Finite} or \\spad{S} has \\spadtype{OrderedSet} according to lexicographical order of entries in cycles of equal length.")) (|odd?| (((|Boolean|) $) "\\spad{odd?(p)} returns \\spad{true} if and only if \\spad{p} is an odd permutation \\spadignore{i.e.} {\\em sign(p)} is {\\em -1}.")) (|even?| (((|Boolean|) $) "\\spad{even?(p)} returns \\spad{true} if and only if \\spad{p} is an even permutation,{} \\spadignore{i.e.} {\\em sign(p)} is 1.")) (|sign| (((|Integer|) $) "\\spad{sign(p)} returns the signum of the permutation \\spad{p},{} \\spad{+1} or \\spad{-1}.")) (|numberOfCycles| (((|NonNegativeInteger|) $) "\\spad{numberOfCycles(p)} returns the number of non-trivial cycles of the permutation \\spad{p}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of a permutation \\spad{p} as a group element.")) (|cyclePartition| (((|Partition|) $) "\\spad{cyclePartition(p)} returns the cycle structure of a permutation \\spad{p} including cycles of length 1 only if \\spad{S} is finite.")) (|movedPoints| (((|Set| |#1|) $) "\\spad{movedPoints(p)} returns the set of points moved by the permutation \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} retuns the number of points moved by the permutation \\spad{p}.")) (|coerceListOfPairs| (($ (|List| (|List| |#1|))) "\\spad{coerceListOfPairs(lls)} coerces a list of pairs {\\em lls} to a permutation. Error: if not consistent,{} \\spadignore{i.e.} the set of the first elements coincides with the set of second elements. coerce(\\spad{p}) generates output of the permutation \\spad{p} with domain OutputForm.")) (|coerce| (($ (|List| |#1|)) "\\spad{coerce(ls)} coerces a cycle {\\em ls},{} \\spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur.") (($ (|List| (|List| |#1|))) "\\spad{coerce(lls)} coerces a list of cycles {\\em lls} to a permutation,{} each cycle being a list with no repetitions,{} is coerced to the permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.")) (|coercePreimagesImages| (($ (|List| (|List| |#1|))) "\\spad{coercePreimagesImages(lls)} coerces the representation {\\em lls} of a permutation as a list of preimages and images to a permutation. We assume that both preimage and image do not contain repetitions.")) (|listRepresentation| (((|Record| (|:| |preimage| (|List| |#1|)) (|:| |image| (|List| |#1|))) $) "\\spad{listRepresentation(p)} produces a representation {\\em rep} of the permutation \\spad{p} as a list of preimages and images,{} \\spad{i}.\\spad{e} \\spad{p} maps {\\em (rep.preimage).k} to {\\em (rep.image).k} for all indices \\spad{k}. Elements of \\spad{S} not in {\\em (rep.preimage).k} are fixed points,{} and these are the only fixed points of the permutation."))) 
-((-4424 . T)) 
-((-3962 (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-853)))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-853)))) 
-(-907 |n| R) 
+((-4431 . T)) 
+((-3969 (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-855)))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-855)))) 
+(-909 |n| R) 
 ((|constructor| (NIL "Permanent implements the functions {\\em permanent},{} the permanent for square matrices.")) (|permanent| ((|#2| (|SquareMatrix| |#1| |#2|)) "\\spad{permanent(x)} computes the permanent of a square matrix \\spad{x}. The {\\em permanent} is equivalent to the \\spadfun{determinant} except that coefficients have no change of sign. This function is much more difficult to compute than the {\\em determinant}. The formula used is by \\spad{H}.\\spad{J}. Ryser,{} improved by [Nijenhuis and Wilf,{} \\spad{Ch}. 19]. Note: permanent(\\spad{x}) choose one of three algorithms,{} depending on the underlying ring \\spad{R} and on \\spad{n},{} the number of rows (and columns) of \\spad{x:}\\begin{items} \\item 1. if 2 has an inverse in \\spad{R} we can use the algorithm of \\indented{3}{[Nijenhuis and Wilf,{} \\spad{ch}.19,{}\\spad{p}.158]; if 2 has no inverse,{}} \\indented{3}{some modifications are necessary:} \\item 2. if {\\em n > 6} and \\spad{R} is an integral domain with characteristic \\indented{3}{different from 2 (the algorithm works if and only 2 is not a} \\indented{3}{zero-divisor of \\spad{R} and {\\em characteristic()\\$R ~= 2},{}} \\indented{3}{but how to check that for any given \\spad{R} ?),{}} \\indented{3}{the local function {\\em permanent2} is called;} \\item 3. else,{} the local function {\\em permanent3} is called \\indented{3}{(works for all commutative rings \\spad{R}).} \\end{items}"))) 
 NIL 
 NIL 
-(-908 S) 
+(-910 S) 
 ((|constructor| (NIL "PermutationCategory provides a categorial environment \\indented{1}{for subgroups of bijections of a set (\\spadignore{i.e.} permutations)}")) (< (((|Boolean|) $ $) "\\spad{p < q} is an order relation on permutations. Note: this order is only total if and only if \\spad{S} is totally ordered or \\spad{S} is finite.")) (|orbit| (((|Set| |#1|) $ |#1|) "\\spad{orbit(p, el)} returns the orbit of {\\em el} under the permutation \\spad{p},{} \\spadignore{i.e.} the set which is given by applications of the powers of \\spad{p} to {\\em el}.")) (|elt| ((|#1| $ |#1|) "\\spad{elt(p, el)} returns the image of {\\em el} under the permutation \\spad{p}.")) (|eval| ((|#1| $ |#1|) "\\spad{eval(p, el)} returns the image of {\\em el} under the permutation \\spad{p}.")) (|cycles| (($ (|List| (|List| |#1|))) "\\spad{cycles(lls)} coerces a list list of cycles {\\em lls} to a permutation,{} each cycle being a list with not repetitions,{} is coerced to the permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.")) (|cycle| (($ (|List| |#1|)) "\\spad{cycle(ls)} coerces a cycle {\\em ls},{} \\spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-909 S) 
+(-911 S) 
 ((|constructor| (NIL "PermutationGroup implements permutation groups acting on a set \\spad{S},{} \\spadignore{i.e.} all subgroups of the symmetric group of \\spad{S},{} represented as a list of permutations (generators). Note that therefore the objects are not members of the \\Language category \\spadtype{Group}. Using the idea of base and strong generators by Sims,{} basic routines and algorithms are implemented so that the word problem for permutation groups can be solved.")) (|initializeGroupForWordProblem| (((|Void|) $ (|Integer|) (|Integer|)) "\\spad{initializeGroupForWordProblem(gp,m,n)} initializes the group {\\em gp} for the word problem. Notes: (1) with a small integer you get shorter words,{} but the routine takes longer than the standard routine for longer words. (2) be careful: invoking this routine will destroy the possibly stored information about your group (but will recompute it again). (3) users need not call this function normally for the soultion of the word problem.") (((|Void|) $) "\\spad{initializeGroupForWordProblem(gp)} initializes the group {\\em gp} for the word problem. Notes: it calls the other function of this name with parameters 0 and 1: {\\em initializeGroupForWordProblem(gp,0,1)}. Notes: (1) be careful: invoking this routine will destroy the possibly information about your group (but will recompute it again) (2) users need not call this function normally for the soultion of the word problem.")) (<= (((|Boolean|) $ $) "\\spad{gp1 <= gp2} returns \\spad{true} if and only if {\\em gp1} is a subgroup of {\\em gp2}. Note: because of a bug in the parser you have to call this function explicitly by {\\em gp1 <=\\$(PERMGRP S) gp2}.")) (< (((|Boolean|) $ $) "\\spad{gp1 < gp2} returns \\spad{true} if and only if {\\em gp1} is a proper subgroup of {\\em gp2}.")) (|movedPoints| (((|Set| |#1|) $) "\\spad{movedPoints(gp)} returns the points moved by the group {\\em gp}.")) (|wordInGenerators| (((|List| (|NonNegativeInteger|)) (|Permutation| |#1|) $) "\\spad{wordInGenerators(p,gp)} returns the word for the permutation \\spad{p} in the original generators of the group {\\em gp},{} represented by the indices of the list,{} given by {\\em generators}.")) (|wordInStrongGenerators| (((|List| (|NonNegativeInteger|)) (|Permutation| |#1|) $) "\\spad{wordInStrongGenerators(p,gp)} returns the word for the permutation \\spad{p} in the strong generators of the group {\\em gp},{} represented by the indices of the list,{} given by {\\em strongGenerators}.")) (|member?| (((|Boolean|) (|Permutation| |#1|) $) "\\spad{member?(pp,gp)} answers the question,{} whether the permutation {\\em pp} is in the group {\\em gp} or not.")) (|orbits| (((|Set| (|Set| |#1|)) $) "\\spad{orbits(gp)} returns the orbits of the group {\\em gp},{} \\spadignore{i.e.} it partitions the (finite) of all moved points.")) (|orbit| (((|Set| (|List| |#1|)) $ (|List| |#1|)) "\\spad{orbit(gp,ls)} returns the orbit of the ordered list {\\em ls} under the group {\\em gp}. Note: return type is \\spad{L} \\spad{L} \\spad{S} temporarily because FSET \\spad{L} \\spad{S} has an error.") (((|Set| (|Set| |#1|)) $ (|Set| |#1|)) "\\spad{orbit(gp,els)} returns the orbit of the unordered set {\\em els} under the group {\\em gp}.") (((|Set| |#1|) $ |#1|) "\\spad{orbit(gp,el)} returns the orbit of the element {\\em el} under the group {\\em gp},{} \\spadignore{i.e.} the set of all points gained by applying each group element to {\\em el}.")) (|permutationGroup| (($ (|List| (|Permutation| |#1|))) "\\spad{permutationGroup(ls)} coerces a list of permutations {\\em ls} to the group generated by this list.")) (|wordsForStrongGenerators| (((|List| (|List| (|NonNegativeInteger|))) $) "\\spad{wordsForStrongGenerators(gp)} returns the words for the strong generators of the group {\\em gp} in the original generators of {\\em gp},{} represented by their indices in the list,{} given by {\\em generators}.")) (|strongGenerators| (((|List| (|Permutation| |#1|)) $) "\\spad{strongGenerators(gp)} returns strong generators for the group {\\em gp}.")) (|base| (((|List| |#1|) $) "\\spad{base(gp)} returns a base for the group {\\em gp}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(gp)} returns the number of points moved by all permutations of the group {\\em gp}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(gp)} returns the order of the group {\\em gp}.")) (|random| (((|Permutation| |#1|) $) "\\spad{random(gp)} returns a random product of maximal 20 generators of the group {\\em gp}. Note: {\\em random(gp)=random(gp,20)}.") (((|Permutation| |#1|) $ (|Integer|)) "\\spad{random(gp,i)} returns a random product of maximal \\spad{i} generators of the group {\\em gp}.")) (|elt| (((|Permutation| |#1|) $ (|NonNegativeInteger|)) "\\spad{elt(gp,i)} returns the \\spad{i}-th generator of the group {\\em gp}.")) (|generators| (((|List| (|Permutation| |#1|)) $) "\\spad{generators(gp)} returns the generators of the group {\\em gp}.")) (|coerce| (($ (|List| (|Permutation| |#1|))) "\\spad{coerce(ls)} coerces a list of permutations {\\em ls} to the group generated by this list.") (((|List| (|Permutation| |#1|)) $) "\\spad{coerce(gp)} returns the generators of the group {\\em gp}."))) 
 NIL 
 NIL 
-(-910 |p|) 
+(-912 |p|) 
 ((|constructor| (NIL "PrimeField(\\spad{p}) implements the field with \\spad{p} elements if \\spad{p} is a prime number. Error: if \\spad{p} is not prime. Note: this domain does not check that argument is a prime."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| $ (QUOTE (-147))) (|HasCategory| $ (QUOTE (-145))) (|HasCategory| $ (QUOTE (-371)))) 
-(-911 R E |VarSet| S) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| $ (QUOTE (-147))) (|HasCategory| $ (QUOTE (-145))) (|HasCategory| $ (QUOTE (-372)))) 
+(-913 R E |VarSet| S) 
 ((|constructor| (NIL "PolynomialFactorizationByRecursion(\\spad{R},{}\\spad{E},{}\\spad{VarSet},{}\\spad{S}) is used for factorization of sparse univariate polynomials over a domain \\spad{S} of multivariate polynomials over \\spad{R}.")) (|factorSFBRlcUnit| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|List| |#3|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorSFBRlcUnit(p)} returns the square free factorization of polynomial \\spad{p} (see \\spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate}) in the case where the leading coefficient of \\spad{p} is a unit.")) (|bivariateSLPEBR| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|List| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|) |#3|) "\\spad{bivariateSLPEBR(lp,p,v)} implements the bivariate case of \\spadfunFrom{solveLinearPolynomialEquationByRecursion}{PolynomialFactorizationByRecursionUnivariate}; its implementation depends on \\spad{R}")) (|randomR| ((|#1|) "\\spad{randomR produces} a random element of \\spad{R}")) (|factorSquareFreeByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorSquareFreeByRecursion(p)} returns the square free factorization of \\spad{p}. This functions performs the recursion step for factorSquareFreePolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorSquareFreePolynomial}).")) (|factorByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorByRecursion(p)} factors polynomial \\spad{p}. This function performs the recursion step for factorPolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorPolynomial})")) (|solveLinearPolynomialEquationByRecursion| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|List| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{solveLinearPolynomialEquationByRecursion([p1,...,pn],p)} returns the list of polynomials \\spad{[q1,...,qn]} such that \\spad{sum qi/pi = p / prod pi},{} a recursion step for solveLinearPolynomialEquation as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{solveLinearPolynomialEquation}). If no such list of \\spad{qi} exists,{} then \"failed\" is returned."))) 
 NIL 
 NIL 
-(-912 R S) 
+(-914 R S) 
 ((|constructor| (NIL "\\indented{1}{PolynomialFactorizationByRecursionUnivariate} \\spad{R} is a \\spadfun{PolynomialFactorizationExplicit} domain,{} \\spad{S} is univariate polynomials over \\spad{R} We are interested in handling SparseUnivariatePolynomials over \\spad{S},{} is a variable we shall call \\spad{z}")) (|factorSFBRlcUnit| (((|Factored| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{factorSFBRlcUnit(p)} returns the square free factorization of polynomial \\spad{p} (see \\spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate}) in the case where the leading coefficient of \\spad{p} is a unit.")) (|randomR| ((|#1|) "\\spad{randomR()} produces a random element of \\spad{R}")) (|factorSquareFreeByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{factorSquareFreeByRecursion(p)} returns the square free factorization of \\spad{p}. This functions performs the recursion step for factorSquareFreePolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorSquareFreePolynomial}).")) (|factorByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{factorByRecursion(p)} factors polynomial \\spad{p}. This function performs the recursion step for factorPolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorPolynomial})")) (|solveLinearPolynomialEquationByRecursion| (((|Union| (|List| (|SparseUnivariatePolynomial| |#2|)) "failed") (|List| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{solveLinearPolynomialEquationByRecursion([p1,...,pn],p)} returns the list of polynomials \\spad{[q1,...,qn]} such that \\spad{sum qi/pi = p / prod pi},{} a recursion step for solveLinearPolynomialEquation as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{solveLinearPolynomialEquation}). If no such list of \\spad{qi} exists,{} then \"failed\" is returned."))) 
 NIL 
 NIL 
-(-913 S) 
+(-915 S) 
 ((|constructor| (NIL "This is the category of domains that know \"enough\" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.")) (|charthRoot| (((|Union| $ "failed") $) "\\spad{charthRoot(r)} returns the \\spad{p}\\spad{-}th root of \\spad{r},{} or \"failed\" if none exists in the domain.")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(m)} returns a vector of elements,{} not all zero,{} whose \\spad{p}\\spad{-}th powers (\\spad{p} is the characteristic of the domain) are a solution of the homogenous linear system represented by \\spad{m},{} or \"failed\" is there is no such vector.")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| $)) "failed") (|List| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}\\spad{'s} exists.")) (|gcdPolynomial| (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $)) "\\spad{gcdPolynomial(p,q)} returns the \\spad{gcd} of the univariate polynomials \\spad{p} \\spad{qnd} \\spad{q}.")) (|factorSquareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorSquareFreePolynomial(p)} factors the univariate polynomial \\spad{p} into irreducibles where \\spad{p} is known to be square free and primitive with respect to its main variable.")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} returns the factorization into irreducibles of the univariate polynomial \\spad{p}.")) (|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} returns the square-free factorization of the univariate polynomial \\spad{p}."))) 
 NIL 
 ((|HasCategory| |#1| (QUOTE (-145)))) 
-(-914) 
+(-916) 
 ((|constructor| (NIL "This is the category of domains that know \"enough\" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.")) (|charthRoot| (((|Union| $ "failed") $) "\\spad{charthRoot(r)} returns the \\spad{p}\\spad{-}th root of \\spad{r},{} or \"failed\" if none exists in the domain.")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(m)} returns a vector of elements,{} not all zero,{} whose \\spad{p}\\spad{-}th powers (\\spad{p} is the characteristic of the domain) are a solution of the homogenous linear system represented by \\spad{m},{} or \"failed\" is there is no such vector.")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| $)) "failed") (|List| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}\\spad{'s} exists.")) (|gcdPolynomial| (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $)) "\\spad{gcdPolynomial(p,q)} returns the \\spad{gcd} of the univariate polynomials \\spad{p} \\spad{qnd} \\spad{q}.")) (|factorSquareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorSquareFreePolynomial(p)} factors the univariate polynomial \\spad{p} into irreducibles where \\spad{p} is known to be square free and primitive with respect to its main variable.")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} returns the factorization into irreducibles of the univariate polynomial \\spad{p}.")) (|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} returns the square-free factorization of the univariate polynomial \\spad{p}."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-915 R0 -3498 UP UPUP R) 
+(-917 R0 -3505 UP UPUP R) 
 ((|constructor| (NIL "This package provides function for testing whether a divisor on a curve is a torsion divisor.")) (|torsionIfCan| (((|Union| (|Record| (|:| |order| (|NonNegativeInteger|)) (|:| |function| |#5|)) "failed") (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{torsionIfCan(f)}\\\\ undocumented")) (|torsion?| (((|Boolean|) (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{torsion?(f)} \\undocumented")) (|order| (((|Union| (|NonNegativeInteger|) "failed") (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{order(f)} \\undocumented"))) 
 NIL 
 NIL 
-(-916 UP UPUP R) 
+(-918 UP UPUP R) 
 ((|constructor| (NIL "This package provides function for testing whether a divisor on a curve is a torsion divisor.")) (|torsionIfCan| (((|Union| (|Record| (|:| |order| (|NonNegativeInteger|)) (|:| |function| |#3|)) "failed") (|FiniteDivisor| (|Fraction| (|Integer|)) |#1| |#2| |#3|)) "\\spad{torsionIfCan(f)} \\undocumented")) (|torsion?| (((|Boolean|) (|FiniteDivisor| (|Fraction| (|Integer|)) |#1| |#2| |#3|)) "\\spad{torsion?(f)} \\undocumented")) (|order| (((|Union| (|NonNegativeInteger|) "failed") (|FiniteDivisor| (|Fraction| (|Integer|)) |#1| |#2| |#3|)) "\\spad{order(f)} \\undocumented"))) 
 NIL 
 NIL 
-(-917 UP UPUP) 
+(-919 UP UPUP) 
 ((|constructor| (NIL "\\indented{1}{Utilities for PFOQ and PFO} Author: Manuel Bronstein Date Created: 25 Aug 1988 Date Last Updated: 11 Jul 1990")) (|polyred| ((|#2| |#2|) "\\spad{polyred(u)} \\undocumented")) (|doubleDisc| (((|Integer|) |#2|) "\\spad{doubleDisc(u)} \\undocumented")) (|mix| (((|Integer|) (|List| (|Record| (|:| |den| (|Integer|)) (|:| |gcdnum| (|Integer|))))) "\\spad{mix(l)} \\undocumented")) (|badNum| (((|Integer|) |#2|) "\\spad{badNum(u)} \\undocumented") (((|Record| (|:| |den| (|Integer|)) (|:| |gcdnum| (|Integer|))) |#1|) "\\spad{badNum(p)} \\undocumented")) (|getGoodPrime| (((|PositiveInteger|) (|Integer|)) "\\spad{getGoodPrime n} returns the smallest prime not dividing \\spad{n}"))) 
 NIL 
 NIL 
-(-918 R) 
+(-920 R) 
 ((|constructor| (NIL "The domain \\spadtype{PartialFraction} implements partial fractions over a euclidean domain \\spad{R}. This requirement on the argument domain allows us to normalize the fractions. Of particular interest are the 2 forms for these fractions. The ``compact\\spad{''} form has only one fractional term per prime in the denominator,{} while the \\spad{``p}-adic\\spad{''} form expands each numerator \\spad{p}-adically via the prime \\spad{p} in the denominator. For computational efficiency,{} the compact form is used,{} though the \\spad{p}-adic form may be gotten by calling the function \\spadfunFrom{padicFraction}{PartialFraction}. For a general euclidean domain,{} it is not known how to factor the denominator. Thus the function \\spadfunFrom{partialFraction}{PartialFraction} takes as its second argument an element of \\spadtype{Factored(R)}.")) (|wholePart| ((|#1| $) "\\spad{wholePart(p)} extracts the whole part of the partial fraction \\spad{p}.")) (|partialFraction| (($ |#1| (|Factored| |#1|)) "\\spad{partialFraction(numer,denom)} is the main function for constructing partial fractions. The second argument is the denominator and should be factored.")) (|padicFraction| (($ $) "\\spad{padicFraction(q)} expands the fraction \\spad{p}-adically in the primes \\spad{p} in the denominator of \\spad{q}. For example,{} \\spad{padicFraction(3/(2**2)) = 1/2 + 1/(2**2)}. Use \\spadfunFrom{compactFraction}{PartialFraction} to return to compact form.")) (|padicallyExpand| (((|SparseUnivariatePolynomial| |#1|) |#1| |#1|) "\\spad{padicallyExpand(p,x)} is a utility function that expands the second argument \\spad{x} \\spad{``p}-adically\\spad{''} in the first.")) (|numberOfFractionalTerms| (((|Integer|) $) "\\spad{numberOfFractionalTerms(p)} computes the number of fractional terms in \\spad{p}. This returns 0 if there is no fractional part.")) (|nthFractionalTerm| (($ $ (|Integer|)) "\\spad{nthFractionalTerm(p,n)} extracts the \\spad{n}th fractional term from the partial fraction \\spad{p}. This returns 0 if the index \\spad{n} is out of range.")) (|firstNumer| ((|#1| $) "\\spad{firstNumer(p)} extracts the numerator of the first fractional term. This returns 0 if there is no fractional part (use \\spadfunFrom{wholePart}{PartialFraction} to get the whole part).")) (|firstDenom| (((|Factored| |#1|) $) "\\spad{firstDenom(p)} extracts the denominator of the first fractional term. This returns 1 if there is no fractional part (use \\spadfunFrom{wholePart}{PartialFraction} to get the whole part).")) (|compactFraction| (($ $) "\\spad{compactFraction(p)} normalizes the partial fraction \\spad{p} to the compact representation. In this form,{} the partial fraction has only one fractional term per prime in the denominator.")) (|coerce| (($ (|Fraction| (|Factored| |#1|))) "\\spad{coerce(f)} takes a fraction with numerator and denominator in factored form and creates a partial fraction. It is necessary for the parts to be factored because it is not known in general how to factor elements of \\spad{R} and this is needed to decompose into partial fractions.") (((|Fraction| |#1|) $) "\\spad{coerce(p)} sums up the components of the partial fraction and returns a single fraction."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-919 R) 
+(-921 R) 
 ((|constructor| (NIL "The package \\spadtype{PartialFractionPackage} gives an easier to use interfact the domain \\spadtype{PartialFraction}. The user gives a fraction of polynomials,{} and a variable and the package converts it to the proper datatype for the \\spadtype{PartialFraction} domain.")) (|partialFraction| (((|Any|) (|Polynomial| |#1|) (|Factored| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{partialFraction(num, facdenom, var)} returns the partial fraction decomposition of the rational function whose numerator is \\spad{num} and whose factored denominator is \\spad{facdenom} with respect to the variable var.") (((|Any|) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{partialFraction(rf, var)} returns the partial fraction decomposition of the rational function \\spad{rf} with respect to the variable var."))) 
 NIL 
 NIL 
-(-920 E OV R P) 
+(-922 E OV R P) 
 ((|gcdPrimitive| ((|#4| (|List| |#4|)) "\\spad{gcdPrimitive lp} computes the \\spad{gcd} of the list of primitive polynomials \\spad{lp}.") (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{gcdPrimitive(p,q)} computes the \\spad{gcd} of the primitive polynomials \\spad{p} and \\spad{q}.") ((|#4| |#4| |#4|) "\\spad{gcdPrimitive(p,q)} computes the \\spad{gcd} of the primitive polynomials \\spad{p} and \\spad{q}.")) (|gcd| (((|SparseUnivariatePolynomial| |#4|) (|List| (|SparseUnivariatePolynomial| |#4|))) "\\spad{gcd(lp)} computes the \\spad{gcd} of the list of polynomials \\spad{lp}.") (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{gcd(p,q)} computes the \\spad{gcd} of the two polynomials \\spad{p} and \\spad{q}.") ((|#4| (|List| |#4|)) "\\spad{gcd(lp)} computes the \\spad{gcd} of the list of polynomials \\spad{lp}.") ((|#4| |#4| |#4|) "\\spad{gcd(p,q)} computes the \\spad{gcd} of the two polynomials \\spad{p} and \\spad{q}."))) 
 NIL 
 NIL 
-(-921) 
+(-923) 
 ((|constructor| (NIL "PermutationGroupExamples provides permutation groups for some classes of groups: symmetric,{} alternating,{} dihedral,{} cyclic,{} direct products of cyclic,{} which are in fact the finite abelian groups of symmetric groups called Young subgroups. Furthermore,{} Rubik\\spad{'s} group as permutation group of 48 integers and a list of sporadic simple groups derived from the atlas of finite groups.")) (|youngGroup| (((|PermutationGroup| (|Integer|)) (|Partition|)) "\\spad{youngGroup(lambda)} constructs the direct product of the symmetric groups given by the parts of the partition {\\em lambda}.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{youngGroup([n1,...,nk])} constructs the direct product of the symmetric groups {\\em Sn1},{}...,{}{\\em Snk}.")) (|rubiksGroup| (((|PermutationGroup| (|Integer|))) "\\spad{rubiksGroup constructs} the permutation group representing Rubic\\spad{'s} Cube acting on integers {\\em 10*i+j} for {\\em 1 <= i <= 6},{} {\\em 1 <= j <= 8}. The faces of Rubik\\spad{'s} Cube are labelled in the obvious way Front,{} Right,{} Up,{} Down,{} Left,{} Back and numbered from 1 to 6 in this given ordering,{} the pieces on each face (except the unmoveable center piece) are clockwise numbered from 1 to 8 starting with the piece in the upper left corner. The moves of the cube are represented as permutations on these pieces,{} represented as a two digit integer {\\em ij} where \\spad{i} is the numer of theface (1 to 6) and \\spad{j} is the number of the piece on this face. The remaining ambiguities are resolved by looking at the 6 generators,{} which represent a 90 degree turns of the faces,{} or from the following pictorial description. Permutation group representing Rubic\\spad{'s} Cube acting on integers 10*i+j for 1 \\spad{<=} \\spad{i} \\spad{<=} 6,{} 1 \\spad{<=} \\spad{j} \\spad{<=8}. \\blankline\\begin{verbatim}Rubik's Cube:   +-----+ +-- B   where: marks Side # :               / U   /|/              /     / |         F(ront)    <->    1      L -->  +-----+ R|         R(ight)    <->    2             |     |  +         U(p)       <->    3             |  F  | /          D(own)     <->    4             |     |/           L(eft)     <->    5             +-----+            B(ack)     <->    6                ^                |                DThe Cube's surface:                               The pieces on each side            +---+              (except the unmoveable center            |567|              piece) are clockwise numbered            |4U8|              from 1 to 8 starting with the            |321|              piece in the upper left        +---+---+---+          corner (see figure on the        |781|123|345|          left).  The moves of the cube        |6L2|8F4|2R6|          are represented as        |543|765|187|          permutations on these pieces.        +---+---+---+          Each of the pieces is            |123|              represented as a two digit            |8D4|              integer ij where i is the            |765|              # of the side ( 1 to 6 for            +---+              F to B (see table above ))            |567|              and j is the # of the piece.            |4B8|            |321|            +---+\\end{verbatim}")) (|janko2| (((|PermutationGroup| (|Integer|))) "\\spad{janko2 constructs} the janko group acting on the integers 1,{}...,{}100.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{janko2(li)} constructs the janko group acting on the 100 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 100 different entries")) (|mathieu24| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu24 constructs} the mathieu group acting on the integers 1,{}...,{}24.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu24(li)} constructs the mathieu group acting on the 24 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 24 different entries.")) (|mathieu23| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu23 constructs} the mathieu group acting on the integers 1,{}...,{}23.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu23(li)} constructs the mathieu group acting on the 23 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 23 different entries.")) (|mathieu22| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu22 constructs} the mathieu group acting on the integers 1,{}...,{}22.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu22(li)} constructs the mathieu group acting on the 22 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 22 different entries.")) (|mathieu12| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu12 constructs} the mathieu group acting on the integers 1,{}...,{}12.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu12(li)} constructs the mathieu group acting on the 12 integers given in the list {\\em li}. Note: duplicates in the list will be removed Error: if {\\em li} has less or more than 12 different entries.")) (|mathieu11| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu11 constructs} the mathieu group acting on the integers 1,{}...,{}11.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu11(li)} constructs the mathieu group acting on the 11 integers given in the list {\\em li}. Note: duplicates in the list will be removed. error,{} if {\\em li} has less or more than 11 different entries.")) (|dihedralGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{dihedralGroup([i1,...,ik])} constructs the dihedral group of order 2k acting on the integers out of {\\em i1},{}...,{}{\\em ik}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{dihedralGroup(n)} constructs the dihedral group of order 2n acting on integers 1,{}...,{}\\spad{N}.")) (|cyclicGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{cyclicGroup([i1,...,ik])} constructs the cyclic group of order \\spad{k} acting on the integers {\\em i1},{}...,{}{\\em ik}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{cyclicGroup(n)} constructs the cyclic group of order \\spad{n} acting on the integers 1,{}...,{}\\spad{n}.")) (|abelianGroup| (((|PermutationGroup| (|Integer|)) (|List| (|PositiveInteger|))) "\\spad{abelianGroup([n1,...,nk])} constructs the abelian group that is the direct product of cyclic groups with order {\\em ni}.")) (|alternatingGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{alternatingGroup(li)} constructs the alternating group acting on the integers in the list {\\em li},{} generators are in general the {\\em n-2}-cycle {\\em (li.3,...,li.n)} and the 3-cycle {\\em (li.1,li.2,li.3)},{} if \\spad{n} is odd and product of the 2-cycle {\\em (li.1,li.2)} with {\\em n-2}-cycle {\\em (li.3,...,li.n)} and the 3-cycle {\\em (li.1,li.2,li.3)},{} if \\spad{n} is even. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{alternatingGroup(n)} constructs the alternating group {\\em An} acting on the integers 1,{}...,{}\\spad{n},{} generators are in general the {\\em n-2}-cycle {\\em (3,...,n)} and the 3-cycle {\\em (1,2,3)} if \\spad{n} is odd and the product of the 2-cycle {\\em (1,2)} with {\\em n-2}-cycle {\\em (3,...,n)} and the 3-cycle {\\em (1,2,3)} if \\spad{n} is even.")) (|symmetricGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{symmetricGroup(li)} constructs the symmetric group acting on the integers in the list {\\em li},{} generators are the cycle given by {\\em li} and the 2-cycle {\\em (li.1,li.2)}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{symmetricGroup(n)} constructs the symmetric group {\\em Sn} acting on the integers 1,{}...,{}\\spad{n},{} generators are the {\\em n}-cycle {\\em (1,...,n)} and the 2-cycle {\\em (1,2)}."))) 
 NIL 
 NIL 
-(-922 -3498) 
+(-924 -3505) 
 ((|constructor| (NIL "Groebner functions for \\spad{P} \\spad{F} \\indented{2}{This package is an interface package to the groebner basis} package which allows you to compute groebner bases for polynomials in either lexicographic ordering or total degree ordering refined by reverse lex. The input is the ordinary polynomial type which is internally converted to a type with the required ordering. The resulting grobner basis is converted back to ordinary polynomials. The ordering among the variables is controlled by an explicit list of variables which is passed as a second argument. The coefficient domain is allowed to be any \\spad{gcd} domain,{} but the groebner basis is computed as if the polynomials were over a field.")) (|totalGroebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{totalGroebner(lp,lv)} computes Groebner basis for the list of polynomials \\spad{lp} with the terms ordered first by total degree and then refined by reverse lexicographic ordering. The variables are ordered by their position in the list \\spad{lv}.")) (|lexGroebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{lexGroebner(lp,lv)} computes Groebner basis for the list of polynomials \\spad{lp} in lexicographic order. The variables are ordered by their position in the list \\spad{lv}."))) 
 NIL 
 NIL 
-(-923) 
+(-925) 
 ((|constructor| (NIL "\\spadtype{PositiveInteger} provides functions for \\indented{2}{positive integers.}")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} means multiplication is commutative : x*y = \\spad{y*x}")) (|gcd| (($ $ $) "\\spad{gcd(a,b)} computes the greatest common divisor of two positive integers \\spad{a} and \\spad{b}."))) 
-(((-4429 "*") . T)) 
+(((-4436 "*") . T)) 
 NIL 
-(-924 R) 
+(-926 R) 
 ((|constructor| (NIL "\\indented{1}{Provides a coercion from the symbolic fractions in \\%\\spad{pi} with} integer coefficients to any Expression type. Date Created: 21 Feb 1990 Date Last Updated: 21 Feb 1990")) (|coerce| (((|Expression| |#1|) (|Pi|)) "\\spad{coerce(f)} returns \\spad{f} as an Expression(\\spad{R})."))) 
 NIL 
 NIL 
-(-925) 
+(-927) 
 ((|constructor| (NIL "The category of constructive principal ideal domains,{} \\spadignore{i.e.} where a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.")) (|expressIdealMember| (((|Union| (|List| $) "failed") (|List| $) $) "\\spad{expressIdealMember([f1,...,fn],h)} returns a representation of \\spad{h} as a linear combination of the \\spad{fi} or \"failed\" if \\spad{h} is not in the ideal generated by the \\spad{fi}.")) (|principalIdeal| (((|Record| (|:| |coef| (|List| $)) (|:| |generator| $)) (|List| $)) "\\spad{principalIdeal([f1,...,fn])} returns a record whose generator component is a generator of the ideal generated by \\spad{[f1,...,fn]} whose coef component satisfies \\spad{generator = sum (input.i * coef.i)}"))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-926 |xx| -3498) 
+(-928 |xx| -3505) 
 ((|constructor| (NIL "This package exports interpolation algorithms")) (|interpolate| (((|SparseUnivariatePolynomial| |#2|) (|List| |#2|) (|List| |#2|)) "\\spad{interpolate(lf,lg)} \\undocumented") (((|UnivariatePolynomial| |#1| |#2|) (|UnivariatePolynomial| |#1| |#2|) (|List| |#2|) (|List| |#2|)) "\\spad{interpolate(u,lf,lg)} \\undocumented"))) 
 NIL 
 NIL 
-(-927 -3498 P) 
+(-929 -3505 P) 
 ((|constructor| (NIL "This package exports interpolation algorithms")) (|LagrangeInterpolation| ((|#2| (|List| |#1|) (|List| |#1|)) "\\spad{LagrangeInterpolation(l1,l2)} \\undocumented"))) 
 NIL 
 NIL 
-(-928 R |Var| |Expon| GR) 
+(-930 R |Var| |Expon| GR) 
 ((|constructor| (NIL "Author: William Sit,{} spring 89")) (|inconsistent?| (((|Boolean|) (|List| (|Polynomial| |#1|))) "inconsistant?(\\spad{pl}) returns \\spad{true} if the system of equations \\spad{p} = 0 for \\spad{p} in \\spad{pl} is inconsistent. It is assumed that \\spad{pl} is a groebner basis.") (((|Boolean|) (|List| |#4|)) "inconsistant?(\\spad{pl}) returns \\spad{true} if the system of equations \\spad{p} = 0 for \\spad{p} in \\spad{pl} is inconsistent. It is assumed that \\spad{pl} is a groebner basis.")) (|sqfree| ((|#4| |#4|) "\\spad{sqfree(p)} returns the product of square free factors of \\spad{p}")) (|regime| (((|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))))) (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))) (|Matrix| |#4|) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|List| |#4|)) (|NonNegativeInteger|) (|NonNegativeInteger|) (|Integer|)) "\\spad{regime(y,c, w, p, r, rm, m)} returns a regime,{} a list of polynomials specifying the consistency conditions,{} a particular solution and basis representing the general solution of the parametric linear system \\spad{c} \\spad{z} = \\spad{w} on that regime. The regime returned depends on the subdeterminant \\spad{y}.det and the row and column indices. The solutions are simplified using the assumption that the system has rank \\spad{r} and maximum rank \\spad{rm}. The list \\spad{p} represents a list of list of factors of polynomials in a groebner basis of the ideal generated by higher order subdeterminants,{} and ius used for the simplification. The mode \\spad{m} distinguishes the cases when the system is homogeneous,{} or the right hand side is arbitrary,{} or when there is no new right hand side variables.")) (|redmat| (((|Matrix| |#4|) (|Matrix| |#4|) (|List| |#4|)) "\\spad{redmat(m,g)} returns a matrix whose entries are those of \\spad{m} modulo the ideal generated by the groebner basis \\spad{g}")) (|ParCond| (((|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|))))) (|Matrix| |#4|) (|NonNegativeInteger|)) "\\spad{ParCond(m,k)} returns the list of all \\spad{k} by \\spad{k} subdeterminants in the matrix \\spad{m}")) (|overset?| (((|Boolean|) (|List| |#4|) (|List| (|List| |#4|))) "\\spad{overset?(s,sl)} returns \\spad{true} if \\spad{s} properly a sublist of a member of \\spad{sl}; otherwise it returns \\spad{false}")) (|nextSublist| (((|List| (|List| (|Integer|))) (|Integer|) (|Integer|)) "\\spad{nextSublist(n,k)} returns a list of \\spad{k}-subsets of {1,{} ...,{} \\spad{n}}.")) (|minset| (((|List| (|List| |#4|)) (|List| (|List| |#4|))) "\\spad{minset(sl)} returns the sublist of \\spad{sl} consisting of the minimal lists (with respect to inclusion) in the list \\spad{sl} of lists")) (|minrank| (((|NonNegativeInteger|) (|List| (|Record| (|:| |rank| (|NonNegativeInteger|)) (|:| |eqns| (|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))))) (|:| |fgb| (|List| |#4|))))) "\\spad{minrank(r)} returns the minimum rank in the list \\spad{r} of regimes")) (|maxrank| (((|NonNegativeInteger|) (|List| (|Record| (|:| |rank| (|NonNegativeInteger|)) (|:| |eqns| (|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))))) (|:| |fgb| (|List| |#4|))))) "\\spad{maxrank(r)} returns the maximum rank in the list \\spad{r} of regimes")) (|factorset| (((|List| |#4|) |#4|) "\\spad{factorset(p)} returns the set of irreducible factors of \\spad{p}.")) (|B1solve| (((|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))) (|Record| (|:| |mat| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|:| |vec| (|List| (|Fraction| (|Polynomial| |#1|)))) (|:| |rank| (|NonNegativeInteger|)) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|))))) "\\spad{B1solve(s)} solves the system (\\spad{s}.mat) \\spad{z} = \\spad{s}.vec for the variables given by the column indices of \\spad{s}.cols in terms of the other variables and the right hand side \\spad{s}.vec by assuming that the rank is \\spad{s}.rank,{} that the system is consistent,{} with the linearly independent equations indexed by the given row indices \\spad{s}.rows; the coefficients in \\spad{s}.mat involving parameters are treated as polynomials. B1solve(\\spad{s}) returns a particular solution to the system and a basis of the homogeneous system (\\spad{s}.mat) \\spad{z} = 0.")) (|redpps| (((|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))) (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))) (|List| |#4|)) "\\spad{redpps(s,g)} returns the simplified form of \\spad{s} after reducing modulo a groebner basis \\spad{g}")) (|ParCondList| (((|List| (|Record| (|:| |rank| (|NonNegativeInteger|)) (|:| |eqns| (|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))))) (|:| |fgb| (|List| |#4|)))) (|Matrix| |#4|) (|NonNegativeInteger|)) "\\spad{ParCondList(c,r)} computes a list of subdeterminants of each rank \\spad{>=} \\spad{r} of the matrix \\spad{c} and returns a groebner basis for the ideal they generate")) (|hasoln| (((|Record| (|:| |sysok| (|Boolean|)) (|:| |z0| (|List| |#4|)) (|:| |n0| (|List| |#4|))) (|List| |#4|) (|List| |#4|)) "\\spad{hasoln(g, l)} tests whether the quasi-algebraic set defined by \\spad{p} = 0 for \\spad{p} in \\spad{g} and \\spad{q} \\spad{~=} 0 for \\spad{q} in \\spad{l} is empty or not and returns a simplified definition of the quasi-algebraic set")) (|pr2dmp| ((|#4| (|Polynomial| |#1|)) "\\spad{pr2dmp(p)} converts \\spad{p} to target domain")) (|se2rfi| (((|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{se2rfi(l)} converts \\spad{l} to target domain")) (|dmp2rfi| (((|List| (|Fraction| (|Polynomial| |#1|))) (|List| |#4|)) "\\spad{dmp2rfi(l)} converts \\spad{l} to target domain") (((|Matrix| (|Fraction| (|Polynomial| |#1|))) (|Matrix| |#4|)) "\\spad{dmp2rfi(m)} converts \\spad{m} to target domain") (((|Fraction| (|Polynomial| |#1|)) |#4|) "\\spad{dmp2rfi(p)} converts \\spad{p} to target domain")) (|bsolve| (((|Record| (|:| |rgl| (|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))))))) (|:| |rgsz| (|Integer|))) (|Matrix| |#4|) (|List| (|Fraction| (|Polynomial| |#1|))) (|NonNegativeInteger|) (|String|) (|Integer|)) "\\spad{bsolve(c, w, r, s, m)} returns a list of regimes and solutions of the system \\spad{c} \\spad{z} = \\spad{w} for ranks at least \\spad{r}; depending on the mode \\spad{m} chosen,{} it writes the output to a file given by the string \\spad{s}.")) (|rdregime| (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|String|)) "\\spad{rdregime(s)} reads in a list from a file with name \\spad{s}")) (|wrregime| (((|Integer|) (|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|String|)) "\\spad{wrregime(l,s)} writes a list of regimes to a file named \\spad{s} and returns the number of regimes written")) (|psolve| (((|Integer|) (|Matrix| |#4|) (|PositiveInteger|) (|String|)) "\\spad{psolve(c,k,s)} solves \\spad{c} \\spad{z} = 0 for all possible ranks \\spad{>=} \\spad{k} of the matrix \\spad{c},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| (|Symbol|)) (|PositiveInteger|) (|String|)) "\\spad{psolve(c,w,k,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks \\spad{>=} \\spad{k} of the matrix \\spad{c} and indeterminate right hand side \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| |#4|) (|PositiveInteger|) (|String|)) "\\spad{psolve(c,w,k,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks \\spad{>=} \\spad{k} of the matrix \\spad{c} and given right hand side \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|String|)) "\\spad{psolve(c,s)} solves \\spad{c} \\spad{z} = 0 for all possible ranks of the matrix \\spad{c} and given right hand side vector \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| (|Symbol|)) (|String|)) "\\spad{psolve(c,w,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and indeterminate right hand side \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| |#4|) (|String|)) "\\spad{psolve(c,w,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and given right hand side vector \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|PositiveInteger|)) "\\spad{psolve(c)} solves the homogeneous linear system \\spad{c} \\spad{z} = 0 for all possible ranks \\spad{>=} \\spad{k} of the matrix \\spad{c}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| (|Symbol|)) (|PositiveInteger|)) "\\spad{psolve(c,w,k)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks \\spad{>=} \\spad{k} of the matrix \\spad{c} and indeterminate right hand side \\spad{w}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| |#4|) (|PositiveInteger|)) "\\spad{psolve(c,w,k)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks \\spad{>=} \\spad{k} of the matrix \\spad{c} and given right hand side vector \\spad{w}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|)) "\\spad{psolve(c)} solves the homogeneous linear system \\spad{c} \\spad{z} = 0 for all possible ranks of the matrix \\spad{c}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| (|Symbol|))) "\\spad{psolve(c,w)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and indeterminate right hand side \\spad{w}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| |#4|)) "\\spad{psolve(c,w)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and given right hand side vector \\spad{w}"))) 
 NIL 
 NIL 
-(-929) 
+(-931) 
 ((|constructor| (NIL "The Plot domain supports plotting of functions defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example floating point numbers and infinite continued fractions. The facilities at this point are limited to 2-dimensional plots or either a single function or a parametric function.")) (|debug| (((|Boolean|) (|Boolean|)) "\\spad{debug(true)} turns debug mode on \\spad{debug(false)} turns debug mode off")) (|numFunEvals| (((|Integer|)) "\\spad{numFunEvals()} returns the number of points computed")) (|setAdaptive| (((|Boolean|) (|Boolean|)) "\\spad{setAdaptive(true)} turns adaptive plotting on \\spad{setAdaptive(false)} turns adaptive plotting off")) (|adaptive?| (((|Boolean|)) "\\spad{adaptive?()} determines whether plotting be done adaptively")) (|setScreenResolution| (((|Integer|) (|Integer|)) "\\spad{setScreenResolution(i)} sets the screen resolution to \\spad{i}")) (|screenResolution| (((|Integer|)) "\\spad{screenResolution()} returns the screen resolution")) (|setMaxPoints| (((|Integer|) (|Integer|)) "\\spad{setMaxPoints(i)} sets the maximum number of points in a plot to \\spad{i}")) (|maxPoints| (((|Integer|)) "\\spad{maxPoints()} returns the maximum number of points in a plot")) (|setMinPoints| (((|Integer|) (|Integer|)) "\\spad{setMinPoints(i)} sets the minimum number of points in a plot to \\spad{i}")) (|minPoints| (((|Integer|)) "\\spad{minPoints()} returns the minimum number of points in a plot")) (|tRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{tRange(p)} returns the range of the parameter in a parametric plot \\spad{p}")) (|refine| (($ $) "\\spad{refine(p)} performs a refinement on the plot \\spad{p}") (($ $ (|Segment| (|DoubleFloat|))) "\\spad{refine(x,r)} \\undocumented")) (|zoom| (($ $ (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{zoom(x,r,s)} \\undocumented") (($ $ (|Segment| (|DoubleFloat|))) "\\spad{zoom(x,r)} \\undocumented")) (|parametric?| (((|Boolean|) $) "\\spad{parametric? determines} whether it is a parametric plot?")) (|plotPolar| (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) "\\spad{plotPolar(f)} plots the polar curve \\spad{r = f(theta)} as theta ranges over the interval \\spad{[0,2*\\%pi]}; this is the same as the parametric curve \\spad{x = f(t) * cos(t)},{} \\spad{y = f(t) * sin(t)}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plotPolar(f,a..b)} plots the polar curve \\spad{r = f(theta)} as theta ranges over the interval \\spad{[a,b]}; this is the same as the parametric curve \\spad{x = f(t) * cos(t)},{} \\spad{y = f(t) * sin(t)}.")) (|pointPlot| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(t +-> (f(t),g(t)),a..b,c..d,e..f)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}; \\spad{x}-range of \\spad{[c,d]} and \\spad{y}-range of \\spad{[e,f]} are noted in Plot object.") (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(t +-> (f(t),g(t)),a..b)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}.")) (|plot| (($ $ (|Segment| (|DoubleFloat|))) "\\spad{plot(x,r)} \\undocumented") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,a..b,c..d,e..f)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}; \\spad{x}-range of \\spad{[c,d]} and \\spad{y}-range of \\spad{[e,f]} are noted in Plot object.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,a..b)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}.") (($ (|List| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot([f1,...,fm],a..b,c..d)} plots the functions \\spad{y = f1(x)},{}...,{} \\spad{y = fm(x)} on the interval \\spad{a..b}; \\spad{y}-range of \\spad{[c,d]} is noted in Plot object.") (($ (|List| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|DoubleFloat|))) "\\spad{plot([f1,...,fm],a..b)} plots the functions \\spad{y = f1(x)},{}...,{} \\spad{y = fm(x)} on the interval \\spad{a..b}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,a..b,c..d)} plots the function \\spad{f(x)} on the interval \\spad{[a,b]}; \\spad{y}-range of \\spad{[c,d]} is noted in Plot object.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,a..b)} plots the function \\spad{f(x)} on the interval \\spad{[a,b]}."))) 
 NIL 
 NIL 
-(-930 S) 
+(-932 S) 
 ((|constructor| (NIL "PlotFunctions1 provides facilities for plotting curves where functions \\spad{SF} \\spad{->} \\spad{SF} are specified by giving an expression")) (|plotPolar| (((|Plot|) |#1| (|Symbol|)) "\\spad{plotPolar(f,theta)} plots the graph of \\spad{r = f(theta)} as \\spad{theta} ranges from 0 to 2 \\spad{pi}") (((|Plot|) |#1| (|Symbol|) (|Segment| (|DoubleFloat|))) "\\spad{plotPolar(f,theta,seg)} plots the graph of \\spad{r = f(theta)} as \\spad{theta} ranges over an interval")) (|plot| (((|Plot|) |#1| |#1| (|Symbol|) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,t,seg)} plots the graph of \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over an interval.") (((|Plot|) |#1| (|Symbol|) (|Segment| (|DoubleFloat|))) "\\spad{plot(fcn,x,seg)} plots the graph of \\spad{y = f(x)} on a interval"))) 
 NIL 
 NIL 
-(-931) 
+(-933) 
 ((|constructor| (NIL "Plot3D supports parametric plots defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example,{} floating point numbers and infinite continued fractions are real number systems. The facilities at this point are limited to 3-dimensional parametric plots.")) (|debug3D| (((|Boolean|) (|Boolean|)) "\\spad{debug3D(true)} turns debug mode on; debug3D(\\spad{false}) turns debug mode off.")) (|numFunEvals3D| (((|Integer|)) "\\spad{numFunEvals3D()} returns the number of points computed.")) (|setAdaptive3D| (((|Boolean|) (|Boolean|)) "\\spad{setAdaptive3D(true)} turns adaptive plotting on; setAdaptive3D(\\spad{false}) turns adaptive plotting off.")) (|adaptive3D?| (((|Boolean|)) "\\spad{adaptive3D?()} determines whether plotting be done adaptively.")) (|setScreenResolution3D| (((|Integer|) (|Integer|)) "\\spad{setScreenResolution3D(i)} sets the screen resolution for a 3d graph to \\spad{i}.")) (|screenResolution3D| (((|Integer|)) "\\spad{screenResolution3D()} returns the screen resolution for a 3d graph.")) (|setMaxPoints3D| (((|Integer|) (|Integer|)) "\\spad{setMaxPoints3D(i)} sets the maximum number of points in a plot to \\spad{i}.")) (|maxPoints3D| (((|Integer|)) "\\spad{maxPoints3D()} returns the maximum number of points in a plot.")) (|setMinPoints3D| (((|Integer|) (|Integer|)) "\\spad{setMinPoints3D(i)} sets the minimum number of points in a plot to \\spad{i}.")) (|minPoints3D| (((|Integer|)) "\\spad{minPoints3D()} returns the minimum number of points in a plot.")) (|tValues| (((|List| (|List| (|DoubleFloat|))) $) "\\spad{tValues(p)} returns a list of lists of the values of the parameter for which a point is computed,{} one list for each curve in the plot \\spad{p}.")) (|tRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{tRange(p)} returns the range of the parameter in a parametric plot \\spad{p}.")) (|refine| (($ $) "\\spad{refine(x)} \\undocumented") (($ $ (|Segment| (|DoubleFloat|))) "\\spad{refine(x,r)} \\undocumented")) (|zoom| (($ $ (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{zoom(x,r,s,t)} \\undocumented")) (|plot| (($ $ (|Segment| (|DoubleFloat|))) "\\spad{plot(x,r)} \\undocumented") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f1,f2,f3,f4,x,y,z,w)} \\undocumented") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,h,a..b)} plots {/emx = \\spad{f}(\\spad{t}),{} \\spad{y} = \\spad{g}(\\spad{t}),{} \\spad{z} = \\spad{h}(\\spad{t})} as \\spad{t} ranges over {/em[a,{}\\spad{b}]}.")) (|pointPlot| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(f,x,y,z,w)} \\undocumented") (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(f,g,h,a..b)} plots {/emx = \\spad{f}(\\spad{t}),{} \\spad{y} = \\spad{g}(\\spad{t}),{} \\spad{z} = \\spad{h}(\\spad{t})} as \\spad{t} ranges over {/em[a,{}\\spad{b}]}."))) 
 NIL 
 NIL 
-(-932) 
+(-934) 
 ((|constructor| (NIL "This package exports plotting tools")) (|calcRanges| (((|List| (|Segment| (|DoubleFloat|))) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{calcRanges(l)} \\undocumented"))) 
 NIL 
 NIL 
-(-933) 
+(-935) 
 ((|constructor| (NIL "Attaching assertions to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|multiple| (((|Expression| (|Integer|)) (|Symbol|)) "\\spad{multiple(x)} tells the pattern matcher that \\spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\\spad{x}) tells the pattern matcher that \\spad{x} should match a list instead of an element of a list.")) (|optional| (((|Expression| (|Integer|)) (|Symbol|)) "\\spad{optional(x)} tells the pattern matcher that \\spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation)..")) (|constant| (((|Expression| (|Integer|)) (|Symbol|)) "\\spad{constant(x)} tells the pattern matcher that \\spad{x} should match only the symbol \\spad{'x} and no other quantity.")) (|assert| (((|Expression| (|Integer|)) (|Symbol|) (|Identifier|)) "\\spad{assert(x, s)} makes the assertion \\spad{s} about \\spad{x}."))) 
 NIL 
 NIL 
-(-934 R -3498) 
+(-936 R -3505) 
 ((|constructor| (NIL "Attaching assertions to symbols for pattern matching; Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|multiple| ((|#2| |#2|) "\\spad{multiple(x)} tells the pattern matcher that \\spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\\spad{x}) tells the pattern matcher that \\spad{x} should match a list instead of an element of a list. Error: if \\spad{x} is not a symbol.")) (|optional| ((|#2| |#2|) "\\spad{optional(x)} tells the pattern matcher that \\spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation). Error: if \\spad{x} is not a symbol.")) (|constant| ((|#2| |#2|) "\\spad{constant(x)} tells the pattern matcher that \\spad{x} should match only the symbol \\spad{'x} and no other quantity. Error: if \\spad{x} is not a symbol.")) (|assert| ((|#2| |#2| (|Identifier|)) "\\spad{assert(x, s)} makes the assertion \\spad{s} about \\spad{x}. Error: if \\spad{x} is not a symbol."))) 
 NIL 
 NIL 
-(-935 S A B) 
+(-937 S A B) 
 ((|constructor| (NIL "This packages provides tools for matching recursively in type towers.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#2| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches. Note: this function handles type towers by changing the predicates and calling the matching function provided by \\spad{A}.")) (|fixPredicate| (((|Mapping| (|Boolean|) |#2|) (|Mapping| (|Boolean|) |#3|)) "\\spad{fixPredicate(f)} returns \\spad{g} defined by \\spad{g}(a) = \\spad{f}(a::B)."))) 
 NIL 
 NIL 
-(-936 S R -3498) 
+(-938 S R -3505) 
 ((|constructor| (NIL "This package provides pattern matching functions on function spaces.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches."))) 
 NIL 
 NIL 
-(-937 I) 
+(-939 I) 
 ((|constructor| (NIL "This package provides pattern matching functions on integers.")) (|patternMatch| (((|PatternMatchResult| (|Integer|) |#1|) |#1| (|Pattern| (|Integer|)) (|PatternMatchResult| (|Integer|) |#1|)) "\\spad{patternMatch(n, pat, res)} matches the pattern \\spad{pat} to the integer \\spad{n}; res contains the variables of \\spad{pat} which are already matched and their matches."))) 
 NIL 
 NIL 
-(-938 S E) 
+(-940 S E) 
 ((|constructor| (NIL "This package provides pattern matching functions on kernels.")) (|patternMatch| (((|PatternMatchResult| |#1| |#2|) (|Kernel| |#2|) (|Pattern| |#1|) (|PatternMatchResult| |#1| |#2|)) "\\spad{patternMatch(f(e1,...,en), pat, res)} matches the pattern \\spad{pat} to \\spad{f(e1,...,en)}; res contains the variables of \\spad{pat} which are already matched and their matches."))) 
 NIL 
 NIL 
-(-939 S R L) 
+(-941 S R L) 
 ((|constructor| (NIL "This package provides pattern matching functions on lists.")) (|patternMatch| (((|PatternMatchListResult| |#1| |#2| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchListResult| |#1| |#2| |#3|)) "\\spad{patternMatch(l, pat, res)} matches the pattern \\spad{pat} to the list \\spad{l}; res contains the variables of \\spad{pat} which are already matched and their matches."))) 
 NIL 
 NIL 
-(-940 S E V R P) 
+(-942 S E V R P) 
 ((|constructor| (NIL "This package provides pattern matching functions on polynomials.")) (|patternMatch| (((|PatternMatchResult| |#1| |#5|) |#5| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|)) "\\spad{patternMatch(p, pat, res)} matches the pattern \\spad{pat} to the polynomial \\spad{p}; res contains the variables of \\spad{pat} which are already matched and their matches.") (((|PatternMatchResult| |#1| |#5|) |#5| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|) (|Mapping| (|PatternMatchResult| |#1| |#5|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|))) "\\spad{patternMatch(p, pat, res, vmatch)} matches the pattern \\spad{pat} to the polynomial \\spad{p}. \\spad{res} contains the variables of \\spad{pat} which are already matched and their matches; vmatch is the matching function to use on the variables."))) 
 NIL 
-((|HasCategory| |#3| (LIST (QUOTE -890) (|devaluate| |#1|)))) 
-(-941 -3074) 
+((|HasCategory| |#3| (LIST (QUOTE -892) (|devaluate| |#1|)))) 
+(-943 -3081) 
 ((|constructor| (NIL "Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|suchThat| (((|Expression| (|Integer|)) (|Symbol|) (|List| (|Mapping| (|Boolean|) |#1|))) "\\spad{suchThat(x, [f1, f2, ..., fn])} attaches the predicate \\spad{f1} and \\spad{f2} and ... and \\spad{fn} to \\spad{x}.") (((|Expression| (|Integer|)) (|Symbol|) (|Mapping| (|Boolean|) |#1|)) "\\spad{suchThat(x, foo)} attaches the predicate foo to \\spad{x}."))) 
 NIL 
 NIL 
-(-942 R -3498 -3074) 
+(-944 R -3505 -3081) 
 ((|constructor| (NIL "Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|suchThat| ((|#2| |#2| (|List| (|Mapping| (|Boolean|) |#3|))) "\\spad{suchThat(x, [f1, f2, ..., fn])} attaches the predicate \\spad{f1} and \\spad{f2} and ... and \\spad{fn} to \\spad{x}. Error: if \\spad{x} is not a symbol.") ((|#2| |#2| (|Mapping| (|Boolean|) |#3|)) "\\spad{suchThat(x, foo)} attaches the predicate foo to \\spad{x}; error if \\spad{x} is not a symbol."))) 
 NIL 
 NIL 
-(-943 S R Q) 
+(-945 S R Q) 
 ((|constructor| (NIL "This package provides pattern matching functions on quotients.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(a/b, pat, res)} matches the pattern \\spad{pat} to the quotient \\spad{a/b}; res contains the variables of \\spad{pat} which are already matched and their matches."))) 
 NIL 
 NIL 
-(-944 S) 
+(-946 S) 
 ((|constructor| (NIL "This package provides pattern matching functions on symbols.")) (|patternMatch| (((|PatternMatchResult| |#1| (|Symbol|)) (|Symbol|) (|Pattern| |#1|) (|PatternMatchResult| |#1| (|Symbol|))) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches (necessary for recursion)."))) 
 NIL 
 NIL 
-(-945 S R P) 
+(-947 S R P) 
 ((|constructor| (NIL "This package provides tools for the pattern matcher.")) (|patternMatchTimes| (((|PatternMatchResult| |#1| |#3|) (|List| |#3|) (|List| (|Pattern| |#1|)) (|PatternMatchResult| |#1| |#3|) (|Mapping| (|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|))) "\\spad{patternMatchTimes(lsubj, lpat, res, match)} matches the product of patterns \\spad{reduce(*,lpat)} to the product of subjects \\spad{reduce(*,lsubj)}; \\spad{r} contains the previous matches and match is a pattern-matching function on \\spad{P}.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) (|List| |#3|) (|List| (|Pattern| |#1|)) (|Mapping| |#3| (|List| |#3|)) (|PatternMatchResult| |#1| |#3|) (|Mapping| (|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|))) "\\spad{patternMatch(lsubj, lpat, op, res, match)} matches the list of patterns \\spad{lpat} to the list of subjects \\spad{lsubj},{} allowing for commutativity; \\spad{op} is the operator such that \\spad{op}(\\spad{lpat}) should match \\spad{op}(\\spad{lsubj}) at the end,{} \\spad{r} contains the previous matches,{} and match is a pattern-matching function on \\spad{P}."))) 
 NIL 
 NIL 
-(-946) 
+(-948) 
 ((|constructor| (NIL "This package provides various polynomial number theoretic functions over the integers.")) (|legendre| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{legendre(n)} returns the \\spad{n}th Legendre polynomial \\spad{P[n](x)}. Note: Legendre polynomials,{} denoted \\spad{P[n](x)},{} are computed from the two term recurrence. The generating function is: \\spad{1/sqrt(1-2*t*x+t**2) = sum(P[n](x)*t**n, n=0..infinity)}.")) (|laguerre| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{laguerre(n)} returns the \\spad{n}th Laguerre polynomial \\spad{L[n](x)}. Note: Laguerre polynomials,{} denoted \\spad{L[n](x)},{} are computed from the two term recurrence. The generating function is: \\spad{exp(x*t/(t-1))/(1-t) = sum(L[n](x)*t**n/n!, n=0..infinity)}.")) (|hermite| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{hermite(n)} returns the \\spad{n}th Hermite polynomial \\spad{H[n](x)}. Note: Hermite polynomials,{} denoted \\spad{H[n](x)},{} are computed from the two term recurrence. The generating function is: \\spad{exp(2*t*x-t**2) = sum(H[n](x)*t**n/n!, n=0..infinity)}.")) (|fixedDivisor| (((|Integer|) (|SparseUnivariatePolynomial| (|Integer|))) "\\spad{fixedDivisor(a)} for \\spad{a(x)} in \\spad{Z[x]} is the largest integer \\spad{f} such that \\spad{f} divides \\spad{a(x=k)} for all integers \\spad{k}. Note: fixed divisor of \\spad{a} is \\spad{reduce(gcd,[a(x=k) for k in 0..degree(a)])}.")) (|euler| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{euler(n)} returns the \\spad{n}th Euler polynomial \\spad{E[n](x)}. Note: Euler polynomials denoted \\spad{E(n,x)} computed by solving the differential equation \\spad{differentiate(E(n,x),x) = n E(n-1,x)} where \\spad{E(0,x) = 1} and initial condition comes from \\spad{E(n) = 2**n E(n,1/2)}.")) (|cyclotomic| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{cyclotomic(n)} returns the \\spad{n}th cyclotomic polynomial \\spad{phi[n](x)}. Note: \\spad{phi[n](x)} is the factor of \\spad{x**n - 1} whose roots are the primitive \\spad{n}th roots of unity.")) (|chebyshevU| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{chebyshevU(n)} returns the \\spad{n}th Chebyshev polynomial \\spad{U[n](x)}. Note: Chebyshev polynomials of the second kind,{} denoted \\spad{U[n](x)},{} computed from the two term recurrence. The generating function \\spad{1/(1-2*t*x+t**2) = sum(T[n](x)*t**n, n=0..infinity)}.")) (|chebyshevT| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{chebyshevT(n)} returns the \\spad{n}th Chebyshev polynomial \\spad{T[n](x)}. Note: Chebyshev polynomials of the first kind,{} denoted \\spad{T[n](x)},{} computed from the two term recurrence. The generating function \\spad{(1-t*x)/(1-2*t*x+t**2) = sum(T[n](x)*t**n, n=0..infinity)}.")) (|bernoulli| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{bernoulli(n)} returns the \\spad{n}th Bernoulli polynomial \\spad{B[n](x)}. Note: Bernoulli polynomials denoted \\spad{B(n,x)} computed by solving the differential equation \\spad{differentiate(B(n,x),x) = n B(n-1,x)} where \\spad{B(0,x) = 1} and initial condition comes from \\spad{B(n) = B(n,0)}."))) 
 NIL 
 NIL 
-(-947 R) 
+(-949 R) 
 ((|constructor| (NIL "This domain implements points in coordinate space"))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#1| (QUOTE (-1053))) (-12 (|HasCategory| |#1| (QUOTE (-1006))) (|HasCategory| |#1| (QUOTE (-1053)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-948 |lv| R) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-731))) (|HasCategory| |#1| (QUOTE (-1055))) (-12 (|HasCategory| |#1| (QUOTE (-1008))) (|HasCategory| |#1| (QUOTE (-1055)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-950 |lv| R) 
 ((|constructor| (NIL "Package with the conversion functions among different kind of polynomials")) (|pToDmp| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|Polynomial| |#2|)) "\\spad{pToDmp(p)} converts \\spad{p} from a \\spadtype{POLY} to a \\spadtype{DMP}.")) (|dmpToP| (((|Polynomial| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{dmpToP(p)} converts \\spad{p} from a \\spadtype{DMP} to a \\spadtype{POLY}.")) (|hdmpToP| (((|Polynomial| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{hdmpToP(p)} converts \\spad{p} from a \\spadtype{HDMP} to a \\spadtype{POLY}.")) (|pToHdmp| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|Polynomial| |#2|)) "\\spad{pToHdmp(p)} converts \\spad{p} from a \\spadtype{POLY} to a \\spadtype{HDMP}.")) (|hdmpToDmp| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{hdmpToDmp(p)} converts \\spad{p} from a \\spadtype{HDMP} to a \\spadtype{DMP}.")) (|dmpToHdmp| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{dmpToHdmp(p)} converts \\spad{p} from a \\spadtype{DMP} to a \\spadtype{HDMP}."))) 
 NIL 
 NIL 
-(-949 |TheField| |ThePols|) 
+(-951 |TheField| |ThePols|) 
 ((|constructor| (NIL "\\axiomType{RealPolynomialUtilitiesPackage} provides common functions used by interval coding.")) (|lazyVariations| (((|NonNegativeInteger|) (|List| |#1|) (|Integer|) (|Integer|)) "\\axiom{lazyVariations(\\spad{l},{}\\spad{s1},{}\\spad{sn})} is the number of sign variations in the list of non null numbers [s1::l]\\spad{@sn},{}")) (|sturmVariationsOf| (((|NonNegativeInteger|) (|List| |#1|)) "\\axiom{sturmVariationsOf(\\spad{l})} is the number of sign variations in the list of numbers \\spad{l},{} note that the first term counts as a sign")) (|boundOfCauchy| ((|#1| |#2|) "\\axiom{boundOfCauchy(\\spad{p})} bounds the roots of \\spad{p}")) (|sturmSequence| (((|List| |#2|) |#2|) "\\axiom{sturmSequence(\\spad{p}) = sylvesterSequence(\\spad{p},{}\\spad{p'})}")) (|sylvesterSequence| (((|List| |#2|) |#2| |#2|) "\\axiom{sylvesterSequence(\\spad{p},{}\\spad{q})} is the negated remainder sequence of \\spad{p} and \\spad{q} divided by the last computed term"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-851)))) 
-(-950 R) 
+((|HasCategory| |#1| (QUOTE (-853)))) 
+(-952 R) 
 ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are arbitrary symbols. The ordering is alphabetic determined by the Symbol type. The coefficient ring may be non commutative,{} but the variables are assumed to commute.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(p,x)} computes the integral of \\spad{p*dx},{} \\spadignore{i.e.} integrates the polynomial \\spad{p} with respect to the variable \\spad{x}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-914))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-914)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-1181) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-1181) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-1181) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-1181) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-1181) (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-366))) (|HasAttribute| |#1| (QUOTE -4425)) (|HasCategory| |#1| (QUOTE (-456))) (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
-(-951 R S) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-1183) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-1183) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-1183) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-1183) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-1183) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-367))) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(-953 R S) 
 ((|constructor| (NIL "\\indented{2}{This package takes a mapping between coefficient rings,{} and lifts} it to a mapping between polynomials over those rings.")) (|map| (((|Polynomial| |#2|) (|Mapping| |#2| |#1|) (|Polynomial| |#1|)) "\\spad{map(f, p)} produces a new polynomial as a result of applying the function \\spad{f} to every coefficient of the polynomial \\spad{p}."))) 
 NIL 
 NIL 
-(-952 |x| R) 
+(-954 |x| R) 
 ((|constructor| (NIL "This package is primarily to help the interpreter do coercions. It allows you to view a polynomial as a univariate polynomial in one of its variables with coefficients which are again a polynomial in all the other variables.")) (|univariate| (((|UnivariatePolynomial| |#1| (|Polynomial| |#2|)) (|Polynomial| |#2|) (|Variable| |#1|)) "\\spad{univariate(p, x)} converts the polynomial \\spad{p} to a one of type \\spad{UnivariatePolynomial(x,Polynomial(R))},{} ie. as a member of \\spad{R[...][x]}."))) 
 NIL 
 NIL 
-(-953 S R E |VarSet|) 
+(-955 S R E |VarSet|) 
 ((|constructor| (NIL "The category for general multi-variate polynomials over a ring \\spad{R},{} in variables from VarSet,{} with exponents from the \\spadtype{OrderedAbelianMonoidSup}.")) (|canonicalUnitNormal| ((|attribute|) "we can choose a unique representative for each associate class. This normalization is chosen to be normalization of leading coefficient (by default).")) (|squareFreePart| (($ $) "\\spad{squareFreePart(p)} returns product of all the irreducible factors of polynomial \\spad{p} each taken with multiplicity one.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(p)} returns the square free factorization of the polynomial \\spad{p}.")) (|primitivePart| (($ $ |#4|) "\\spad{primitivePart(p,v)} returns the unitCanonical associate of the polynomial \\spad{p} with its content with respect to the variable \\spad{v} divided out.") (($ $) "\\spad{primitivePart(p)} returns the unitCanonical associate of the polynomial \\spad{p} with its content divided out.")) (|content| (($ $ |#4|) "\\spad{content(p,v)} is the \\spad{gcd} of the coefficients of the polynomial \\spad{p} when \\spad{p} is viewed as a univariate polynomial with respect to the variable \\spad{v}. Thus,{} for polynomial 7*x**2*y + 14*x*y**2,{} the \\spad{gcd} of the coefficients with respect to \\spad{x} is 7*y.")) (|discriminant| (($ $ |#4|) "\\spad{discriminant(p,v)} returns the disriminant of the polynomial \\spad{p} with respect to the variable \\spad{v}.")) (|resultant| (($ $ $ |#4|) "\\spad{resultant(p,q,v)} returns the resultant of the polynomials \\spad{p} and \\spad{q} with respect to the variable \\spad{v}.")) (|primitiveMonomials| (((|List| $) $) "\\spad{primitiveMonomials(p)} gives the list of monomials of the polynomial \\spad{p} with their coefficients removed. Note: \\spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),...,X^(n)]}.")) (|variables| (((|List| |#4|) $) "\\spad{variables(p)} returns the list of those variables actually appearing in the polynomial \\spad{p}.")) (|totalDegree| (((|NonNegativeInteger|) $ (|List| |#4|)) "\\spad{totalDegree(p, lv)} returns the maximum sum (over all monomials of polynomial \\spad{p}) of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $) "\\spad{totalDegree(p)} returns the largest sum over all monomials of all exponents of a monomial.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#4|) (|:| |exponent| (|NonNegativeInteger|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if polynomial \\spad{p} has the form \\spad{x**n} and \\spad{n > 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if polynomial \\spad{p = a1 ... an} and \\spad{n >= 2},{} and,{} for each \\spad{i},{} \\spad{ai} is either a nontrivial constant in \\spad{R} or else of the form \\spad{x**e},{} where \\spad{e > 0} is an integer and \\spad{x} in a member of VarSet.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if polynomial \\spad{p = m1 + ... + mn} and \\spad{n >= 2} and each \\spad{mi} is a nonzero monomial.")) (|multivariate| (($ (|SparseUnivariatePolynomial| $) |#4|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.") (($ (|SparseUnivariatePolynomial| |#2|) |#4|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.")) (|monomial| (($ $ (|List| |#4|) (|List| (|NonNegativeInteger|))) "\\spad{monomial(a,[v1..vn],[e1..en])} returns \\spad{a*prod(vi**ei)}.") (($ $ |#4| (|NonNegativeInteger|)) "\\spad{monomial(a,x,n)} creates the monomial \\spad{a*x**n} where \\spad{a} is a polynomial,{} \\spad{x} is a variable and \\spad{n} is a nonnegative integer.")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $ |#4|) "\\spad{monicDivide(a,b,v)} divides the polynomial a by the polynomial \\spad{b},{} with each viewed as a univariate polynomial in \\spad{v} returning both the quotient and remainder. Error: if \\spad{b} is not monic with respect to \\spad{v}.")) (|minimumDegree| (((|List| (|NonNegativeInteger|)) $ (|List| |#4|)) "\\spad{minimumDegree(p, lv)} gives the list of minimum degrees of the polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}") (((|NonNegativeInteger|) $ |#4|) "\\spad{minimumDegree(p,v)} gives the minimum degree of polynomial \\spad{p} with respect to \\spad{v},{} \\spadignore{i.e.} viewed a univariate polynomial in \\spad{v}")) (|mainVariable| (((|Union| |#4| "failed") $) "\\spad{mainVariable(p)} returns the biggest variable which actually occurs in the polynomial \\spad{p},{} or \"failed\" if no variables are present. fails precisely if polynomial satisfies ground?")) (|univariate| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{univariate(p)} converts the multivariate polynomial \\spad{p},{} which should actually involve only one variable,{} into a univariate polynomial in that variable,{} whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate") (((|SparseUnivariatePolynomial| $) $ |#4|) "\\spad{univariate(p,v)} converts the multivariate polynomial \\spad{p} into a univariate polynomial in \\spad{v},{} whose coefficients are still multivariate polynomials (in all the other variables).")) (|monomials| (((|List| $) $) "\\spad{monomials(p)} returns the list of non-zero monomials of polynomial \\spad{p},{} \\spadignore{i.e.} \\spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}.")) (|coefficient| (($ $ (|List| |#4|) (|List| (|NonNegativeInteger|))) "\\spad{coefficient(p, lv, ln)} views the polynomial \\spad{p} as a polynomial in the variables of \\spad{lv} and returns the coefficient of the term \\spad{lv**ln},{} \\spadignore{i.e.} \\spad{prod(lv_i ** ln_i)}.") (($ $ |#4| (|NonNegativeInteger|)) "\\spad{coefficient(p,v,n)} views the polynomial \\spad{p} as a univariate polynomial in \\spad{v} and returns the coefficient of the \\spad{v**n} term.")) (|degree| (((|List| (|NonNegativeInteger|)) $ (|List| |#4|)) "\\spad{degree(p,lv)} gives the list of degrees of polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $ |#4|) "\\spad{degree(p,v)} gives the degree of polynomial \\spad{p} with respect to the variable \\spad{v}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-914))) (|HasAttribute| |#2| (QUOTE -4425)) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#4| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| |#4| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#4| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| |#4| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#4| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539))))) 
-(-954 R E |VarSet|) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasAttribute| |#2| (QUOTE -4432)) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#4| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| |#4| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#4| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| |#4| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540))))) 
+(-956 R E |VarSet|) 
 ((|constructor| (NIL "The category for general multi-variate polynomials over a ring \\spad{R},{} in variables from VarSet,{} with exponents from the \\spadtype{OrderedAbelianMonoidSup}.")) (|canonicalUnitNormal| ((|attribute|) "we can choose a unique representative for each associate class. This normalization is chosen to be normalization of leading coefficient (by default).")) (|squareFreePart| (($ $) "\\spad{squareFreePart(p)} returns product of all the irreducible factors of polynomial \\spad{p} each taken with multiplicity one.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(p)} returns the square free factorization of the polynomial \\spad{p}.")) (|primitivePart| (($ $ |#3|) "\\spad{primitivePart(p,v)} returns the unitCanonical associate of the polynomial \\spad{p} with its content with respect to the variable \\spad{v} divided out.") (($ $) "\\spad{primitivePart(p)} returns the unitCanonical associate of the polynomial \\spad{p} with its content divided out.")) (|content| (($ $ |#3|) "\\spad{content(p,v)} is the \\spad{gcd} of the coefficients of the polynomial \\spad{p} when \\spad{p} is viewed as a univariate polynomial with respect to the variable \\spad{v}. Thus,{} for polynomial 7*x**2*y + 14*x*y**2,{} the \\spad{gcd} of the coefficients with respect to \\spad{x} is 7*y.")) (|discriminant| (($ $ |#3|) "\\spad{discriminant(p,v)} returns the disriminant of the polynomial \\spad{p} with respect to the variable \\spad{v}.")) (|resultant| (($ $ $ |#3|) "\\spad{resultant(p,q,v)} returns the resultant of the polynomials \\spad{p} and \\spad{q} with respect to the variable \\spad{v}.")) (|primitiveMonomials| (((|List| $) $) "\\spad{primitiveMonomials(p)} gives the list of monomials of the polynomial \\spad{p} with their coefficients removed. Note: \\spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),...,X^(n)]}.")) (|variables| (((|List| |#3|) $) "\\spad{variables(p)} returns the list of those variables actually appearing in the polynomial \\spad{p}.")) (|totalDegree| (((|NonNegativeInteger|) $ (|List| |#3|)) "\\spad{totalDegree(p, lv)} returns the maximum sum (over all monomials of polynomial \\spad{p}) of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $) "\\spad{totalDegree(p)} returns the largest sum over all monomials of all exponents of a monomial.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#3|) (|:| |exponent| (|NonNegativeInteger|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if polynomial \\spad{p} has the form \\spad{x**n} and \\spad{n > 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if polynomial \\spad{p = a1 ... an} and \\spad{n >= 2},{} and,{} for each \\spad{i},{} \\spad{ai} is either a nontrivial constant in \\spad{R} or else of the form \\spad{x**e},{} where \\spad{e > 0} is an integer and \\spad{x} in a member of VarSet.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if polynomial \\spad{p = m1 + ... + mn} and \\spad{n >= 2} and each \\spad{mi} is a nonzero monomial.")) (|multivariate| (($ (|SparseUnivariatePolynomial| $) |#3|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.") (($ (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.")) (|monomial| (($ $ (|List| |#3|) (|List| (|NonNegativeInteger|))) "\\spad{monomial(a,[v1..vn],[e1..en])} returns \\spad{a*prod(vi**ei)}.") (($ $ |#3| (|NonNegativeInteger|)) "\\spad{monomial(a,x,n)} creates the monomial \\spad{a*x**n} where \\spad{a} is a polynomial,{} \\spad{x} is a variable and \\spad{n} is a nonnegative integer.")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $ |#3|) "\\spad{monicDivide(a,b,v)} divides the polynomial a by the polynomial \\spad{b},{} with each viewed as a univariate polynomial in \\spad{v} returning both the quotient and remainder. Error: if \\spad{b} is not monic with respect to \\spad{v}.")) (|minimumDegree| (((|List| (|NonNegativeInteger|)) $ (|List| |#3|)) "\\spad{minimumDegree(p, lv)} gives the list of minimum degrees of the polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}") (((|NonNegativeInteger|) $ |#3|) "\\spad{minimumDegree(p,v)} gives the minimum degree of polynomial \\spad{p} with respect to \\spad{v},{} \\spadignore{i.e.} viewed a univariate polynomial in \\spad{v}")) (|mainVariable| (((|Union| |#3| "failed") $) "\\spad{mainVariable(p)} returns the biggest variable which actually occurs in the polynomial \\spad{p},{} or \"failed\" if no variables are present. fails precisely if polynomial satisfies ground?")) (|univariate| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{univariate(p)} converts the multivariate polynomial \\spad{p},{} which should actually involve only one variable,{} into a univariate polynomial in that variable,{} whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate") (((|SparseUnivariatePolynomial| $) $ |#3|) "\\spad{univariate(p,v)} converts the multivariate polynomial \\spad{p} into a univariate polynomial in \\spad{v},{} whose coefficients are still multivariate polynomials (in all the other variables).")) (|monomials| (((|List| $) $) "\\spad{monomials(p)} returns the list of non-zero monomials of polynomial \\spad{p},{} \\spadignore{i.e.} \\spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}.")) (|coefficient| (($ $ (|List| |#3|) (|List| (|NonNegativeInteger|))) "\\spad{coefficient(p, lv, ln)} views the polynomial \\spad{p} as a polynomial in the variables of \\spad{lv} and returns the coefficient of the term \\spad{lv**ln},{} \\spadignore{i.e.} \\spad{prod(lv_i ** ln_i)}.") (($ $ |#3| (|NonNegativeInteger|)) "\\spad{coefficient(p,v,n)} views the polynomial \\spad{p} as a univariate polynomial in \\spad{v} and returns the coefficient of the \\spad{v**n} term.")) (|degree| (((|List| (|NonNegativeInteger|)) $ (|List| |#3|)) "\\spad{degree(p,lv)} gives the list of degrees of polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $ |#3|) "\\spad{degree(p,v)} gives the degree of polynomial \\spad{p} with respect to the variable \\spad{v}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
-(-955 E V R P -3498) 
+(-957 E V R P -3505) 
 ((|constructor| (NIL "This package transforms multivariate polynomials or fractions into univariate polynomials or fractions,{} and back.")) (|isPower| (((|Union| (|Record| (|:| |val| |#5|) (|:| |exponent| (|Integer|))) "failed") |#5|) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0},{} \"failed\" otherwise.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#2|) (|:| |exponent| (|Integer|))) "failed") |#5|) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0},{} \"failed\" otherwise.")) (|isTimes| (((|Union| (|List| |#5|) "failed") |#5|) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1 ... an} and \\spad{n > 1},{} \"failed\" otherwise.")) (|isPlus| (((|Union| (|List| |#5|) "failed") |#5|) "\\spad{isPlus(p)} returns [\\spad{m1},{}...,{}\\spad{mn}] if \\spad{p = m1 + ... + mn} and \\spad{n > 1},{} \"failed\" otherwise.")) (|multivariate| ((|#5| (|Fraction| (|SparseUnivariatePolynomial| |#5|)) |#2|) "\\spad{multivariate(f, v)} applies both the numerator and denominator of \\spad{f} to \\spad{v}.")) (|univariate| (((|SparseUnivariatePolynomial| |#5|) |#5| |#2| (|SparseUnivariatePolynomial| |#5|)) "\\spad{univariate(f, x, p)} returns \\spad{f} viewed as a univariate polynomial in \\spad{x},{} using the side-condition \\spad{p(x) = 0}.") (((|Fraction| (|SparseUnivariatePolynomial| |#5|)) |#5| |#2|) "\\spad{univariate(f, v)} returns \\spad{f} viewed as a univariate rational function in \\spad{v}.")) (|mainVariable| (((|Union| |#2| "failed") |#5|) "\\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \\spad{f},{} \"failed\" if \\spad{f} has no variables.")) (|variables| (((|List| |#2|) |#5|) "\\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \\spad{f}."))) 
 NIL 
 NIL 
-(-956 E |Vars| R P S) 
+(-958 E |Vars| R P S) 
 ((|constructor| (NIL "This package provides a very general map function,{} which given a set \\spad{S} and polynomials over \\spad{R} with maps from the variables into \\spad{S} and the coefficients into \\spad{S},{} maps polynomials into \\spad{S}. \\spad{S} is assumed to support \\spad{+},{} \\spad{*} and \\spad{**}.")) (|map| ((|#5| (|Mapping| |#5| |#2|) (|Mapping| |#5| |#3|) |#4|) "\\spad{map(varmap, coefmap, p)} takes a \\spad{varmap},{} a mapping from the variables of polynomial \\spad{p} into \\spad{S},{} \\spad{coefmap},{} a mapping from coefficients of \\spad{p} into \\spad{S},{} and \\spad{p},{} and produces a member of \\spad{S} using the corresponding arithmetic. in \\spad{S}"))) 
 NIL 
 NIL 
-(-957 E V R P -3498) 
+(-959 E V R P -3505) 
 ((|constructor| (NIL "computes \\spad{n}-th roots of quotients of multivariate polynomials")) (|nthr| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#4|) (|:| |radicand| (|List| |#4|))) |#4| (|NonNegativeInteger|)) "\\spad{nthr(p,n)} should be local but conditional")) (|froot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) |#5| (|NonNegativeInteger|)) "\\spad{froot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|qroot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) (|Fraction| (|Integer|)) (|NonNegativeInteger|)) "\\spad{qroot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|rroot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) |#3| (|NonNegativeInteger|)) "\\spad{rroot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|denom| ((|#4| $) "\\spad{denom(x)} \\undocumented")) (|numer| ((|#4| $) "\\spad{numer(x)} \\undocumented"))) 
 NIL 
-((|HasCategory| |#3| (QUOTE (-456)))) 
-(-958) 
+((|HasCategory| |#3| (QUOTE (-457)))) 
+(-960) 
 ((|constructor| (NIL "This domain represents network port numbers (notable \\spad{TCP} and UDP).")) (|port| (($ (|SingleInteger|)) "\\spad{port(n)} constructs a PortNumber from the integer \\spad{`n'}."))) 
 NIL 
 NIL 
-(-959) 
+(-961) 
 ((|constructor| (NIL "PlottablePlaneCurveCategory is the category of curves in the plane which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the \\spad{x}-coordinates and \\spad{y}-coordinates of the points on the curve.")) (|yRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{yRange(c)} returns the range of the \\spad{y}-coordinates of the points on the curve \\spad{c}.")) (|xRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{xRange(c)} returns the range of the \\spad{x}-coordinates of the points on the curve \\spad{c}.")) (|listBranches| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{listBranches(c)} returns a list of lists of points,{} representing the branches of the curve \\spad{c}."))) 
 NIL 
 NIL 
-(-960 R E) 
+(-962 R E) 
 ((|constructor| (NIL "This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and terms indexed by their exponents (from an arbitrary ordered abelian monoid). This type is used,{} for example,{} by the \\spadtype{DistributedMultivariatePolynomial} domain where the exponent domain is a direct product of non negative integers.")) (|canonicalUnitNormal| ((|attribute|) "canonicalUnitNormal guarantees that the function unitCanonical returns the same representative for all associates of any particular element.")) (|fmecg| (($ $ |#2| |#1| $) "\\spad{fmecg(p1,e,r,p2)} finds \\spad{X} : \\spad{p1} - \\spad{r} * X**e * \\spad{p2}"))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-456))) (-12 (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-131)))) (|HasAttribute| |#1| (QUOTE -4425))) 
-(-961 R L) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4428 . T) (-4429 . T) (-4431 . T)) 
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+(-963 R L) 
 ((|constructor| (NIL "\\spadtype{PrecomputedAssociatedEquations} stores some generic precomputations which speed up the computations of the associated equations needed for factoring operators.")) (|firstUncouplingMatrix| (((|Union| (|Matrix| |#1|) "failed") |#2| (|PositiveInteger|)) "\\spad{firstUncouplingMatrix(op, m)} returns the matrix A such that \\spad{A w = (W',W'',...,W^N)} in the corresponding associated equations for right-factors of order \\spad{m} of \\spad{op}. Returns \"failed\" if the matrix A has not been precomputed for the particular combination \\spad{degree(L), m}."))) 
 NIL 
 NIL 
-(-962 S) 
+(-964 S) 
 ((|constructor| (NIL "\\indented{1}{This provides a fast array type with no bound checking on elt\\spad{'s}.} Minimum index is 0 in this type,{} cannot be changed"))) 
-((-4428 . T) (-4427 . T)) 
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-(-963 A B) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-965 A B) 
 ((|constructor| (NIL "\\indented{1}{This package provides tools for operating on primitive arrays} with unary and binary functions involving different underlying types")) (|map| (((|PrimitiveArray| |#2|) (|Mapping| |#2| |#1|) (|PrimitiveArray| |#1|)) "\\spad{map(f,a)} applies function \\spad{f} to each member of primitive array \\spad{a} resulting in a new primitive array over a possibly different underlying domain.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|PrimitiveArray| |#1|) |#2|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the primitive array \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as the identity element for the function \\spad{f}.")) (|scan| (((|PrimitiveArray| |#2|) (|Mapping| |#2| |#1| |#2|) (|PrimitiveArray| |#1|) |#2|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-arrays \\spad{x} of primitive array \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}."))) 
 NIL 
 NIL 
-(-964) 
+(-966) 
 ((|constructor| (NIL "Category for the functions defined by integrals.")) (|integral| (($ $ (|SegmentBinding| $)) "\\spad{integral(f, x = a..b)} returns the formal definite integral of \\spad{f} \\spad{dx} for \\spad{x} between \\spad{a} and \\spad{b}.") (($ $ (|Symbol|)) "\\spad{integral(f, x)} returns the formal integral of \\spad{f} \\spad{dx}."))) 
 NIL 
 NIL 
-(-965 -3498) 
+(-967 -3505) 
 ((|constructor| (NIL "PrimitiveElement provides functions to compute primitive elements in algebraic extensions.")) (|primitiveElement| (((|Record| (|:| |coef| (|List| (|Integer|))) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#1|))) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|)) (|Symbol|)) "\\spad{primitiveElement([p1,...,pn], [a1,...,an], a)} returns \\spad{[[c1,...,cn], [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} where \\spad{a = a1 c1 + ... + an cn},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The \\spad{pi}\\spad{'s} are the defining polynomials for the \\spad{ai}\\spad{'s}. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}.") (((|Record| (|:| |coef| (|List| (|Integer|))) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#1|))) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{primitiveElement([p1,...,pn], [a1,...,an])} returns \\spad{[[c1,...,cn], [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} where \\spad{a = a1 c1 + ... + an cn},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The \\spad{pi}\\spad{'s} are the defining polynomials for the \\spad{ai}\\spad{'s}. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}.") (((|Record| (|:| |coef1| (|Integer|)) (|:| |coef2| (|Integer|)) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|Polynomial| |#1|) (|Symbol|) (|Polynomial| |#1|) (|Symbol|)) "\\spad{primitiveElement(p1, a1, p2, a2)} returns \\spad{[c1, c2, q]} such that \\spad{k(a1, a2) = k(a)} where \\spad{a = c1 a1 + c2 a2, and q(a) = 0}. The \\spad{pi}\\spad{'s} are the defining polynomials for the \\spad{ai}\\spad{'s}. The \\spad{p2} may involve \\spad{a1},{} but \\spad{p1} must not involve a2. This operation uses \\spadfun{resultant}."))) 
 NIL 
 NIL 
-(-966 I) 
+(-968 I) 
 ((|constructor| (NIL "The \\spadtype{IntegerPrimesPackage} implements a modification of Rabin\\spad{'s} probabilistic primality test and the utility functions \\spadfun{nextPrime},{} \\spadfun{prevPrime} and \\spadfun{primes}.")) (|primes| (((|List| |#1|) |#1| |#1|) "\\spad{primes(a,b)} returns a list of all primes \\spad{p} with \\spad{a <= p <= b}")) (|prevPrime| ((|#1| |#1|) "\\spad{prevPrime(n)} returns the largest prime strictly smaller than \\spad{n}")) (|nextPrime| ((|#1| |#1|) "\\spad{nextPrime(n)} returns the smallest prime strictly larger than \\spad{n}")) (|prime?| (((|Boolean|) |#1|) "\\spad{prime?(n)} returns \\spad{true} if \\spad{n} is prime and \\spad{false} if not. The algorithm used is Rabin\\spad{'s} probabilistic primality test (reference: Knuth Volume 2 Semi Numerical Algorithms). If \\spad{prime? n} returns \\spad{false},{} \\spad{n} is proven composite. If \\spad{prime? n} returns \\spad{true},{} prime? may be in error however,{} the probability of error is very low. and is zero below 25*10**9 (due to a result of Pomerance et al),{} below 10**12 and 10**13 due to results of Pinch,{} and below 341550071728321 due to a result of Jaeschke. Specifically,{} this implementation does at least 10 pseudo prime tests and so the probability of error is \\spad{< 4**(-10)}. The running time of this method is cubic in the length of the input \\spad{n},{} that is \\spad{O( (log n)**3 )},{} for n<10**20. beyond that,{} the algorithm is quartic,{} \\spad{O( (log n)**4 )}. Two improvements due to Davenport have been incorporated which catches some trivial strong pseudo-primes,{} such as [Jaeschke,{} 1991] 1377161253229053 * 413148375987157,{} which the original algorithm regards as prime"))) 
 NIL 
 NIL 
-(-967) 
+(-969) 
 ((|constructor| (NIL "PrintPackage provides a print function for output forms.")) (|print| (((|Void|) (|OutputForm|)) "\\spad{print(o)} writes the output form \\spad{o} on standard output using the two-dimensional formatter."))) 
 NIL 
 NIL 
-(-968 A B) 
+(-970 A B) 
 ((|constructor| (NIL "This domain implements cartesian product")) (|selectsecond| ((|#2| $) "\\spad{selectsecond(x)} \\undocumented")) (|selectfirst| ((|#1| $) "\\spad{selectfirst(x)} \\undocumented")) (|makeprod| (($ |#1| |#2|) "\\spad{makeprod(a,b)} \\undocumented"))) 
-((-4424 -12 (|has| |#2| (-477)) (|has| |#1| (-477)))) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-796)))) (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-853))))) (-12 (|HasCategory| |#1| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-796)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-131)))) (-12 (|HasCategory| |#1| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-796)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21))))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-131)))) (-12 (|HasCategory| |#1| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-796)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23))))) (-12 (|HasCategory| |#1| (QUOTE (-477))) (|HasCategory| |#2| (QUOTE (-477)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-477))) (|HasCategory| |#2| (QUOTE (-477)))) (-12 (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#2| (QUOTE (-729))))) (-12 (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#2| (QUOTE (-371)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-131)))) (-12 (|HasCategory| |#1| (QUOTE (-796))) (|HasCategory| |#2| (QUOTE (-796)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-477))) (|HasCategory| |#2| (QUOTE (-477)))) (-12 (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#2| (QUOTE (-729))))) (-12 (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#2| (QUOTE (-729)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-131))) (|HasCategory| |#2| (QUOTE (-131)))) (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-853))))) 
-(-969) 
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+(-971) 
 ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. An `Property' is a pair of name and value.")) (|property| (($ (|Identifier|) (|SExpression|)) "\\spad{property(n,val)} constructs a property with name \\spad{`n'} and value `val'.")) (|value| (((|SExpression|) $) "\\spad{value(p)} returns value of property \\spad{p}")) (|name| (((|Identifier|) $) "\\spad{name(p)} returns the name of property \\spad{p}"))) 
 NIL 
 NIL 
-(-970 T$) 
+(-972 T$) 
 ((|constructor| (NIL "This domain implements propositional formula build over a term domain,{} that itself belongs to PropositionalLogic")) (|disjunction| (($ $ $) "\\spad{disjunction(p,q)} returns a formula denoting the disjunction of \\spad{p} and \\spad{q}.")) (|conjunction| (($ $ $) "\\spad{conjunction(p,q)} returns a formula denoting the conjunction of \\spad{p} and \\spad{q}.")) (|isEquiv| (((|Maybe| (|Pair| $ $)) $) "\\spad{isEquiv f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is an equivalence formula.")) (|isImplies| (((|Maybe| (|Pair| $ $)) $) "\\spad{isImplies f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is an implication formula.")) (|isOr| (((|Maybe| (|Pair| $ $)) $) "\\spad{isOr f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is a disjunction formula.")) (|isAnd| (((|Maybe| (|Pair| $ $)) $) "\\spad{isAnd f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is a conjunction formula.")) (|isNot| (((|Maybe| $) $) "\\spad{isNot f} returns a value \\spad{v} such that \\spad{v case \\%} holds if the formula \\spad{f} is a negation.")) (|isTerm| (((|Maybe| |#1|) $) "\\spad{isTerm f} returns a value \\spad{v} such that \\spad{v case T} holds if the formula \\spad{f} is a term."))) 
 NIL 
 NIL 
-(-971) 
+(-973) 
 ((|constructor| (NIL "This category declares the connectives of Propositional Logic.")) (|equiv| (($ $ $) "\\spad{equiv(p,q)} returns the logical equivalence of \\spad{`p'},{} \\spad{`q'}.")) (|implies| (($ $ $) "\\spad{implies(p,q)} returns the logical implication of \\spad{`q'} by \\spad{`p'}.")) (|or| (($ $ $) "\\spad{p or q} returns the logical disjunction of \\spad{`p'},{} \\spad{`q'}.")) (|and| (($ $ $) "\\spad{p and q} returns the logical conjunction of \\spad{`p'},{} \\spad{`q'}.")) (|not| (($ $) "\\spad{not p} returns the logical negation of \\spad{`p'}.")) (|false| (($) "\\spad{false} is a logical constant.")) (|true| (($) "\\spad{true} is a logical constant."))) 
 NIL 
 NIL 
-(-972 S) 
+(-974 S) 
 ((|constructor| (NIL "A priority queue is a bag of items from an ordered set where the item extracted is always the maximum element.")) (|merge!| (($ $ $) "\\spad{merge!(q,q1)} destructively changes priority queue \\spad{q} to include the values from priority queue \\spad{q1}.")) (|merge| (($ $ $) "\\spad{merge(q1,q2)} returns combines priority queues \\spad{q1} and \\spad{q2} to return a single priority queue \\spad{q}.")) (|max| ((|#1| $) "\\spad{max(q)} returns the maximum element of priority queue \\spad{q}."))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
-(-973 R |polR|) 
+(-975 R |polR|) 
 ((|constructor| (NIL "This package contains some functions: \\axiomOpFrom{discriminant}{PseudoRemainderSequence},{} \\axiomOpFrom{resultant}{PseudoRemainderSequence},{} \\axiomOpFrom{subResultantGcd}{PseudoRemainderSequence},{} \\axiomOpFrom{chainSubResultants}{PseudoRemainderSequence},{} \\axiomOpFrom{degreeSubResultant}{PseudoRemainderSequence},{} \\axiomOpFrom{lastSubResultant}{PseudoRemainderSequence},{} \\axiomOpFrom{resultantEuclidean}{PseudoRemainderSequence},{} \\axiomOpFrom{subResultantGcdEuclidean}{PseudoRemainderSequence},{} \\axiomOpFrom{semiSubResultantGcdEuclidean1}{PseudoRemainderSequence},{} \\axiomOpFrom{semiSubResultantGcdEuclidean2}{PseudoRemainderSequence},{} etc. This procedures are coming from improvements of the subresultants algorithm. \\indented{2}{Version : 7} \\indented{2}{References : Lionel Ducos \"Optimizations of the subresultant algorithm\"} \\indented{2}{to appear in the Journal of Pure and Applied Algebra.} \\indented{2}{Author : Ducos Lionel \\axiom{Lionel.Ducos@mathlabo.univ-poitiers.\\spad{fr}}}")) (|semiResultantEuclideannaif| (((|Record| (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{resultantEuclidean_naif(\\spad{P},{}\\spad{Q})} returns the semi-extended resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}} computed by means of the naive algorithm.")) (|resultantEuclideannaif| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{resultantEuclidean_naif(\\spad{P},{}\\spad{Q})} returns the extended resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}} computed by means of the naive algorithm.")) (|resultantnaif| ((|#1| |#2| |#2|) "\\axiom{resultantEuclidean_naif(\\spad{P},{}\\spad{Q})} returns the resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}} computed by means of the naive algorithm.")) (|nextsousResultant2| ((|#2| |#2| |#2| |#2| |#1|) "\\axiom{nextsousResultant2(\\spad{P},{} \\spad{Q},{} \\spad{Z},{} \\spad{s})} returns the subresultant \\axiom{\\spad{S_}{\\spad{e}-1}} where \\axiom{\\spad{P} ~ \\spad{S_d},{} \\spad{Q} = \\spad{S_}{\\spad{d}-1},{} \\spad{Z} = S_e,{} \\spad{s} = \\spad{lc}(\\spad{S_d})}")) (|Lazard2| ((|#2| |#2| |#1| |#1| (|NonNegativeInteger|)) "\\axiom{Lazard2(\\spad{F},{} \\spad{x},{} \\spad{y},{} \\spad{n})} computes \\axiom{(x/y)\\spad{**}(\\spad{n}-1) * \\spad{F}}")) (|Lazard| ((|#1| |#1| |#1| (|NonNegativeInteger|)) "\\axiom{Lazard(\\spad{x},{} \\spad{y},{} \\spad{n})} computes \\axiom{x**n/y**(\\spad{n}-1)}")) (|divide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2|) "\\axiom{divide(\\spad{F},{}\\spad{G})} computes quotient and rest of the exact euclidean division of \\axiom{\\spad{F}} by \\axiom{\\spad{G}}.")) (|pseudoDivide| (((|Record| (|:| |coef| |#1|) (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2|) "\\axiom{pseudoDivide(\\spad{P},{}\\spad{Q})} computes the pseudoDivide of \\axiom{\\spad{P}} by \\axiom{\\spad{Q}}.")) (|exquo| (((|Vector| |#2|) (|Vector| |#2|) |#1|) "\\axiom{\\spad{v} exquo \\spad{r}} computes the exact quotient of \\axiom{\\spad{v}} by \\axiom{\\spad{r}}")) (* (((|Vector| |#2|) |#1| (|Vector| |#2|)) "\\axiom{\\spad{r} * \\spad{v}} computes the product of \\axiom{\\spad{r}} and \\axiom{\\spad{v}}")) (|gcd| ((|#2| |#2| |#2|) "\\axiom{\\spad{gcd}(\\spad{P},{} \\spad{Q})} returns the \\spad{gcd} of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiResultantReduitEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |resultantReduit| |#1|)) |#2| |#2|) "\\axiom{semiResultantReduitEuclidean(\\spad{P},{}\\spad{Q})} returns the \"reduce resultant\" and carries out the equality \\axiom{...\\spad{P} + coef2*Q = resultantReduit(\\spad{P},{}\\spad{Q})}.")) (|resultantReduitEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |resultantReduit| |#1|)) |#2| |#2|) "\\axiom{resultantReduitEuclidean(\\spad{P},{}\\spad{Q})} returns the \"reduce resultant\" and carries out the equality \\axiom{coef1*P + coef2*Q = resultantReduit(\\spad{P},{}\\spad{Q})}.")) (|resultantReduit| ((|#1| |#2| |#2|) "\\axiom{resultantReduit(\\spad{P},{}\\spad{Q})} returns the \"reduce resultant\" of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|schema| (((|List| (|NonNegativeInteger|)) |#2| |#2|) "\\axiom{schema(\\spad{P},{}\\spad{Q})} returns the list of degrees of non zero subresultants of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|chainSubResultants| (((|List| |#2|) |#2| |#2|) "\\axiom{chainSubResultants(\\spad{P},{} \\spad{Q})} computes the list of non zero subresultants of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiDiscriminantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |discriminant| |#1|)) |#2|) "\\axiom{discriminantEuclidean(\\spad{P})} carries out the equality \\axiom{...\\spad{P} + coef2 * \\spad{D}(\\spad{P}) = discriminant(\\spad{P})}. Warning: \\axiom{degree(\\spad{P}) \\spad{>=} degree(\\spad{Q})}.")) (|discriminantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |discriminant| |#1|)) |#2|) "\\axiom{discriminantEuclidean(\\spad{P})} carries out the equality \\axiom{coef1 * \\spad{P} + coef2 * \\spad{D}(\\spad{P}) = discriminant(\\spad{P})}.")) (|discriminant| ((|#1| |#2|) "\\axiom{discriminant(\\spad{P},{} \\spad{Q})} returns the discriminant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiSubResultantGcdEuclidean1| (((|Record| (|:| |coef1| |#2|) (|:| |gcd| |#2|)) |#2| |#2|) "\\axiom{semiSubResultantGcdEuclidean1(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{coef1*P + ? \\spad{Q} = \\spad{+/-} S_i(\\spad{P},{}\\spad{Q})} where the degree (not the indice) of the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} is the smaller as possible.")) (|semiSubResultantGcdEuclidean2| (((|Record| (|:| |coef2| |#2|) (|:| |gcd| |#2|)) |#2| |#2|) "\\axiom{semiSubResultantGcdEuclidean2(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{...\\spad{P} + coef2*Q = \\spad{+/-} S_i(\\spad{P},{}\\spad{Q})} where the degree (not the indice) of the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} is the smaller as possible. Warning: \\axiom{degree(\\spad{P}) \\spad{>=} degree(\\spad{Q})}.")) (|subResultantGcdEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |gcd| |#2|)) |#2| |#2|) "\\axiom{subResultantGcdEuclidean(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{coef1*P + coef2*Q = \\spad{+/-} S_i(\\spad{P},{}\\spad{Q})} where the degree (not the indice) of the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} is the smaller as possible.")) (|subResultantGcd| ((|#2| |#2| |#2|) "\\axiom{subResultantGcd(\\spad{P},{} \\spad{Q})} returns the \\spad{gcd} of two primitive polynomials \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiLastSubResultantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2|) "\\axiom{semiLastSubResultantEuclidean(\\spad{P},{} \\spad{Q})} computes the last non zero subresultant \\axiom{\\spad{S}} and carries out the equality \\axiom{...\\spad{P} + coef2*Q = \\spad{S}}. Warning: \\axiom{degree(\\spad{P}) \\spad{>=} degree(\\spad{Q})}.")) (|lastSubResultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2|) "\\axiom{lastSubResultantEuclidean(\\spad{P},{} \\spad{Q})} computes the last non zero subresultant \\axiom{\\spad{S}} and carries out the equality \\axiom{coef1*P + coef2*Q = \\spad{S}}.")) (|lastSubResultant| ((|#2| |#2| |#2|) "\\axiom{lastSubResultant(\\spad{P},{} \\spad{Q})} computes the last non zero subresultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}")) (|semiDegreeSubResultantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns a subresultant \\axiom{\\spad{S}} of degree \\axiom{\\spad{d}} and carries out the equality \\axiom{...\\spad{P} + coef2*Q = S_i}. Warning: \\axiom{degree(\\spad{P}) \\spad{>=} degree(\\spad{Q})}.")) (|degreeSubResultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns a subresultant \\axiom{\\spad{S}} of degree \\axiom{\\spad{d}} and carries out the equality \\axiom{coef1*P + coef2*Q = S_i}.")) (|degreeSubResultant| ((|#2| |#2| |#2| (|NonNegativeInteger|)) "\\axiom{degreeSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{d})} computes a subresultant of degree \\axiom{\\spad{d}}.")) (|semiIndiceSubResultantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{semiIndiceSubResultantEuclidean(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} and carries out the equality \\axiom{...\\spad{P} + coef2*Q = S_i(\\spad{P},{}\\spad{Q})} Warning: \\axiom{degree(\\spad{P}) \\spad{>=} degree(\\spad{Q})}.")) (|indiceSubResultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} and carries out the equality \\axiom{coef1*P + coef2*Q = S_i(\\spad{P},{}\\spad{Q})}")) (|indiceSubResultant| ((|#2| |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns the subresultant of indice \\axiom{\\spad{i}}")) (|semiResultantEuclidean1| (((|Record| (|:| |coef1| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{semiResultantEuclidean1(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{coef1.\\spad{P} + ? \\spad{Q} = resultant(\\spad{P},{}\\spad{Q})}.")) (|semiResultantEuclidean2| (((|Record| (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{semiResultantEuclidean2(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{...\\spad{P} + coef2*Q = resultant(\\spad{P},{}\\spad{Q})}. Warning: \\axiom{degree(\\spad{P}) \\spad{>=} degree(\\spad{Q})}.")) (|resultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{resultantEuclidean(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{coef1*P + coef2*Q = resultant(\\spad{P},{}\\spad{Q})}")) (|resultant| ((|#1| |#2| |#2|) "\\axiom{resultant(\\spad{P},{} \\spad{Q})} returns the resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-456)))) 
-(-974) 
+((|HasCategory| |#1| (QUOTE (-457)))) 
+(-976) 
 ((|constructor| (NIL "This domain represents `pretend' expressions.")) (|target| (((|TypeAst|) $) "\\spad{target(e)} returns the target type of the conversion..")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression being converted."))) 
 NIL 
 NIL 
-(-975) 
+(-977) 
 ((|constructor| (NIL "\\indented{1}{Partition is an OrderedCancellationAbelianMonoid which is used} as the basis for symmetric polynomial representation of the sums of powers in SymmetricPolynomial. Thus,{} \\spad{(5 2 2 1)} will represent \\spad{s5 * s2**2 * s1}.")) (|conjugate| (($ $) "\\spad{conjugate(p)} returns the conjugate partition of a partition \\spad{p}")) (|pdct| (((|Integer|) $) "\\spad{pdct(a1**n1 a2**n2 ...)} returns \\spad{n1! * a1**n1 * n2! * a2**n2 * ...}. This function is used in the package \\spadtype{CycleIndicators}.")) (|powers| (((|List| (|List| (|Integer|))) (|List| (|Integer|))) "\\spad{powers(li)} returns a list of 2-element lists. For each 2-element list,{} the first element is an entry of \\spad{li} and the second element is the multiplicity with which the first element occurs in \\spad{li}. There is a 2-element list for each value occurring in \\spad{l}.")) (|partition| (($ (|List| (|Integer|))) "\\spad{partition(li)} converts a list of integers \\spad{li} to a partition"))) 
 NIL 
 NIL 
-(-976 S |Coef| |Expon| |Var|) 
+(-978 S |Coef| |Expon| |Var|) 
 ((|constructor| (NIL "\\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.")) (|complete| (($ $) "\\spad{complete(f)} causes all terms of \\spad{f} to be computed. Note: this results in an infinite loop if \\spad{f} has infinitely many terms.")) (|pole?| (((|Boolean|) $) "\\spad{pole?(f)} determines if the power series \\spad{f} has a pole.")) (|variables| (((|List| |#4|) $) "\\spad{variables(f)} returns a list of the variables occuring in the power series \\spad{f}.")) (|degree| ((|#3| $) "\\spad{degree(f)} returns the exponent of the lowest order term of \\spad{f}.")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(f)} returns the coefficient of the lowest order term of \\spad{f}")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(f)} returns the monomial of \\spad{f} of lowest order.")) (|monomial| (($ $ (|List| |#4|) (|List| |#3|)) "\\spad{monomial(a,[x1,..,xk],[n1,..,nk])} computes \\spad{a * x1**n1 * .. * xk**nk}.") (($ $ |#4| |#3|) "\\spad{monomial(a,x,n)} computes \\spad{a*x**n}."))) 
 NIL 
 NIL 
-(-977 |Coef| |Expon| |Var|) 
+(-979 |Coef| |Expon| |Var|) 
 ((|constructor| (NIL "\\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.")) (|complete| (($ $) "\\spad{complete(f)} causes all terms of \\spad{f} to be computed. Note: this results in an infinite loop if \\spad{f} has infinitely many terms.")) (|pole?| (((|Boolean|) $) "\\spad{pole?(f)} determines if the power series \\spad{f} has a pole.")) (|variables| (((|List| |#3|) $) "\\spad{variables(f)} returns a list of the variables occuring in the power series \\spad{f}.")) (|degree| ((|#2| $) "\\spad{degree(f)} returns the exponent of the lowest order term of \\spad{f}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(f)} returns the coefficient of the lowest order term of \\spad{f}")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(f)} returns the monomial of \\spad{f} of lowest order.")) (|monomial| (($ $ (|List| |#3|) (|List| |#2|)) "\\spad{monomial(a,[x1,..,xk],[n1,..,nk])} computes \\spad{a * x1**n1 * .. * xk**nk}.") (($ $ |#3| |#2|) "\\spad{monomial(a,x,n)} computes \\spad{a*x**n}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-978) 
+(-980) 
 ((|constructor| (NIL "PlottableSpaceCurveCategory is the category of curves in 3-space which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the \\spad{x-},{} \\spad{y-},{} and \\spad{z}-coordinates of the points on the curve.")) (|zRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{zRange(c)} returns the range of the \\spad{z}-coordinates of the points on the curve \\spad{c}.")) (|yRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{yRange(c)} returns the range of the \\spad{y}-coordinates of the points on the curve \\spad{c}.")) (|xRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{xRange(c)} returns the range of the \\spad{x}-coordinates of the points on the curve \\spad{c}.")) (|listBranches| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{listBranches(c)} returns a list of lists of points,{} representing the branches of the curve \\spad{c}."))) 
 NIL 
 NIL 
-(-979 S R E |VarSet| P) 
+(-981 S R E |VarSet| P) 
 ((|constructor| (NIL "A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \\spad{R} being an integral domain,{} a set of polynomials may be viewed as a representation of the ideal it generates in the polynomial ring \\spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.")) (|triangular?| (((|Boolean|) $) "\\axiom{triangular?(\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{ps}} is a triangular set,{} \\spadignore{i.e.} two distinct polynomials have distinct main variables and no constant lies in \\axiom{\\spad{ps}}.")) (|rewriteIdealWithRemainder| (((|List| |#5|) (|List| |#5|) $) "\\axiom{rewriteIdealWithRemainder(\\spad{lp},{}\\spad{cs})} returns \\axiom{\\spad{lr}} such that every polynomial in \\axiom{\\spad{lr}} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{cs}} and \\axiom{(\\spad{lp},{}\\spad{cs})} and \\axiom{(\\spad{lr},{}\\spad{cs})} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|rewriteIdealWithHeadRemainder| (((|List| |#5|) (|List| |#5|) $) "\\axiom{rewriteIdealWithHeadRemainder(\\spad{lp},{}\\spad{cs})} returns \\axiom{\\spad{lr}} such that the leading monomial of every polynomial in \\axiom{\\spad{lr}} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{cs}} and \\axiom{(\\spad{lp},{}\\spad{cs})} and \\axiom{(\\spad{lr},{}\\spad{cs})} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|remainder| (((|Record| (|:| |rnum| |#2|) (|:| |polnum| |#5|) (|:| |den| |#2|)) |#5| $) "\\axiom{remainder(a,{}\\spad{ps})} returns \\axiom{[\\spad{c},{}\\spad{b},{}\\spad{r}]} such that \\axiom{\\spad{b}} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ps}},{} \\axiom{r*a - \\spad{c*b}} lies in the ideal generated by \\axiom{\\spad{ps}}. Furthermore,{} if \\axiom{\\spad{R}} is a \\spad{gcd}-domain,{} \\axiom{\\spad{b}} is primitive.")) (|headRemainder| (((|Record| (|:| |num| |#5|) (|:| |den| |#2|)) |#5| $) "\\axiom{headRemainder(a,{}\\spad{ps})} returns \\axiom{[\\spad{b},{}\\spad{r}]} such that the leading monomial of \\axiom{\\spad{b}} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ps}} and \\axiom{r*a - \\spad{b}} lies in the ideal generated by \\axiom{\\spad{ps}}.")) (|roughUnitIdeal?| (((|Boolean|) $) "\\axiom{roughUnitIdeal?(\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{ps}} contains some non null element lying in the base ring \\axiom{\\spad{R}}.")) (|roughEqualIdeals?| (((|Boolean|) $ $) "\\axiom{roughEqualIdeals?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that \\axiom{\\spad{ps1}} and \\axiom{\\spad{ps2}} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}} without computing Groebner bases.")) (|roughSubIdeal?| (((|Boolean|) $ $) "\\axiom{roughSubIdeal?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that all polynomials in \\axiom{\\spad{ps1}} lie in the ideal generated by \\axiom{\\spad{ps2}} in \\axiom{\\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}} without computing Groebner bases.")) (|roughBase?| (((|Boolean|) $) "\\axiom{roughBase?(\\spad{ps})} returns \\spad{true} iff for every pair \\axiom{{\\spad{p},{}\\spad{q}}} of polynomials in \\axiom{\\spad{ps}} their leading monomials are relatively prime.")) (|trivialIdeal?| (((|Boolean|) $) "\\axiom{trivialIdeal?(\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{ps}} does not contain non-zero elements.")) (|sort| (((|Record| (|:| |under| $) (|:| |floor| $) (|:| |upper| $)) $ |#4|) "\\axiom{sort(\\spad{v},{}\\spad{ps})} returns \\axiom{us,{}\\spad{vs},{}\\spad{ws}} such that \\axiom{us} is \\axiom{collectUnder(\\spad{ps},{}\\spad{v})},{} \\axiom{\\spad{vs}} is \\axiom{collect(\\spad{ps},{}\\spad{v})} and \\axiom{\\spad{ws}} is \\axiom{collectUpper(\\spad{ps},{}\\spad{v})}.")) (|collectUpper| (($ $ |#4|) "\\axiom{collectUpper(\\spad{ps},{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{\\spad{ps}} with main variable greater than \\axiom{\\spad{v}}.")) (|collect| (($ $ |#4|) "\\axiom{collect(\\spad{ps},{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{\\spad{ps}} with \\axiom{\\spad{v}} as main variable.")) (|collectUnder| (($ $ |#4|) "\\axiom{collectUnder(\\spad{ps},{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{\\spad{ps}} with main variable less than \\axiom{\\spad{v}}.")) (|mainVariable?| (((|Boolean|) |#4| $) "\\axiom{mainVariable?(\\spad{v},{}\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{\\spad{ps}}.")) (|mainVariables| (((|List| |#4|) $) "\\axiom{mainVariables(\\spad{ps})} returns the decreasingly sorted list of the variables which are main variables of some polynomial in \\axiom{\\spad{ps}}.")) (|variables| (((|List| |#4|) $) "\\axiom{variables(\\spad{ps})} returns the decreasingly sorted list of the variables which are variables of some polynomial in \\axiom{\\spad{ps}}.")) (|mvar| ((|#4| $) "\\axiom{mvar(\\spad{ps})} returns the main variable of the non constant polynomial with the greatest main variable,{} if any,{} else an error is returned.")) (|retract| (($ (|List| |#5|)) "\\axiom{retract(\\spad{lp})} returns an element of the domain whose elements are the members of \\axiom{\\spad{lp}} if such an element exists,{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|List| |#5|)) "\\axiom{retractIfCan(\\spad{lp})} returns an element of the domain whose elements are the members of \\axiom{\\spad{lp}} if such an element exists,{} otherwise \\axiom{\"failed\"} is returned."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-561)))) 
-(-980 R E |VarSet| P) 
+((|HasCategory| |#2| (QUOTE (-562)))) 
+(-982 R E |VarSet| P) 
 ((|constructor| (NIL "A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \\spad{R} being an integral domain,{} a set of polynomials may be viewed as a representation of the ideal it generates in the polynomial ring \\spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.")) (|triangular?| (((|Boolean|) $) "\\axiom{triangular?(\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{ps}} is a triangular set,{} \\spadignore{i.e.} two distinct polynomials have distinct main variables and no constant lies in \\axiom{\\spad{ps}}.")) (|rewriteIdealWithRemainder| (((|List| |#4|) (|List| |#4|) $) "\\axiom{rewriteIdealWithRemainder(\\spad{lp},{}\\spad{cs})} returns \\axiom{\\spad{lr}} such that every polynomial in \\axiom{\\spad{lr}} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{cs}} and \\axiom{(\\spad{lp},{}\\spad{cs})} and \\axiom{(\\spad{lr},{}\\spad{cs})} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|rewriteIdealWithHeadRemainder| (((|List| |#4|) (|List| |#4|) $) "\\axiom{rewriteIdealWithHeadRemainder(\\spad{lp},{}\\spad{cs})} returns \\axiom{\\spad{lr}} such that the leading monomial of every polynomial in \\axiom{\\spad{lr}} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{cs}} and \\axiom{(\\spad{lp},{}\\spad{cs})} and \\axiom{(\\spad{lr},{}\\spad{cs})} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|remainder| (((|Record| (|:| |rnum| |#1|) (|:| |polnum| |#4|) (|:| |den| |#1|)) |#4| $) "\\axiom{remainder(a,{}\\spad{ps})} returns \\axiom{[\\spad{c},{}\\spad{b},{}\\spad{r}]} such that \\axiom{\\spad{b}} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ps}},{} \\axiom{r*a - \\spad{c*b}} lies in the ideal generated by \\axiom{\\spad{ps}}. Furthermore,{} if \\axiom{\\spad{R}} is a \\spad{gcd}-domain,{} \\axiom{\\spad{b}} is primitive.")) (|headRemainder| (((|Record| (|:| |num| |#4|) (|:| |den| |#1|)) |#4| $) "\\axiom{headRemainder(a,{}\\spad{ps})} returns \\axiom{[\\spad{b},{}\\spad{r}]} such that the leading monomial of \\axiom{\\spad{b}} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ps}} and \\axiom{r*a - \\spad{b}} lies in the ideal generated by \\axiom{\\spad{ps}}.")) (|roughUnitIdeal?| (((|Boolean|) $) "\\axiom{roughUnitIdeal?(\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{ps}} contains some non null element lying in the base ring \\axiom{\\spad{R}}.")) (|roughEqualIdeals?| (((|Boolean|) $ $) "\\axiom{roughEqualIdeals?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that \\axiom{\\spad{ps1}} and \\axiom{\\spad{ps2}} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}} without computing Groebner bases.")) (|roughSubIdeal?| (((|Boolean|) $ $) "\\axiom{roughSubIdeal?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that all polynomials in \\axiom{\\spad{ps1}} lie in the ideal generated by \\axiom{\\spad{ps2}} in \\axiom{\\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}} without computing Groebner bases.")) (|roughBase?| (((|Boolean|) $) "\\axiom{roughBase?(\\spad{ps})} returns \\spad{true} iff for every pair \\axiom{{\\spad{p},{}\\spad{q}}} of polynomials in \\axiom{\\spad{ps}} their leading monomials are relatively prime.")) (|trivialIdeal?| (((|Boolean|) $) "\\axiom{trivialIdeal?(\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{ps}} does not contain non-zero elements.")) (|sort| (((|Record| (|:| |under| $) (|:| |floor| $) (|:| |upper| $)) $ |#3|) "\\axiom{sort(\\spad{v},{}\\spad{ps})} returns \\axiom{us,{}\\spad{vs},{}\\spad{ws}} such that \\axiom{us} is \\axiom{collectUnder(\\spad{ps},{}\\spad{v})},{} \\axiom{\\spad{vs}} is \\axiom{collect(\\spad{ps},{}\\spad{v})} and \\axiom{\\spad{ws}} is \\axiom{collectUpper(\\spad{ps},{}\\spad{v})}.")) (|collectUpper| (($ $ |#3|) "\\axiom{collectUpper(\\spad{ps},{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{\\spad{ps}} with main variable greater than \\axiom{\\spad{v}}.")) (|collect| (($ $ |#3|) "\\axiom{collect(\\spad{ps},{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{\\spad{ps}} with \\axiom{\\spad{v}} as main variable.")) (|collectUnder| (($ $ |#3|) "\\axiom{collectUnder(\\spad{ps},{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{\\spad{ps}} with main variable less than \\axiom{\\spad{v}}.")) (|mainVariable?| (((|Boolean|) |#3| $) "\\axiom{mainVariable?(\\spad{v},{}\\spad{ps})} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{\\spad{ps}}.")) (|mainVariables| (((|List| |#3|) $) "\\axiom{mainVariables(\\spad{ps})} returns the decreasingly sorted list of the variables which are main variables of some polynomial in \\axiom{\\spad{ps}}.")) (|variables| (((|List| |#3|) $) "\\axiom{variables(\\spad{ps})} returns the decreasingly sorted list of the variables which are variables of some polynomial in \\axiom{\\spad{ps}}.")) (|mvar| ((|#3| $) "\\axiom{mvar(\\spad{ps})} returns the main variable of the non constant polynomial with the greatest main variable,{} if any,{} else an error is returned.")) (|retract| (($ (|List| |#4|)) "\\axiom{retract(\\spad{lp})} returns an element of the domain whose elements are the members of \\axiom{\\spad{lp}} if such an element exists,{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|List| |#4|)) "\\axiom{retractIfCan(\\spad{lp})} returns an element of the domain whose elements are the members of \\axiom{\\spad{lp}} if such an element exists,{} otherwise \\axiom{\"failed\"} is returned."))) 
-((-4427 . T)) 
+((-4434 . T)) 
 NIL 
-(-981 R E V P) 
+(-983 R E V P) 
 ((|constructor| (NIL "This package provides modest routines for polynomial system solving. The aim of many of the operations of this package is to remove certain factors in some polynomials in order to avoid unnecessary computations in algorithms involving splitting techniques by partial factorization.")) (|removeIrreducibleRedundantFactors| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeIrreducibleRedundantFactors(\\spad{lp},{}\\spad{lq})} returns the same as \\axiom{irreducibleFactors(concat(\\spad{lp},{}\\spad{lq}))} assuming that \\axiom{irreducibleFactors(\\spad{lp})} returns \\axiom{\\spad{lp}} up to replacing some polynomial \\axiom{\\spad{pj}} in \\axiom{\\spad{lp}} by some polynomial \\axiom{\\spad{qj}} associated to \\axiom{\\spad{pj}}.")) (|lazyIrreducibleFactors| (((|List| |#4|) (|List| |#4|)) "\\axiom{lazyIrreducibleFactors(\\spad{lp})} returns \\axiom{\\spad{lf}} such that if \\axiom{\\spad{lp} = [\\spad{p1},{}...,{}\\spad{pn}]} and \\axiom{\\spad{lf} = [\\spad{f1},{}...,{}\\spad{fm}]} then \\axiom{p1*p2*...*pn=0} means \\axiom{f1*f2*...*fm=0},{} and the \\axiom{\\spad{fi}} are irreducible over \\axiom{\\spad{R}} and are pairwise distinct. The algorithm tries to avoid factorization into irreducible factors as far as possible and makes previously use of \\spad{gcd} techniques over \\axiom{\\spad{R}}.")) (|irreducibleFactors| (((|List| |#4|) (|List| |#4|)) "\\axiom{irreducibleFactors(\\spad{lp})} returns \\axiom{\\spad{lf}} such that if \\axiom{\\spad{lp} = [\\spad{p1},{}...,{}\\spad{pn}]} and \\axiom{\\spad{lf} = [\\spad{f1},{}...,{}\\spad{fm}]} then \\axiom{p1*p2*...*pn=0} means \\axiom{f1*f2*...*fm=0},{} and the \\axiom{\\spad{fi}} are irreducible over \\axiom{\\spad{R}} and are pairwise distinct.")) (|removeRedundantFactorsInPols| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactorsInPols(\\spad{lp},{}\\spad{lf})} returns \\axiom{newlp} where \\axiom{newlp} is obtained from \\axiom{\\spad{lp}} by removing in every polynomial \\axiom{\\spad{p}} of \\axiom{\\spad{lp}} any non trivial factor of any polynomial \\axiom{\\spad{f}} in \\axiom{\\spad{lf}}. Moreover,{} squares over \\axiom{\\spad{R}} are first removed in every polynomial \\axiom{\\spad{lp}}.")) (|removeRedundantFactorsInContents| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactorsInContents(\\spad{lp},{}\\spad{lf})} returns \\axiom{newlp} where \\axiom{newlp} is obtained from \\axiom{\\spad{lp}} by removing in the content of every polynomial of \\axiom{\\spad{lp}} any non trivial factor of any polynomial \\axiom{\\spad{f}} in \\axiom{\\spad{lf}}. Moreover,{} squares over \\axiom{\\spad{R}} are first removed in the content of every polynomial of \\axiom{\\spad{lp}}.")) (|removeRoughlyRedundantFactorsInContents| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRoughlyRedundantFactorsInContents(\\spad{lp},{}\\spad{lf})} returns \\axiom{newlp}where \\axiom{newlp} is obtained from \\axiom{\\spad{lp}} by removing in the content of every polynomial of \\axiom{\\spad{lp}} any occurence of a polynomial \\axiom{\\spad{f}} in \\axiom{\\spad{lf}}. Moreover,{} squares over \\axiom{\\spad{R}} are first removed in the content of every polynomial of \\axiom{\\spad{lp}}.")) (|univariatePolynomialsGcds| (((|List| |#4|) (|List| |#4|) (|Boolean|)) "\\axiom{univariatePolynomialsGcds(\\spad{lp},{}opt)} returns the same as \\axiom{univariatePolynomialsGcds(\\spad{lp})} if \\axiom{opt} is \\axiom{\\spad{false}} and if the previous operation does not return any non null and constant polynomial,{} else return \\axiom{[1]}.") (((|List| |#4|) (|List| |#4|)) "\\axiom{univariatePolynomialsGcds(\\spad{lp})} returns \\axiom{\\spad{lg}} where \\axiom{\\spad{lg}} is a list of the gcds of every pair in \\axiom{\\spad{lp}} of univariate polynomials in the same main variable.")) (|squareFreeFactors| (((|List| |#4|) |#4|) "\\axiom{squareFreeFactors(\\spad{p})} returns the square-free factors of \\axiom{\\spad{p}} over \\axiom{\\spad{R}}")) (|rewriteIdealWithQuasiMonicGenerators| (((|List| |#4|) (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{rewriteIdealWithQuasiMonicGenerators(\\spad{lp},{}redOp?,{}redOp)} returns \\axiom{\\spad{lq}} where \\axiom{\\spad{lq}} and \\axiom{\\spad{lp}} generate the same ideal in \\axiom{\\spad{R^}(\\spad{-1}) \\spad{P}} and \\axiom{\\spad{lq}} has rank not higher than the one of \\axiom{\\spad{lp}}. Moreover,{} \\axiom{\\spad{lq}} is computed by reducing \\axiom{\\spad{lp}} \\spad{w}.\\spad{r}.\\spad{t}. some basic set of the ideal generated by the quasi-monic polynomials in \\axiom{\\spad{lp}}.")) (|rewriteSetByReducingWithParticularGenerators| (((|List| |#4|) (|List| |#4|) (|Mapping| (|Boolean|) |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{rewriteSetByReducingWithParticularGenerators(\\spad{lp},{}pred?,{}redOp?,{}redOp)} returns \\axiom{\\spad{lq}} where \\axiom{\\spad{lq}} is computed by the following algorithm. Chose a basic set \\spad{w}.\\spad{r}.\\spad{t}. the reduction-test \\axiom{redOp?} among the polynomials satisfying property \\axiom{pred?},{} if it is empty then leave,{} else reduce the other polynomials by this basic set \\spad{w}.\\spad{r}.\\spad{t}. the reduction-operation \\axiom{redOp}. Repeat while another basic set with smaller rank can be computed. See code. If \\axiom{pred?} is \\axiom{quasiMonic?} the ideal is unchanged.")) (|crushedSet| (((|List| |#4|) (|List| |#4|)) "\\axiom{crushedSet(\\spad{lp})} returns \\axiom{\\spad{lq}} such that \\axiom{\\spad{lp}} and and \\axiom{\\spad{lq}} generate the same ideal and no rough basic sets reduce (in the sense of Groebner bases) the other polynomials in \\axiom{\\spad{lq}}.")) (|roughBasicSet| (((|Union| (|Record| (|:| |bas| (|GeneralTriangularSet| |#1| |#2| |#3| |#4|)) (|:| |top| (|List| |#4|))) "failed") (|List| |#4|)) "\\axiom{roughBasicSet(\\spad{lp})} returns the smallest (with Ritt-Wu ordering) triangular set contained in \\axiom{\\spad{lp}}.")) (|interReduce| (((|List| |#4|) (|List| |#4|)) "\\axiom{interReduce(\\spad{lp})} returns \\axiom{\\spad{lq}} such that \\axiom{\\spad{lp}} and \\axiom{\\spad{lq}} generate the same ideal and no polynomial in \\axiom{\\spad{lq}} is reducuble by the others in the sense of Groebner bases. Since no assumptions are required the result may depend on the ordering the reductions are performed.")) (|removeRoughlyRedundantFactorsInPol| ((|#4| |#4| (|List| |#4|)) "\\axiom{removeRoughlyRedundantFactorsInPol(\\spad{p},{}\\spad{lf})} returns the same as removeRoughlyRedundantFactorsInPols([\\spad{p}],{}\\spad{lf},{}\\spad{true})")) (|removeRoughlyRedundantFactorsInPols| (((|List| |#4|) (|List| |#4|) (|List| |#4|) (|Boolean|)) "\\axiom{removeRoughlyRedundantFactorsInPols(\\spad{lp},{}\\spad{lf},{}opt)} returns the same as \\axiom{removeRoughlyRedundantFactorsInPols(\\spad{lp},{}\\spad{lf})} if \\axiom{opt} is \\axiom{\\spad{false}} and if the previous operation does not return any non null and constant polynomial,{} else return \\axiom{[1]}.") (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRoughlyRedundantFactorsInPols(\\spad{lp},{}\\spad{lf})} returns \\axiom{newlp}where \\axiom{newlp} is obtained from \\axiom{\\spad{lp}} by removing in every polynomial \\axiom{\\spad{p}} of \\axiom{\\spad{lp}} any occurence of a polynomial \\axiom{\\spad{f}} in \\axiom{\\spad{lf}}. This may involve a lot of exact-quotients computations.")) (|bivariatePolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{bivariatePolynomials(\\spad{lp})} returns \\axiom{\\spad{bps},{}nbps} where \\axiom{\\spad{bps}} is a list of the bivariate polynomials,{} and \\axiom{nbps} are the other ones.")) (|bivariate?| (((|Boolean|) |#4|) "\\axiom{bivariate?(\\spad{p})} returns \\spad{true} iff \\axiom{\\spad{p}} involves two and only two variables.")) (|linearPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{linearPolynomials(\\spad{lp})} returns \\axiom{\\spad{lps},{}nlps} where \\axiom{\\spad{lps}} is a list of the linear polynomials in \\spad{lp},{} and \\axiom{nlps} are the other ones.")) (|linear?| (((|Boolean|) |#4|) "\\axiom{linear?(\\spad{p})} returns \\spad{true} iff \\axiom{\\spad{p}} does not lie in the base ring \\axiom{\\spad{R}} and has main degree \\axiom{1}.")) (|univariatePolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{univariatePolynomials(\\spad{lp})} returns \\axiom{ups,{}nups} where \\axiom{ups} is a list of the univariate polynomials,{} and \\axiom{nups} are the other ones.")) (|univariate?| (((|Boolean|) |#4|) "\\axiom{univariate?(\\spad{p})} returns \\spad{true} iff \\axiom{\\spad{p}} involves one and only one variable.")) (|quasiMonicPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{quasiMonicPolynomials(\\spad{lp})} returns \\axiom{qmps,{}nqmps} where \\axiom{qmps} is a list of the quasi-monic polynomials in \\axiom{\\spad{lp}} and \\axiom{nqmps} are the other ones.")) (|selectAndPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| (|Mapping| (|Boolean|) |#4|)) (|List| |#4|)) "\\axiom{selectAndPolynomials(lpred?,{}\\spad{ps})} returns \\axiom{\\spad{gps},{}\\spad{bps}} where \\axiom{\\spad{gps}} is a list of the polynomial \\axiom{\\spad{p}} in \\axiom{\\spad{ps}} such that \\axiom{pred?(\\spad{p})} holds for every \\axiom{pred?} in \\axiom{lpred?} and \\axiom{\\spad{bps}} are the other ones.")) (|selectOrPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| (|Mapping| (|Boolean|) |#4|)) (|List| |#4|)) "\\axiom{selectOrPolynomials(lpred?,{}\\spad{ps})} returns \\axiom{\\spad{gps},{}\\spad{bps}} where \\axiom{\\spad{gps}} is a list of the polynomial \\axiom{\\spad{p}} in \\axiom{\\spad{ps}} such that \\axiom{pred?(\\spad{p})} holds for some \\axiom{pred?} in \\axiom{lpred?} and \\axiom{\\spad{bps}} are the other ones.")) (|selectPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|Mapping| (|Boolean|) |#4|) (|List| |#4|)) "\\axiom{selectPolynomials(pred?,{}\\spad{ps})} returns \\axiom{\\spad{gps},{}\\spad{bps}} where \\axiom{\\spad{gps}} is a list of the polynomial \\axiom{\\spad{p}} in \\axiom{\\spad{ps}} such that \\axiom{pred?(\\spad{p})} holds and \\axiom{\\spad{bps}} are the other ones.")) (|probablyZeroDim?| (((|Boolean|) (|List| |#4|)) "\\axiom{probablyZeroDim?(\\spad{lp})} returns \\spad{true} iff the number of polynomials in \\axiom{\\spad{lp}} is not smaller than the number of variables occurring in these polynomials.")) (|possiblyNewVariety?| (((|Boolean|) (|List| |#4|) (|List| (|List| |#4|))) "\\axiom{possiblyNewVariety?(newlp,{}\\spad{llp})} returns \\spad{true} iff for every \\axiom{\\spad{lp}} in \\axiom{\\spad{llp}} certainlySubVariety?(newlp,{}\\spad{lp}) does not hold.")) (|certainlySubVariety?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{certainlySubVariety?(newlp,{}\\spad{lp})} returns \\spad{true} iff for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}} the remainder of \\axiom{\\spad{p}} by \\axiom{newlp} using the division algorithm of Groebner techniques is zero.")) (|unprotectedRemoveRedundantFactors| (((|List| |#4|) |#4| |#4|) "\\axiom{unprotectedRemoveRedundantFactors(\\spad{p},{}\\spad{q})} returns the same as \\axiom{removeRedundantFactors(\\spad{p},{}\\spad{q})} but does assume that neither \\axiom{\\spad{p}} nor \\axiom{\\spad{q}} lie in the base ring \\axiom{\\spad{R}} and assumes that \\axiom{infRittWu?(\\spad{p},{}\\spad{q})} holds. Moreover,{} if \\axiom{\\spad{R}} is \\spad{gcd}-domain,{} then \\axiom{\\spad{p}} and \\axiom{\\spad{q}} are assumed to be square free.")) (|removeSquaresIfCan| (((|List| |#4|) (|List| |#4|)) "\\axiom{removeSquaresIfCan(\\spad{lp})} returns \\axiom{removeDuplicates [squareFreePart(\\spad{p})\\$\\spad{P} for \\spad{p} in \\spad{lp}]} if \\axiom{\\spad{R}} is \\spad{gcd}-domain else returns \\axiom{\\spad{lp}}.")) (|removeRedundantFactors| (((|List| |#4|) (|List| |#4|) (|List| |#4|) (|Mapping| (|List| |#4|) (|List| |#4|))) "\\axiom{removeRedundantFactors(\\spad{lp},{}\\spad{lq},{}remOp)} returns the same as \\axiom{concat(remOp(removeRoughlyRedundantFactorsInPols(\\spad{lp},{}\\spad{lq})),{}\\spad{lq})} assuming that \\axiom{remOp(\\spad{lq})} returns \\axiom{\\spad{lq}} up to similarity.") (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactors(\\spad{lp},{}\\spad{lq})} returns the same as \\axiom{removeRedundantFactors(concat(\\spad{lp},{}\\spad{lq}))} assuming that \\axiom{removeRedundantFactors(\\spad{lp})} returns \\axiom{\\spad{lp}} up to replacing some polynomial \\axiom{\\spad{pj}} in \\axiom{\\spad{lp}} by some polynomial \\axiom{\\spad{qj}} associated to \\axiom{\\spad{pj}}.") (((|List| |#4|) (|List| |#4|) |#4|) "\\axiom{removeRedundantFactors(\\spad{lp},{}\\spad{q})} returns the same as \\axiom{removeRedundantFactors(cons(\\spad{q},{}\\spad{lp}))} assuming that \\axiom{removeRedundantFactors(\\spad{lp})} returns \\axiom{\\spad{lp}} up to replacing some polynomial \\axiom{\\spad{pj}} in \\axiom{\\spad{lp}} by some some polynomial \\axiom{\\spad{qj}} associated to \\axiom{\\spad{pj}}.") (((|List| |#4|) |#4| |#4|) "\\axiom{removeRedundantFactors(\\spad{p},{}\\spad{q})} returns the same as \\axiom{removeRedundantFactors([\\spad{p},{}\\spad{q}])}") (((|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactors(\\spad{lp})} returns \\axiom{\\spad{lq}} such that if \\axiom{\\spad{lp} = [\\spad{p1},{}...,{}\\spad{pn}]} and \\axiom{\\spad{lq} = [\\spad{q1},{}...,{}\\spad{qm}]} then the product \\axiom{p1*p2*...\\spad{*pn}} vanishes iff the product \\axiom{q1*q2*...\\spad{*qm}} vanishes,{} and the product of degrees of the \\axiom{\\spad{qi}} is not greater than the one of the \\axiom{\\spad{pj}},{} and no polynomial in \\axiom{\\spad{lq}} divides another polynomial in \\axiom{\\spad{lq}}. In particular,{} polynomials lying in the base ring \\axiom{\\spad{R}} are removed. Moreover,{} \\axiom{\\spad{lq}} is sorted \\spad{w}.\\spad{r}.\\spad{t} \\axiom{infRittWu?}. Furthermore,{} if \\spad{R} is \\spad{gcd}-domain,{} the polynomials in \\axiom{\\spad{lq}} are pairwise without common non trivial factor."))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-309)))) (|HasCategory| |#1| (QUOTE (-456)))) 
-(-982 K) 
+((-12 (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-310)))) (|HasCategory| |#1| (QUOTE (-457)))) 
+(-984 K) 
 ((|constructor| (NIL "PseudoLinearNormalForm provides a function for computing a block-companion form for pseudo-linear operators.")) (|companionBlocks| (((|List| (|Record| (|:| C (|Matrix| |#1|)) (|:| |g| (|Vector| |#1|)))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{companionBlocks(m, v)} returns \\spad{[[C_1, g_1],...,[C_k, g_k]]} such that each \\spad{C_i} is a companion block and \\spad{m = diagonal(C_1,...,C_k)}.")) (|changeBase| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{changeBase(M, A, sig, der)}: computes the new matrix of a pseudo-linear transform given by the matrix \\spad{M} under the change of base A")) (|normalForm| (((|Record| (|:| R (|Matrix| |#1|)) (|:| A (|Matrix| |#1|)) (|:| |Ainv| (|Matrix| |#1|))) (|Matrix| |#1|) (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{normalForm(M, sig, der)} returns \\spad{[R, A, A^{-1}]} such that the pseudo-linear operator whose matrix in the basis \\spad{y} is \\spad{M} had matrix \\spad{R} in the basis \\spad{z = A y}. \\spad{der} is a \\spad{sig}-derivation."))) 
 NIL 
 NIL 
-(-983 |VarSet| E RC P) 
+(-985 |VarSet| E RC P) 
 ((|constructor| (NIL "This package computes square-free decomposition of multivariate polynomials over a coefficient ring which is an arbitrary \\spad{gcd} domain. The requirement on the coefficient domain guarantees that the \\spadfun{content} can be removed so that factors will be primitive as well as square-free. Over an infinite ring of finite characteristic,{}it may not be possible to guarantee that the factors are square-free.")) (|squareFree| (((|Factored| |#4|) |#4|) "\\spad{squareFree(p)} returns the square-free factorization of the polynomial \\spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime."))) 
 NIL 
 NIL 
-(-984 R) 
+(-986 R) 
 ((|constructor| (NIL "PointCategory is the category of points in space which may be plotted via the graphics facilities. Functions are provided for defining points and handling elements of points.")) (|extend| (($ $ (|List| |#1|)) "\\spad{extend(x,l,r)} \\undocumented")) (|cross| (($ $ $) "\\spad{cross(p,q)} computes the cross product of the two points \\spad{p} and \\spad{q}. Error if the \\spad{p} and \\spad{q} are not 3 dimensional")) (|dimension| (((|PositiveInteger|) $) "\\spad{dimension(s)} returns the dimension of the point category \\spad{s}.")) (|point| (($ (|List| |#1|)) "\\spad{point(l)} returns a point category defined by a list \\spad{l} of elements from the domain \\spad{R}."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-985 R1 R2) 
+(-987 R1 R2) 
 ((|constructor| (NIL "This package \\undocumented")) (|map| (((|Point| |#2|) (|Mapping| |#2| |#1|) (|Point| |#1|)) "\\spad{map(f,p)} \\undocumented"))) 
 NIL 
 NIL 
-(-986 R) 
+(-988 R) 
 ((|constructor| (NIL "This package \\undocumented")) (|shade| ((|#1| (|Point| |#1|)) "\\spad{shade(pt)} returns the fourth element of the two dimensional point,{} \\spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} shade to express a fourth dimension.")) (|hue| ((|#1| (|Point| |#1|)) "\\spad{hue(pt)} returns the third element of the two dimensional point,{} \\spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} hue to express a third dimension.")) (|color| ((|#1| (|Point| |#1|)) "\\spad{color(pt)} returns the fourth element of the point,{} \\spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} color to express a fourth dimension.")) (|phiCoord| ((|#1| (|Point| |#1|)) "\\spad{phiCoord(pt)} returns the third element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical coordinate system.")) (|thetaCoord| ((|#1| (|Point| |#1|)) "\\spad{thetaCoord(pt)} returns the second element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical or a cylindrical coordinate system.")) (|rCoord| ((|#1| (|Point| |#1|)) "\\spad{rCoord(pt)} returns the first element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical or a cylindrical coordinate system.")) (|zCoord| ((|#1| (|Point| |#1|)) "\\spad{zCoord(pt)} returns the third element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian or a cylindrical coordinate system.")) (|yCoord| ((|#1| (|Point| |#1|)) "\\spad{yCoord(pt)} returns the second element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian coordinate system.")) (|xCoord| ((|#1| (|Point| |#1|)) "\\spad{xCoord(pt)} returns the first element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian coordinate system."))) 
 NIL 
 NIL 
-(-987 K) 
+(-989 K) 
 ((|constructor| (NIL "This is the description of any package which provides partial functions on a domain belonging to TranscendentalFunctionCategory.")) (|acschIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acschIfCan(z)} returns acsch(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asechIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asechIfCan(z)} returns asech(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acothIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acothIfCan(z)} returns acoth(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|atanhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{atanhIfCan(z)} returns atanh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acoshIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acoshIfCan(z)} returns acosh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asinhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asinhIfCan(z)} returns asinh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cschIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cschIfCan(z)} returns csch(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|sechIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{sechIfCan(z)} returns sech(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cothIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cothIfCan(z)} returns coth(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|tanhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{tanhIfCan(z)} returns tanh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|coshIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{coshIfCan(z)} returns cosh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|sinhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{sinhIfCan(z)} returns sinh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acscIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acscIfCan(z)} returns acsc(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asecIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asecIfCan(z)} returns asec(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acotIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acotIfCan(z)} returns acot(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|atanIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{atanIfCan(z)} returns atan(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acosIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acosIfCan(z)} returns acos(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asinIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asinIfCan(z)} returns asin(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cscIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cscIfCan(z)} returns \\spad{csc}(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|secIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{secIfCan(z)} returns sec(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cotIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cotIfCan(z)} returns cot(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|tanIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{tanIfCan(z)} returns tan(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cosIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cosIfCan(z)} returns cos(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|sinIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{sinIfCan(z)} returns sin(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|logIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{logIfCan(z)} returns log(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|expIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{expIfCan(z)} returns exp(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|nthRootIfCan| (((|Union| |#1| "failed") |#1| (|NonNegativeInteger|)) "\\spad{nthRootIfCan(z,n)} returns the \\spad{n}th root of \\spad{z} if possible,{} and \"failed\" otherwise."))) 
 NIL 
 NIL 
-(-988 R E OV PPR) 
+(-990 R E OV PPR) 
 ((|constructor| (NIL "This package \\undocumented{}")) (|map| ((|#4| (|Mapping| |#4| (|Polynomial| |#1|)) |#4|) "\\spad{map(f,p)} \\undocumented{}")) (|pushup| ((|#4| |#4| (|List| |#3|)) "\\spad{pushup(p,lv)} \\undocumented{}") ((|#4| |#4| |#3|) "\\spad{pushup(p,v)} \\undocumented{}")) (|pushdown| ((|#4| |#4| (|List| |#3|)) "\\spad{pushdown(p,lv)} \\undocumented{}") ((|#4| |#4| |#3|) "\\spad{pushdown(p,v)} \\undocumented{}")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} makes an element from symbol \\spad{s} or fails")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) 
 NIL 
 NIL 
-(-989 K R UP -3498) 
+(-991 K R UP -3505) 
 ((|constructor| (NIL "In this package \\spad{K} is a finite field,{} \\spad{R} is a ring of univariate polynomials over \\spad{K},{} and \\spad{F} is a monogenic algebra over \\spad{R}. We require that \\spad{F} is monogenic,{} \\spadignore{i.e.} that \\spad{F = K[x,y]/(f(x,y))},{} because the integral basis algorithm used will factor the polynomial \\spad{f(x,y)}. The package provides a function to compute the integral closure of \\spad{R} in the quotient field of \\spad{F} as well as a function to compute a \"local integral basis\" at a specific prime.")) (|reducedDiscriminant| ((|#2| |#3|) "\\spad{reducedDiscriminant(up)} \\undocumented")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) |#2|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv] } containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of the framed algebra \\spad{F}. \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If 'basis' is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix 'basisInv' contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if 'basisInv' is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv] } containing information regarding the integral closure of \\spad{R} in the quotient field of the framed algebra \\spad{F}. \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If 'basis' is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix 'basisInv' contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if 'basisInv' is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}."))) 
 NIL 
 NIL 
-(-990 R |Var| |Expon| |Dpoly|) 
+(-992 R |Var| |Expon| |Dpoly|) 
 ((|constructor| (NIL "\\spadtype{QuasiAlgebraicSet} constructs a domain representing quasi-algebraic sets,{} which is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the common zeros of a polynomial \\spad{f} (the defining polynomial for the inequation). This domain provides simplification of a user-given representation using groebner basis computations. There are two simplification routines: the first function \\spadfun{idealSimplify} uses groebner basis of ideals alone,{} while the second,{} \\spadfun{simplify} uses both groebner basis and factorization. The resulting defining equations \\spad{L} always form a groebner basis,{} and the resulting defining inequation \\spad{f} is always reduced. The function \\spadfun{simplify} may be applied several times if desired. A third simplification routine \\spadfun{radicalSimplify} is provided in \\spadtype{QuasiAlgebraicSet2} for comparison study only,{} as it is inefficient compared to the other two,{} as well as is restricted to only certain coefficient domains. For detail analysis and a comparison of the three methods,{} please consult the reference cited. \\blankline A polynomial function \\spad{q} defined on the quasi-algebraic set is equivalent to its reduced form with respect to \\spad{L}. While this may be obtained using the usual normal form algorithm,{} there is no canonical form for \\spad{q}. \\blankline The ordering in groebner basis computation is determined by the data type of the input polynomials. If it is possible we suggest to use refinements of total degree orderings.")) (|simplify| (($ $) "\\spad{simplify(s)} returns a different and presumably simpler representation of \\spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using a heuristic algorithm based on factoring.")) (|idealSimplify| (($ $) "\\spad{idealSimplify(s)} returns a different and presumably simpler representation of \\spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using Buchberger\\spad{'s} algorithm.")) (|definingInequation| ((|#4| $) "\\spad{definingInequation(s)} returns a single defining polynomial for the inequation,{} that is,{} the Zariski open part of \\spad{s}.")) (|definingEquations| (((|List| |#4|) $) "\\spad{definingEquations(s)} returns a list of defining polynomials for equations,{} that is,{} for the Zariski closed part of \\spad{s}.")) (|empty?| (((|Boolean|) $) "\\spad{empty?(s)} returns \\spad{true} if the quasialgebraic set \\spad{s} has no points,{} and \\spad{false} otherwise.")) (|setStatus| (($ $ (|Union| (|Boolean|) #1="failed")) "\\spad{setStatus(s,t)} returns the same representation for \\spad{s},{} but asserts the following: if \\spad{t} is \\spad{true},{} then \\spad{s} is empty,{} if \\spad{t} is \\spad{false},{} then \\spad{s} is non-empty,{} and if \\spad{t} = \"failed\",{} then no assertion is made (that is,{} \"don\\spad{'t} know\"). Note: for internal use only,{} with care.")) (|status| (((|Union| (|Boolean|) #1#) $) "\\spad{status(s)} returns \\spad{true} if the quasi-algebraic set is empty,{} \\spad{false} if it is not,{} and \"failed\" if not yet known")) (|quasiAlgebraicSet| (($ (|List| |#4|) |#4|) "\\spad{quasiAlgebraicSet(pl,q)} returns the quasi-algebraic set with defining equations \\spad{p} = 0 for \\spad{p} belonging to the list \\spad{pl},{} and defining inequation \\spad{q} \\spad{~=} 0.")) (|empty| (($) "\\spad{empty()} returns the empty quasi-algebraic set"))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-309))))) 
-(-991 |vl| |nv|) 
+((-12 (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-310))))) 
+(-993 |vl| |nv|) 
 ((|constructor| (NIL "\\spadtype{QuasiAlgebraicSet2} adds a function \\spadfun{radicalSimplify} which uses \\spadtype{IdealDecompositionPackage} to simplify the representation of a quasi-algebraic set. A quasi-algebraic set is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the common zeros of a polynomial \\spad{f} (the defining polynomial for the inequation). Quasi-algebraic sets are implemented in the domain \\spadtype{QuasiAlgebraicSet},{} where two simplification routines are provided: \\spadfun{idealSimplify} and \\spadfun{simplify}. The function \\spadfun{radicalSimplify} is added for comparison study only. Because the domain \\spadtype{IdealDecompositionPackage} provides facilities for computing with radical ideals,{} it is necessary to restrict the ground ring to the domain \\spadtype{Fraction Integer},{} and the polynomial ring to be of type \\spadtype{DistributedMultivariatePolynomial}. The routine \\spadfun{radicalSimplify} uses these to compute groebner basis of radical ideals and is inefficient and restricted when compared to the two in \\spadtype{QuasiAlgebraicSet}.")) (|radicalSimplify| (((|QuasiAlgebraicSet| (|Fraction| (|Integer|)) (|OrderedVariableList| |#1|) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|QuasiAlgebraicSet| (|Fraction| (|Integer|)) (|OrderedVariableList| |#1|) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{radicalSimplify(s)} returns a different and presumably simpler representation of \\spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using using groebner basis of radical ideals"))) 
 NIL 
 NIL 
-(-992 R E V P TS) 
+(-994 R E V P TS) 
 ((|constructor| (NIL "A package for removing redundant quasi-components and redundant branches when decomposing a variety by means of quasi-components of regular triangular sets. \\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|branchIfCan| (((|Union| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|))) "failed") (|List| |#4|) |#5| (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{branchIfCan(leq,{}\\spad{ts},{}lineq,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement.")) (|prepareDecompose| (((|List| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|)))) (|List| |#4|) (|List| |#5|) (|Boolean|) (|Boolean|)) "\\axiom{prepareDecompose(\\spad{lp},{}\\spad{lts},{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousCases| (((|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) (|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)))) "\\axiom{removeSuperfluousCases(llpwt)} is an internal subroutine,{} exported only for developement.")) (|subCase?| (((|Boolean|) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) "\\axiom{subCase?(lpwt1,{}lpwt2)} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousQuasiComponents| (((|List| |#5|) (|List| |#5|)) "\\axiom{removeSuperfluousQuasiComponents(\\spad{lts})} removes from \\axiom{\\spad{lts}} any \\spad{ts} such that \\axiom{subQuasiComponent?(\\spad{ts},{}us)} holds for another \\spad{us} in \\axiom{\\spad{lts}}.")) (|subQuasiComponent?| (((|Boolean|) |#5| (|List| |#5|)) "\\axiom{subQuasiComponent?(\\spad{ts},{}lus)} returns \\spad{true} iff \\axiom{subQuasiComponent?(\\spad{ts},{}us)} holds for one \\spad{us} in \\spad{lus}.") (((|Boolean|) |#5| |#5|) "\\axiom{subQuasiComponent?(\\spad{ts},{}us)} returns \\spad{true} iff \\axiomOpFrom{internalSubQuasiComponent?}{QuasiComponentPackage} returs \\spad{true}.")) (|internalSubQuasiComponent?| (((|Union| (|Boolean|) "failed") |#5| |#5|) "\\axiom{internalSubQuasiComponent?(\\spad{ts},{}us)} returns a boolean \\spad{b} value if the fact that the regular zero set of \\axiom{us} contains that of \\axiom{\\spad{ts}} can be decided (and in that case \\axiom{\\spad{b}} gives this inclusion) otherwise returns \\axiom{\"failed\"}.")) (|infRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{infRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalInfRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalInfRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalSubPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalSubPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}} assuming that these lists are sorted increasingly \\spad{w}.\\spad{r}.\\spad{t}. \\axiomOpFrom{infRittWu?}{RecursivePolynomialCategory}.")) (|subPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{subPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}}.")) (|subTriSet?| (((|Boolean|) |#5| |#5|) "\\axiom{subTriSet?(\\spad{ts},{}us)} returns \\spad{true} iff \\axiom{\\spad{ts}} is a sub-set of \\axiom{us}.")) (|moreAlgebraic?| (((|Boolean|) |#5| |#5|) "\\axiom{moreAlgebraic?(\\spad{ts},{}us)} returns \\spad{false} iff \\axiom{\\spad{ts}} and \\axiom{us} are both empty,{} or \\axiom{\\spad{ts}} has less elements than \\axiom{us},{} or some variable is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{us} and is not \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ts}}.")) (|algebraicSort| (((|List| |#5|) (|List| |#5|)) "\\axiom{algebraicSort(\\spad{lts})} sorts \\axiom{\\spad{lts}} \\spad{w}.\\spad{r}.\\spad{t} \\axiomOpFrom{supDimElseRittWu?}{QuasiComponentPackage}.")) (|supDimElseRittWu?| (((|Boolean|) |#5| |#5|) "\\axiom{supDimElseRittWu(\\spad{ts},{}us)} returns \\spad{true} iff \\axiom{\\spad{ts}} has less elements than \\axiom{us} otherwise if \\axiom{\\spad{ts}} has higher rank than \\axiom{us} \\spad{w}.\\spad{r}.\\spad{t}. Riit and Wu ordering.")) (|stopTable!| (((|Void|)) "\\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.")) (|startTable!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableGcd!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement."))) 
 NIL 
 NIL 
-(-993) 
+(-995) 
 ((|constructor| (NIL "This domain implements simple database queries")) (|value| (((|String|) $) "\\spad{value(q)} returns the value (\\spadignore{i.e.} right hand side) of \\axiom{\\spad{q}}.")) (|variable| (((|Symbol|) $) "\\spad{variable(q)} returns the variable (\\spadignore{i.e.} left hand side) of \\axiom{\\spad{q}}.")) (|equation| (($ (|Symbol|) (|String|)) "\\spad{equation(s,\"a\")} creates a new equation."))) 
 NIL 
 NIL 
-(-994 A S) 
+(-996 A S) 
 ((|constructor| (NIL "QuotientField(\\spad{S}) is the category of fractions of an Integral Domain \\spad{S}.")) (|floor| ((|#2| $) "\\spad{floor(x)} returns the largest integral element below \\spad{x}.")) (|ceiling| ((|#2| $) "\\spad{ceiling(x)} returns the smallest integral element above \\spad{x}.")) (|random| (($) "\\spad{random()} returns a random fraction.")) (|fractionPart| (($ $) "\\spad{fractionPart(x)} returns the fractional part of \\spad{x}. \\spad{x} = wholePart(\\spad{x}) + fractionPart(\\spad{x})")) (|wholePart| ((|#2| $) "\\spad{wholePart(x)} returns the whole part of the fraction \\spad{x} \\spadignore{i.e.} the truncated quotient of the numerator by the denominator.")) (|denominator| (($ $) "\\spad{denominator(x)} is the denominator of the fraction \\spad{x} converted to \\%.")) (|numerator| (($ $) "\\spad{numerator(x)} is the numerator of the fraction \\spad{x} converted to \\%.")) (|denom| ((|#2| $) "\\spad{denom(x)} returns the denominator of the fraction \\spad{x}.")) (|numer| ((|#2| $) "\\spad{numer(x)} returns the numerator of the fraction \\spad{x}.")) (/ (($ |#2| |#2|) "\\spad{d1 / d2} returns the fraction \\spad{d1} divided by \\spad{d2}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| |#2| (QUOTE (-549))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (QUOTE (-1024))) (|HasCategory| |#2| (QUOTE (-823))) (|HasCategory| |#2| (QUOTE (-853))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-1155)))) 
-(-995 S) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-310))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (QUOTE (-1026))) (|HasCategory| |#2| (QUOTE (-825))) (|HasCategory| |#2| (QUOTE (-855))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-1157)))) 
+(-997 S) 
 ((|constructor| (NIL "QuotientField(\\spad{S}) is the category of fractions of an Integral Domain \\spad{S}.")) (|floor| ((|#1| $) "\\spad{floor(x)} returns the largest integral element below \\spad{x}.")) (|ceiling| ((|#1| $) "\\spad{ceiling(x)} returns the smallest integral element above \\spad{x}.")) (|random| (($) "\\spad{random()} returns a random fraction.")) (|fractionPart| (($ $) "\\spad{fractionPart(x)} returns the fractional part of \\spad{x}. \\spad{x} = wholePart(\\spad{x}) + fractionPart(\\spad{x})")) (|wholePart| ((|#1| $) "\\spad{wholePart(x)} returns the whole part of the fraction \\spad{x} \\spadignore{i.e.} the truncated quotient of the numerator by the denominator.")) (|denominator| (($ $) "\\spad{denominator(x)} is the denominator of the fraction \\spad{x} converted to \\%.")) (|numerator| (($ $) "\\spad{numerator(x)} is the numerator of the fraction \\spad{x} converted to \\%.")) (|denom| ((|#1| $) "\\spad{denom(x)} returns the denominator of the fraction \\spad{x}.")) (|numer| ((|#1| $) "\\spad{numer(x)} returns the numerator of the fraction \\spad{x}.")) (/ (($ |#1| |#1|) "\\spad{d1 / d2} returns the fraction \\spad{d1} divided by \\spad{d2}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-996 A B R S) 
+(-998 A B R S) 
 ((|constructor| (NIL "This package extends a function between integral domains to a mapping between their quotient fields.")) (|map| ((|#4| (|Mapping| |#2| |#1|) |#3|) "\\spad{map(func,frac)} applies the function \\spad{func} to the numerator and denominator of \\spad{frac}."))) 
 NIL 
 NIL 
-(-997 |n| K) 
+(-999 |n| K) 
 ((|constructor| (NIL "This domain provides modest support for quadratic forms.")) (|elt| ((|#2| $ (|DirectProduct| |#1| |#2|)) "\\spad{elt(qf,v)} evaluates the quadratic form \\spad{qf} on the vector \\spad{v},{} producing a scalar.")) (|matrix| (((|SquareMatrix| |#1| |#2|) $) "\\spad{matrix(qf)} creates a square matrix from the quadratic form \\spad{qf}.")) (|quadraticForm| (($ (|SquareMatrix| |#1| |#2|)) "\\spad{quadraticForm(m)} creates a quadratic form from a symmetric,{} square matrix \\spad{m}."))) 
 NIL 
 NIL 
-(-998) 
+(-1000) 
 ((|constructor| (NIL "This domain represents the syntax of a quasiquote \\indented{2}{expression.}")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the syntax for the expression being quoted."))) 
 NIL 
 NIL 
-(-999 S) 
+(-1001 S) 
 ((|constructor| (NIL "A queue is a bag where the first item inserted is the first item extracted.")) (|back| ((|#1| $) "\\spad{back(q)} returns the element at the back of the queue. The queue \\spad{q} is unchanged by this operation. Error: if \\spad{q} is empty.")) (|front| ((|#1| $) "\\spad{front(q)} returns the element at the front of the queue. The queue \\spad{q} is unchanged by this operation. Error: if \\spad{q} is empty.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length(q)} returns the number of elements in the queue. Note: \\axiom{length(\\spad{q}) = \\spad{#q}}.")) (|rotate!| (($ $) "\\spad{rotate! q} rotates queue \\spad{q} so that the element at the front of the queue goes to the back of the queue. Note: rotate! \\spad{q} is equivalent to enqueue!(dequeue!(\\spad{q})).")) (|dequeue!| ((|#1| $) "\\spad{dequeue! s} destructively extracts the first (top) element from queue \\spad{q}. The element previously second in the queue becomes the first element. Error: if \\spad{q} is empty.")) (|enqueue!| ((|#1| |#1| $) "\\spad{enqueue!(x,q)} inserts \\spad{x} into the queue \\spad{q} at the back end."))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
-(-1000 R) 
+(-1002 R) 
 ((|constructor| (NIL "\\spadtype{Quaternion} implements quaternions over a \\indented{2}{commutative ring. The main constructor function is \\spadfun{quatern}} \\indented{2}{which takes 4 arguments: the real part,{} the \\spad{i} imaginary part,{} the \\spad{j}} \\indented{2}{imaginary part and the \\spad{k} imaginary part.}"))) 
-((-4420 |has| |#1| (-292)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (|HasCategory| |#1| (QUOTE (-292))) (|HasCategory| |#1| (QUOTE (-366)))) (|HasCategory| |#1| (QUOTE (-292))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#1| (LIST (QUOTE -518) (QUOTE (-1181)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -288) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-1064))) (|HasCategory| |#1| (QUOTE (-549)))) 
-(-1001 S R) 
+((-4427 |has| |#1| (-293)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (QUOTE (-367))) (-3969 (|HasCategory| |#1| (QUOTE (-293))) (|HasCategory| |#1| (QUOTE (-367)))) (|HasCategory| |#1| (QUOTE (-293))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (LIST (QUOTE -519) (QUOTE (-1183)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -289) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-1066))) (|HasCategory| |#1| (QUOTE (-550)))) 
+(-1003 S R) 
 ((|constructor| (NIL "\\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(q)} returns \\spad{q} as a rational number,{} or \"failed\" if this is not possible. Note: if \\spad{rational?(q)} is \\spad{true},{} the conversion can be done and the rational number will be returned.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(q)} tries to convert \\spad{q} into a rational number. Error: if this is not possible. If \\spad{rational?(q)} is \\spad{true},{} the conversion will be done and the rational number returned.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(q)} returns {\\it \\spad{true}} if all the imaginary parts of \\spad{q} are zero and the real part can be converted into a rational number,{} and {\\it \\spad{false}} otherwise.")) (|abs| ((|#2| $) "\\spad{abs(q)} computes the absolute value of quaternion \\spad{q} (sqrt of norm).")) (|real| ((|#2| $) "\\spad{real(q)} extracts the real part of quaternion \\spad{q}.")) (|quatern| (($ |#2| |#2| |#2| |#2|) "\\spad{quatern(r,i,j,k)} constructs a quaternion from scalars.")) (|norm| ((|#2| $) "\\spad{norm(q)} computes the norm of \\spad{q} (the sum of the squares of the components).")) (|imagK| ((|#2| $) "\\spad{imagK(q)} extracts the imaginary \\spad{k} part of quaternion \\spad{q}.")) (|imagJ| ((|#2| $) "\\spad{imagJ(q)} extracts the imaginary \\spad{j} part of quaternion \\spad{q}.")) (|imagI| ((|#2| $) "\\spad{imagI(q)} extracts the imaginary \\spad{i} part of quaternion \\spad{q}.")) (|conjugate| (($ $) "\\spad{conjugate(q)} negates the imaginary parts of quaternion \\spad{q}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-549))) (|HasCategory| |#2| (QUOTE (-1064))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-292)))) 
-(-1002 R) 
+((|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (QUOTE (-1066))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-293)))) 
+(-1004 R) 
 ((|constructor| (NIL "\\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(q)} returns \\spad{q} as a rational number,{} or \"failed\" if this is not possible. Note: if \\spad{rational?(q)} is \\spad{true},{} the conversion can be done and the rational number will be returned.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(q)} tries to convert \\spad{q} into a rational number. Error: if this is not possible. If \\spad{rational?(q)} is \\spad{true},{} the conversion will be done and the rational number returned.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(q)} returns {\\it \\spad{true}} if all the imaginary parts of \\spad{q} are zero and the real part can be converted into a rational number,{} and {\\it \\spad{false}} otherwise.")) (|abs| ((|#1| $) "\\spad{abs(q)} computes the absolute value of quaternion \\spad{q} (sqrt of norm).")) (|real| ((|#1| $) "\\spad{real(q)} extracts the real part of quaternion \\spad{q}.")) (|quatern| (($ |#1| |#1| |#1| |#1|) "\\spad{quatern(r,i,j,k)} constructs a quaternion from scalars.")) (|norm| ((|#1| $) "\\spad{norm(q)} computes the norm of \\spad{q} (the sum of the squares of the components).")) (|imagK| ((|#1| $) "\\spad{imagK(q)} extracts the imaginary \\spad{k} part of quaternion \\spad{q}.")) (|imagJ| ((|#1| $) "\\spad{imagJ(q)} extracts the imaginary \\spad{j} part of quaternion \\spad{q}.")) (|imagI| ((|#1| $) "\\spad{imagI(q)} extracts the imaginary \\spad{i} part of quaternion \\spad{q}.")) (|conjugate| (($ $) "\\spad{conjugate(q)} negates the imaginary parts of quaternion \\spad{q}."))) 
-((-4420 |has| |#1| (-292)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 |has| |#1| (-293)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1003 QR R QS S) 
+(-1005 QR R QS S) 
 ((|constructor| (NIL "\\spadtype{QuaternionCategoryFunctions2} implements functions between two quaternion domains. The function \\spadfun{map} is used by the system interpreter to coerce between quaternion types.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the component parts of the quaternion \\spad{u}."))) 
 NIL 
 NIL 
-(-1004 S) 
+(-1006 S) 
 ((|constructor| (NIL "Linked List implementation of a Queue")) (|queue| (($ (|List| |#1|)) "\\spad{queue([x,y,...,z])} creates a queue with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom) element \\spad{z}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1005 S) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1007 S) 
 ((|constructor| (NIL "The \\spad{RadicalCategory} is a model for the rational numbers.")) (** (($ $ (|Fraction| (|Integer|))) "\\spad{x ** y} is the rational exponentiation of \\spad{x} by the power \\spad{y}.")) (|nthRoot| (($ $ (|Integer|)) "\\spad{nthRoot(x,n)} returns the \\spad{n}th root of \\spad{x}.")) (|sqrt| (($ $) "\\spad{sqrt(x)} returns the square root of \\spad{x}."))) 
 NIL 
 NIL 
-(-1006) 
+(-1008) 
 ((|constructor| (NIL "The \\spad{RadicalCategory} is a model for the rational numbers.")) (** (($ $ (|Fraction| (|Integer|))) "\\spad{x ** y} is the rational exponentiation of \\spad{x} by the power \\spad{y}.")) (|nthRoot| (($ $ (|Integer|)) "\\spad{nthRoot(x,n)} returns the \\spad{n}th root of \\spad{x}.")) (|sqrt| (($ $) "\\spad{sqrt(x)} returns the square root of \\spad{x}."))) 
 NIL 
 NIL 
-(-1007 -3498 UP UPUP |radicnd| |n|) 
+(-1009 -3505 UP UPUP |radicnd| |n|) 
 ((|constructor| (NIL "Function field defined by y**n = \\spad{f}(\\spad{x})."))) 
-((-4420 |has| (-411 |#2|) (-366)) (-4425 |has| (-411 |#2|) (-366)) (-4419 |has| (-411 |#2|) (-366)) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-411 |#2|) (QUOTE (-145))) (|HasCategory| (-411 |#2|) (QUOTE (-147))) (|HasCategory| (-411 |#2|) (QUOTE (-353))) (-3962 (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (QUOTE (-353)))) (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (QUOTE (-371))) (-3962 (-12 (|HasCategory| (-411 |#2|) (QUOTE (-234))) (|HasCategory| (-411 |#2|) (QUOTE (-366)))) (|HasCategory| (-411 |#2|) (QUOTE (-353)))) (-3962 (-12 (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| (-411 |#2|) (QUOTE (-353))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -904) (QUOTE (-1181)))))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -642) (QUOTE (-550)))) (-3962 (|HasCategory| (-411 |#2|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-411 |#2|) (QUOTE (-366)))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-371))) (-12 (|HasCategory| (-411 |#2|) (QUOTE (-366))) (|HasCategory| (-411 |#2|) (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| (-411 |#2|) (QUOTE (-234))) (|HasCategory| (-411 |#2|) (QUOTE (-366))))) 
-(-1008 |bb|) 
+((-4427 |has| (-412 |#2|) (-367)) (-4432 |has| (-412 |#2|) (-367)) (-4426 |has| (-412 |#2|) (-367)) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-412 |#2|) (QUOTE (-145))) (|HasCategory| (-412 |#2|) (QUOTE (-147))) (|HasCategory| (-412 |#2|) (QUOTE (-354))) (-3969 (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (QUOTE (-354)))) (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (QUOTE (-372))) (-3969 (-12 (|HasCategory| (-412 |#2|) (QUOTE (-234))) (|HasCategory| (-412 |#2|) (QUOTE (-367)))) (|HasCategory| (-412 |#2|) (QUOTE (-354)))) (-3969 (-12 (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -906) (QUOTE (-1183))))) (-12 (|HasCategory| (-412 |#2|) (QUOTE (-354))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -906) (QUOTE (-1183)))))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -644) (QUOTE (-551)))) (-3969 (|HasCategory| (-412 |#2|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-412 |#2|) (QUOTE (-367)))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-372))) (-12 (|HasCategory| (-412 |#2|) (QUOTE (-367))) (|HasCategory| (-412 |#2|) (LIST (QUOTE -906) (QUOTE (-1183))))) (-12 (|HasCategory| (-412 |#2|) (QUOTE (-234))) (|HasCategory| (-412 |#2|) (QUOTE (-367))))) 
+(-1010 |bb|) 
 ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating decimal expansions or more generally as repeating expansions in any base.")) (|fractRadix| (($ (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{fractRadix(pre,cyc)} creates a fractional radix expansion from a list of prefix ragits and a list of cyclic ragits. For example,{} \\spad{fractRadix([1],[6])} will return \\spad{0.16666666...}.")) (|wholeRadix| (($ (|List| (|Integer|))) "\\spad{wholeRadix(l)} creates an integral radix expansion from a list of ragits. For example,{} \\spad{wholeRadix([1,3,4])} will return \\spad{134}.")) (|cycleRagits| (((|List| (|Integer|)) $) "\\spad{cycleRagits(rx)} returns the cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \\spad{x = 3/28 = 0.10 714285 714285 ...},{} then \\spad{cycleRagits(x) = [7,1,4,2,8,5]}.")) (|prefixRagits| (((|List| (|Integer|)) $) "\\spad{prefixRagits(rx)} returns the non-cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \\spad{x = 3/28 = 0.10 714285 714285 ...},{} then \\spad{prefixRagits(x)=[1,0]}.")) (|fractRagits| (((|Stream| (|Integer|)) $) "\\spad{fractRagits(rx)} returns the ragits of the fractional part of a radix expansion.")) (|wholeRagits| (((|List| (|Integer|)) $) "\\spad{wholeRagits(rx)} returns the ragits of the integer part of a radix expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(rx)} returns the fractional part of a radix expansion."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-550) (QUOTE (-914))) (|HasCategory| (-550) (LIST (QUOTE -1042) (QUOTE (-1181)))) (|HasCategory| (-550) (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-147))) (|HasCategory| (-550) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-550) (QUOTE (-1024))) (|HasCategory| (-550) (QUOTE (-823))) (-3962 (|HasCategory| (-550) (QUOTE (-823))) (|HasCategory| (-550) (QUOTE (-853)))) (|HasCategory| (-550) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-550) (QUOTE (-1155))) (|HasCategory| (-550) (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-550) (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-550) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-550) (QUOTE (-234))) (|HasCategory| (-550) (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| (-550) (LIST (QUOTE -518) (QUOTE (-1181)) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -311) (QUOTE (-550)))) (|HasCategory| (-550) (LIST (QUOTE -288) (QUOTE (-550)) (QUOTE (-550)))) (|HasCategory| (-550) (QUOTE (-309))) (|HasCategory| (-550) (QUOTE (-549))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-550) (LIST (QUOTE -642) (QUOTE (-550)))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-914)))) (-3962 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-550) (QUOTE (-914)))) (|HasCategory| (-550) (QUOTE (-145))))) 
-(-1009) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-551) (QUOTE (-916))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-1183)))) (|HasCategory| (-551) (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-147))) (|HasCategory| (-551) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-551) (QUOTE (-1026))) (|HasCategory| (-551) (QUOTE (-825))) (-3969 (|HasCategory| (-551) (QUOTE (-825))) (|HasCategory| (-551) (QUOTE (-855)))) (|HasCategory| (-551) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-1157))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-551) (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-551) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-551) (QUOTE (-234))) (|HasCategory| (-551) (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| (-551) (LIST (QUOTE -519) (QUOTE (-1183)) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -312) (QUOTE (-551)))) (|HasCategory| (-551) (LIST (QUOTE -289) (QUOTE (-551)) (QUOTE (-551)))) (|HasCategory| (-551) (QUOTE (-310))) (|HasCategory| (-551) (QUOTE (-550))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-551) (LIST (QUOTE -644) (QUOTE (-551)))) (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (-3969 (-12 (|HasCategory| $ (QUOTE (-145))) (|HasCategory| (-551) (QUOTE (-916)))) (|HasCategory| (-551) (QUOTE (-145))))) 
+(-1011) 
 ((|constructor| (NIL "This package provides tools for creating radix expansions.")) (|radix| (((|Any|) (|Fraction| (|Integer|)) (|Integer|)) "\\spad{radix(x,b)} converts \\spad{x} to a radix expansion in base \\spad{b}."))) 
 NIL 
 NIL 
-(-1010) 
+(-1012) 
 ((|constructor| (NIL "Random number generators \\indented{2}{All random numbers used in the system should originate from} \\indented{2}{the same generator.\\space{2}This package is intended to be the source.}")) (|seed| (((|Integer|)) "\\spad{seed()} returns the current seed value.")) (|reseed| (((|Void|) (|Integer|)) "\\spad{reseed(n)} restarts the random number generator at \\spad{n}.")) (|size| (((|Integer|)) "\\spad{size()} is the base of the random number generator")) (|randnum| (((|Integer|) (|Integer|)) "\\spad{randnum(n)} is a random number between 0 and \\spad{n}.") (((|Integer|)) "\\spad{randnum()} is a random number between 0 and size()."))) 
 NIL 
 NIL 
-(-1011 RP) 
+(-1013 RP) 
 ((|factorSquareFree| (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(p)} factors an extended squareFree polynomial \\spad{p} over the rational numbers.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} factors an extended polynomial \\spad{p} over the rational numbers."))) 
 NIL 
 NIL 
-(-1012 S) 
+(-1014 S) 
 ((|constructor| (NIL "rational number testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") |#1|) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} \"failed\" if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) |#1|) "\\spad{rational?(x)} returns \\spad{true} if \\spad{x} is a rational number,{} \\spad{false} otherwise.")) (|rational| (((|Fraction| (|Integer|)) |#1|) "\\spad{rational(x)} returns \\spad{x} as a rational number; error if \\spad{x} is not a rational number."))) 
 NIL 
 NIL 
-(-1013 A S) 
+(-1015 A S) 
 ((|constructor| (NIL "A recursive aggregate over a type \\spad{S} is a model for a a directed graph containing values of type \\spad{S}. Recursively,{} a recursive aggregate is a {\\em node} consisting of a \\spadfun{value} from \\spad{S} and 0 or more \\spadfun{children} which are recursive aggregates. A node with no children is called a \\spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infinite loop.")) (|setvalue!| ((|#2| $ |#2|) "\\spad{setvalue!(u,x)} sets the value of node \\spad{u} to \\spad{x}.")) (|setelt| ((|#2| $ "value" |#2|) "\\spad{setelt(a,\"value\",x)} (also written \\axiom{a . value \\spad{:=} \\spad{x}}) is equivalent to \\axiom{setvalue!(a,{}\\spad{x})}")) (|setchildren!| (($ $ (|List| $)) "\\spad{setchildren!(u,v)} replaces the current children of node \\spad{u} with the members of \\spad{v} in left-to-right order.")) (|node?| (((|Boolean|) $ $) "\\spad{node?(u,v)} tests if node \\spad{u} is contained in node \\spad{v} (either as a child,{} a child of a child,{} etc.).")) (|child?| (((|Boolean|) $ $) "\\spad{child?(u,v)} tests if node \\spad{u} is a child of node \\spad{v}.")) (|distance| (((|Integer|) $ $) "\\spad{distance(u,v)} returns the path length (an integer) from node \\spad{u} to \\spad{v}.")) (|leaves| (((|List| |#2|) $) "\\spad{leaves(t)} returns the list of values in obtained by visiting the nodes of tree \\axiom{\\spad{t}} in left-to-right order.")) (|cyclic?| (((|Boolean|) $) "\\spad{cyclic?(u)} tests if \\spad{u} has a cycle.")) (|elt| ((|#2| $ "value") "\\spad{elt(u,\"value\")} (also written: \\axiom{a. value}) is equivalent to \\axiom{value(a)}.")) (|value| ((|#2| $) "\\spad{value(u)} returns the value of the node \\spad{u}.")) (|leaf?| (((|Boolean|) $) "\\spad{leaf?(u)} tests if \\spad{u} is a terminal node.")) (|nodes| (((|List| $) $) "\\spad{nodes(u)} returns a list of all of the nodes of aggregate \\spad{u}.")) (|children| (((|List| $) $) "\\spad{children(u)} returns a list of the children of aggregate \\spad{u}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4428)) (|HasCategory| |#2| (QUOTE (-1105)))) 
-(-1014 S) 
+((|HasAttribute| |#1| (QUOTE -4435)) (|HasCategory| |#2| (QUOTE (-1107)))) 
+(-1016 S) 
 ((|constructor| (NIL "A recursive aggregate over a type \\spad{S} is a model for a a directed graph containing values of type \\spad{S}. Recursively,{} a recursive aggregate is a {\\em node} consisting of a \\spadfun{value} from \\spad{S} and 0 or more \\spadfun{children} which are recursive aggregates. A node with no children is called a \\spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infinite loop.")) (|setvalue!| ((|#1| $ |#1|) "\\spad{setvalue!(u,x)} sets the value of node \\spad{u} to \\spad{x}.")) (|setelt| ((|#1| $ "value" |#1|) "\\spad{setelt(a,\"value\",x)} (also written \\axiom{a . value \\spad{:=} \\spad{x}}) is equivalent to \\axiom{setvalue!(a,{}\\spad{x})}")) (|setchildren!| (($ $ (|List| $)) "\\spad{setchildren!(u,v)} replaces the current children of node \\spad{u} with the members of \\spad{v} in left-to-right order.")) (|node?| (((|Boolean|) $ $) "\\spad{node?(u,v)} tests if node \\spad{u} is contained in node \\spad{v} (either as a child,{} a child of a child,{} etc.).")) (|child?| (((|Boolean|) $ $) "\\spad{child?(u,v)} tests if node \\spad{u} is a child of node \\spad{v}.")) (|distance| (((|Integer|) $ $) "\\spad{distance(u,v)} returns the path length (an integer) from node \\spad{u} to \\spad{v}.")) (|leaves| (((|List| |#1|) $) "\\spad{leaves(t)} returns the list of values in obtained by visiting the nodes of tree \\axiom{\\spad{t}} in left-to-right order.")) (|cyclic?| (((|Boolean|) $) "\\spad{cyclic?(u)} tests if \\spad{u} has a cycle.")) (|elt| ((|#1| $ "value") "\\spad{elt(u,\"value\")} (also written: \\axiom{a. value}) is equivalent to \\axiom{value(a)}.")) (|value| ((|#1| $) "\\spad{value(u)} returns the value of the node \\spad{u}.")) (|leaf?| (((|Boolean|) $) "\\spad{leaf?(u)} tests if \\spad{u} is a terminal node.")) (|nodes| (((|List| $) $) "\\spad{nodes(u)} returns a list of all of the nodes of aggregate \\spad{u}.")) (|children| (((|List| $) $) "\\spad{children(u)} returns a list of the children of aggregate \\spad{u}."))) 
 NIL 
 NIL 
-(-1015 S) 
+(-1017 S) 
 ((|constructor| (NIL "\\axiomType{RealClosedField} provides common acces functions for all real closed fields.")) (|approximate| (((|Fraction| (|Integer|)) $ $) "\\axiom{approximate(\\spad{n},{}\\spad{p})} gives an approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|rename| (($ $ (|OutputForm|)) "\\axiom{rename(\\spad{x},{}name)} gives a new number that prints as name")) (|rename!| (($ $ (|OutputForm|)) "\\axiom{rename!(\\spad{x},{}name)} changes the way \\axiom{\\spad{x}} is printed")) (|sqrt| (($ (|Integer|)) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ (|Fraction| (|Integer|))) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ $) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ $ (|PositiveInteger|)) "\\axiom{sqrt(\\spad{x},{}\\spad{n})} is \\axiom{\\spad{x} \\spad{**} (1/n)}")) (|allRootsOf| (((|List| $) (|Polynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely")) (|rootOf| (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} creates the \\spad{n}th root for the order of \\axiom{pol} and gives it unique name") (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|) (|OutputForm|)) "\\axiom{rootOf(pol,{}\\spad{n},{}name)} creates the \\spad{n}th root for the order of \\axiom{pol} and names it \\axiom{name}")) (|mainValue| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainValue(\\spad{x})} is the expression of \\axiom{\\spad{x}} in terms of \\axiom{SparseUnivariatePolynomial(\\$)}")) (|mainDefiningPolynomial| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainDefiningPolynomial(\\spad{x})} is the defining polynomial for the main algebraic quantity of \\axiom{\\spad{x}}")) (|mainForm| (((|Union| (|OutputForm|) "failed") $) "\\axiom{mainForm(\\spad{x})} is the main algebraic quantity name of \\axiom{\\spad{x}}"))) 
 NIL 
 NIL 
-(-1016) 
+(-1018) 
 ((|constructor| (NIL "\\axiomType{RealClosedField} provides common acces functions for all real closed fields.")) (|approximate| (((|Fraction| (|Integer|)) $ $) "\\axiom{approximate(\\spad{n},{}\\spad{p})} gives an approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|rename| (($ $ (|OutputForm|)) "\\axiom{rename(\\spad{x},{}name)} gives a new number that prints as name")) (|rename!| (($ $ (|OutputForm|)) "\\axiom{rename!(\\spad{x},{}name)} changes the way \\axiom{\\spad{x}} is printed")) (|sqrt| (($ (|Integer|)) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ (|Fraction| (|Integer|))) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ $) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} \\spad{**} (1/2)}") (($ $ (|PositiveInteger|)) "\\axiom{sqrt(\\spad{x},{}\\spad{n})} is \\axiom{\\spad{x} \\spad{**} (1/n)}")) (|allRootsOf| (((|List| $) (|Polynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely")) (|rootOf| (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} creates the \\spad{n}th root for the order of \\axiom{pol} and gives it unique name") (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|) (|OutputForm|)) "\\axiom{rootOf(pol,{}\\spad{n},{}name)} creates the \\spad{n}th root for the order of \\axiom{pol} and names it \\axiom{name}")) (|mainValue| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainValue(\\spad{x})} is the expression of \\axiom{\\spad{x}} in terms of \\axiom{SparseUnivariatePolynomial(\\$)}")) (|mainDefiningPolynomial| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainDefiningPolynomial(\\spad{x})} is the defining polynomial for the main algebraic quantity of \\axiom{\\spad{x}}")) (|mainForm| (((|Union| (|OutputForm|) "failed") $) "\\axiom{mainForm(\\spad{x})} is the main algebraic quantity name of \\axiom{\\spad{x}}"))) 
-((-4420 . T) (-4425 . T) (-4419 . T) (-4422 . T) (-4421 . T) ((-4429 "*") . T) (-4424 . T)) 
+((-4427 . T) (-4432 . T) (-4426 . T) (-4429 . T) (-4428 . T) ((-4436 "*") . T) (-4431 . T)) 
 NIL 
-(-1017 R -3498) 
+(-1019 R -3505) 
 ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 1 February 1988 Date Last Updated: 2 November 1995 Keywords: elementary,{} function,{} integration.")) (|rischDE| (((|Record| (|:| |ans| |#2|) (|:| |right| |#2|) (|:| |sol?| (|Boolean|))) (|Integer|) |#2| |#2| (|Symbol|) (|Mapping| (|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|List| |#2|)) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| |#2|)) "\\spad{rischDE(n, f, g, x, lim, ext)} returns \\spad{[y, h, b]} such that \\spad{dy/dx + n df/dx y = h} and \\spad{b := h = g}. The equation \\spad{dy/dx + n df/dx y = g} has no solution if \\spad{h \\~~= g} (\\spad{y} is a partial solution in that case). Notes: \\spad{lim} is a limited integration function,{} and ext is an extended integration function."))) 
 NIL 
 NIL 
-(-1018 R -3498) 
+(-1020 R -3505) 
 ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 12 August 1992 Date Last Updated: 17 August 1992 Keywords: elementary,{} function,{} integration.")) (|rischDEsys| (((|Union| (|List| |#2|) "failed") (|Integer|) |#2| |#2| |#2| (|Symbol|) (|Mapping| (|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|List| |#2|)) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| |#2|)) "\\spad{rischDEsys(n, f, g_1, g_2, x,lim,ext)} returns \\spad{y_1.y_2} such that \\spad{(dy1/dx,dy2/dx) + ((0, - n df/dx),(n df/dx,0)) (y1,y2) = (g1,g2)} if \\spad{y_1,y_2} exist,{} \"failed\" otherwise. \\spad{lim} is a limited integration function,{} \\spad{ext} is an extended integration function."))) 
 NIL 
 NIL 
-(-1019 -3498 UP) 
+(-1021 -3505 UP) 
 ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} transcendental case.} Author: Manuel Bronstein Date Created: Jan 1988 Date Last Updated: 2 November 1995")) (|polyRDE| (((|Union| (|:| |ans| (|Record| (|:| |ans| |#2|) (|:| |nosol| (|Boolean|)))) (|:| |eq| (|Record| (|:| |b| |#2|) (|:| |c| |#2|) (|:| |m| (|Integer|)) (|:| |alpha| |#2|) (|:| |beta| |#2|)))) |#2| |#2| |#2| (|Integer|) (|Mapping| |#2| |#2|)) "\\spad{polyRDE(a, B, C, n, D)} returns either: 1. \\spad{[Q, b]} such that \\spad{degree(Q) <= n} and \\indented{3}{\\spad{a Q'+ B Q = C} if \\spad{b = true},{} \\spad{Q} is a partial solution} \\indented{3}{otherwise.} 2. \\spad{[B1, C1, m, \\alpha, \\beta]} such that any polynomial solution \\indented{3}{of degree at most \\spad{n} of \\spad{A Q' + BQ = C} must be of the form} \\indented{3}{\\spad{Q = \\alpha H + \\beta} where \\spad{degree(H) <= m} and} \\indented{3}{\\spad{H} satisfies \\spad{H' + B1 H = C1}.} \\spad{D} is the derivation to use.")) (|baseRDE| (((|Record| (|:| |ans| (|Fraction| |#2|)) (|:| |nosol| (|Boolean|))) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{baseRDE(f, g)} returns a \\spad{[y, b]} such that \\spad{y' + fy = g} if \\spad{b = true},{} \\spad{y} is a partial solution otherwise (no solution in that case). \\spad{D} is the derivation to use.")) (|monomRDE| (((|Union| (|Record| (|:| |a| |#2|) (|:| |b| (|Fraction| |#2|)) (|:| |c| (|Fraction| |#2|)) (|:| |t| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomRDE(f,g,D)} returns \\spad{[A, B, C, T]} such that \\spad{y' + f y = g} has a solution if and only if \\spad{y = Q / T},{} where \\spad{Q} satisfies \\spad{A Q' + B Q = C} and has no normal pole. A and \\spad{T} are polynomials and \\spad{B} and \\spad{C} have no normal poles. \\spad{D} is the derivation to use."))) 
 NIL 
 NIL 
-(-1020 -3498 UP) 
+(-1022 -3505 UP) 
 ((|constructor| (NIL "\\indented{1}{Risch differential equation system,{} transcendental case.} Author: Manuel Bronstein Date Created: 17 August 1992 Date Last Updated: 3 February 1994")) (|baseRDEsys| (((|Union| (|List| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{baseRDEsys(f, g1, g2)} returns fractions \\spad{y_1.y_2} such that \\spad{(y1', y2') + ((0, -f), (f, 0)) (y1,y2) = (g1,g2)} if \\spad{y_1,y_2} exist,{} \"failed\" otherwise.")) (|monomRDEsys| (((|Union| (|Record| (|:| |a| |#2|) (|:| |b| (|Fraction| |#2|)) (|:| |h| |#2|) (|:| |c1| (|Fraction| |#2|)) (|:| |c2| (|Fraction| |#2|)) (|:| |t| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomRDEsys(f,g1,g2,D)} returns \\spad{[A, B, H, C1, C2, T]} such that \\spad{(y1', y2') + ((0, -f), (f, 0)) (y1,y2) = (g1,g2)} has a solution if and only if \\spad{y1 = Q1 / T, y2 = Q2 / T},{} where \\spad{B,C1,C2,Q1,Q2} have no normal poles and satisfy A \\spad{(Q1', Q2') + ((H, -B), (B, H)) (Q1,Q2) = (C1,C2)} \\spad{D} is the derivation to use."))) 
 NIL 
 NIL 
-(-1021 S) 
+(-1023 S) 
 ((|constructor| (NIL "This package exports random distributions")) (|rdHack1| (((|Mapping| |#1|) (|Vector| |#1|) (|Vector| (|Integer|)) (|Integer|)) "\\spad{rdHack1(v,u,n)} \\undocumented")) (|weighted| (((|Mapping| |#1|) (|List| (|Record| (|:| |value| |#1|) (|:| |weight| (|Integer|))))) "\\spad{weighted(l)} \\undocumented")) (|uniform| (((|Mapping| |#1|) (|Set| |#1|)) "\\spad{uniform(s)} \\undocumented"))) 
 NIL 
 NIL 
-(-1022 F1 UP UPUP R F2) 
+(-1024 F1 UP UPUP R F2) 
 ((|constructor| (NIL "\\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 8 November 1994")) (|order| (((|NonNegativeInteger|) (|FiniteDivisor| |#1| |#2| |#3| |#4|) |#3| (|Mapping| |#5| |#1|)) "\\spad{order(f,u,g)} \\undocumented"))) 
 NIL 
 NIL 
-(-1023) 
+(-1025) 
 ((|constructor| (NIL "This domain represents list reduction syntax.")) (|body| (((|SpadAst|) $) "\\spad{body(e)} return the list of expressions being redcued.")) (|operator| (((|SpadAst|) $) "\\spad{operator(e)} returns the magma operation being applied."))) 
 NIL 
 NIL 
-(-1024) 
+(-1026) 
 ((|constructor| (NIL "The category of real numeric domains,{} \\spadignore{i.e.} convertible to floats."))) 
 NIL 
 NIL 
-(-1025 |Pol|) 
+(-1027 |Pol|) 
 ((|constructor| (NIL "\\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the integers to arbitrary user-specified precision. The results are returned as a list of isolating intervals which are expressed as records with \"left\" and \"right\" rational number components.")) (|midpoints| (((|List| (|Fraction| (|Integer|))) (|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))))) "\\spad{midpoints(isolist)} returns the list of midpoints for the list of intervals \\spad{isolist}.")) (|midpoint| (((|Fraction| (|Integer|)) (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{midpoint(int)} returns the midpoint of the interval \\spad{int}.")) (|refine| (((|Union| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) "failed") |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{refine(pol, int, range)} takes a univariate polynomial \\spad{pol} and and isolating interval \\spad{int} containing exactly one real root of \\spad{pol}; the operation returns an isolating interval which is contained within range,{} or \"failed\" if no such isolating interval exists.") (((|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{refine(pol, int, eps)} refines the interval \\spad{int} containing exactly one root of the univariate polynomial \\spad{pol} to size less than the rational number eps.")) (|realZeros| (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{realZeros(pol, int, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol} which lie in the interval expressed by the record \\spad{int}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Fraction| (|Integer|))) "\\spad{realZeros(pol, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{realZeros(pol, range)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol} which lie in the interval expressed by the record range.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1|) "\\spad{realZeros(pol)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol}."))) 
 NIL 
 NIL 
-(-1026 |Pol|) 
+(-1028 |Pol|) 
 ((|constructor| (NIL "\\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the rational numbers to arbitrary user-specified precision. The results are returned as a list of isolating intervals,{} expressed as records with \"left\" and \"right\" rational number components.")) (|refine| (((|Union| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) "failed") |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{refine(pol, int, range)} takes a univariate polynomial \\spad{pol} and and isolating interval \\spad{int} which must contain exactly one real root of \\spad{pol},{} and returns an isolating interval which is contained within range,{} or \"failed\" if no such isolating interval exists.") (((|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{refine(pol, int, eps)} refines the interval \\spad{int} containing exactly one root of the univariate polynomial \\spad{pol} to size less than the rational number eps.")) (|realZeros| (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{realZeros(pol, int, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol} which lie in the interval expressed by the record \\spad{int}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Fraction| (|Integer|))) "\\spad{realZeros(pol, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{realZeros(pol, range)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol} which lie in the interval expressed by the record range.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1|) "\\spad{realZeros(pol)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol}."))) 
 NIL 
 NIL 
-(-1027) 
+(-1029) 
 ((|constructor| (NIL "\\indented{1}{This package provides numerical solutions of systems of polynomial} equations for use in ACPLOT.")) (|realSolve| (((|List| (|List| (|Float|))) (|List| (|Polynomial| (|Integer|))) (|List| (|Symbol|)) (|Float|)) "\\spad{realSolve(lp,lv,eps)} = compute the list of the real solutions of the list \\spad{lp} of polynomials with integer coefficients with respect to the variables in \\spad{lv},{} with precision \\spad{eps}.")) (|solve| (((|List| (|Float|)) (|Polynomial| (|Integer|)) (|Float|)) "\\spad{solve(p,eps)} finds the real zeroes of a univariate integer polynomial \\spad{p} with precision \\spad{eps}.") (((|List| (|Float|)) (|Polynomial| (|Fraction| (|Integer|))) (|Float|)) "\\spad{solve(p,eps)} finds the real zeroes of a univariate rational polynomial \\spad{p} with precision \\spad{eps}."))) 
 NIL 
 NIL 
-(-1028 |TheField|) 
+(-1030 |TheField|) 
 ((|constructor| (NIL "This domain implements the real closure of an ordered field.")) (|relativeApprox| (((|Fraction| (|Integer|)) $ $) "\\axiom{relativeApprox(\\spad{n},{}\\spad{p})} gives a relative approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|mainCharacterization| (((|Union| (|RightOpenIntervalRootCharacterization| $ (|SparseUnivariatePolynomial| $)) "failed") $) "\\axiom{mainCharacterization(\\spad{x})} is the main algebraic quantity of \\axiom{\\spad{x}} (\\axiom{SEG})")) (|algebraicOf| (($ (|RightOpenIntervalRootCharacterization| $ (|SparseUnivariatePolynomial| $)) (|OutputForm|)) "\\axiom{algebraicOf(char)} is the external number"))) 
-((-4420 . T) (-4425 . T) (-4419 . T) (-4422 . T) (-4421 . T) ((-4429 "*") . T) (-4424 . T)) 
-((-3962 (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-411 (-550)) (LIST (QUOTE -1042) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-411 (-550)) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-411 (-550)) (LIST (QUOTE -1042) (QUOTE (-550))))) 
-(-1029 -3498 L) 
+((-4427 . T) (-4432 . T) (-4426 . T) (-4429 . T) (-4428 . T) ((-4436 "*") . T) (-4431 . T)) 
+((-3969 (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-412 (-551)) (LIST (QUOTE -1044) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-412 (-551)) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-412 (-551)) (LIST (QUOTE -1044) (QUOTE (-551))))) 
+(-1031 -3505 L) 
 ((|constructor| (NIL "\\spadtype{ReductionOfOrder} provides functions for reducing the order of linear ordinary differential equations once some solutions are known.")) (|ReduceOrder| (((|Record| (|:| |eq| |#2|) (|:| |op| (|List| |#1|))) |#2| (|List| |#1|)) "\\spad{ReduceOrder(op, [f1,...,fk])} returns \\spad{[op1,[g1,...,gk]]} such that for any solution \\spad{z} of \\spad{op1 z = 0},{} \\spad{y = gk \\int(g_{k-1} \\int(... \\int(g1 \\int z)...)} is a solution of \\spad{op y = 0}. Each \\spad{fi} must satisfy \\spad{op fi = 0}.") ((|#2| |#2| |#1|) "\\spad{ReduceOrder(op, s)} returns \\spad{op1} such that for any solution \\spad{z} of \\spad{op1 z = 0},{} \\spad{y = s \\int z} is a solution of \\spad{op y = 0}. \\spad{s} must satisfy \\spad{op s = 0}."))) 
 NIL 
 NIL 
-(-1030 S) 
+(-1032 S) 
 ((|constructor| (NIL "\\indented{1}{\\spadtype{Reference} is for making a changeable instance} of something.")) (= (((|Boolean|) $ $) "\\spad{a=b} tests if \\spad{a} and \\spad{b} are equal.")) (|setref| ((|#1| $ |#1|) "\\spad{setref(n,m)} same as \\spad{setelt(n,m)}.")) (|deref| ((|#1| $) "\\spad{deref(n)} is equivalent to \\spad{elt(n)}.")) (|setelt| ((|#1| $ |#1|) "\\spad{setelt(n,m)} changes the value of the object \\spad{n} to \\spad{m}.")) (|elt| ((|#1| $) "\\spad{elt(n)} returns the object \\spad{n}.")) (|ref| (($ |#1|) "\\spad{ref(n)} creates a pointer (reference) to the object \\spad{n}."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-1105)))) 
-(-1031 R E V P) 
+((|HasCategory| |#1| (QUOTE (-1107)))) 
+(-1033 R E V P) 
 ((|constructor| (NIL "This domain provides an implementation of regular chains. Moreover,{} the operation \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|preprocess| (((|Record| (|:| |val| (|List| |#4|)) (|:| |towers| (|List| $))) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{pre_process(\\spad{lp},{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|internalZeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalZeroSetSplit(\\spad{lp},{}\\spad{b1},{}\\spad{b2},{}\\spad{b3})} is an internal subroutine,{} exported only for developement.")) (|zeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(\\spad{lp},{}\\spad{b1},{}\\spad{b2}.\\spad{b3},{}\\spad{b4})} is an internal subroutine,{} exported only for developement.") (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(\\spad{lp},{}clos?,{}info?)} has the same specifications as \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory}. Moreover,{} if \\axiom{clos?} then solves in the sense of the Zariski closure else solves in the sense of the regular zeros. If \\axiom{info?} then do print messages during the computations.")) (|internalAugment| (((|List| $) |#4| $ (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalAugment(\\spad{p},{}\\spad{ts},{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement."))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#4| (LIST (QUOTE -311) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#3| (QUOTE (-371))) (|HasCategory| |#4| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1032) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#4| (LIST (QUOTE -312) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-372))) (|HasCategory| |#4| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1034) 
 ((|constructor| (NIL "Package for the computation of eigenvalues and eigenvectors. This package works for matrices with coefficients which are rational functions over the integers. (see \\spadtype{Fraction Polynomial Integer}). The eigenvalues and eigenvectors are expressed in terms of radicals.")) (|orthonormalBasis| (((|List| (|Matrix| (|Expression| (|Integer|)))) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{orthonormalBasis(m)} returns the orthogonal matrix \\spad{b} such that \\spad{b*m*(inverse b)} is diagonal. Error: if \\spad{m} is not a symmetric matrix.")) (|gramschmidt| (((|List| (|Matrix| (|Expression| (|Integer|)))) (|List| (|Matrix| (|Expression| (|Integer|))))) "\\spad{gramschmidt(lv)} converts the list of column vectors \\spad{lv} into a set of orthogonal column vectors of euclidean length 1 using the Gram-Schmidt algorithm.")) (|normalise| (((|Matrix| (|Expression| (|Integer|))) (|Matrix| (|Expression| (|Integer|)))) "\\spad{normalise(v)} returns the column vector \\spad{v} divided by its euclidean norm; when possible,{} the vector \\spad{v} is expressed in terms of radicals.")) (|eigenMatrix| (((|Union| (|Matrix| (|Expression| (|Integer|))) "failed") (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{eigenMatrix(m)} returns the matrix \\spad{b} such that \\spad{b*m*(inverse b)} is diagonal,{} or \"failed\" if no such \\spad{b} exists.")) (|radicalEigenvalues| (((|List| (|Expression| (|Integer|))) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{radicalEigenvalues(m)} computes the eigenvalues of the matrix \\spad{m}; when possible,{} the eigenvalues are expressed in terms of radicals.")) (|radicalEigenvector| (((|List| (|Matrix| (|Expression| (|Integer|)))) (|Expression| (|Integer|)) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{radicalEigenvector(c,m)} computes the eigenvector(\\spad{s}) of the matrix \\spad{m} corresponding to the eigenvalue \\spad{c}; when possible,{} values are expressed in terms of radicals.")) (|radicalEigenvectors| (((|List| (|Record| (|:| |radval| (|Expression| (|Integer|))) (|:| |radmult| (|Integer|)) (|:| |radvect| (|List| (|Matrix| (|Expression| (|Integer|))))))) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{radicalEigenvectors(m)} computes the eigenvalues and the corresponding eigenvectors of the matrix \\spad{m}; when possible,{} values are expressed in terms of radicals."))) 
 NIL 
 NIL 
-(-1033 R) 
+(-1035 R) 
 ((|constructor| (NIL "RepresentationPackage1 provides functions for representation theory for finite groups and algebras. The package creates permutation representations and uses tensor products and its symmetric and antisymmetric components to create new representations of larger degree from given ones. Note: instead of having parameters from \\spadtype{Permutation} this package allows list notation of permutations as well: \\spadignore{e.g.} \\spad{[1,4,3,2]} denotes permutes 2 and 4 and fixes 1 and 3.")) (|permutationRepresentation| (((|List| (|Matrix| (|Integer|))) (|List| (|List| (|Integer|)))) "\\spad{permutationRepresentation([pi1,...,pik],n)} returns the list of matrices {\\em [(deltai,pi1(i)),...,(deltai,pik(i))]} if the permutations {\\em pi1},{}...,{}{\\em pik} are in list notation and are permuting {\\em {1,2,...,n}}.") (((|List| (|Matrix| (|Integer|))) (|List| (|Permutation| (|Integer|))) (|Integer|)) "\\spad{permutationRepresentation([pi1,...,pik],n)} returns the list of matrices {\\em [(deltai,pi1(i)),...,(deltai,pik(i))]} (Kronecker delta) for the permutations {\\em pi1,...,pik} of {\\em {1,2,...,n}}.") (((|Matrix| (|Integer|)) (|List| (|Integer|))) "\\spad{permutationRepresentation(pi,n)} returns the matrix {\\em (deltai,pi(i))} (Kronecker delta) if the permutation {\\em pi} is in list notation and permutes {\\em {1,2,...,n}}.") (((|Matrix| (|Integer|)) (|Permutation| (|Integer|)) (|Integer|)) "\\spad{permutationRepresentation(pi,n)} returns the matrix {\\em (deltai,pi(i))} (Kronecker delta) for a permutation {\\em pi} of {\\em {1,2,...,n}}.")) (|tensorProduct| (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{tensorProduct([a1,...ak])} calculates the list of Kronecker products of each matrix {\\em ai} with itself for {1 \\spad{<=} \\spad{i} \\spad{<=} \\spad{k}}. Note: If the list of matrices corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the representation with itself.") (((|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{tensorProduct(a)} calculates the Kronecker product of the matrix {\\em a} with itself.") (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{tensorProduct([a1,...,ak],[b1,...,bk])} calculates the list of Kronecker products of the matrices {\\em ai} and {\\em bi} for {1 \\spad{<=} \\spad{i} \\spad{<=} \\spad{k}}. Note: If each list of matrices corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the two representations.") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{tensorProduct(a,b)} calculates the Kronecker product of the matrices {\\em a} and \\spad{b}. Note: if each matrix corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the two representations.")) (|symmetricTensors| (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|PositiveInteger|)) "\\spad{symmetricTensors(la,n)} applies to each \\spad{m}-by-\\spad{m} square matrix in the list {\\em la} the irreducible,{} polynomial representation of the general linear group {\\em GLm} which corresponds to the partition {\\em (n,0,...,0)} of \\spad{n}. Error: if the matrices in {\\em la} are not square matrices. Note: this corresponds to the symmetrization of the representation with the trivial representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the symmetric tensors of the \\spad{n}-fold tensor product.") (((|Matrix| |#1|) (|Matrix| |#1|) (|PositiveInteger|)) "\\spad{symmetricTensors(a,n)} applies to the \\spad{m}-by-\\spad{m} square matrix {\\em a} the irreducible,{} polynomial representation of the general linear group {\\em GLm} which corresponds to the partition {\\em (n,0,...,0)} of \\spad{n}. Error: if {\\em a} is not a square matrix. Note: this corresponds to the symmetrization of the representation with the trivial representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the symmetric tensors of the \\spad{n}-fold tensor product.")) (|createGenericMatrix| (((|Matrix| (|Polynomial| |#1|)) (|NonNegativeInteger|)) "\\spad{createGenericMatrix(m)} creates a square matrix of dimension \\spad{k} whose entry at the \\spad{i}-th row and \\spad{j}-th column is the indeterminate {\\em x[i,j]} (double subscripted).")) (|antisymmetricTensors| (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|PositiveInteger|)) "\\spad{antisymmetricTensors(la,n)} applies to each \\spad{m}-by-\\spad{m} square matrix in the list {\\em la} the irreducible,{} polynomial representation of the general linear group {\\em GLm} which corresponds to the partition {\\em (1,1,...,1,0,0,...,0)} of \\spad{n}. Error: if \\spad{n} is greater than \\spad{m}. Note: this corresponds to the symmetrization of the representation with the sign representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the antisymmetric tensors of the \\spad{n}-fold tensor product.") (((|Matrix| |#1|) (|Matrix| |#1|) (|PositiveInteger|)) "\\spad{antisymmetricTensors(a,n)} applies to the square matrix {\\em a} the irreducible,{} polynomial representation of the general linear group {\\em GLm},{} where \\spad{m} is the number of rows of {\\em a},{} which corresponds to the partition {\\em (1,1,...,1,0,0,...,0)} of \\spad{n}. Error: if \\spad{n} is greater than \\spad{m}. Note: this corresponds to the symmetrization of the representation with the sign representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the antisymmetric tensors of the \\spad{n}-fold tensor product."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE (-4429 "*")))) 
-(-1034 R) 
+((|HasAttribute| |#1| (QUOTE (-4436 "*")))) 
+(-1036 R) 
 ((|constructor| (NIL "RepresentationPackage2 provides functions for working with modular representations of finite groups and algebra. The routines in this package are created,{} using ideas of \\spad{R}. Parker,{} (the meat-Axe) to get smaller representations from bigger ones,{} \\spadignore{i.e.} finding sub- and factormodules,{} or to show,{} that such the representations are irreducible. Note: most functions are randomized functions of Las Vegas type \\spadignore{i.e.} every answer is correct,{} but with small probability the algorithm fails to get an answer.")) (|scanOneDimSubspaces| (((|Vector| |#1|) (|List| (|Vector| |#1|)) (|Integer|)) "\\spad{scanOneDimSubspaces(basis,n)} gives a canonical representative of the {\\em n}\\spad{-}th one-dimensional subspace of the vector space generated by the elements of {\\em basis},{} all from {\\em R**n}. The coefficients of the representative are of shape {\\em (0,...,0,1,*,...,*)},{} {\\em *} in \\spad{R}. If the size of \\spad{R} is \\spad{q},{} then there are {\\em (q**n-1)/(q-1)} of them. We first reduce \\spad{n} modulo this number,{} then find the largest \\spad{i} such that {\\em +/[q**i for i in 0..i-1] <= n}. Subtracting this sum of powers from \\spad{n} results in an \\spad{i}-digit number to \\spad{basis} \\spad{q}. This fills the positions of the stars.")) (|meatAxe| (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|PositiveInteger|)) "\\spad{meatAxe(aG, numberOfTries)} calls {\\em meatAxe(aG,true,numberOfTries,7)}. Notes: 7 covers the case of three-dimensional kernels over the field with 2 elements.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Boolean|)) "\\spad{meatAxe(aG, randomElements)} calls {\\em meatAxe(aG,false,6,7)},{} only using Parker\\spad{'s} fingerprints,{} if {\\em randomElemnts} is \\spad{false}. If it is \\spad{true},{} it calls {\\em meatAxe(aG,true,25,7)},{} only using random elements. Note: the choice of 25 was rather arbitrary. Also,{} 7 covers the case of three-dimensional kernels over the field with 2 elements.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|))) "\\spad{meatAxe(aG)} calls {\\em meatAxe(aG,false,25,7)} returns a 2-list of representations as follows. All matrices of argument \\spad{aG} are assumed to be square and of equal size. Then \\spad{aG} generates a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an A-module in the usual way. meatAxe(\\spad{aG}) creates at most 25 random elements of the algebra,{} tests them for singularity. If singular,{} it tries at most 7 elements of its kernel to generate a proper submodule. If successful a list which contains first the list of the representations of the submodule,{} then a list of the representations of the factor module is returned. Otherwise,{} if we know that all the kernel is already scanned,{} Norton\\spad{'s} irreducibility test can be used either to prove irreducibility or to find the splitting. Notes: the first 6 tries use Parker\\spad{'s} fingerprints. Also,{} 7 covers the case of three-dimensional kernels over the field with 2 elements.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Boolean|) (|Integer|) (|Integer|)) "\\spad{meatAxe(aG,randomElements,numberOfTries, maxTests)} returns a 2-list of representations as follows. All matrices of argument \\spad{aG} are assumed to be square and of equal size. Then \\spad{aG} generates a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an A-module in the usual way. meatAxe(\\spad{aG},{}\\spad{numberOfTries},{} maxTests) creates at most {\\em numberOfTries} random elements of the algebra,{} tests them for singularity. If singular,{} it tries at most {\\em maxTests} elements of its kernel to generate a proper submodule. If successful,{} a 2-list is returned: first,{} a list containing first the list of the representations of the submodule,{} then a list of the representations of the factor module. Otherwise,{} if we know that all the kernel is already scanned,{} Norton\\spad{'s} irreducibility test can be used either to prove irreducibility or to find the splitting. If {\\em randomElements} is {\\em false},{} the first 6 tries use Parker\\spad{'s} fingerprints.")) (|split| (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Vector| (|Vector| |#1|))) "\\spad{split(aG,submodule)} uses a proper \\spad{submodule} of {\\em R**n} to create the representations of the \\spad{submodule} and of the factor module.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Vector| |#1|)) "\\spad{split(aG, vector)} returns a subalgebra \\spad{A} of all square matrix of dimension \\spad{n} as a list of list of matrices,{} generated by the list of matrices \\spad{aG},{} where \\spad{n} denotes both the size of vector as well as the dimension of each of the square matrices. {\\em V R} is an A-module in the natural way. split(\\spad{aG},{} vector) then checks whether the cyclic submodule generated by {\\em vector} is a proper submodule of {\\em V R}. If successful,{} it returns a two-element list,{} which contains first the list of the representations of the submodule,{} then the list of the representations of the factor module. If the vector generates the whole module,{} a one-element list of the old representation is given. Note: a later version this should call the other split.")) (|isAbsolutelyIrreducible?| (((|Boolean|) (|List| (|Matrix| |#1|))) "\\spad{isAbsolutelyIrreducible?(aG)} calls {\\em isAbsolutelyIrreducible?(aG,25)}. Note: the choice of 25 was rather arbitrary.") (((|Boolean|) (|List| (|Matrix| |#1|)) (|Integer|)) "\\spad{isAbsolutelyIrreducible?(aG, numberOfTries)} uses Norton\\spad{'s} irreducibility test to check for absolute irreduciblity,{} assuming if a one-dimensional kernel is found. As no field extension changes create \"new\" elements in a one-dimensional space,{} the criterium stays \\spad{true} for every extension. The method looks for one-dimensionals only by creating random elements (no fingerprints) since a run of {\\em meatAxe} would have proved absolute irreducibility anyway.")) (|areEquivalent?| (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|Integer|)) "\\spad{areEquivalent?(aG0,aG1,numberOfTries)} calls {\\em areEquivalent?(aG0,aG1,true,25)}. Note: the choice of 25 was rather arbitrary.") (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{areEquivalent?(aG0,aG1)} calls {\\em areEquivalent?(aG0,aG1,true,25)}. Note: the choice of 25 was rather arbitrary.") (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|Boolean|) (|Integer|)) "\\spad{areEquivalent?(aG0,aG1,randomelements,numberOfTries)} tests whether the two lists of matrices,{} all assumed of same square shape,{} can be simultaneously conjugated by a non-singular matrix. If these matrices represent the same group generators,{} the representations are equivalent. The algorithm tries {\\em numberOfTries} times to create elements in the generated algebras in the same fashion. If their ranks differ,{} they are not equivalent. If an isomorphism is assumed,{} then the kernel of an element of the first algebra is mapped to the kernel of the corresponding element in the second algebra. Now consider the one-dimensional ones. If they generate the whole space (\\spadignore{e.g.} irreducibility !) we use {\\em standardBasisOfCyclicSubmodule} to create the only possible transition matrix. The method checks whether the matrix conjugates all corresponding matrices from {\\em aGi}. The way to choose the singular matrices is as in {\\em meatAxe}. If the two representations are equivalent,{} this routine returns the transformation matrix {\\em TM} with {\\em aG0.i * TM = TM * aG1.i} for all \\spad{i}. If the representations are not equivalent,{} a small 0-matrix is returned. Note: the case with different sets of group generators cannot be handled.")) (|standardBasisOfCyclicSubmodule| (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|Vector| |#1|)) "\\spad{standardBasisOfCyclicSubmodule(lm,v)} returns a matrix as follows. It is assumed that the size \\spad{n} of the vector equals the number of rows and columns of the matrices. Then the matrices generate a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an \\spad{A}-module in the natural way. standardBasisOfCyclicSubmodule(\\spad{lm},{}\\spad{v}) calculates a matrix whose non-zero column vectors are the \\spad{R}-Basis of {\\em Av} achieved in the way as described in section 6 of \\spad{R}. A. Parker\\spad{'s} \"The Meat-Axe\". Note: in contrast to {\\em cyclicSubmodule},{} the result is not in echelon form.")) (|cyclicSubmodule| (((|Vector| (|Vector| |#1|)) (|List| (|Matrix| |#1|)) (|Vector| |#1|)) "\\spad{cyclicSubmodule(lm,v)} generates a basis as follows. It is assumed that the size \\spad{n} of the vector equals the number of rows and columns of the matrices. Then the matrices generate a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an \\spad{A}-module in the natural way. cyclicSubmodule(\\spad{lm},{}\\spad{v}) generates the \\spad{R}-Basis of {\\em Av} as described in section 6 of \\spad{R}. A. Parker\\spad{'s} \"The Meat-Axe\". Note: in contrast to the description in \"The Meat-Axe\" and to {\\em standardBasisOfCyclicSubmodule} the result is in echelon form.")) (|createRandomElement| (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|Matrix| |#1|)) "\\spad{createRandomElement(aG,x)} creates a random element of the group algebra generated by {\\em aG}.")) (|completeEchelonBasis| (((|Matrix| |#1|) (|Vector| (|Vector| |#1|))) "\\spad{completeEchelonBasis(lv)} completes the basis {\\em lv} assumed to be in echelon form of a subspace of {\\em R**n} (\\spad{n} the length of all the vectors in {\\em lv}) with unit vectors to a basis of {\\em R**n}. It is assumed that the argument is not an empty vector and that it is not the basis of the 0-subspace. Note: the rows of the result correspond to the vectors of the basis."))) 
 NIL 
-((-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-371)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-309)))) 
-(-1035 S) 
+((-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-372)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-310)))) 
+(-1037 S) 
 ((|constructor| (NIL "Implements multiplication by repeated addition")) (|double| ((|#1| (|PositiveInteger|) |#1|) "\\spad{double(i, r)} multiplies \\spad{r} by \\spad{i} using repeated doubling.")) (+ (($ $ $) "\\spad{x+y} returns the sum of \\spad{x} and \\spad{y}"))) 
 NIL 
 NIL 
-(-1036 S) 
+(-1038 S) 
 ((|constructor| (NIL "Implements exponentiation by repeated squaring")) (|expt| ((|#1| |#1| (|PositiveInteger|)) "\\spad{expt(r, i)} computes r**i by repeated squaring")) (* (($ $ $) "\\spad{x*y} returns the product of \\spad{x} and \\spad{y}"))) 
 NIL 
 NIL 
-(-1037 S) 
+(-1039 S) 
 ((|constructor| (NIL "This package provides coercions for the special types \\spadtype{Exit} and \\spadtype{Void}.")) (|coerce| ((|#1| (|Exit|)) "\\spad{coerce(e)} is never really evaluated. This coercion is used for formal type correctness when a function will not return directly to its caller.") (((|Void|) |#1|) "\\spad{coerce(s)} throws all information about \\spad{s} away. This coercion allows values of any type to appear in contexts where they will not be used. For example,{} it allows the resolution of different types in the \\spad{then} and \\spad{else} branches when an \\spad{if} is in a context where the resulting value is not used."))) 
 NIL 
 NIL 
-(-1038 -3498 |Expon| |VarSet| |FPol| |LFPol|) 
+(-1040 -3505 |Expon| |VarSet| |FPol| |LFPol|) 
 ((|constructor| (NIL "ResidueRing is the quotient of a polynomial ring by an ideal. The ideal is given as a list of generators. The elements of the domain are equivalence classes expressed in terms of reduced elements")) (|lift| ((|#4| $) "\\spad{lift(x)} return the canonical representative of the equivalence class \\spad{x}")) (|coerce| (($ |#4|) "\\spad{coerce(f)} produces the equivalence class of \\spad{f} in the residue ring")) (|reduce| (($ |#4|) "\\spad{reduce(f)} produces the equivalence class of \\spad{f} in the residue ring"))) 
-(((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1039) 
+(-1041) 
 ((|constructor| (NIL "A domain used to return the results from a call to the NAG Library. It prints as a list of names and types,{} though the user may choose to display values automatically if he or she wishes.")) (|showArrayValues| (((|Boolean|) (|Boolean|)) "\\spad{showArrayValues(true)} forces the values of array components to be \\indented{1}{displayed rather than just their types.}")) (|showScalarValues| (((|Boolean|) (|Boolean|)) "\\spad{showScalarValues(true)} forces the values of scalar components to be \\indented{1}{displayed rather than just their types.}"))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (QUOTE (-1181))) (LIST (QUOTE |:|) (QUOTE -2256) (QUOTE (-51)))))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105)))) (-3962 (|HasCategory| (-51) (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-51) (QUOTE (-1105))) (|HasCategory| (-51) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| (-51) (QUOTE (-1105))) (|HasCategory| (-51) (LIST (QUOTE -311) (QUOTE (-51))))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105))) (|HasCategory| (-1181) (QUOTE (-853))) (|HasCategory| (-51) (QUOTE (-1105))) (-3962 (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-51) (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| (-51) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1040) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (QUOTE (-1183))) (LIST (QUOTE |:|) (QUOTE -2263) (QUOTE (-51)))))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107)))) (-3969 (|HasCategory| (-51) (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-51) (QUOTE (-1107))) (|HasCategory| (-51) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| (-51) (QUOTE (-1107))) (|HasCategory| (-51) (LIST (QUOTE -312) (QUOTE (-51))))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107))) (|HasCategory| (-1183) (QUOTE (-855))) (|HasCategory| (-51) (QUOTE (-1107))) (-3969 (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-51) (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| (-51) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1042) 
 ((|constructor| (NIL "This domain represents `return' expressions.")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression returned by `e'."))) 
 NIL 
 NIL 
-(-1041 A S) 
+(-1043 A S) 
 ((|constructor| (NIL "A is retractable to \\spad{B} means that some elementsif A can be converted into elements of \\spad{B} and any element of \\spad{B} can be converted into an element of A.")) (|retract| ((|#2| $) "\\spad{retract(a)} transforms a into an element of \\spad{S} if possible. Error: if a cannot be made into an element of \\spad{S}.")) (|retractIfCan| (((|Union| |#2| "failed") $) "\\spad{retractIfCan(a)} transforms a into an element of \\spad{S} if possible. Returns \"failed\" if a cannot be made into an element of \\spad{S}."))) 
 NIL 
 NIL 
-(-1042 S) 
+(-1044 S) 
 ((|constructor| (NIL "A is retractable to \\spad{B} means that some elementsif A can be converted into elements of \\spad{B} and any element of \\spad{B} can be converted into an element of A.")) (|retract| ((|#1| $) "\\spad{retract(a)} transforms a into an element of \\spad{S} if possible. Error: if a cannot be made into an element of \\spad{S}.")) (|retractIfCan| (((|Union| |#1| "failed") $) "\\spad{retractIfCan(a)} transforms a into an element of \\spad{S} if possible. Returns \"failed\" if a cannot be made into an element of \\spad{S}."))) 
 NIL 
 NIL 
-(-1043 Q R) 
+(-1045 Q R) 
 ((|constructor| (NIL "RetractSolvePackage is an interface to \\spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving.")) (|solveRetract| (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#2|))))) (|List| (|Polynomial| |#2|)) (|List| (|Symbol|))) "\\spad{solveRetract(lp,lv)} finds the solutions of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}. The function tries to retract all the coefficients of the equations to \\spad{Q} before solving if possible."))) 
 NIL 
 NIL 
-(-1044 R) 
+(-1046 R) 
 ((|constructor| (NIL "Utilities that provide the same top-level manipulations on fractions than on polynomials.")) (|coerce| (((|Fraction| (|Polynomial| |#1|)) |#1|) "\\spad{coerce(r)} returns \\spad{r} viewed as a rational function over \\spad{R}.")) (|eval| (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) "\\spad{eval(f, [v1 = g1,...,vn = gn])} returns \\spad{f} with each \\spad{vi} replaced by \\spad{gi} in parallel,{} \\spadignore{i.e.} \\spad{vi}\\spad{'s} appearing inside the \\spad{gi}\\spad{'s} are not replaced. Error: if any \\spad{vi} is not a symbol.") (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eval(f, v = g)} returns \\spad{f} with \\spad{v} replaced by \\spad{g}. Error: if \\spad{v} is not a symbol.") (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|List| (|Symbol|)) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eval(f, [v1,...,vn], [g1,...,gn])} returns \\spad{f} with each \\spad{vi} replaced by \\spad{gi} in parallel,{} \\spadignore{i.e.} \\spad{vi}\\spad{'s} appearing inside the \\spad{gi}\\spad{'s} are not replaced.") (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|Fraction| (|Polynomial| |#1|))) "\\spad{eval(f, v, g)} returns \\spad{f} with \\spad{v} replaced by \\spad{g}.")) (|multivariate| (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) (|Symbol|)) "\\spad{multivariate(f, v)} applies both the numerator and denominator of \\spad{f} to \\spad{v}.")) (|univariate| (((|Fraction| (|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{univariate(f, v)} returns \\spad{f} viewed as a univariate rational function in \\spad{v}.")) (|mainVariable| (((|Union| (|Symbol|) "failed") (|Fraction| (|Polynomial| |#1|))) "\\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \\spad{f},{} \"failed\" if \\spad{f} has no variables.")) (|variables| (((|List| (|Symbol|)) (|Fraction| (|Polynomial| |#1|))) "\\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \\spad{f}."))) 
 NIL 
 NIL 
-(-1045) 
+(-1047) 
 ((|t| (((|Mapping| (|Float|)) (|NonNegativeInteger|)) "\\spad{t(n)} \\undocumented")) (F (((|Mapping| (|Float|)) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{F(n,m)} \\undocumented")) (|Beta| (((|Mapping| (|Float|)) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{Beta(n,m)} \\undocumented")) (|chiSquare| (((|Mapping| (|Float|)) (|NonNegativeInteger|)) "\\spad{chiSquare(n)} \\undocumented")) (|exponential| (((|Mapping| (|Float|)) (|Float|)) "\\spad{exponential(f)} \\undocumented")) (|normal| (((|Mapping| (|Float|)) (|Float|) (|Float|)) "\\spad{normal(f,g)} \\undocumented")) (|uniform| (((|Mapping| (|Float|)) (|Float|) (|Float|)) "\\spad{uniform(f,g)} \\undocumented")) (|chiSquare1| (((|Float|) (|NonNegativeInteger|)) "\\spad{chiSquare1(n)} \\undocumented")) (|exponential1| (((|Float|)) "\\spad{exponential1()} \\undocumented")) (|normal01| (((|Float|)) "\\spad{normal01()} \\undocumented")) (|uniform01| (((|Float|)) "\\spad{uniform01()} \\undocumented"))) 
 NIL 
 NIL 
-(-1046 UP) 
+(-1048 UP) 
 ((|constructor| (NIL "Factorization of univariate polynomials with coefficients which are rational functions with integer coefficients.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p}."))) 
 NIL 
 NIL 
-(-1047 R) 
+(-1049 R) 
 ((|constructor| (NIL "\\spadtype{RationalFunctionFactorizer} contains the factor function (called factorFraction) which factors fractions of polynomials by factoring the numerator and denominator. Since any non zero fraction is a unit the usual factor operation will just return the original fraction.")) (|factorFraction| (((|Fraction| (|Factored| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|))) "\\spad{factorFraction(r)} factors the numerator and the denominator of the polynomial fraction \\spad{r}."))) 
 NIL 
 NIL 
-(-1048 T$) 
+(-1050 T$) 
 ((|constructor| (NIL "This category defines the common interface for \\spad{RGB} color models.")) (|componentUpperBound| ((|#1|) "componentUpperBound is an upper bound for all component values.")) (|blue| ((|#1| $) "\\spad{blue(c)} returns the `blue' component of \\spad{`c'}.")) (|green| ((|#1| $) "\\spad{green(c)} returns the `green' component of \\spad{`c'}.")) (|red| ((|#1| $) "\\spad{red(c)} returns the `red' component of \\spad{`c'}."))) 
 NIL 
 NIL 
-(-1049 T$) 
+(-1051 T$) 
 ((|constructor| (NIL "This category defines the common interface for \\spad{RGB} color spaces.")) (|whitePoint| (($) "whitePoint is the contant indicating the white point of this color space."))) 
 NIL 
 NIL 
-(-1050 R |ls|) 
+(-1052 R |ls|) 
 ((|constructor| (NIL "A domain for regular chains (\\spadignore{i.e.} regular triangular sets) over a \\spad{Gcd}-Domain and with a fix list of variables. This is just a front-end for the \\spadtype{RegularTriangularSet} domain constructor.")) (|zeroSetSplit| (((|List| $) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|) (|Boolean|)) "\\spad{zeroSetSplit(lp,clos?,info?)} returns a list \\spad{lts} of regular chains such that the union of the closures of their regular zero sets equals the affine variety associated with \\spad{lp}. Moreover,{} if \\spad{clos?} is \\spad{false} then the union of the regular zero set of the \\spad{ts} (for \\spad{ts} in \\spad{lts}) equals this variety. If \\spad{info?} is \\spad{true} then some information is displayed during the computations. See \\axiomOpFrom{zeroSetSplit}{RegularTriangularSet}."))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| (-783 |#1| (-867 |#2|)) (QUOTE (-1105))) (|HasCategory| (-783 |#1| (-867 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -783) (|devaluate| |#1|) (LIST (QUOTE -867) (|devaluate| |#2|)))))) (|HasCategory| (-783 |#1| (-867 |#2|)) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-783 |#1| (-867 |#2|)) (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| (-867 |#2|) (QUOTE (-371))) (|HasCategory| (-783 |#1| (-867 |#2|)) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1051) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| (-785 |#1| (-869 |#2|)) (QUOTE (-1107))) (|HasCategory| (-785 |#1| (-869 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -785) (|devaluate| |#1|) (LIST (QUOTE -869) (|devaluate| |#2|)))))) (|HasCategory| (-785 |#1| (-869 |#2|)) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-785 |#1| (-869 |#2|)) (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| (-869 |#2|) (QUOTE (-372))) (|HasCategory| (-785 |#1| (-869 |#2|)) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1053) 
 ((|constructor| (NIL "This package exports integer distributions")) (|ridHack1| (((|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{ridHack1(i,j,k,l)} \\undocumented")) (|geometric| (((|Mapping| (|Integer|)) |RationalNumber|) "\\spad{geometric(f)} \\undocumented")) (|poisson| (((|Mapping| (|Integer|)) |RationalNumber|) "\\spad{poisson(f)} \\undocumented")) (|binomial| (((|Mapping| (|Integer|)) (|Integer|) |RationalNumber|) "\\spad{binomial(n,f)} \\undocumented")) (|uniform| (((|Mapping| (|Integer|)) (|Segment| (|Integer|))) "\\spad{uniform(s)} \\undocumented"))) 
 NIL 
 NIL 
-(-1052 S) 
+(-1054 S) 
 ((|constructor| (NIL "The category of rings with unity,{} always associative,{} but not necessarily commutative.")) (|unitsKnown| ((|attribute|) "recip truly yields reciprocal or \"failed\" if not a unit. Note: \\spad{recip(0) = \"failed\"}.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring this is the smallest positive integer \\spad{n} such that \\spad{n*x=0} for all \\spad{x} in the ring,{} or zero if no such \\spad{n} exists."))) 
 NIL 
 NIL 
-(-1053) 
+(-1055) 
 ((|constructor| (NIL "The category of rings with unity,{} always associative,{} but not necessarily commutative.")) (|unitsKnown| ((|attribute|) "recip truly yields reciprocal or \"failed\" if not a unit. Note: \\spad{recip(0) = \"failed\"}.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring this is the smallest positive integer \\spad{n} such that \\spad{n*x=0} for all \\spad{x} in the ring,{} or zero if no such \\spad{n} exists."))) 
-((-4424 . T)) 
+((-4431 . T)) 
 NIL 
-(-1054 |xx| -3498) 
+(-1056 |xx| -3505) 
 ((|constructor| (NIL "This package exports rational interpolation algorithms"))) 
 NIL 
 NIL 
-(-1055 R) 
+(-1057 R) 
 ((|constructor| (NIL "\\indented{2}{A set is an \\spad{R}-right linear set if it is stable by right-dilation} \\indented{2}{by elements in the ring \\spad{R}.\\space{2}This category differs from} \\indented{2}{\\spad{RightModule} in that no other assumption (such as addition)} \\indented{2}{is made about the underlying set.} See Also: LeftLinearSet.")) (* (($ $ |#1|) "\\spad{r*x} is the left-dilation of \\spad{x} by \\spad{r}.")) (|zero?| (((|Boolean|) $) "\\spad{zero? x} holds is \\spad{x} is the origin.")) ((|Zero|) (($) "\\spad{0} represents the origin of the linear set"))) 
 NIL 
 NIL 
-(-1056 S |m| |n| R |Row| |Col|) 
+(-1058 S |m| |n| R |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \\spad{R}-modules and will be non-mutable.")) (|nullSpace| (((|List| |#6|) $) "\\spad{nullSpace(m)}+ returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#4|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#4|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (|map| (($ (|Mapping| |#4| |#4| |#4|) $ $) "\\spad{map(f,a,b)} returns \\spad{c},{} where \\spad{c} is such that \\spad{c(i,j) = f(a(i,j),b(i,j))} for all \\spad{i},{} \\spad{j}.") (($ (|Mapping| |#4| |#4|) $) "\\spad{map(f,a)} returns \\spad{b},{} where \\spad{b(i,j) = a(i,j)} for all \\spad{i},{} \\spad{j}.")) (|column| ((|#6| $ (|Integer|)) "\\spad{column(m,j)} returns the \\spad{j}th column of the matrix \\spad{m}. Error: if the index outside the proper range.")) (|row| ((|#5| $ (|Integer|)) "\\spad{row(m,i)} returns the \\spad{i}th row of the matrix \\spad{m}. Error: if the index is outside the proper range.")) (|qelt| ((|#4| $ (|Integer|) (|Integer|)) "\\spad{qelt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Note: there is NO error check to determine if indices are in the proper ranges.")) (|elt| ((|#4| $ (|Integer|) (|Integer|) |#4|) "\\spad{elt(m,i,j,r)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{} and returns \\spad{r} otherwise.") ((|#4| $ (|Integer|) (|Integer|)) "\\spad{elt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Error: if indices are outside the proper ranges.")) (|listOfLists| (((|List| (|List| |#4|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|ncols| (((|NonNegativeInteger|) $) "\\spad{ncols(m)} returns the number of columns in the matrix \\spad{m}.")) (|nrows| (((|NonNegativeInteger|) $) "\\spad{nrows(m)} returns the number of rows in the matrix \\spad{m}.")) (|maxColIndex| (((|Integer|) $) "\\spad{maxColIndex(m)} returns the index of the 'last' column of the matrix \\spad{m}.")) (|minColIndex| (((|Integer|) $) "\\spad{minColIndex(m)} returns the index of the 'first' column of the matrix \\spad{m}.")) (|maxRowIndex| (((|Integer|) $) "\\spad{maxRowIndex(m)} returns the index of the 'last' row of the matrix \\spad{m}.")) (|minRowIndex| (((|Integer|) $) "\\spad{minRowIndex(m)} returns the index of the 'first' row of the matrix \\spad{m}.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|matrix| (($ (|List| (|List| |#4|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|finiteAggregate| ((|attribute|) "matrices are finite"))) 
 NIL 
-((|HasCategory| |#4| (QUOTE (-309))) (|HasCategory| |#4| (QUOTE (-366))) (|HasCategory| |#4| (QUOTE (-561))) (|HasCategory| |#4| (QUOTE (-173)))) 
-(-1057 |m| |n| R |Row| |Col|) 
+((|HasCategory| |#4| (QUOTE (-310))) (|HasCategory| |#4| (QUOTE (-367))) (|HasCategory| |#4| (QUOTE (-562))) (|HasCategory| |#4| (QUOTE (-173)))) 
+(-1059 |m| |n| R |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \\spad{R}-modules and will be non-mutable.")) (|nullSpace| (((|List| |#5|) $) "\\spad{nullSpace(m)}+ returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#3|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#3|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (|map| (($ (|Mapping| |#3| |#3| |#3|) $ $) "\\spad{map(f,a,b)} returns \\spad{c},{} where \\spad{c} is such that \\spad{c(i,j) = f(a(i,j),b(i,j))} for all \\spad{i},{} \\spad{j}.") (($ (|Mapping| |#3| |#3|) $) "\\spad{map(f,a)} returns \\spad{b},{} where \\spad{b(i,j) = a(i,j)} for all \\spad{i},{} \\spad{j}.")) (|column| ((|#5| $ (|Integer|)) "\\spad{column(m,j)} returns the \\spad{j}th column of the matrix \\spad{m}. Error: if the index outside the proper range.")) (|row| ((|#4| $ (|Integer|)) "\\spad{row(m,i)} returns the \\spad{i}th row of the matrix \\spad{m}. Error: if the index is outside the proper range.")) (|qelt| ((|#3| $ (|Integer|) (|Integer|)) "\\spad{qelt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Note: there is NO error check to determine if indices are in the proper ranges.")) (|elt| ((|#3| $ (|Integer|) (|Integer|) |#3|) "\\spad{elt(m,i,j,r)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{} and returns \\spad{r} otherwise.") ((|#3| $ (|Integer|) (|Integer|)) "\\spad{elt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Error: if indices are outside the proper ranges.")) (|listOfLists| (((|List| (|List| |#3|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|ncols| (((|NonNegativeInteger|) $) "\\spad{ncols(m)} returns the number of columns in the matrix \\spad{m}.")) (|nrows| (((|NonNegativeInteger|) $) "\\spad{nrows(m)} returns the number of rows in the matrix \\spad{m}.")) (|maxColIndex| (((|Integer|) $) "\\spad{maxColIndex(m)} returns the index of the 'last' column of the matrix \\spad{m}.")) (|minColIndex| (((|Integer|) $) "\\spad{minColIndex(m)} returns the index of the 'first' column of the matrix \\spad{m}.")) (|maxRowIndex| (((|Integer|) $) "\\spad{maxRowIndex(m)} returns the index of the 'last' row of the matrix \\spad{m}.")) (|minRowIndex| (((|Integer|) $) "\\spad{minRowIndex(m)} returns the index of the 'first' row of the matrix \\spad{m}.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|matrix| (($ (|List| (|List| |#3|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|finiteAggregate| ((|attribute|) "matrices are finite"))) 
-((-4427 . T) (-4422 . T) (-4421 . T)) 
+((-4434 . T) (-4429 . T) (-4428 . T)) 
 NIL 
-(-1058 |m| |n| R) 
+(-1060 |m| |n| R) 
 ((|constructor| (NIL "\\spadtype{RectangularMatrix} is a matrix domain where the number of rows and the number of columns are parameters of the domain.")) (|rectangularMatrix| (($ (|Matrix| |#3|)) "\\spad{rectangularMatrix(m)} converts a matrix of type \\spadtype{Matrix} to a matrix of type \\spad{RectangularMatrix}."))) 
-((-4427 . T) (-4422 . T) (-4421 . T)) 
-((|HasCategory| |#3| (QUOTE (-173))) (-3962 (-12 (|HasCategory| |#3| (QUOTE (-173))) (|HasCategory| |#3| (LIST (QUOTE -311) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-366))) (|HasCategory| |#3| (LIST (QUOTE -311) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (LIST (QUOTE -311) (|devaluate| |#3|))))) (|HasCategory| |#3| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#3| (QUOTE (-173))) (|HasCategory| |#3| (QUOTE (-366)))) (|HasCategory| |#3| (QUOTE (-366))) (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (QUOTE (-309))) (|HasCategory| |#3| (QUOTE (-561))) (-12 (|HasCategory| |#3| (QUOTE (-1105))) (|HasCategory| |#3| (LIST (QUOTE -311) (|devaluate| |#3|)))) (|HasCategory| |#3| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1059 |m| |n| R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) 
+((-4434 . T) (-4429 . T) (-4428 . T)) 
+((|HasCategory| |#3| (QUOTE (-173))) (-3969 (-12 (|HasCategory| |#3| (QUOTE (-173))) (|HasCategory| |#3| (LIST (QUOTE -312) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-367))) (|HasCategory| |#3| (LIST (QUOTE -312) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -312) (|devaluate| |#3|))))) (|HasCategory| |#3| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#3| (QUOTE (-173))) (|HasCategory| |#3| (QUOTE (-367)))) (|HasCategory| |#3| (QUOTE (-367))) (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (QUOTE (-310))) (|HasCategory| |#3| (QUOTE (-562))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -312) (|devaluate| |#3|)))) (|HasCategory| |#3| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1061 |m| |n| R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) 
 ((|constructor| (NIL "\\spadtype{RectangularMatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \\spadfun{map} and \\spadfun{reduce}.")) (|reduce| ((|#7| (|Mapping| |#7| |#3| |#7|) |#6| |#7|) "\\spad{reduce(f,m,r)} returns a matrix \\spad{n} where \\spad{n[i,j] = f(m[i,j],r)} for all indices spad{\\spad{i}} and \\spad{j}.")) (|map| ((|#10| (|Mapping| |#7| |#3|) |#6|) "\\spad{map(f,m)} applies the function \\spad{f} to the elements of the matrix \\spad{m}."))) 
 NIL 
 NIL 
-(-1060 R) 
+(-1062 R) 
 ((|constructor| (NIL "The category of right modules over an \\spad{rng} (ring not necessarily with unit). This is an abelian group which supports right multiplation by elements of the \\spad{rng}. \\blankline"))) 
 NIL 
 NIL 
-(-1061) 
+(-1063) 
 ((|constructor| (NIL "The category of associative rings,{} not necessarily commutative,{} and not necessarily with a 1. This is a combination of an abelian group and a semigroup,{} with multiplication distributing over addition. \\blankline"))) 
 NIL 
 NIL 
-(-1062 S T$) 
+(-1064 S T$) 
 ((|constructor| (NIL "This domain represents the notion of binding a variable to range over a specific segment (either bounded,{} or half bounded).")) (|segment| ((|#1| $) "\\spad{segment(x)} returns the segment from the right hand side of the \\spadtype{RangeBinding}. For example,{} if \\spad{x} is \\spad{v=s},{} then \\spad{segment(x)} returns \\spad{s}.")) (|variable| (((|Symbol|) $) "\\spad{variable(x)} returns the variable from the left hand side of the \\spadtype{RangeBinding}. For example,{} if \\spad{x} is \\spad{v=s},{} then \\spad{variable(x)} returns \\spad{v}.")) (|equation| (($ (|Symbol|) |#1|) "\\spad{equation(v,s)} creates a segment binding value with variable \\spad{v} and segment \\spad{s}. Note that the interpreter parses \\spad{v=s} to this form."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-1105)))) 
-(-1063 S) 
+((|HasCategory| |#1| (QUOTE (-1107)))) 
+(-1065 S) 
 ((|constructor| (NIL "The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \\spadtype{DifferentialRing} or the elementary functions (see \\spadtype{TranscendentalFunctionCategory}) in the definition.")) (|abs| (($ $) "\\spad{abs x} returns the absolute value of \\spad{x}.")) (|round| (($ $) "\\spad{round x} computes the integer closest to \\spad{x}.")) (|truncate| (($ $) "\\spad{truncate x} returns the integer between \\spad{x} and 0 closest to \\spad{x}.")) (|fractionPart| (($ $) "\\spad{fractionPart x} returns the fractional part of \\spad{x}.")) (|wholePart| (((|Integer|) $) "\\spad{wholePart x} returns the integer part of \\spad{x}.")) (|floor| (($ $) "\\spad{floor x} returns the largest integer \\spad{<= x}.")) (|ceiling| (($ $) "\\spad{ceiling x} returns the small integer \\spad{>= x}.")) (|norm| (($ $) "\\spad{norm x} returns the same as absolute value."))) 
 NIL 
 NIL 
-(-1064) 
+(-1066) 
 ((|constructor| (NIL "The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \\spadtype{DifferentialRing} or the elementary functions (see \\spadtype{TranscendentalFunctionCategory}) in the definition.")) (|abs| (($ $) "\\spad{abs x} returns the absolute value of \\spad{x}.")) (|round| (($ $) "\\spad{round x} computes the integer closest to \\spad{x}.")) (|truncate| (($ $) "\\spad{truncate x} returns the integer between \\spad{x} and 0 closest to \\spad{x}.")) (|fractionPart| (($ $) "\\spad{fractionPart x} returns the fractional part of \\spad{x}.")) (|wholePart| (((|Integer|) $) "\\spad{wholePart x} returns the integer part of \\spad{x}.")) (|floor| (($ $) "\\spad{floor x} returns the largest integer \\spad{<= x}.")) (|ceiling| (($ $) "\\spad{ceiling x} returns the small integer \\spad{>= x}.")) (|norm| (($ $) "\\spad{norm x} returns the same as absolute value."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1065 |TheField| |ThePolDom|) 
+(-1067 |TheField| |ThePolDom|) 
 ((|constructor| (NIL "\\axiomType{RightOpenIntervalRootCharacterization} provides work with interval root coding.")) (|relativeApprox| ((|#1| |#2| $ |#1|) "\\axiom{relativeApprox(exp,{}\\spad{c},{}\\spad{p}) = a} is relatively close to exp as a polynomial in \\spad{c} ip to precision \\spad{p}")) (|mightHaveRoots| (((|Boolean|) |#2| $) "\\axiom{mightHaveRoots(\\spad{p},{}\\spad{r})} is \\spad{false} if \\axiom{\\spad{p}.\\spad{r}} is not 0")) (|refine| (($ $) "\\axiom{refine(rootChar)} shrinks isolating interval around \\axiom{rootChar}")) (|middle| ((|#1| $) "\\axiom{middle(rootChar)} is the middle of the isolating interval")) (|size| ((|#1| $) "The size of the isolating interval")) (|right| ((|#1| $) "\\axiom{right(rootChar)} is the right bound of the isolating interval")) (|left| ((|#1| $) "\\axiom{left(rootChar)} is the left bound of the isolating interval"))) 
 NIL 
 NIL 
-(-1066) 
+(-1068) 
 ((|constructor| (NIL "\\spadtype{RomanNumeral} provides functions for converting \\indented{1}{integers to roman numerals.}")) (|roman| (($ (|Integer|)) "\\spad{roman(n)} creates a roman numeral for \\spad{n}.") (($ (|Symbol|)) "\\spad{roman(n)} creates a roman numeral for symbol \\spad{n}.")) (|noetherian| ((|attribute|) "ascending chain condition on ideals.")) (|canonicalsClosed| ((|attribute|) "two positives multiply to give positive.")) (|canonical| ((|attribute|) "mathematical equality is data structure equality."))) 
-((-4415 . T) (-4419 . T) (-4414 . T) (-4425 . T) (-4426 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4422 . T) (-4426 . T) (-4421 . T) (-4432 . T) (-4433 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1067) 
+(-1069) 
 ((|constructor| (NIL "\\axiomType{RoutinesTable} implements a database and associated tuning mechanisms for a set of known NAG routines")) (|recoverAfterFail| (((|Union| (|String|) "failed") $ (|String|) (|Integer|)) "\\spad{recoverAfterFail(routs,routineName,ifailValue)} acts on the instructions given by the ifail list")) (|showTheRoutinesTable| (($) "\\spad{showTheRoutinesTable()} returns the current table of NAG routines.")) (|deleteRoutine!| (($ $ (|Symbol|)) "\\spad{deleteRoutine!(R,s)} destructively deletes the given routine from the current database of NAG routines")) (|getExplanations| (((|List| (|String|)) $ (|String|)) "\\spad{getExplanations(R,s)} gets the explanations of the output parameters for the given NAG routine.")) (|getMeasure| (((|Float|) $ (|Symbol|)) "\\spad{getMeasure(R,s)} gets the current value of the maximum measure for the given NAG routine.")) (|changeMeasure| (($ $ (|Symbol|) (|Float|)) "\\spad{changeMeasure(R,s,newValue)} changes the maximum value for a measure of the given NAG routine.")) (|changeThreshhold| (($ $ (|Symbol|) (|Float|)) "\\spad{changeThreshhold(R,s,newValue)} changes the value below which,{} given a NAG routine generating a higher measure,{} the routines will make no attempt to generate a measure.")) (|selectMultiDimensionalRoutines| (($ $) "\\spad{selectMultiDimensionalRoutines(R)} chooses only those routines from the database which are designed for use with multi-dimensional expressions")) (|selectNonFiniteRoutines| (($ $) "\\spad{selectNonFiniteRoutines(R)} chooses only those routines from the database which are designed for use with non-finite expressions.")) (|selectSumOfSquaresRoutines| (($ $) "\\spad{selectSumOfSquaresRoutines(R)} chooses only those routines from the database which are designed for use with sums of squares")) (|selectFiniteRoutines| (($ $) "\\spad{selectFiniteRoutines(R)} chooses only those routines from the database which are designed for use with finite expressions")) (|selectODEIVPRoutines| (($ $) "\\spad{selectODEIVPRoutines(R)} chooses only those routines from the database which are for the solution of ODE\\spad{'s}")) (|selectPDERoutines| (($ $) "\\spad{selectPDERoutines(R)} chooses only those routines from the database which are for the solution of PDE\\spad{'s}")) (|selectOptimizationRoutines| (($ $) "\\spad{selectOptimizationRoutines(R)} chooses only those routines from the database which are for integration")) (|selectIntegrationRoutines| (($ $) "\\spad{selectIntegrationRoutines(R)} chooses only those routines from the database which are for integration")) (|routines| (($) "\\spad{routines()} initialises a database of known NAG routines")) (|concat| (($ $ $) "\\spad{concat(x,y)} merges two tables \\spad{x} and \\spad{y}"))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (QUOTE (-1181))) (LIST (QUOTE |:|) (QUOTE -2256) (QUOTE (-51)))))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105)))) (-3962 (|HasCategory| (-51) (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-51) (QUOTE (-1105))) (|HasCategory| (-51) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| (-51) (QUOTE (-1105))) (|HasCategory| (-51) (LIST (QUOTE -311) (QUOTE (-51))))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (QUOTE (-1105))) (|HasCategory| (-1181) (QUOTE (-853))) (|HasCategory| (-51) (QUOTE (-1105))) (-3962 (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-51) (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| (-51) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 (-1181)) (|:| -2256 (-51))) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1068 S R E V) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (QUOTE (-1183))) (LIST (QUOTE |:|) (QUOTE -2263) (QUOTE (-51)))))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107)))) (-3969 (|HasCategory| (-51) (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-51) (QUOTE (-1107))) (|HasCategory| (-51) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| (-51) (QUOTE (-1107))) (|HasCategory| (-51) (LIST (QUOTE -312) (QUOTE (-51))))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (QUOTE (-1107))) (|HasCategory| (-1183) (QUOTE (-855))) (|HasCategory| (-51) (QUOTE (-1107))) (-3969 (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-51) (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| (-51) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 (-1183)) (|:| -2263 (-51))) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1070 S R E V) 
 ((|constructor| (NIL "A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \\axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.")) (|mainSquareFreePart| (($ $) "\\axiom{mainSquareFreePart(\\spad{p})} returns the square free part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainPrimitivePart| (($ $) "\\axiom{mainPrimitivePart(\\spad{p})} returns the primitive part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainContent| (($ $) "\\axiom{mainContent(\\spad{p})} returns the content of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|primitivePart!| (($ $) "\\axiom{primitivePart!(\\spad{p})} replaces \\axiom{\\spad{p}} by its primitive part.")) (|gcd| ((|#2| |#2| $) "\\axiom{\\spad{gcd}(\\spad{r},{}\\spad{p})} returns the \\spad{gcd} of \\axiom{\\spad{r}} and the content of \\axiom{\\spad{p}}.")) (|nextsubResultant2| (($ $ $ $ $) "\\axiom{nextsubResultant2(\\spad{p},{}\\spad{q},{}\\spad{z},{}\\spad{s})} is the multivariate version of the operation \\axiomOpFrom{next_sousResultant2}{PseudoRemainderSequence} from the \\axiomType{PseudoRemainderSequence} constructor.")) (|LazardQuotient2| (($ $ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient2(\\spad{p},{}a,{}\\spad{b},{}\\spad{n})} returns \\axiom{(a**(\\spad{n}-1) * \\spad{p}) exquo \\spad{b**}(\\spad{n}-1)} assuming that this quotient does not fail.")) (|LazardQuotient| (($ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient(a,{}\\spad{b},{}\\spad{n})} returns \\axiom{a**n exquo \\spad{b**}(\\spad{n}-1)} assuming that this quotient does not fail.")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns the last non-zero subresultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|subResultantChain| (((|List| $) $ $) "\\axiom{subResultantChain(a,{}\\spad{b})},{} where \\axiom{a} and \\axiom{\\spad{b}} are not contant polynomials with the same main variable,{} returns the subresultant chain of \\axiom{a} and \\axiom{\\spad{b}}.")) (|resultant| (($ $ $) "\\axiom{resultant(a,{}\\spad{b})} computes the resultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{halfExtendedSubResultantGcd2(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}\\spad{cb}]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}\\spad{cb}]} otherwise produces an error.")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{halfExtendedSubResultantGcd1(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}\\spad{cb}]} otherwise produces an error.")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[ca,{}\\spad{cb},{}\\spad{r}]} such that \\axiom{\\spad{r}} is \\axiom{subResultantGcd(a,{}\\spad{b})} and we have \\axiom{ca * a + \\spad{cb} * \\spad{cb} = \\spad{r}} .")) (|subResultantGcd| (($ $ $) "\\axiom{subResultantGcd(a,{}\\spad{b})} computes a \\spad{gcd} of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}} with coefficients in the fraction field of the polynomial ring generated by their other variables over \\axiom{\\spad{R}}.")) (|exactQuotient!| (($ $ $) "\\axiom{exactQuotient!(a,{}\\spad{b})} replaces \\axiom{a} by \\axiom{exactQuotient(a,{}\\spad{b})}") (($ $ |#2|) "\\axiom{exactQuotient!(\\spad{p},{}\\spad{r})} replaces \\axiom{\\spad{p}} by \\axiom{exactQuotient(\\spad{p},{}\\spad{r})}.")) (|exactQuotient| (($ $ $) "\\axiom{exactQuotient(a,{}\\spad{b})} computes the exact quotient of \\axiom{a} by \\axiom{\\spad{b}},{} which is assumed to be a divisor of \\axiom{a}. No error is returned if this exact quotient fails!") (($ $ |#2|) "\\axiom{exactQuotient(\\spad{p},{}\\spad{r})} computes the exact quotient of \\axiom{\\spad{p}} by \\axiom{\\spad{r}},{} which is assumed to be a divisor of \\axiom{\\spad{p}}. No error is returned if this exact quotient fails!")) (|primPartElseUnitCanonical!| (($ $) "\\axiom{primPartElseUnitCanonical!(\\spad{p})} replaces \\axiom{\\spad{p}} by \\axiom{primPartElseUnitCanonical(\\spad{p})}.")) (|primPartElseUnitCanonical| (($ $) "\\axiom{primPartElseUnitCanonical(\\spad{p})} returns \\axiom{primitivePart(\\spad{p})} if \\axiom{\\spad{R}} is a \\spad{gcd}-domain,{} otherwise \\axiom{unitCanonical(\\spad{p})}.")) (|convert| (($ (|Polynomial| |#2|)) "\\axiom{convert(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}},{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.")) (|retract| (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.")) (|initiallyReduce| (($ $ $) "\\axiom{initiallyReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|headReduce| (($ $ $) "\\axiom{headReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| $) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{p},{}\\spad{q},{}\\spad{n}]} where \\axiom{\\spad{p} / q**n} represents the residue class of \\axiom{a} modulo \\axiom{\\spad{b}} and \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{q}} is \\axiom{init(\\spad{b})}.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} computes \\axiom{a mod \\spad{b}},{} if \\axiom{\\spad{b}} is monic as univariate polynomial in its main variable.")) (|pseudoDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{pseudoDivide(a,{}\\spad{b})} computes \\axiom{[pquo(a,{}\\spad{b}),{}prem(a,{}\\spad{b})]},{} both polynomials viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}},{} if \\axiom{\\spad{b}} is not a constant polynomial.")) (|lazyPseudoDivide| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $ |#4|) "\\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})},{} \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}] = lazyPremWithDefault(a,{}\\spad{b})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.")) (|lazyPremWithDefault| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $ |#4|) "\\axiom{lazyPremWithDefault(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b},{}\\spad{v})}.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $) "\\axiom{lazyPremWithDefault(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b})} and \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b})}.")) (|lazyPquo| (($ $ $ |#4|) "\\axiom{lazyPquo(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.") (($ $ $) "\\axiom{lazyPquo(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.")) (|lazyPrem| (($ $ $ |#4|) "\\axiom{lazyPrem(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} viewed as univariate polynomials in the variable \\axiom{\\spad{v}} such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.") (($ $ $) "\\axiom{lazyPrem(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.")) (|pquo| (($ $ $ |#4|) "\\axiom{pquo(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{pquo(a,{}\\spad{b})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|prem| (($ $ $ |#4|) "\\axiom{prem(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{prem(a,{}\\spad{b})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|normalized?| (((|Boolean|) $ (|List| $)) "\\axiom{normalized?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{normalized?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{normalized?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{a} and its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variable of \\axiom{\\spad{b}}")) (|initiallyReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{initiallyReduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{initiallyReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{initiallyReduced?(a,{}\\spad{b})} returns \\spad{false} iff there exists an iterated initial of \\axiom{a} which is not reduced \\spad{w}.\\spad{r}.\\spad{t} \\axiom{\\spad{b}}.")) (|headReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{headReduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{headReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{headReduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(head(a),{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|reduced?| (((|Boolean|) $ (|List| $)) "\\axiom{reduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{reduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{reduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(a,{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|supRittWu?| (((|Boolean|) $ $) "\\axiom{supRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is greater than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is less than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|RittWuCompare| (((|Union| (|Boolean|) "failed") $ $) "\\axiom{RittWuCompare(a,{}\\spad{b})} returns \\axiom{\"failed\"} if \\axiom{a} and \\axiom{\\spad{b}} have same rank \\spad{w}.\\spad{r}.\\spad{t}. Ritt and Wu Wen Tsun ordering using the refinement of Lazard,{} otherwise returns \\axiom{infRittWu?(a,{}\\spad{b})}.")) (|mainMonomials| (((|List| $) $) "\\axiom{mainMonomials(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [1],{} otherwise returns the list of the monomials of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainCoefficients| (((|List| $) $) "\\axiom{mainCoefficients(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [\\spad{p}],{} otherwise returns the list of the coefficients of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|leastMonomial| (($ $) "\\axiom{leastMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} the monomial of \\axiom{\\spad{p}} with lowest degree,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainMonomial| (($ $) "\\axiom{mainMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} \\axiom{mvar(\\spad{p})} raised to the power \\axiom{mdeg(\\spad{p})}.")) (|quasiMonic?| (((|Boolean|) $) "\\axiom{quasiMonic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff the initial of \\axiom{\\spad{p}} lies in the base ring \\axiom{\\spad{R}}.")) (|monic?| (((|Boolean|) $) "\\axiom{monic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff \\axiom{\\spad{p}} is monic as a univariate polynomial in its main variable.")) (|reductum| (($ $ |#4|) "\\axiom{reductum(\\spad{p},{}\\spad{v})} returns the reductum of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in \\axiom{\\spad{v}}.")) (|leadingCoefficient| (($ $ |#4|) "\\axiom{leadingCoefficient(\\spad{p},{}\\spad{v})} returns the leading coefficient of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as A univariate polynomial in \\axiom{\\spad{v}}.")) (|deepestInitial| (($ $) "\\axiom{deepestInitial(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the last term of \\axiom{iteratedInitials(\\spad{p})}.")) (|iteratedInitials| (((|List| $) $) "\\axiom{iteratedInitials(\\spad{p})} returns \\axiom{[]} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the list of the iterated initials of \\axiom{\\spad{p}}.")) (|deepestTail| (($ $) "\\axiom{deepestTail(\\spad{p})} returns \\axiom{0} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns tail(\\spad{p}),{} if \\axiom{tail(\\spad{p})} belongs to \\axiom{\\spad{R}} or \\axiom{mvar(tail(\\spad{p})) < mvar(\\spad{p})},{} otherwise returns \\axiom{deepestTail(tail(\\spad{p}))}.")) (|tail| (($ $) "\\axiom{tail(\\spad{p})} returns its reductum,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|head| (($ $) "\\axiom{head(\\spad{p})} returns \\axiom{\\spad{p}} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading term (monomial in the AXIOM sense),{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|init| (($ $) "\\axiom{init(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading coefficient,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mdeg| (((|NonNegativeInteger|) $) "\\axiom{mdeg(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{0},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{0},{} otherwise,{} returns the degree of \\axiom{\\spad{p}} in its main variable.")) (|mvar| ((|#4| $) "\\axiom{mvar(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in \\axiom{\\spad{V}}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-549))) (|HasCategory| |#2| (LIST (QUOTE -38) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -995) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#4| (LIST (QUOTE -617) (QUOTE (-1181))))) 
-(-1069 R E V) 
+((|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-550))) (|HasCategory| |#2| (LIST (QUOTE -38) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -997) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-1183))))) 
+(-1071 R E V) 
 ((|constructor| (NIL "A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \\axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.")) (|mainSquareFreePart| (($ $) "\\axiom{mainSquareFreePart(\\spad{p})} returns the square free part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainPrimitivePart| (($ $) "\\axiom{mainPrimitivePart(\\spad{p})} returns the primitive part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainContent| (($ $) "\\axiom{mainContent(\\spad{p})} returns the content of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|primitivePart!| (($ $) "\\axiom{primitivePart!(\\spad{p})} replaces \\axiom{\\spad{p}} by its primitive part.")) (|gcd| ((|#1| |#1| $) "\\axiom{\\spad{gcd}(\\spad{r},{}\\spad{p})} returns the \\spad{gcd} of \\axiom{\\spad{r}} and the content of \\axiom{\\spad{p}}.")) (|nextsubResultant2| (($ $ $ $ $) "\\axiom{nextsubResultant2(\\spad{p},{}\\spad{q},{}\\spad{z},{}\\spad{s})} is the multivariate version of the operation \\axiomOpFrom{next_sousResultant2}{PseudoRemainderSequence} from the \\axiomType{PseudoRemainderSequence} constructor.")) (|LazardQuotient2| (($ $ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient2(\\spad{p},{}a,{}\\spad{b},{}\\spad{n})} returns \\axiom{(a**(\\spad{n}-1) * \\spad{p}) exquo \\spad{b**}(\\spad{n}-1)} assuming that this quotient does not fail.")) (|LazardQuotient| (($ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient(a,{}\\spad{b},{}\\spad{n})} returns \\axiom{a**n exquo \\spad{b**}(\\spad{n}-1)} assuming that this quotient does not fail.")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns the last non-zero subresultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|subResultantChain| (((|List| $) $ $) "\\axiom{subResultantChain(a,{}\\spad{b})},{} where \\axiom{a} and \\axiom{\\spad{b}} are not contant polynomials with the same main variable,{} returns the subresultant chain of \\axiom{a} and \\axiom{\\spad{b}}.")) (|resultant| (($ $ $) "\\axiom{resultant(a,{}\\spad{b})} computes the resultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{halfExtendedSubResultantGcd2(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}\\spad{cb}]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}\\spad{cb}]} otherwise produces an error.")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{halfExtendedSubResultantGcd1(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}\\spad{cb}]} otherwise produces an error.")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[ca,{}\\spad{cb},{}\\spad{r}]} such that \\axiom{\\spad{r}} is \\axiom{subResultantGcd(a,{}\\spad{b})} and we have \\axiom{ca * a + \\spad{cb} * \\spad{cb} = \\spad{r}} .")) (|subResultantGcd| (($ $ $) "\\axiom{subResultantGcd(a,{}\\spad{b})} computes a \\spad{gcd} of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}} with coefficients in the fraction field of the polynomial ring generated by their other variables over \\axiom{\\spad{R}}.")) (|exactQuotient!| (($ $ $) "\\axiom{exactQuotient!(a,{}\\spad{b})} replaces \\axiom{a} by \\axiom{exactQuotient(a,{}\\spad{b})}") (($ $ |#1|) "\\axiom{exactQuotient!(\\spad{p},{}\\spad{r})} replaces \\axiom{\\spad{p}} by \\axiom{exactQuotient(\\spad{p},{}\\spad{r})}.")) (|exactQuotient| (($ $ $) "\\axiom{exactQuotient(a,{}\\spad{b})} computes the exact quotient of \\axiom{a} by \\axiom{\\spad{b}},{} which is assumed to be a divisor of \\axiom{a}. No error is returned if this exact quotient fails!") (($ $ |#1|) "\\axiom{exactQuotient(\\spad{p},{}\\spad{r})} computes the exact quotient of \\axiom{\\spad{p}} by \\axiom{\\spad{r}},{} which is assumed to be a divisor of \\axiom{\\spad{p}}. No error is returned if this exact quotient fails!")) (|primPartElseUnitCanonical!| (($ $) "\\axiom{primPartElseUnitCanonical!(\\spad{p})} replaces \\axiom{\\spad{p}} by \\axiom{primPartElseUnitCanonical(\\spad{p})}.")) (|primPartElseUnitCanonical| (($ $) "\\axiom{primPartElseUnitCanonical(\\spad{p})} returns \\axiom{primitivePart(\\spad{p})} if \\axiom{\\spad{R}} is a \\spad{gcd}-domain,{} otherwise \\axiom{unitCanonical(\\spad{p})}.")) (|convert| (($ (|Polynomial| |#1|)) "\\axiom{convert(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}},{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.")) (|retract| (($ (|Polynomial| |#1|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#1|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#1|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|Polynomial| |#1|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#1|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#1|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.")) (|initiallyReduce| (($ $ $) "\\axiom{initiallyReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|headReduce| (($ $ $) "\\axiom{headReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| $) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{p},{}\\spad{q},{}\\spad{n}]} where \\axiom{\\spad{p} / q**n} represents the residue class of \\axiom{a} modulo \\axiom{\\spad{b}} and \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{q}} is \\axiom{init(\\spad{b})}.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} computes \\axiom{a mod \\spad{b}},{} if \\axiom{\\spad{b}} is monic as univariate polynomial in its main variable.")) (|pseudoDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{pseudoDivide(a,{}\\spad{b})} computes \\axiom{[pquo(a,{}\\spad{b}),{}prem(a,{}\\spad{b})]},{} both polynomials viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}},{} if \\axiom{\\spad{b}} is not a constant polynomial.")) (|lazyPseudoDivide| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $ |#3|) "\\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})},{} \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}] = lazyPremWithDefault(a,{}\\spad{b})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.")) (|lazyPremWithDefault| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $ |#3|) "\\axiom{lazyPremWithDefault(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b},{}\\spad{v})}.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $) "\\axiom{lazyPremWithDefault(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b})} and \\axiom{(c**g)\\spad{*r} = prem(a,{}\\spad{b})}.")) (|lazyPquo| (($ $ $ |#3|) "\\axiom{lazyPquo(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.") (($ $ $) "\\axiom{lazyPquo(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.")) (|lazyPrem| (($ $ $ |#3|) "\\axiom{lazyPrem(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} viewed as univariate polynomials in the variable \\axiom{\\spad{v}} such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.") (($ $ $) "\\axiom{lazyPrem(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.")) (|pquo| (($ $ $ |#3|) "\\axiom{pquo(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{pquo(a,{}\\spad{b})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|prem| (($ $ $ |#3|) "\\axiom{prem(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{prem(a,{}\\spad{b})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|normalized?| (((|Boolean|) $ (|List| $)) "\\axiom{normalized?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{normalized?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{normalized?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{a} and its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variable of \\axiom{\\spad{b}}")) (|initiallyReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{initiallyReduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{initiallyReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{initiallyReduced?(a,{}\\spad{b})} returns \\spad{false} iff there exists an iterated initial of \\axiom{a} which is not reduced \\spad{w}.\\spad{r}.\\spad{t} \\axiom{\\spad{b}}.")) (|headReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{headReduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{headReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{headReduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(head(a),{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|reduced?| (((|Boolean|) $ (|List| $)) "\\axiom{reduced?(\\spad{q},{}\\spad{lp})} returns \\spad{true} iff \\axiom{reduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{\\spad{lp}}.") (((|Boolean|) $ $) "\\axiom{reduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(a,{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|supRittWu?| (((|Boolean|) $ $) "\\axiom{supRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is greater than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is less than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|RittWuCompare| (((|Union| (|Boolean|) "failed") $ $) "\\axiom{RittWuCompare(a,{}\\spad{b})} returns \\axiom{\"failed\"} if \\axiom{a} and \\axiom{\\spad{b}} have same rank \\spad{w}.\\spad{r}.\\spad{t}. Ritt and Wu Wen Tsun ordering using the refinement of Lazard,{} otherwise returns \\axiom{infRittWu?(a,{}\\spad{b})}.")) (|mainMonomials| (((|List| $) $) "\\axiom{mainMonomials(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [1],{} otherwise returns the list of the monomials of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainCoefficients| (((|List| $) $) "\\axiom{mainCoefficients(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [\\spad{p}],{} otherwise returns the list of the coefficients of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|leastMonomial| (($ $) "\\axiom{leastMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} the monomial of \\axiom{\\spad{p}} with lowest degree,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainMonomial| (($ $) "\\axiom{mainMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} \\axiom{mvar(\\spad{p})} raised to the power \\axiom{mdeg(\\spad{p})}.")) (|quasiMonic?| (((|Boolean|) $) "\\axiom{quasiMonic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff the initial of \\axiom{\\spad{p}} lies in the base ring \\axiom{\\spad{R}}.")) (|monic?| (((|Boolean|) $) "\\axiom{monic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff \\axiom{\\spad{p}} is monic as a univariate polynomial in its main variable.")) (|reductum| (($ $ |#3|) "\\axiom{reductum(\\spad{p},{}\\spad{v})} returns the reductum of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in \\axiom{\\spad{v}}.")) (|leadingCoefficient| (($ $ |#3|) "\\axiom{leadingCoefficient(\\spad{p},{}\\spad{v})} returns the leading coefficient of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as A univariate polynomial in \\axiom{\\spad{v}}.")) (|deepestInitial| (($ $) "\\axiom{deepestInitial(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the last term of \\axiom{iteratedInitials(\\spad{p})}.")) (|iteratedInitials| (((|List| $) $) "\\axiom{iteratedInitials(\\spad{p})} returns \\axiom{[]} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the list of the iterated initials of \\axiom{\\spad{p}}.")) (|deepestTail| (($ $) "\\axiom{deepestTail(\\spad{p})} returns \\axiom{0} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns tail(\\spad{p}),{} if \\axiom{tail(\\spad{p})} belongs to \\axiom{\\spad{R}} or \\axiom{mvar(tail(\\spad{p})) < mvar(\\spad{p})},{} otherwise returns \\axiom{deepestTail(tail(\\spad{p}))}.")) (|tail| (($ $) "\\axiom{tail(\\spad{p})} returns its reductum,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|head| (($ $) "\\axiom{head(\\spad{p})} returns \\axiom{\\spad{p}} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading term (monomial in the AXIOM sense),{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|init| (($ $) "\\axiom{init(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading coefficient,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mdeg| (((|NonNegativeInteger|) $) "\\axiom{mdeg(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{0},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{0},{} otherwise,{} returns the degree of \\axiom{\\spad{p}} in its main variable.")) (|mvar| ((|#3| $) "\\axiom{mvar(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in \\axiom{\\spad{V}}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
-(-1070) 
+(-1072) 
 ((|constructor| (NIL "This domain represents the `repeat' iterator syntax.")) (|body| (((|SpadAst|) $) "\\spad{body(e)} returns the body of the loop `e'.")) (|iterators| (((|List| (|SpadAst|)) $) "\\spad{iterators(e)} returns the list of iterators controlling the loop `e'."))) 
 NIL 
 NIL 
-(-1071 S |TheField| |ThePols|) 
+(-1073 S |TheField| |ThePols|) 
 ((|constructor| (NIL "\\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.")) (|relativeApprox| ((|#2| |#3| $ |#2|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|approximate| ((|#2| |#3| $ |#2|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|rootOf| (((|Union| $ "failed") |#3| (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} gives the \\spad{n}th root for the order of the Real Closure")) (|allRootsOf| (((|List| $) |#3|) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} in the Real Closure,{} assumed in order.")) (|definingPolynomial| ((|#3| $) "\\axiom{definingPolynomial(aRoot)} gives a polynomial such that \\axiom{definingPolynomial(aRoot).aRoot = 0}")) (|recip| (((|Union| |#3| "failed") |#3| $) "\\axiom{recip(pol,{}aRoot)} tries to inverse \\axiom{pol} interpreted as \\axiom{aRoot}")) (|positive?| (((|Boolean|) |#3| $) "\\axiom{positive?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is positive")) (|negative?| (((|Boolean|) |#3| $) "\\axiom{negative?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is negative")) (|zero?| (((|Boolean|) |#3| $) "\\axiom{zero?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is \\axiom{0}")) (|sign| (((|Integer|) |#3| $) "\\axiom{sign(pol,{}aRoot)} gives the sign of \\axiom{pol} interpreted as \\axiom{aRoot}"))) 
 NIL 
 NIL 
-(-1072 |TheField| |ThePols|) 
+(-1074 |TheField| |ThePols|) 
 ((|constructor| (NIL "\\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.")) (|relativeApprox| ((|#1| |#2| $ |#1|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|approximate| ((|#1| |#2| $ |#1|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|rootOf| (((|Union| $ "failed") |#2| (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} gives the \\spad{n}th root for the order of the Real Closure")) (|allRootsOf| (((|List| $) |#2|) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} in the Real Closure,{} assumed in order.")) (|definingPolynomial| ((|#2| $) "\\axiom{definingPolynomial(aRoot)} gives a polynomial such that \\axiom{definingPolynomial(aRoot).aRoot = 0}")) (|recip| (((|Union| |#2| "failed") |#2| $) "\\axiom{recip(pol,{}aRoot)} tries to inverse \\axiom{pol} interpreted as \\axiom{aRoot}")) (|positive?| (((|Boolean|) |#2| $) "\\axiom{positive?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is positive")) (|negative?| (((|Boolean|) |#2| $) "\\axiom{negative?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is negative")) (|zero?| (((|Boolean|) |#2| $) "\\axiom{zero?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is \\axiom{0}")) (|sign| (((|Integer|) |#2| $) "\\axiom{sign(pol,{}aRoot)} gives the sign of \\axiom{pol} interpreted as \\axiom{aRoot}"))) 
 NIL 
 NIL 
-(-1073 R E V P TS) 
+(-1075 R E V P TS) 
 ((|constructor| (NIL "A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are proposed: in the sense of Zariski closure (like in Kalkbrener\\spad{'s} algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard methods). This algorithm is valid for nay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \\spad{TS} and the packages \\axiomType{QCMPACK}(\\spad{R},{}\\spad{E},{}\\spad{V},{}\\spad{P},{}\\spad{TS}) and \\axiomType{RSETGCD}(\\spad{R},{}\\spad{E},{}\\spad{V},{}\\spad{P},{}\\spad{TS}). The same way it does not care about the way univariate polynomial \\spad{gcd} (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these \\spad{gcd} need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \\axiom{\\spad{TS}}. Thus,{} the operations of this package are not documented.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) 
 NIL 
 NIL 
-(-1074 S R E V P) 
+(-1076 S R E V P) 
 ((|constructor| (NIL "The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polynomial ring \\spad{k[x1,...,xn]} where \\spad{k} is the fraction field of \\spad{R}. The triangular set \\spad{[t1,...,tm]} is regular iff for every \\spad{i} the initial of \\spad{ti+1} is invertible in the tower of simple extensions associated with \\spad{[t1,...,ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \\spad{I} iff the radical of \\spad{I} is equal to the intersection of the radical ideals generated by the saturated ideals of the \\spad{[T1,...,Ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \\spad{T} iff it is a split of Kalkbrener of the saturated ideal of \\spad{T}. Let \\spad{K} be an algebraic closure of \\spad{k}. Assume that \\spad{V} is finite with cardinality \\spad{n} and let \\spad{A} be the affine space \\spad{K^n}. For a regular triangular set \\spad{T} let denote by \\spad{W(T)} the set of regular zeros of \\spad{T}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given subset \\spad{S} of \\spad{A} iff the union of the \\spad{W(Ti)} contains \\spad{S} and is contained in the closure of \\spad{S} (\\spad{w}.\\spad{r}.\\spad{t}. Zariski topology). A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given triangular set \\spad{T} if it is a split of Lazard of \\spad{W(T)}. Note that if \\spad{[T1,...,Ts]} is a split of Lazard of \\spad{T} then it is also a split of Kalkbrener of \\spad{T}. The converse is \\spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \\newline References : \\indented{1}{[1] \\spad{M}. KALKBRENER \"Three contributions to elimination theory\"} \\indented{5}{\\spad{Phd} Thesis,{} University of Linz,{} Austria,{} 1991.} \\indented{1}{[2] \\spad{M}. KALKBRENER \"Algorithmic properties of polynomial rings\"} \\indented{5}{Journal of Symbol. Comp. 1998} \\indented{1}{[3] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)} \\indented{1}{[4] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|zeroSetSplit| (((|List| $) (|List| |#5|) (|Boolean|)) "\\spad{zeroSetSplit(lp,clos?)} returns \\spad{lts} a split of Kalkbrener of the radical ideal associated with \\spad{lp}. If \\spad{clos?} is \\spad{false},{} it is also a decomposition of the variety associated with \\spad{lp} into the regular zero set of the \\spad{ts} in \\spad{lts} (or,{} in other words,{} a split of Lazard of this variety). See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets.")) (|extend| (((|List| $) (|List| |#5|) (|List| $)) "\\spad{extend(lp,lts)} returns the same as \\spad{concat([extend(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#5|) $) "\\spad{extend(lp,ts)} returns \\spad{ts} if \\spad{empty? lp} \\spad{extend(p,ts)} if \\spad{lp = [p]} else \\spad{extend(first lp, extend(rest lp, ts))}") (((|List| $) |#5| (|List| $)) "\\spad{extend(p,lts)} returns the same as \\spad{concat([extend(p,ts) for ts in lts])|}") (((|List| $) |#5| $) "\\spad{extend(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is not a regular triangular set.")) (|internalAugment| (($ (|List| |#5|) $) "\\spad{internalAugment(lp,ts)} returns \\spad{ts} if \\spad{lp} is empty otherwise returns \\spad{internalAugment(rest lp, internalAugment(first lp, ts))}") (($ |#5| $) "\\spad{internalAugment(p,ts)} assumes that \\spad{augment(p,ts)} returns a singleton and returns it.")) (|augment| (((|List| $) (|List| |#5|) (|List| $)) "\\spad{augment(lp,lts)} returns the same as \\spad{concat([augment(lp,ts) for ts in lts])}") (((|List| $) (|List| |#5|) $) "\\spad{augment(lp,ts)} returns \\spad{ts} if \\spad{empty? lp},{} \\spad{augment(p,ts)} if \\spad{lp = [p]},{} otherwise \\spad{augment(first lp, augment(rest lp, ts))}") (((|List| $) |#5| (|List| $)) "\\spad{augment(p,lts)} returns the same as \\spad{concat([augment(p,ts) for ts in lts])}") (((|List| $) |#5| $) "\\spad{augment(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. This operation assumes also that if \\spad{p} is added to \\spad{ts} the resulting set,{} say \\spad{ts+p},{} is a regular triangular set. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is required to be square-free.")) (|intersect| (((|List| $) |#5| (|List| $)) "\\spad{intersect(p,lts)} returns the same as \\spad{intersect([p],lts)}") (((|List| $) (|List| |#5|) (|List| $)) "\\spad{intersect(lp,lts)} returns the same as \\spad{concat([intersect(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#5|) $) "\\spad{intersect(lp,ts)} returns \\spad{lts} a split of Lazard of the intersection of the affine variety associated with \\spad{lp} and the regular zero set of \\spad{ts}.") (((|List| $) |#5| $) "\\spad{intersect(p,ts)} returns the same as \\spad{intersect([p],ts)}")) (|squareFreePart| (((|List| (|Record| (|:| |val| |#5|) (|:| |tower| $))) |#5| $) "\\spad{squareFreePart(p,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a square-free polynomial \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} this polynomial being associated with \\spad{p} modulo \\spad{lpwt.i.tower},{} for every \\spad{i}. Moreover,{} the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. WARNING: This assumes that \\spad{p} is a non-constant polynomial such that if \\spad{p} is added to \\spad{ts},{} then the resulting set is a regular triangular set.")) (|lastSubResultant| (((|List| (|Record| (|:| |val| |#5|) (|:| |tower| $))) |#5| |#5| $) "\\spad{lastSubResultant(p1,p2,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a quasi-monic \\spad{gcd} of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} for every \\spad{i},{} and such that the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. Moreover,{} if \\spad{p1} and \\spad{p2} do not have a non-trivial \\spad{gcd} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower} then \\spad{lpwt.i.val} is the resultant of these polynomials \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|lastSubResultantElseSplit| (((|Union| |#5| (|List| $)) |#5| |#5| $) "\\spad{lastSubResultantElseSplit(p1,p2,ts)} returns either \\spad{g} a quasi-monic \\spad{gcd} of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. the \\spad{ts} or a split of Kalkbrener of \\spad{ts}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|invertibleSet| (((|List| $) |#5| $) "\\spad{invertibleSet(p,ts)} returns a split of Kalkbrener of the quotient ideal of the ideal \\axiom{\\spad{I}} by \\spad{p} where \\spad{I} is the radical of saturated of \\spad{ts}.")) (|invertible?| (((|Boolean|) |#5| $) "\\spad{invertible?(p,ts)} returns \\spad{true} iff \\spad{p} is invertible in the tower associated with \\spad{ts}.") (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| $))) |#5| $) "\\spad{invertible?(p,ts)} returns \\spad{lbwt} where \\spad{lbwt.i} is the result of \\spad{invertibleElseSplit?(p,lbwt.i.tower)} and the list of the \\spad{(lqrwt.i).tower} is a split of Kalkbrener of \\spad{ts}.")) (|invertibleElseSplit?| (((|Union| (|Boolean|) (|List| $)) |#5| $) "\\spad{invertibleElseSplit?(p,ts)} returns \\spad{true} (resp. \\spad{false}) if \\spad{p} is invertible in the tower associated with \\spad{ts} or returns a split of Kalkbrener of \\spad{ts}.")) (|purelyAlgebraicLeadingMonomial?| (((|Boolean|) |#5| $) "\\spad{purelyAlgebraicLeadingMonomial?(p,ts)} returns \\spad{true} iff the main variable of any non-constant iterarted initial of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|algebraicCoefficients?| (((|Boolean|) |#5| $) "\\spad{algebraicCoefficients?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} which is not the main one of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|purelyTranscendental?| (((|Boolean|) |#5| $) "\\spad{purelyTranscendental?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is not algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}")) (|purelyAlgebraic?| (((|Boolean|) $) "\\spad{purelyAlgebraic?(ts)} returns \\spad{true} iff for every algebraic variable \\spad{v} of \\spad{ts} we have \\spad{algebraicCoefficients?(t_v,ts_v_-)} where \\spad{ts_v} is \\axiomOpFrom{select}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}) and \\spad{ts_v_-} is \\axiomOpFrom{collectUnder}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}).") (((|Boolean|) |#5| $) "\\spad{purelyAlgebraic?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}."))) 
 NIL 
 NIL 
-(-1075 R E V P) 
+(-1077 R E V P) 
 ((|constructor| (NIL "The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polynomial ring \\spad{k[x1,...,xn]} where \\spad{k} is the fraction field of \\spad{R}. The triangular set \\spad{[t1,...,tm]} is regular iff for every \\spad{i} the initial of \\spad{ti+1} is invertible in the tower of simple extensions associated with \\spad{[t1,...,ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \\spad{I} iff the radical of \\spad{I} is equal to the intersection of the radical ideals generated by the saturated ideals of the \\spad{[T1,...,Ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \\spad{T} iff it is a split of Kalkbrener of the saturated ideal of \\spad{T}. Let \\spad{K} be an algebraic closure of \\spad{k}. Assume that \\spad{V} is finite with cardinality \\spad{n} and let \\spad{A} be the affine space \\spad{K^n}. For a regular triangular set \\spad{T} let denote by \\spad{W(T)} the set of regular zeros of \\spad{T}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given subset \\spad{S} of \\spad{A} iff the union of the \\spad{W(Ti)} contains \\spad{S} and is contained in the closure of \\spad{S} (\\spad{w}.\\spad{r}.\\spad{t}. Zariski topology). A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given triangular set \\spad{T} if it is a split of Lazard of \\spad{W(T)}. Note that if \\spad{[T1,...,Ts]} is a split of Lazard of \\spad{T} then it is also a split of Kalkbrener of \\spad{T}. The converse is \\spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \\newline References : \\indented{1}{[1] \\spad{M}. KALKBRENER \"Three contributions to elimination theory\"} \\indented{5}{\\spad{Phd} Thesis,{} University of Linz,{} Austria,{} 1991.} \\indented{1}{[2] \\spad{M}. KALKBRENER \"Algorithmic properties of polynomial rings\"} \\indented{5}{Journal of Symbol. Comp. 1998} \\indented{1}{[3] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)} \\indented{1}{[4] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|zeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|)) "\\spad{zeroSetSplit(lp,clos?)} returns \\spad{lts} a split of Kalkbrener of the radical ideal associated with \\spad{lp}. If \\spad{clos?} is \\spad{false},{} it is also a decomposition of the variety associated with \\spad{lp} into the regular zero set of the \\spad{ts} in \\spad{lts} (or,{} in other words,{} a split of Lazard of this variety). See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets.")) (|extend| (((|List| $) (|List| |#4|) (|List| $)) "\\spad{extend(lp,lts)} returns the same as \\spad{concat([extend(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#4|) $) "\\spad{extend(lp,ts)} returns \\spad{ts} if \\spad{empty? lp} \\spad{extend(p,ts)} if \\spad{lp = [p]} else \\spad{extend(first lp, extend(rest lp, ts))}") (((|List| $) |#4| (|List| $)) "\\spad{extend(p,lts)} returns the same as \\spad{concat([extend(p,ts) for ts in lts])|}") (((|List| $) |#4| $) "\\spad{extend(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is not a regular triangular set.")) (|internalAugment| (($ (|List| |#4|) $) "\\spad{internalAugment(lp,ts)} returns \\spad{ts} if \\spad{lp} is empty otherwise returns \\spad{internalAugment(rest lp, internalAugment(first lp, ts))}") (($ |#4| $) "\\spad{internalAugment(p,ts)} assumes that \\spad{augment(p,ts)} returns a singleton and returns it.")) (|augment| (((|List| $) (|List| |#4|) (|List| $)) "\\spad{augment(lp,lts)} returns the same as \\spad{concat([augment(lp,ts) for ts in lts])}") (((|List| $) (|List| |#4|) $) "\\spad{augment(lp,ts)} returns \\spad{ts} if \\spad{empty? lp},{} \\spad{augment(p,ts)} if \\spad{lp = [p]},{} otherwise \\spad{augment(first lp, augment(rest lp, ts))}") (((|List| $) |#4| (|List| $)) "\\spad{augment(p,lts)} returns the same as \\spad{concat([augment(p,ts) for ts in lts])}") (((|List| $) |#4| $) "\\spad{augment(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. This operation assumes also that if \\spad{p} is added to \\spad{ts} the resulting set,{} say \\spad{ts+p},{} is a regular triangular set. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is required to be square-free.")) (|intersect| (((|List| $) |#4| (|List| $)) "\\spad{intersect(p,lts)} returns the same as \\spad{intersect([p],lts)}") (((|List| $) (|List| |#4|) (|List| $)) "\\spad{intersect(lp,lts)} returns the same as \\spad{concat([intersect(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#4|) $) "\\spad{intersect(lp,ts)} returns \\spad{lts} a split of Lazard of the intersection of the affine variety associated with \\spad{lp} and the regular zero set of \\spad{ts}.") (((|List| $) |#4| $) "\\spad{intersect(p,ts)} returns the same as \\spad{intersect([p],ts)}")) (|squareFreePart| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| $))) |#4| $) "\\spad{squareFreePart(p,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a square-free polynomial \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} this polynomial being associated with \\spad{p} modulo \\spad{lpwt.i.tower},{} for every \\spad{i}. Moreover,{} the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. WARNING: This assumes that \\spad{p} is a non-constant polynomial such that if \\spad{p} is added to \\spad{ts},{} then the resulting set is a regular triangular set.")) (|lastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| $))) |#4| |#4| $) "\\spad{lastSubResultant(p1,p2,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a quasi-monic \\spad{gcd} of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} for every \\spad{i},{} and such that the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. Moreover,{} if \\spad{p1} and \\spad{p2} do not have a non-trivial \\spad{gcd} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower} then \\spad{lpwt.i.val} is the resultant of these polynomials \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|lastSubResultantElseSplit| (((|Union| |#4| (|List| $)) |#4| |#4| $) "\\spad{lastSubResultantElseSplit(p1,p2,ts)} returns either \\spad{g} a quasi-monic \\spad{gcd} of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. the \\spad{ts} or a split of Kalkbrener of \\spad{ts}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|invertibleSet| (((|List| $) |#4| $) "\\spad{invertibleSet(p,ts)} returns a split of Kalkbrener of the quotient ideal of the ideal \\axiom{\\spad{I}} by \\spad{p} where \\spad{I} is the radical of saturated of \\spad{ts}.")) (|invertible?| (((|Boolean|) |#4| $) "\\spad{invertible?(p,ts)} returns \\spad{true} iff \\spad{p} is invertible in the tower associated with \\spad{ts}.") (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| $))) |#4| $) "\\spad{invertible?(p,ts)} returns \\spad{lbwt} where \\spad{lbwt.i} is the result of \\spad{invertibleElseSplit?(p,lbwt.i.tower)} and the list of the \\spad{(lqrwt.i).tower} is a split of Kalkbrener of \\spad{ts}.")) (|invertibleElseSplit?| (((|Union| (|Boolean|) (|List| $)) |#4| $) "\\spad{invertibleElseSplit?(p,ts)} returns \\spad{true} (resp. \\spad{false}) if \\spad{p} is invertible in the tower associated with \\spad{ts} or returns a split of Kalkbrener of \\spad{ts}.")) (|purelyAlgebraicLeadingMonomial?| (((|Boolean|) |#4| $) "\\spad{purelyAlgebraicLeadingMonomial?(p,ts)} returns \\spad{true} iff the main variable of any non-constant iterarted initial of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|algebraicCoefficients?| (((|Boolean|) |#4| $) "\\spad{algebraicCoefficients?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} which is not the main one of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|purelyTranscendental?| (((|Boolean|) |#4| $) "\\spad{purelyTranscendental?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is not algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}")) (|purelyAlgebraic?| (((|Boolean|) $) "\\spad{purelyAlgebraic?(ts)} returns \\spad{true} iff for every algebraic variable \\spad{v} of \\spad{ts} we have \\spad{algebraicCoefficients?(t_v,ts_v_-)} where \\spad{ts_v} is \\axiomOpFrom{select}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}) and \\spad{ts_v_-} is \\axiomOpFrom{collectUnder}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}).") (((|Boolean|) |#4| $) "\\spad{purelyAlgebraic?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-1076 R E V P TS) 
+(-1078 R E V P TS) 
 ((|constructor| (NIL "An internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of \\spad{gcd} over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of AAECC11} \\indented{5}{Paris,{} 1995.} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|toseSquareFreePart| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{toseSquareFreePart(\\spad{p},{}\\spad{ts})} has the same specifications as \\axiomOpFrom{squareFreePart}{RegularTriangularSetCategory}.")) (|toseInvertibleSet| (((|List| |#5|) |#4| |#5|) "\\axiom{toseInvertibleSet(\\spad{p1},{}\\spad{p2},{}\\spad{ts})} has the same specifications as \\axiomOpFrom{invertibleSet}{RegularTriangularSetCategory}.")) (|toseInvertible?| (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{toseInvertible?(\\spad{p1},{}\\spad{p2},{}\\spad{ts})} has the same specifications as \\axiomOpFrom{invertible?}{RegularTriangularSetCategory}.") (((|Boolean|) |#4| |#5|) "\\axiom{toseInvertible?(\\spad{p1},{}\\spad{p2},{}\\spad{ts})} has the same specifications as \\axiomOpFrom{invertible?}{RegularTriangularSetCategory}.")) (|toseLastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#4| |#5|) "\\axiom{toseLastSubResultant(\\spad{p1},{}\\spad{p2},{}\\spad{ts})} has the same specifications as \\axiomOpFrom{lastSubResultant}{RegularTriangularSetCategory}.")) (|integralLastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#4| |#5|) "\\axiom{integralLastSubResultant(\\spad{p1},{}\\spad{p2},{}\\spad{ts})} is an internal subroutine,{} exported only for developement.")) (|internalLastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) (|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) |#3| (|Boolean|)) "\\axiom{internalLastSubResultant(lpwt,{}\\spad{v},{}flag)} is an internal subroutine,{} exported only for developement.") (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#4| |#5| (|Boolean|) (|Boolean|)) "\\axiom{internalLastSubResultant(\\spad{p1},{}\\spad{p2},{}\\spad{ts},{}inv?,{}break?)} is an internal subroutine,{} exported only for developement.")) (|prepareSubResAlgo| (((|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) |#4| |#4| |#5|) "\\axiom{prepareSubResAlgo(\\spad{p1},{}\\spad{p2},{}\\spad{ts})} is an internal subroutine,{} exported only for developement.")) (|stopTableInvSet!| (((|Void|)) "\\axiom{stopTableInvSet!()} is an internal subroutine,{} exported only for developement.")) (|startTableInvSet!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableInvSet!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement.")) (|stopTableGcd!| (((|Void|)) "\\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.")) (|startTableGcd!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableGcd!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement."))) 
 NIL 
 NIL 
-(-1077) 
+(-1079) 
 ((|constructor| (NIL "This domain represents `restrict' expressions.")) (|target| (((|TypeAst|) $) "\\spad{target(e)} returns the target type of the conversion..")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression being converted."))) 
 NIL 
 NIL 
-(-1078) 
+(-1080) 
 ((|constructor| (NIL "This is the datatype of OpenAxiom runtime values. It exists solely for internal purposes.")) (|eq| (((|Boolean|) $ $) "\\spad{eq(x,y)} holds if both values \\spad{x} and \\spad{y} resides at the same address in memory."))) 
 NIL 
 NIL 
-(-1079 |Base| R -3498) 
+(-1081 |Base| R -3505) 
 ((|constructor| (NIL "\\indented{1}{Rules for the pattern matcher} Author: Manuel Bronstein Date Created: 24 Oct 1988 Date Last Updated: 26 October 1993 Keywords: pattern,{} matching,{} rule.")) (|quotedOperators| (((|List| (|Symbol|)) $) "\\spad{quotedOperators(r)} returns the list of operators on the right hand side of \\spad{r} that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.")) (|elt| ((|#3| $ |#3| (|PositiveInteger|)) "\\spad{elt(r,f,n)} or \\spad{r}(\\spad{f},{} \\spad{n}) applies the rule \\spad{r} to \\spad{f} at most \\spad{n} times.")) (|rhs| ((|#3| $) "\\spad{rhs(r)} returns the right hand side of the rule \\spad{r}.")) (|lhs| ((|#3| $) "\\spad{lhs(r)} returns the left hand side of the rule \\spad{r}.")) (|pattern| (((|Pattern| |#1|) $) "\\spad{pattern(r)} returns the pattern corresponding to the left hand side of the rule \\spad{r}.")) (|suchThat| (($ $ (|List| (|Symbol|)) (|Mapping| (|Boolean|) (|List| |#3|))) "\\spad{suchThat(r, [a1,...,an], f)} returns the rewrite rule \\spad{r} with the predicate \\spad{f(a1,...,an)} attached to it.")) (|rule| (($ |#3| |#3| (|List| (|Symbol|))) "\\spad{rule(f, g, [f1,...,fn])} creates the rewrite rule \\spad{f == eval(eval(g, g is f), [f1,...,fn])},{} that is a rule with left-hand side \\spad{f} and right-hand side \\spad{g}; The symbols \\spad{f1},{}...,{}\\spad{fn} are the operators that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.") (($ |#3| |#3|) "\\spad{rule(f, g)} creates the rewrite rule: \\spad{f == eval(g, g is f)},{} with left-hand side \\spad{f} and right-hand side \\spad{g}."))) 
 NIL 
 NIL 
-(-1080 |f|) 
+(-1082 |f|) 
 ((|constructor| (NIL "This domain implements named rules")) (|name| (((|Symbol|) $) "\\spad{name(x)} returns the symbol"))) 
 NIL 
 NIL 
-(-1081 |Base| R -3498) 
+(-1083 |Base| R -3505) 
 ((|constructor| (NIL "A ruleset is a set of pattern matching rules grouped together.")) (|elt| ((|#3| $ |#3| (|PositiveInteger|)) "\\spad{elt(r,f,n)} or \\spad{r}(\\spad{f},{} \\spad{n}) applies all the rules of \\spad{r} to \\spad{f} at most \\spad{n} times.")) (|rules| (((|List| (|RewriteRule| |#1| |#2| |#3|)) $) "\\spad{rules(r)} returns the rules contained in \\spad{r}.")) (|ruleset| (($ (|List| (|RewriteRule| |#1| |#2| |#3|))) "\\spad{ruleset([r1,...,rn])} creates the rule set \\spad{{r1,...,rn}}."))) 
 NIL 
 NIL 
-(-1082 R |ls|) 
+(-1084 R |ls|) 
 ((|constructor| (NIL "\\indented{1}{A package for computing the rational univariate representation} \\indented{1}{of a zero-dimensional algebraic variety given by a regular} \\indented{1}{triangular set. This package is essentially an interface for the} \\spadtype{InternalRationalUnivariateRepresentationPackage} constructor. It is used in the \\spadtype{ZeroDimensionalSolvePackage} for solving polynomial systems with finitely many solutions.")) (|rur| (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{rur(lp,univ?,check?)} returns the same as \\spad{rur(lp,true)}. Moreover,{} if \\spad{check?} is \\spad{true} then the result is checked.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{rur(lp)} returns the same as \\spad{rur(lp,true)}") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{rur(lp,univ?)} returns a rational univariate representation of \\spad{lp}. This assumes that \\spad{lp} defines a regular triangular \\spad{ts} whose associated variety is zero-dimensional over \\spad{R}. \\spad{rur(lp,univ?)} returns a list of items \\spad{[u,lc]} where \\spad{u} is an irreducible univariate polynomial and each \\spad{c} in \\spad{lc} involves two variables: one from \\spad{ls},{} called the coordinate of \\spad{c},{} and an extra variable which represents any root of \\spad{u}. Every root of \\spad{u} leads to a tuple of values for the coordinates of \\spad{lc}. Moreover,{} a point \\spad{x} belongs to the variety associated with \\spad{lp} iff there exists an item \\spad{[u,lc]} in \\spad{rur(lp,univ?)} and a root \\spad{r} of \\spad{u} such that \\spad{x} is given by the tuple of values for the coordinates of \\spad{lc} evaluated at \\spad{r}. If \\spad{univ?} is \\spad{true} then each polynomial \\spad{c} will have a constant leading coefficient \\spad{w}.\\spad{r}.\\spad{t}. its coordinate. See the example which illustrates the \\spadtype{ZeroDimensionalSolvePackage} package constructor."))) 
 NIL 
 NIL 
-(-1083 R UP M) 
+(-1085 R UP M) 
 ((|constructor| (NIL "Domain which represents simple algebraic extensions of arbitrary rings. The first argument to the domain,{} \\spad{R},{} is the underlying ring,{} the second argument is a domain of univariate polynomials over \\spad{K},{} while the last argument specifies the defining minimal polynomial. The elements of the domain are canonically represented as polynomials of degree less than that of the minimal polynomial with coefficients in \\spad{R}. The second argument is both the type of the third argument and the underlying representation used by \\spadtype{SAE} itself."))) 
-((-4420 |has| |#1| (-366)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-353))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-353)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-371))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (QUOTE (-366)))) (|HasCategory| |#1| (QUOTE (-353)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#1| (QUOTE (-353))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181))))) (-12 (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (QUOTE (-366))))) 
-(-1084 UP SAE UPA) 
+((-4427 |has| |#1| (-367)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-354))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-372))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (QUOTE (-367)))) (|HasCategory| |#1| (QUOTE (-354)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-12 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183))))) (-12 (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (QUOTE (-367))))) 
+(-1086 UP SAE UPA) 
 ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of the rational numbers (\\spadtype{Fraction Integer}).")) (|factor| (((|Factored| |#3|) |#3|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p}."))) 
 NIL 
 NIL 
-(-1085 UP SAE UPA) 
+(-1087 UP SAE UPA) 
 ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of \\spadtype{Fraction Polynomial Integer}.")) (|factor| (((|Factored| |#3|) |#3|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p}."))) 
 NIL 
 NIL 
-(-1086) 
+(-1088) 
 ((|constructor| (NIL "This trivial domain lets us build Univariate Polynomials in an anonymous variable"))) 
 NIL 
 NIL 
-(-1087) 
+(-1089) 
 ((|constructor| (NIL "This is the category of Spad syntax objects."))) 
 NIL 
 NIL 
-(-1088 S) 
+(-1090 S) 
 ((|constructor| (NIL "\\indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \\indented{2}{A sorted cache of a cachable set \\spad{S} is a dynamic structure that} \\indented{2}{keeps the elements of \\spad{S} sorted and assigns an integer to each} \\indented{2}{element of \\spad{S} once it is in the cache. This way,{} equality and ordering} \\indented{2}{on \\spad{S} are tested directly on the integers associated with the elements} \\indented{2}{of \\spad{S},{} once they have been entered in the cache.}")) (|enterInCache| ((|#1| |#1| (|Mapping| (|Integer|) |#1| |#1|)) "\\spad{enterInCache(x, f)} enters \\spad{x} in the cache,{} calling \\spad{f(x, y)} to determine whether \\spad{x < y (f(x,y) < 0), x = y (f(x,y) = 0)},{} or \\spad{x > y (f(x,y) > 0)}. It returns \\spad{x} with an integer associated with it.") ((|#1| |#1| (|Mapping| (|Boolean|) |#1|)) "\\spad{enterInCache(x, f)} enters \\spad{x} in the cache,{} calling \\spad{f(y)} to determine whether \\spad{x} is equal to \\spad{y}. It returns \\spad{x} with an integer associated with it.")) (|cache| (((|List| |#1|)) "\\spad{cache()} returns the current cache as a list.")) (|clearCache| (((|Void|)) "\\spad{clearCache()} empties the cache."))) 
 NIL 
 NIL 
-(-1089) 
+(-1091) 
 ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. A `Scope' is a sequence of contours.")) (|currentCategoryFrame| (($) "\\spad{currentCategoryFrame()} returns the category frame currently in effect.")) (|currentScope| (($) "\\spad{currentScope()} returns the scope currently in effect")) (|pushNewContour| (($ (|Binding|) $) "\\spad{pushNewContour(b,s)} pushs a new contour with sole binding \\spad{`b'}.")) (|findBinding| (((|Maybe| (|Binding|)) (|Identifier|) $) "\\spad{findBinding(n,s)} returns the first binding of \\spad{`n'} in \\spad{`s'}; otherwise `nothing'.")) (|contours| (((|List| (|Contour|)) $) "\\spad{contours(s)} returns the list of contours in scope \\spad{s}.")) (|empty| (($) "\\spad{empty()} returns an empty scope."))) 
 NIL 
 NIL 
-(-1090 R) 
+(-1092 R) 
 ((|constructor| (NIL "StructuralConstantsPackage provides functions creating structural constants from a multiplication tables or a basis of a matrix algebra and other useful functions in this context.")) (|coordinates| (((|Vector| |#1|) (|Matrix| |#1|) (|List| (|Matrix| |#1|))) "\\spad{coordinates(a,[v1,...,vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|structuralConstants| (((|Vector| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{structuralConstants(basis)} takes the \\spad{basis} of a matrix algebra,{} \\spadignore{e.g.} the result of \\spadfun{basisOfCentroid} and calculates the structural constants. Note,{} that the it is not checked,{} whether \\spad{basis} really is a \\spad{basis} of a matrix algebra.") (((|Vector| (|Matrix| (|Polynomial| |#1|))) (|List| (|Symbol|)) (|Matrix| (|Polynomial| |#1|))) "\\spad{structuralConstants(ls,mt)} determines the structural constants of an algebra with generators \\spad{ls} and multiplication table \\spad{mt},{} the entries of which must be given as linear polynomials in the indeterminates given by \\spad{ls}. The result is in particular useful \\indented{1}{as fourth argument for \\spadtype{AlgebraGivenByStructuralConstants}} \\indented{1}{and \\spadtype{GenericNonAssociativeAlgebra}.}") (((|Vector| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|List| (|Symbol|)) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{structuralConstants(ls,mt)} determines the structural constants of an algebra with generators \\spad{ls} and multiplication table \\spad{mt},{} the entries of which must be given as linear polynomials in the indeterminates given by \\spad{ls}. The result is in particular useful \\indented{1}{as fourth argument for \\spadtype{AlgebraGivenByStructuralConstants}} \\indented{1}{and \\spadtype{GenericNonAssociativeAlgebra}.}"))) 
 NIL 
 NIL 
-(-1091 R) 
+(-1093 R) 
 ((|constructor| (NIL "\\spadtype{SequentialDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is sequential. \\blankline"))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-914))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-914)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-1092 (-1181)) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-1092 (-1181)) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-1092 (-1181)) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-1092 (-1181)) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-1092 (-1181)) (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasAttribute| |#1| (QUOTE -4425)) (|HasCategory| |#1| (QUOTE (-456))) (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
-(-1092 S) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-1094 (-1183)) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-1094 (-1183)) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-1094 (-1183)) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-1094 (-1183)) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-1094 (-1183)) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-234))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(-1094 S) 
 ((|constructor| (NIL "\\spadtype{OrderlyDifferentialVariable} adds a commonly used sequential ranking to the set of derivatives of an ordered list of differential indeterminates. A sequential ranking is a ranking \\spadfun{<} of the derivatives with the property that for any derivative \\spad{v},{} there are only a finite number of derivatives \\spad{u} with \\spad{u} \\spadfun{<} \\spad{v}. This domain belongs to \\spadtype{DifferentialVariableCategory}. It defines \\spadfun{weight} to be just \\spadfun{order},{} and it defines a sequential ranking \\spadfun{<} on derivatives \\spad{u} by the lexicographic order on the pair (\\spadfun{variable}(\\spad{u}),{} \\spadfun{order}(\\spad{u}))."))) 
 NIL 
 NIL 
-(-1093 S) 
+(-1095 S) 
 ((|constructor| (NIL "This type is used to specify a range of values from type \\spad{S}."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-851))) (|HasCategory| |#1| (QUOTE (-1105)))) 
-(-1094 R S) 
+((|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1107)))) 
+(-1096 R S) 
 ((|constructor| (NIL "This package provides operations for mapping functions onto segments.")) (|map| (((|List| |#2|) (|Mapping| |#2| |#1|) (|Segment| |#1|)) "\\spad{map(f,s)} expands the segment \\spad{s},{} applying \\spad{f} to each value. For example,{} if \\spad{s = l..h by k},{} then the list \\spad{[f(l), f(l+k),..., f(lN)]} is computed,{} where \\spad{lN <= h < lN+k}.") (((|Segment| |#2|) (|Mapping| |#2| |#1|) (|Segment| |#1|)) "\\spad{map(f,l..h)} returns a new segment \\spad{f(l)..f(h)}."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-851)))) 
-(-1095) 
+((|HasCategory| |#1| (QUOTE (-853)))) 
+(-1097) 
 ((|constructor| (NIL "This domain represents segement expressions.")) (|bounds| (((|List| (|SpadAst|)) $) "\\spad{bounds(s)} returns the bounds of the segment \\spad{`s'}. If \\spad{`s'} designates an infinite interval,{} then the returns list a singleton list."))) 
 NIL 
 NIL 
-(-1096 S) 
+(-1098 S) 
 ((|constructor| (NIL "This domain is used to provide the function argument syntax \\spad{v=a..b}. This is used,{} for example,{} by the top-level \\spadfun{draw} functions."))) 
 NIL 
-((|HasCategory| (-1093 |#1|) (QUOTE (-1105)))) 
-(-1097 R S) 
+((|HasCategory| (-1095 |#1|) (QUOTE (-1107)))) 
+(-1099 R S) 
 ((|constructor| (NIL "This package provides operations for mapping functions onto \\spadtype{SegmentBinding}\\spad{s}.")) (|map| (((|SegmentBinding| |#2|) (|Mapping| |#2| |#1|) (|SegmentBinding| |#1|)) "\\spad{map(f,v=a..b)} returns the value given by \\spad{v=f(a)..f(b)}."))) 
 NIL 
 NIL 
-(-1098 S) 
+(-1100 S) 
 ((|constructor| (NIL "This category provides operations on ranges,{} or {\\em segments} as they are called.")) (|segment| (($ |#1| |#1|) "\\spad{segment(i,j)} is an alternate way to create the segment \\spad{i..j}.")) (|incr| (((|Integer|) $) "\\spad{incr(s)} returns \\spad{n},{} where \\spad{s} is a segment in which every \\spad{n}\\spad{-}th element is used. Note: \\spad{incr(l..h by n) = n}.")) (|high| ((|#1| $) "\\spad{high(s)} returns the second endpoint of \\spad{s}. Note: \\spad{high(l..h) = h}.")) (|low| ((|#1| $) "\\spad{low(s)} returns the first endpoint of \\spad{s}. Note: \\spad{low(l..h) = l}.")) (|hi| ((|#1| $) "\\spad{hi(s)} returns the second endpoint of \\spad{s}. Note: \\spad{hi(l..h) = h}.")) (|lo| ((|#1| $) "\\spad{lo(s)} returns the first endpoint of \\spad{s}. Note: \\spad{lo(l..h) = l}.")) (BY (($ $ (|Integer|)) "\\spad{s by n} creates a new segment in which only every \\spad{n}\\spad{-}th element is used.")) (SEGMENT (($ |#1| |#1|) "\\spad{l..h} creates a segment with \\spad{l} and \\spad{h} as the endpoints."))) 
 NIL 
 NIL 
-(-1099 S L) 
+(-1101 S L) 
 ((|constructor| (NIL "This category provides an interface for expanding segments to a stream of elements.")) (|map| ((|#2| (|Mapping| |#1| |#1|) $) "\\spad{map(f,l..h by k)} produces a value of type \\spad{L} by applying \\spad{f} to each of the succesive elements of the segment,{} that is,{} \\spad{[f(l), f(l+k), ..., f(lN)]},{} where \\spad{lN <= h < lN+k}.")) (|expand| ((|#2| $) "\\spad{expand(l..h by k)} creates value of type \\spad{L} with elements \\spad{l, l+k, ... lN} where \\spad{lN <= h < lN+k}. For example,{} \\spad{expand(1..5 by 2) = [1,3,5]}.") ((|#2| (|List| $)) "\\spad{expand(l)} creates a new value of type \\spad{L} in which each segment \\spad{l..h by k} is replaced with \\spad{l, l+k, ... lN},{} where \\spad{lN <= h < lN+k}. For example,{} \\spad{expand [1..4, 7..9] = [1,2,3,4,7,8,9]}."))) 
 NIL 
 NIL 
-(-1100) 
+(-1102) 
 ((|constructor| (NIL "This domain represents a block of expressions.")) (|last| (((|SpadAst|) $) "\\spad{last(e)} returns the last instruction in `e'.")) (|body| (((|List| (|SpadAst|)) $) "\\spad{body(e)} returns the list of expressions in the sequence of instruction `e'."))) 
 NIL 
 NIL 
-(-1101 S) 
+(-1103 S) 
 ((|constructor| (NIL "A set over a domain \\spad{D} models the usual mathematical notion of a finite set of elements from \\spad{D}. Sets are unordered collections of distinct elements (that is,{} order and duplication does not matter). The notation \\spad{set [a,b,c]} can be used to create a set and the usual operations such as union and intersection are available to form new sets. In our implementation,{} \\Language{} maintains the entries in sorted order. Specifically,{} the parts function returns the entries as a list in ascending order and the extract operation returns the maximum entry. Given two sets \\spad{s} and \\spad{t} where \\spad{\\#s = m} and \\spad{\\#t = n},{} the complexity of \\indented{2}{\\spad{s = t} is \\spad{O(min(n,m))}} \\indented{2}{\\spad{s < t} is \\spad{O(max(n,m))}} \\indented{2}{\\spad{union(s,t)},{} \\spad{intersect(s,t)},{} \\spad{minus(s,t)},{} \\spad{symmetricDifference(s,t)} is \\spad{O(max(n,m))}} \\indented{2}{\\spad{member(x,t)} is \\spad{O(n log n)}} \\indented{2}{\\spad{insert(x,t)} and \\spad{remove(x,t)} is \\spad{O(n)}}"))) 
-((-4427 . T) (-4417 . T) (-4428 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#1| (QUOTE (-371))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-1102 A S) 
+((-4434 . T) (-4424 . T) (-4435 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#1| (QUOTE (-372))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-1104 A S) 
 ((|constructor| (NIL "A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.")) (|union| (($ |#2| $) "\\spad{union(x,u)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{x},{}\\spad{u})} returns a copy of \\spad{u}.") (($ $ |#2|) "\\spad{union(u,x)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{u},{}\\spad{x})} returns a copy of \\spad{u}.") (($ $ $) "\\spad{union(u,v)} returns the set aggregate of elements which are members of either set aggregate \\spad{u} or \\spad{v}.")) (|subset?| (((|Boolean|) $ $) "\\spad{subset?(u,v)} tests if \\spad{u} is a subset of \\spad{v}. Note: equivalent to \\axiom{reduce(and,{}{member?(\\spad{x},{}\\spad{v}) for \\spad{x} in \\spad{u}},{}\\spad{true},{}\\spad{false})}.")) (|symmetricDifference| (($ $ $) "\\spad{symmetricDifference(u,v)} returns the set aggregate of elements \\spad{x} which are members of set aggregate \\spad{u} or set aggregate \\spad{v} but not both. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{symmetricDifference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: \\axiom{symmetricDifference(\\spad{u},{}\\spad{v}) = union(difference(\\spad{u},{}\\spad{v}),{}difference(\\spad{v},{}\\spad{u}))}")) (|difference| (($ $ |#2|) "\\spad{difference(u,x)} returns the set aggregate \\spad{u} with element \\spad{x} removed. If \\spad{u} does not contain \\spad{x},{} a copy of \\spad{u} is returned. Note: \\axiom{difference(\\spad{s},{} \\spad{x}) = difference(\\spad{s},{} {\\spad{x}})}.") (($ $ $) "\\spad{difference(u,v)} returns the set aggregate \\spad{w} consisting of elements in set aggregate \\spad{u} but not in set aggregate \\spad{v}. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{difference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: equivalent to the notation (not currently supported) \\axiom{{\\spad{x} for \\spad{x} in \\spad{u} | not member?(\\spad{x},{}\\spad{v})}}.")) (|intersect| (($ $ $) "\\spad{intersect(u,v)} returns the set aggregate \\spad{w} consisting of elements common to both set aggregates \\spad{u} and \\spad{v}. Note: equivalent to the notation (not currently supported) {\\spad{x} for \\spad{x} in \\spad{u} | member?(\\spad{x},{}\\spad{v})}.")) (|set| (($ (|List| |#2|)) "\\spad{set([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.") (($) "\\spad{set()}\\$\\spad{D} creates an empty set aggregate of type \\spad{D}.")) (|brace| (($ (|List| |#2|)) "\\spad{brace([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}. This form is considered obsolete. Use \\axiomFun{set} instead.") (($) "\\spad{brace()}\\$\\spad{D} (otherwise written {}\\$\\spad{D}) creates an empty set aggregate of type \\spad{D}. This form is considered obsolete. Use \\axiomFun{set} instead.")) (|part?| (((|Boolean|) $ $) "\\spad{s} < \\spad{t} returns \\spad{true} if all elements of set aggregate \\spad{s} are also elements of set aggregate \\spad{t}."))) 
 NIL 
 NIL 
-(-1103 S) 
+(-1105 S) 
 ((|constructor| (NIL "A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.")) (|union| (($ |#1| $) "\\spad{union(x,u)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{x},{}\\spad{u})} returns a copy of \\spad{u}.") (($ $ |#1|) "\\spad{union(u,x)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{u},{}\\spad{x})} returns a copy of \\spad{u}.") (($ $ $) "\\spad{union(u,v)} returns the set aggregate of elements which are members of either set aggregate \\spad{u} or \\spad{v}.")) (|subset?| (((|Boolean|) $ $) "\\spad{subset?(u,v)} tests if \\spad{u} is a subset of \\spad{v}. Note: equivalent to \\axiom{reduce(and,{}{member?(\\spad{x},{}\\spad{v}) for \\spad{x} in \\spad{u}},{}\\spad{true},{}\\spad{false})}.")) (|symmetricDifference| (($ $ $) "\\spad{symmetricDifference(u,v)} returns the set aggregate of elements \\spad{x} which are members of set aggregate \\spad{u} or set aggregate \\spad{v} but not both. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{symmetricDifference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: \\axiom{symmetricDifference(\\spad{u},{}\\spad{v}) = union(difference(\\spad{u},{}\\spad{v}),{}difference(\\spad{v},{}\\spad{u}))}")) (|difference| (($ $ |#1|) "\\spad{difference(u,x)} returns the set aggregate \\spad{u} with element \\spad{x} removed. If \\spad{u} does not contain \\spad{x},{} a copy of \\spad{u} is returned. Note: \\axiom{difference(\\spad{s},{} \\spad{x}) = difference(\\spad{s},{} {\\spad{x}})}.") (($ $ $) "\\spad{difference(u,v)} returns the set aggregate \\spad{w} consisting of elements in set aggregate \\spad{u} but not in set aggregate \\spad{v}. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{difference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: equivalent to the notation (not currently supported) \\axiom{{\\spad{x} for \\spad{x} in \\spad{u} | not member?(\\spad{x},{}\\spad{v})}}.")) (|intersect| (($ $ $) "\\spad{intersect(u,v)} returns the set aggregate \\spad{w} consisting of elements common to both set aggregates \\spad{u} and \\spad{v}. Note: equivalent to the notation (not currently supported) {\\spad{x} for \\spad{x} in \\spad{u} | member?(\\spad{x},{}\\spad{v})}.")) (|set| (($ (|List| |#1|)) "\\spad{set([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.") (($) "\\spad{set()}\\$\\spad{D} creates an empty set aggregate of type \\spad{D}.")) (|brace| (($ (|List| |#1|)) "\\spad{brace([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}. This form is considered obsolete. Use \\axiomFun{set} instead.") (($) "\\spad{brace()}\\$\\spad{D} (otherwise written {}\\$\\spad{D}) creates an empty set aggregate of type \\spad{D}. This form is considered obsolete. Use \\axiomFun{set} instead.")) (|part?| (((|Boolean|) $ $) "\\spad{s} < \\spad{t} returns \\spad{true} if all elements of set aggregate \\spad{s} are also elements of set aggregate \\spad{t}."))) 
-((-4417 . T)) 
+((-4424 . T)) 
 NIL 
-(-1104 S) 
+(-1106 S) 
 ((|constructor| (NIL "\\spadtype{SetCategory} is the basic category for describing a collection of elements with \\spadop{=} (equality) and \\spadfun{coerce} to output form. \\blankline Conditional Attributes: \\indented{3}{canonical\\tab{15}data structure equality is the same as \\spadop{=}}")) (|before?| (((|Boolean|) $ $) "spad{before?(\\spad{x},{}\\spad{y})} holds if \\spad{x} comes before \\spad{y} in the internal total ordering used by OpenAxiom.")) (|latex| (((|String|) $) "\\spad{latex(s)} returns a LaTeX-printable output representation of \\spad{s}.")) (|hash| (((|SingleInteger|) $) "\\spad{hash(s)} calculates a hash code for \\spad{s}."))) 
 NIL 
 NIL 
-(-1105) 
+(-1107) 
 ((|constructor| (NIL "\\spadtype{SetCategory} is the basic category for describing a collection of elements with \\spadop{=} (equality) and \\spadfun{coerce} to output form. \\blankline Conditional Attributes: \\indented{3}{canonical\\tab{15}data structure equality is the same as \\spadop{=}}")) (|before?| (((|Boolean|) $ $) "spad{before?(\\spad{x},{}\\spad{y})} holds if \\spad{x} comes before \\spad{y} in the internal total ordering used by OpenAxiom.")) (|latex| (((|String|) $) "\\spad{latex(s)} returns a LaTeX-printable output representation of \\spad{s}.")) (|hash| (((|SingleInteger|) $) "\\spad{hash(s)} calculates a hash code for \\spad{s}."))) 
 NIL 
 NIL 
-(-1106 |m| |n|) 
+(-1108 |m| |n|) 
 ((|constructor| (NIL "\\spadtype{SetOfMIntegersInOneToN} implements the subsets of \\spad{M} integers in the interval \\spad{[1..n]}")) (|delta| (((|NonNegativeInteger|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{delta(S,k,p)} returns the number of elements of \\spad{S} which are strictly between \\spad{p} and the \\spad{k^}{th} element of \\spad{S}.")) (|member?| (((|Boolean|) (|PositiveInteger|) $) "\\spad{member?(p, s)} returns \\spad{true} is \\spad{p} is in \\spad{s},{} \\spad{false} otherwise.")) (|enumerate| (((|Vector| $)) "\\spad{enumerate()} returns a vector of all the sets of \\spad{M} integers in \\spad{1..n}.")) (|setOfMinN| (($ (|List| (|PositiveInteger|))) "\\spad{setOfMinN([a_1,...,a_m])} returns the set {a_1,{}...,{}a_m}. Error if {a_1,{}...,{}a_m} is not a set of \\spad{M} integers in \\spad{1..n}.")) (|elements| (((|List| (|PositiveInteger|)) $) "\\spad{elements(S)} returns the list of the elements of \\spad{S} in increasing order.")) (|replaceKthElement| (((|Union| $ #1="failed") $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{replaceKthElement(S,k,p)} replaces the \\spad{k^}{th} element of \\spad{S} by \\spad{p},{} and returns \"failed\" if the result is not a set of \\spad{M} integers in \\spad{1..n} any more.")) (|incrementKthElement| (((|Union| $ #1#) $ (|PositiveInteger|)) "\\spad{incrementKthElement(S,k)} increments the \\spad{k^}{th} element of \\spad{S},{} and returns \"failed\" if the result is not a set of \\spad{M} integers in \\spad{1..n} any more."))) 
 NIL 
 NIL 
-(-1107) 
+(-1109) 
 ((|constructor| (NIL "This domain allows the manipulation of the usual Lisp values."))) 
 NIL 
 NIL 
-(-1108 |Str| |Sym| |Int| |Flt| |Expr|) 
+(-1110 |Str| |Sym| |Int| |Flt| |Expr|) 
 ((|constructor| (NIL "This category allows the manipulation of Lisp values while keeping the grunge fairly localized.")) (|elt| (($ $ (|List| (|Integer|))) "\\spad{elt((a1,...,an), [i1,...,im])} returns \\spad{(a_i1,...,a_im)}.") (($ $ (|Integer|)) "\\spad{elt((a1,...,an), i)} returns \\spad{ai}.")) (|#| (((|Integer|) $) "\\spad{\\#((a1,...,an))} returns \\spad{n}.")) (|cdr| (($ $) "\\spad{cdr((a1,...,an))} returns \\spad{(a2,...,an)}.")) (|car| (($ $) "\\spad{car((a1,...,an))} returns a1.")) (|expr| ((|#5| $) "\\spad{expr(s)} returns \\spad{s} as an element of Expr; Error: if \\spad{s} is not an atom that also belongs to Expr.")) (|float| ((|#4| $) "\\spad{float(s)} returns \\spad{s} as an element of \\spad{Flt}; Error: if \\spad{s} is not an atom that also belongs to \\spad{Flt}.")) (|integer| ((|#3| $) "\\spad{integer(s)} returns \\spad{s} as an element of Int. Error: if \\spad{s} is not an atom that also belongs to Int.")) (|symbol| ((|#2| $) "\\spad{symbol(s)} returns \\spad{s} as an element of \\spad{Sym}. Error: if \\spad{s} is not an atom that also belongs to \\spad{Sym}.")) (|string| ((|#1| $) "\\spad{string(s)} returns \\spad{s} as an element of \\spad{Str}. Error: if \\spad{s} is not an atom that also belongs to \\spad{Str}.")) (|destruct| (((|List| $) $) "\\spad{destruct((a1,...,an))} returns the list [a1,{}...,{}an].")) (|float?| (((|Boolean|) $) "\\spad{float?(s)} is \\spad{true} if \\spad{s} is an atom and belong to \\spad{Flt}.")) (|integer?| (((|Boolean|) $) "\\spad{integer?(s)} is \\spad{true} if \\spad{s} is an atom and belong to Int.")) (|symbol?| (((|Boolean|) $) "\\spad{symbol?(s)} is \\spad{true} if \\spad{s} is an atom and belong to \\spad{Sym}.")) (|string?| (((|Boolean|) $) "\\spad{string?(s)} is \\spad{true} if \\spad{s} is an atom and belong to \\spad{Str}.")) (|list?| (((|Boolean|) $) "\\spad{list?(s)} is \\spad{true} if \\spad{s} is a Lisp list,{} possibly ().")) (|pair?| (((|Boolean|) $) "\\spad{pair?(s)} is \\spad{true} if \\spad{s} has is a non-null Lisp list.")) (|atom?| (((|Boolean|) $) "\\spad{atom?(s)} is \\spad{true} if \\spad{s} is a Lisp atom.")) (|null?| (((|Boolean|) $) "\\spad{null?(s)} is \\spad{true} if \\spad{s} is the \\spad{S}-expression ().")) (|eq| (((|Boolean|) $ $) "\\spad{eq(s, t)} is \\spad{true} if EQ(\\spad{s},{}\\spad{t}) is \\spad{true} in Lisp."))) 
 NIL 
 NIL 
-(-1109 |Str| |Sym| |Int| |Flt| |Expr|) 
+(-1111 |Str| |Sym| |Int| |Flt| |Expr|) 
 ((|constructor| (NIL "This domain allows the manipulation of Lisp values over arbitrary atomic types."))) 
 NIL 
 NIL 
-(-1110 R FS) 
+(-1112 R FS) 
 ((|constructor| (NIL "\\axiomType{SimpleFortranProgram(\\spad{f},{}type)} provides a simple model of some FORTRAN subprograms,{} making it possible to coerce objects of various domains into a FORTRAN subprogram called \\axiom{\\spad{f}}. These can then be translated into legal FORTRAN code.")) (|fortran| (($ (|Symbol|) (|FortranScalarType|) |#2|) "\\spad{fortran(fname,ftype,body)} builds an object of type \\axiomType{FortranProgramCategory}. The three arguments specify the name,{} the type and the \\spad{body} of the program."))) 
 NIL 
 NIL 
-(-1111 R E V P TS) 
+(-1113 R E V P TS) 
 ((|constructor| (NIL "\\indented{2}{A internal package for removing redundant quasi-components and redundant} \\indented{2}{branches when decomposing a variety by means of quasi-components} \\indented{2}{of regular triangular sets. \\newline} References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{5}{Tech. Report (PoSSo project)} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|branchIfCan| (((|Union| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|))) "failed") (|List| |#4|) |#5| (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{branchIfCan(leq,{}\\spad{ts},{}lineq,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement.")) (|prepareDecompose| (((|List| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|)))) (|List| |#4|) (|List| |#5|) (|Boolean|) (|Boolean|)) "\\axiom{prepareDecompose(\\spad{lp},{}\\spad{lts},{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousCases| (((|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) (|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)))) "\\axiom{removeSuperfluousCases(llpwt)} is an internal subroutine,{} exported only for developement.")) (|subCase?| (((|Boolean|) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) "\\axiom{subCase?(lpwt1,{}lpwt2)} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousQuasiComponents| (((|List| |#5|) (|List| |#5|)) "\\axiom{removeSuperfluousQuasiComponents(\\spad{lts})} removes from \\axiom{\\spad{lts}} any \\spad{ts} such that \\axiom{subQuasiComponent?(\\spad{ts},{}us)} holds for another \\spad{us} in \\axiom{\\spad{lts}}.")) (|subQuasiComponent?| (((|Boolean|) |#5| (|List| |#5|)) "\\axiom{subQuasiComponent?(\\spad{ts},{}lus)} returns \\spad{true} iff \\axiom{subQuasiComponent?(\\spad{ts},{}us)} holds for one \\spad{us} in \\spad{lus}.") (((|Boolean|) |#5| |#5|) "\\axiom{subQuasiComponent?(\\spad{ts},{}us)} returns \\spad{true} iff \\axiomOpFrom{internalSubQuasiComponent?(\\spad{ts},{}us)}{QuasiComponentPackage} returs \\spad{true}.")) (|internalSubQuasiComponent?| (((|Union| (|Boolean|) "failed") |#5| |#5|) "\\axiom{internalSubQuasiComponent?(\\spad{ts},{}us)} returns a boolean \\spad{b} value if the fact the regular zero set of \\axiom{us} contains that of \\axiom{\\spad{ts}} can be decided (and in that case \\axiom{\\spad{b}} gives this inclusion) otherwise returns \\axiom{\"failed\"}.")) (|infRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{infRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalInfRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalInfRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalSubPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalSubPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}} assuming that these lists are sorted increasingly \\spad{w}.\\spad{r}.\\spad{t}. \\axiomOpFrom{infRittWu?}{RecursivePolynomialCategory}.")) (|subPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{subPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}}.")) (|subTriSet?| (((|Boolean|) |#5| |#5|) "\\axiom{subTriSet?(\\spad{ts},{}us)} returns \\spad{true} iff \\axiom{\\spad{ts}} is a sub-set of \\axiom{us}.")) (|moreAlgebraic?| (((|Boolean|) |#5| |#5|) "\\axiom{moreAlgebraic?(\\spad{ts},{}us)} returns \\spad{false} iff \\axiom{\\spad{ts}} and \\axiom{us} are both empty,{} or \\axiom{\\spad{ts}} has less elements than \\axiom{us},{} or some variable is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{us} and is not \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ts}}.")) (|algebraicSort| (((|List| |#5|) (|List| |#5|)) "\\axiom{algebraicSort(\\spad{lts})} sorts \\axiom{\\spad{lts}} \\spad{w}.\\spad{r}.\\spad{t} \\axiomOpFrom{supDimElseRittWu}{QuasiComponentPackage}.")) (|supDimElseRittWu?| (((|Boolean|) |#5| |#5|) "\\axiom{supDimElseRittWu(\\spad{ts},{}us)} returns \\spad{true} iff \\axiom{\\spad{ts}} has less elements than \\axiom{us} otherwise if \\axiom{\\spad{ts}} has higher rank than \\axiom{us} \\spad{w}.\\spad{r}.\\spad{t}. Riit and Wu ordering.")) (|stopTable!| (((|Void|)) "\\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.")) (|startTable!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableGcd!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement."))) 
 NIL 
 NIL 
-(-1112 R E V P TS) 
+(-1114 R E V P TS) 
 ((|constructor| (NIL "A internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field. There is no need to use directly this package since its main operations are available from \\spad{TS}. \\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of \\spad{gcd} over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of AAECC11} \\indented{5}{Paris,{} 1995.} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) 
 NIL 
 NIL 
-(-1113 R E V P) 
+(-1115 R E V P) 
 ((|constructor| (NIL "The category of square-free regular triangular sets. A regular triangular set \\spad{ts} is square-free if the \\spad{gcd} of any polynomial \\spad{p} in \\spad{ts} and \\spad{differentiate(p,mvar(p))} \\spad{w}.\\spad{r}.\\spad{t}. \\axiomOpFrom{collectUnder}{TriangularSetCategory}(\\spad{ts},{}\\axiomOpFrom{mvar}{RecursivePolynomialCategory}(\\spad{p})) has degree zero \\spad{w}.\\spad{r}.\\spad{t}. \\spad{mvar(p)}. Thus any square-free regular set defines a tower of square-free simple extensions.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. KALKBRENER \"Algorithmic properties of polynomial rings\"} \\indented{5}{Habilitation Thesis,{} ETZH,{} Zurich,{} 1995.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-1114) 
+(-1116) 
 ((|constructor| (NIL "SymmetricGroupCombinatoricFunctions contains combinatoric functions concerning symmetric groups and representation theory: list young tableaus,{} improper partitions,{} subsets bijection of Coleman.")) (|unrankImproperPartitions1| (((|List| (|Integer|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{unrankImproperPartitions1(n,m,k)} computes the {\\em k}\\spad{-}th improper partition of nonnegative \\spad{n} in at most \\spad{m} nonnegative parts ordered as follows: first,{} in reverse lexicographically according to their non-zero parts,{} then according to their positions (\\spadignore{i.e.} lexicographical order using {\\em subSet}: {\\em [3,0,0] < [0,3,0] < [0,0,3] < [2,1,0] < [2,0,1] < [0,2,1] < [1,2,0] < [1,0,2] < [0,1,2] < [1,1,1]}). Note: counting of subtrees is done by {\\em numberOfImproperPartitionsInternal}.")) (|unrankImproperPartitions0| (((|List| (|Integer|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{unrankImproperPartitions0(n,m,k)} computes the {\\em k}\\spad{-}th improper partition of nonnegative \\spad{n} in \\spad{m} nonnegative parts in reverse lexicographical order. Example: {\\em [0,0,3] < [0,1,2] < [0,2,1] < [0,3,0] < [1,0,2] < [1,1,1] < [1,2,0] < [2,0,1] < [2,1,0] < [3,0,0]}. Error: if \\spad{k} is negative or too big. Note: counting of subtrees is done by \\spadfunFrom{numberOfImproperPartitions}{SymmetricGroupCombinatoricFunctions}.")) (|subSet| (((|List| (|Integer|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{subSet(n,m,k)} calculates the {\\em k}\\spad{-}th {\\em m}-subset of the set {\\em 0,1,...,(n-1)} in the lexicographic order considered as a decreasing map from {\\em 0,...,(m-1)} into {\\em 0,...,(n-1)}. See \\spad{S}.\\spad{G}. Williamson: Theorem 1.60. Error: if not {\\em (0 <= m <= n and 0 < = k < (n choose m))}.")) (|numberOfImproperPartitions| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{numberOfImproperPartitions(n,m)} computes the number of partitions of the nonnegative integer \\spad{n} in \\spad{m} nonnegative parts with regarding the order (improper partitions). Example: {\\em numberOfImproperPartitions (3,3)} is 10,{} since {\\em [0,0,3], [0,1,2], [0,2,1], [0,3,0], [1,0,2], [1,1,1], [1,2,0], [2,0,1], [2,1,0], [3,0,0]} are the possibilities. Note: this operation has a recursive implementation.")) (|nextPartition| (((|Vector| (|Integer|)) (|List| (|Integer|)) (|Vector| (|Integer|)) (|Integer|)) "\\spad{nextPartition(gamma,part,number)} generates the partition of {\\em number} which follows {\\em part} according to the right-to-left lexicographical order. The partition has the property that its components do not exceed the corresponding components of {\\em gamma}. the first partition is achieved by {\\em part=[]}. Also,{} {\\em []} indicates that {\\em part} is the last partition.") (((|Vector| (|Integer|)) (|Vector| (|Integer|)) (|Vector| (|Integer|)) (|Integer|)) "\\spad{nextPartition(gamma,part,number)} generates the partition of {\\em number} which follows {\\em part} according to the right-to-left lexicographical order. The partition has the property that its components do not exceed the corresponding components of {\\em gamma}. The first partition is achieved by {\\em part=[]}. Also,{} {\\em []} indicates that {\\em part} is the last partition.")) (|nextLatticePermutation| (((|List| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|)) (|Boolean|)) "\\spad{nextLatticePermutation(lambda,lattP,constructNotFirst)} generates the lattice permutation according to the proper partition {\\em lambda} succeeding the lattice permutation {\\em lattP} in lexicographical order as long as {\\em constructNotFirst} is \\spad{true}. If {\\em constructNotFirst} is \\spad{false},{} the first lattice permutation is returned. The result {\\em nil} indicates that {\\em lattP} has no successor.")) (|nextColeman| (((|Matrix| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|)) (|Matrix| (|Integer|))) "\\spad{nextColeman(alpha,beta,C)} generates the next Coleman matrix of column sums {\\em alpha} and row sums {\\em beta} according to the lexicographical order from bottom-to-top. The first Coleman matrix is achieved by {\\em C=new(1,1,0)}. Also,{} {\\em new(1,1,0)} indicates that \\spad{C} is the last Coleman matrix.")) (|makeYoungTableau| (((|Matrix| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{makeYoungTableau(lambda,gitter)} computes for a given lattice permutation {\\em gitter} and for an improper partition {\\em lambda} the corresponding standard tableau of shape {\\em lambda}. Notes: see {\\em listYoungTableaus}. The entries are from {\\em 0,...,n-1}.")) (|listYoungTableaus| (((|List| (|Matrix| (|Integer|))) (|List| (|Integer|))) "\\spad{listYoungTableaus(lambda)} where {\\em lambda} is a proper partition generates the list of all standard tableaus of shape {\\em lambda} by means of lattice permutations. The numbers of the lattice permutation are interpreted as column labels. Hence the contents of these lattice permutations are the conjugate of {\\em lambda}. Notes: the functions {\\em nextLatticePermutation} and {\\em makeYoungTableau} are used. The entries are from {\\em 0,...,n-1}.")) (|inverseColeman| (((|List| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|)) (|Matrix| (|Integer|))) "\\spad{inverseColeman(alpha,beta,C)}: there is a bijection from the set of matrices having nonnegative entries and row sums {\\em alpha},{} column sums {\\em beta} to the set of {\\em Salpha - Sbeta} double cosets of the symmetric group {\\em Sn}. ({\\em Salpha} is the Young subgroup corresponding to the improper partition {\\em alpha}). For such a matrix \\spad{C},{} inverseColeman(\\spad{alpha},{}\\spad{beta},{}\\spad{C}) calculates the lexicographical smallest {\\em pi} in the corresponding double coset. Note: the resulting permutation {\\em pi} of {\\em {1,2,...,n}} is given in list form. Notes: the inverse of this map is {\\em coleman}. For details,{} see James/Kerber.")) (|coleman| (((|Matrix| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{coleman(alpha,beta,pi)}: there is a bijection from the set of matrices having nonnegative entries and row sums {\\em alpha},{} column sums {\\em beta} to the set of {\\em Salpha - Sbeta} double cosets of the symmetric group {\\em Sn}. ({\\em Salpha} is the Young subgroup corresponding to the improper partition {\\em alpha}). For a representing element {\\em pi} of such a double coset,{} coleman(\\spad{alpha},{}\\spad{beta},{}\\spad{pi}) generates the Coleman-matrix corresponding to {\\em alpha, beta, pi}. Note: The permutation {\\em pi} of {\\em {1,2,...,n}} has to be given in list form. Note: the inverse of this map is {\\em inverseColeman} (if {\\em pi} is the lexicographical smallest permutation in the coset). For details see James/Kerber."))) 
 NIL 
 NIL 
-(-1115 S) 
+(-1117 S) 
 ((|constructor| (NIL "the class of all multiplicative semigroups,{} \\spadignore{i.e.} a set with an associative operation \\spadop{*}. \\blankline")) (** (($ $ (|PositiveInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (* (($ $ $) "\\spad{x*y} returns the product of \\spad{x} and \\spad{y}."))) 
 NIL 
 NIL 
-(-1116) 
+(-1118) 
 ((|constructor| (NIL "the class of all multiplicative semigroups,{} \\spadignore{i.e.} a set with an associative operation \\spadop{*}. \\blankline")) (** (($ $ (|PositiveInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (* (($ $ $) "\\spad{x*y} returns the product of \\spad{x} and \\spad{y}."))) 
 NIL 
 NIL 
-(-1117 |dimtot| |dim1| S) 
+(-1119 |dimtot| |dim1| S) 
 ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The vectors are ordered as if they were split into two blocks. The dim1 parameter specifies the length of the first block. The ordering is lexicographic between the blocks but acts like \\spadtype{HomogeneousDirectProduct} within each block. This type is a suitable third argument for \\spadtype{GeneralDistributedMultivariatePolynomial}."))) 
-((-4421 |has| |#3| (-1053)) (-4422 |has| |#3| (-1053)) (-4424 |has| |#3| (-6 -4424)) ((-4429 "*") |has| |#3| (-173)) (-4427 . T)) 
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-(-1118 R |x|) 
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(|HasCategory| |#3| (QUOTE (-731))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-798))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-853))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-1055))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551)))))) (|HasCategory| (-551) (QUOTE (-855))) (-12 (|HasCategory| |#3| (QUOTE (-1055))) (|HasCategory| |#3| (LIST (QUOTE -644) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-234))) (|HasCategory| |#3| (QUOTE (-1055)))) (-12 (|HasCategory| |#3| (QUOTE (-1055))) (|HasCategory| |#3| (LIST (QUOTE -906) (QUOTE (-1183))))) (-3969 (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (|HasCategory| |#3| (QUOTE (-1055)))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -1044) (QUOTE (-551))))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasAttribute| |#3| (QUOTE -4431)) (|HasCategory| |#3| (QUOTE (-131))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#3| (QUOTE (-1107))) (|HasCategory| |#3| (LIST (QUOTE -312) (|devaluate| |#3|))))) 
+(-1120 R |x|) 
 ((|constructor| (NIL "This package produces functions for counting etc. real roots of univariate polynomials in \\spad{x} over \\spad{R},{} which must be an OrderedIntegralDomain")) (|countRealRootsMultiple| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{countRealRootsMultiple(p)} says how many real roots \\spad{p} has,{} counted with multiplicity")) (|SturmHabichtMultiple| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtMultiple(p1,p2)} computes \\spad{c_}{+}\\spad{-c_}{-} where \\spad{c_}{+} is the number of real roots of \\spad{p1} with p2>0 and \\spad{c_}{-} is the number of real roots of \\spad{p1} with p2<0. If p2=1 what you get is the number of real roots of \\spad{p1}.")) (|countRealRoots| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{countRealRoots(p)} says how many real roots \\spad{p} has")) (|SturmHabicht| (((|Integer|) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabicht(p1,p2)} computes \\spad{c_}{+}\\spad{-c_}{-} where \\spad{c_}{+} is the number of real roots of \\spad{p1} with p2>0 and \\spad{c_}{-} is the number of real roots of \\spad{p1} with p2<0. If p2=1 what you get is the number of real roots of \\spad{p1}.")) (|SturmHabichtCoefficients| (((|List| |#1|) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtCoefficients(p1,p2)} computes the principal Sturm-Habicht coefficients of \\spad{p1} and \\spad{p2}")) (|SturmHabichtSequence| (((|List| (|UnivariatePolynomial| |#2| |#1|)) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtSequence(p1,p2)} computes the Sturm-Habicht sequence of \\spad{p1} and \\spad{p2}")) (|subresultantSequence| (((|List| (|UnivariatePolynomial| |#2| |#1|)) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{subresultantSequence(p1,p2)} computes the (standard) subresultant sequence of \\spad{p1} and \\spad{p2}"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-456)))) 
-(-1119) 
+((|HasCategory| |#1| (QUOTE (-457)))) 
+(-1121) 
 ((|constructor| (NIL "This is the datatype for operation signatures as \\indented{2}{used by the compiler and the interpreter.\\space{2}Note that this domain} \\indented{2}{differs from SignatureAst.} See also: ConstructorCall,{} Domain.")) (|source| (((|List| (|Syntax|)) $) "\\spad{source(s)} returns the list of parameter types of \\spad{`s'}.")) (|target| (((|Syntax|) $) "\\spad{target(s)} returns the target type of the signature \\spad{`s'}.")) (|signature| (($ (|List| (|Syntax|)) (|Syntax|)) "\\spad{signature(s,t)} constructs a Signature object with parameter types indicaded by \\spad{`s'},{} and return type indicated by \\spad{`t'}."))) 
 NIL 
 NIL 
-(-1120) 
+(-1122) 
 ((|constructor| (NIL "This domain represents a signature AST. A signature AST \\indented{2}{is a description of an exported operation,{} \\spadignore{e.g.} its name,{} result} \\indented{2}{type,{} and the list of its argument types.}")) (|signature| (((|Signature|) $) "\\spad{signature(s)} returns AST of the declared signature for \\spad{`s'}.")) (|name| (((|Identifier|) $) "\\spad{name(s)} returns the name of the signature \\spad{`s'}.")) (|signatureAst| (($ (|Identifier|) (|Signature|)) "\\spad{signatureAst(n,s,t)} builds the signature AST \\spad{n:} \\spad{s} \\spad{->} \\spad{t}"))) 
 NIL 
 NIL 
-(-1121 R -3498) 
+(-1123 R -3505) 
 ((|constructor| (NIL "This package provides functions to determine the sign of an elementary function around a point or infinity.")) (|sign| (((|Union| (|Integer|) #1="failed") |#2| (|Symbol|) |#2| (|String|)) "\\spad{sign(f, x, a, s)} returns the sign of \\spad{f} as \\spad{x} nears \\spad{a} from below if \\spad{s} is \"left\",{} or above if \\spad{s} is \"right\".") (((|Union| (|Integer|) #1#) |#2| (|Symbol|) (|OrderedCompletion| |#2|)) "\\spad{sign(f, x, a)} returns the sign of \\spad{f} as \\spad{x} nears \\spad{a},{} from both sides if \\spad{a} is finite.") (((|Union| (|Integer|) #1#) |#2|) "\\spad{sign(f)} returns the sign of \\spad{f} if it is constant everywhere."))) 
 NIL 
 NIL 
-(-1122 R) 
+(-1124 R) 
 ((|constructor| (NIL "Find the sign of a rational function around a point or infinity.")) (|sign| (((|Union| (|Integer|) #1="failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|Fraction| (|Polynomial| |#1|)) (|String|)) "\\spad{sign(f, x, a, s)} returns the sign of \\spad{f} as \\spad{x} nears \\spad{a} from the left (below) if \\spad{s} is the string \\spad{\"left\"},{} or from the right (above) if \\spad{s} is the string \\spad{\"right\"}.") (((|Union| (|Integer|) #1#) (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|)))) "\\spad{sign(f, x, a)} returns the sign of \\spad{f} as \\spad{x} approaches \\spad{a},{} from both sides if \\spad{a} is finite.") (((|Union| (|Integer|) #1#) (|Fraction| (|Polynomial| |#1|))) "\\spad{sign f} returns the sign of \\spad{f} if it is constant everywhere."))) 
 NIL 
 NIL 
-(-1123) 
+(-1125) 
 ((|constructor| (NIL "\\indented{1}{Package to allow simplify to be called on AlgebraicNumbers} by converting to EXPR(INT)")) (|simplify| (((|Expression| (|Integer|)) (|AlgebraicNumber|)) "\\spad{simplify(an)} applies simplifications to \\spad{an}"))) 
 NIL 
 NIL 
-(-1124) 
+(-1126) 
 ((|constructor| (NIL "SingleInteger is intended to support machine integer arithmetic.")) (|Or| (($ $ $) "\\spad{Or(n,m)} returns the bit-by-bit logical {\\em or} of the single integers \\spad{n} and \\spad{m}.")) (|And| (($ $ $) "\\spad{And(n,m)} returns the bit-by-bit logical {\\em and} of the single integers \\spad{n} and \\spad{m}.")) (|Not| (($ $) "\\spad{Not(n)} returns the bit-by-bit logical {\\em not} of the single integer \\spad{n}.")) (|xor| (($ $ $) "\\spad{xor(n,m)} returns the bit-by-bit logical {\\em xor} of the single integers \\spad{n} and \\spad{m}.")) (|not| (($ $) "\\spad{not(n)} returns the bit-by-bit logical {\\em not} of the single integer \\spad{n}.")) (|noetherian| ((|attribute|) "\\spad{noetherian} all ideals are finitely generated (in fact principal).")) (|canonicalsClosed| ((|attribute|) "\\spad{canonicalClosed} means two positives multiply to give positive.")) (|canonical| ((|attribute|) "\\spad{canonical} means that mathematical equality is implied by data structure equality."))) 
-((-4415 . T) (-4419 . T) (-4414 . T) (-4425 . T) (-4426 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4422 . T) (-4426 . T) (-4421 . T) (-4432 . T) (-4433 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1125 S) 
+(-1127 S) 
 ((|constructor| (NIL "A stack is a bag where the last item inserted is the first item extracted.")) (|depth| (((|NonNegativeInteger|) $) "\\spad{depth(s)} returns the number of elements of stack \\spad{s}. Note: \\axiom{depth(\\spad{s}) = \\spad{#s}}.")) (|top| ((|#1| $) "\\spad{top(s)} returns the top element \\spad{x} from \\spad{s}; \\spad{s} remains unchanged. Note: Use \\axiom{pop!(\\spad{s})} to obtain \\spad{x} and remove it from \\spad{s}.")) (|pop!| ((|#1| $) "\\spad{pop!(s)} returns the top element \\spad{x},{} destructively removing \\spad{x} from \\spad{s}. Note: Use \\axiom{top(\\spad{s})} to obtain \\spad{x} without removing it from \\spad{s}. Error: if \\spad{s} is empty.")) (|push!| ((|#1| |#1| $) "\\spad{push!(x,s)} pushes \\spad{x} onto stack \\spad{s},{} \\spadignore{i.e.} destructively changing \\spad{s} so as to have a new first (top) element \\spad{x}. Afterwards,{} pop!(\\spad{s}) produces \\spad{x} and pop!(\\spad{s}) produces the original \\spad{s}."))) 
-((-4427 . T) (-4428 . T)) 
+((-4434 . T) (-4435 . T)) 
 NIL 
-(-1126 S |ndim| R |Row| |Col|) 
+(-1128 S |ndim| R |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if the matrix is not invertible.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m},{} if that matrix is invertible and returns \"failed\" otherwise.")) (|minordet| ((|#3| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors.")) (|determinant| ((|#3| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}.")) (* ((|#4| |#4| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#5| $ |#5|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.")) (|diagonalProduct| ((|#3| $) "\\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \\spad{m}.")) (|trace| ((|#3| $) "\\spad{trace(m)} returns the trace of the matrix \\spad{m}. this is the sum of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonal| ((|#4| $) "\\spad{diagonal(m)} returns a row consisting of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonalMatrix| (($ (|List| |#3|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ |#3|) "\\spad{scalarMatrix(r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}\\spad{'s} on the diagonal and zeroes elsewhere."))) 
 NIL 
-((|HasCategory| |#3| (QUOTE (-366))) (|HasAttribute| |#3| (QUOTE (-4429 "*"))) (|HasCategory| |#3| (QUOTE (-173)))) 
-(-1127 |ndim| R |Row| |Col|) 
+((|HasCategory| |#3| (QUOTE (-367))) (|HasAttribute| |#3| (QUOTE (-4436 "*"))) (|HasCategory| |#3| (QUOTE (-173)))) 
+(-1129 |ndim| R |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if the matrix is not invertible.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m},{} if that matrix is invertible and returns \"failed\" otherwise.")) (|minordet| ((|#2| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors.")) (|determinant| ((|#2| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}.")) (* ((|#3| |#3| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#4| $ |#4|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.")) (|diagonalProduct| ((|#2| $) "\\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \\spad{m}.")) (|trace| ((|#2| $) "\\spad{trace(m)} returns the trace of the matrix \\spad{m}. this is the sum of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonal| ((|#3| $) "\\spad{diagonal(m)} returns a row consisting of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonalMatrix| (($ (|List| |#2|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ |#2|) "\\spad{scalarMatrix(r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}\\spad{'s} on the diagonal and zeroes elsewhere."))) 
-((-4427 . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4434 . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1128 R |Row| |Col| M) 
+(-1130 R |Row| |Col| M) 
 ((|constructor| (NIL "\\spadtype{SmithNormalForm} is a package which provides some standard canonical forms for matrices.")) (|diophantineSystem| (((|Record| (|:| |particular| (|Union| |#3| "failed")) (|:| |basis| (|List| |#3|))) |#4| |#3|) "\\spad{diophantineSystem(A,B)} returns a particular integer solution and an integer basis of the equation \\spad{AX = B}.")) (|completeSmith| (((|Record| (|:| |Smith| |#4|) (|:| |leftEqMat| |#4|) (|:| |rightEqMat| |#4|)) |#4|) "\\spad{completeSmith} returns a record that contains the Smith normal form \\spad{H} of the matrix and the left and right equivalence matrices \\spad{U} and \\spad{V} such that U*m*v = \\spad{H}")) (|smith| ((|#4| |#4|) "\\spad{smith(m)} returns the Smith Normal form of the matrix \\spad{m}.")) (|completeHermite| (((|Record| (|:| |Hermite| |#4|) (|:| |eqMat| |#4|)) |#4|) "\\spad{completeHermite} returns a record that contains the Hermite normal form \\spad{H} of the matrix and the equivalence matrix \\spad{U} such that U*m = \\spad{H}")) (|hermite| ((|#4| |#4|) "\\spad{hermite(m)} returns the Hermite normal form of the matrix \\spad{m}."))) 
 NIL 
 NIL 
-(-1129 R |VarSet|) 
+(-1131 R |VarSet|) 
 ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials. It is parameterized by the coefficient ring and the variable set which may be infinite. The variable ordering is determined by the variable set parameter. The coefficient ring may be non-commutative,{} but the variables are assumed to commute."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-914))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-914)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-366))) (|HasAttribute| |#1| (QUOTE -4425)) (|HasCategory| |#1| (QUOTE (-456))) (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
-(-1130 |Coef| |Var| SMP) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-367))) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(-1132 |Coef| |Var| SMP) 
 ((|constructor| (NIL "This domain provides multivariate Taylor series with variables from an arbitrary ordered set. A Taylor series is represented by a stream of polynomials from the polynomial domain \\spad{SMP}. The \\spad{n}th element of the stream is a form of degree \\spad{n}. SMTS is an internal domain.")) (|fintegrate| (($ (|Mapping| $) |#2| |#1|) "\\spad{fintegrate(f,v,c)} is the integral of \\spad{f()} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.} \\indented{1}{The evaluation of \\spad{f()} is delayed.}")) (|integrate| (($ $ |#2| |#1|) "\\spad{integrate(s,v,c)} is the integral of \\spad{s} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.}")) (|csubst| (((|Mapping| (|Stream| |#3|) |#3|) (|List| |#2|) (|List| (|Stream| |#3|))) "\\spad{csubst(a,b)} is for internal use only")) (* (($ |#3| $) "\\spad{smp*ts} multiplies a TaylorSeries by a monomial \\spad{SMP}.")) (|coerce| (($ |#3|) "\\spad{coerce(poly)} regroups the terms by total degree and forms a series.") (($ |#2|) "\\spad{coerce(var)} converts a variable to a Taylor series")) (|coefficient| ((|#3| $ (|NonNegativeInteger|)) "\\spad{coefficient(s, n)} gives the terms of total degree \\spad{n}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-1131 R E V P) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-1133 R E V P) 
 ((|constructor| (NIL "The category of square-free and normalized triangular sets. Thus,{} up to the primitivity axiom of [1],{} these sets are Lazard triangular sets.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991}"))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-1132 UP -3498) 
+(-1134 UP -3505) 
 ((|constructor| (NIL "This package factors the formulas out of the general solve code,{} allowing their recursive use over different domains. Care is taken to introduce few radicals so that radical extension domains can more easily simplify the results.")) (|aQuartic| ((|#2| |#2| |#2| |#2| |#2| |#2|) "\\spad{aQuartic(f,g,h,i,k)} \\undocumented")) (|aCubic| ((|#2| |#2| |#2| |#2| |#2|) "\\spad{aCubic(f,g,h,j)} \\undocumented")) (|aQuadratic| ((|#2| |#2| |#2| |#2|) "\\spad{aQuadratic(f,g,h)} \\undocumented")) (|aLinear| ((|#2| |#2| |#2|) "\\spad{aLinear(f,g)} \\undocumented")) (|quartic| (((|List| |#2|) |#2| |#2| |#2| |#2| |#2|) "\\spad{quartic(f,g,h,i,j)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{quartic(u)} \\undocumented")) (|cubic| (((|List| |#2|) |#2| |#2| |#2| |#2|) "\\spad{cubic(f,g,h,i)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{cubic(u)} \\undocumented")) (|quadratic| (((|List| |#2|) |#2| |#2| |#2|) "\\spad{quadratic(f,g,h)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{quadratic(u)} \\undocumented")) (|linear| (((|List| |#2|) |#2| |#2|) "\\spad{linear(f,g)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{linear(u)} \\undocumented")) (|mapSolve| (((|Record| (|:| |solns| (|List| |#2|)) (|:| |maps| (|List| (|Record| (|:| |arg| |#2|) (|:| |res| |#2|))))) |#1| (|Mapping| |#2| |#2|)) "\\spad{mapSolve(u,f)} \\undocumented")) (|particularSolution| ((|#2| |#1|) "\\spad{particularSolution(u)} \\undocumented")) (|solve| (((|List| |#2|) |#1|) "\\spad{solve(u)} \\undocumented"))) 
 NIL 
 NIL 
-(-1133 R) 
+(-1135 R) 
 ((|constructor| (NIL "This package tries to find solutions expressed in terms of radicals for systems of equations of rational functions with coefficients in an integral domain \\spad{R}.")) (|contractSolve| (((|SuchThat| (|List| (|Expression| |#1|)) (|List| (|Equation| (|Expression| |#1|)))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{contractSolve(rf,x)} finds the solutions expressed in terms of radicals of the equation \\spad{rf} = 0 with respect to the symbol \\spad{x},{} where \\spad{rf} is a rational function. The result contains new symbols for common subexpressions in order to reduce the size of the output.") (((|SuchThat| (|List| (|Expression| |#1|)) (|List| (|Equation| (|Expression| |#1|)))) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{contractSolve(eq,x)} finds the solutions expressed in terms of radicals of the equation of rational functions \\spad{eq} with respect to the symbol \\spad{x}. The result contains new symbols for common subexpressions in order to reduce the size of the output.")) (|radicalRoots| (((|List| (|List| (|Expression| |#1|))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{radicalRoots(lrf,lvar)} finds the roots expressed in terms of radicals of the list of rational functions \\spad{lrf} with respect to the list of symbols \\spad{lvar}.") (((|List| (|Expression| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{radicalRoots(rf,x)} finds the roots expressed in terms of radicals of the rational function \\spad{rf} with respect to the symbol \\spad{x}.")) (|radicalSolve| (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) "\\spad{radicalSolve(leq)} finds the solutions expressed in terms of radicals of the system of equations of rational functions \\spad{leq} with respect to the unique symbol \\spad{x} appearing in \\spad{leq}.") (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|List| (|Symbol|))) "\\spad{radicalSolve(leq,lvar)} finds the solutions expressed in terms of radicals of the system of equations of rational functions \\spad{leq} with respect to the list of symbols \\spad{lvar}.") (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{radicalSolve(lrf)} finds the solutions expressed in terms of radicals of the system of equations \\spad{lrf} = 0,{} where \\spad{lrf} is a system of univariate rational functions.") (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{radicalSolve(lrf,lvar)} finds the solutions expressed in terms of radicals of the system of equations \\spad{lrf} = 0 with respect to the list of symbols \\spad{lvar},{} where \\spad{lrf} is a list of rational functions.") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{radicalSolve(eq)} finds the solutions expressed in terms of radicals of the equation of rational functions \\spad{eq} with respect to the unique symbol \\spad{x} appearing in \\spad{eq}.") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{radicalSolve(eq,x)} finds the solutions expressed in terms of radicals of the equation of rational functions \\spad{eq} with respect to the symbol \\spad{x}.") (((|List| (|Equation| (|Expression| |#1|))) (|Fraction| (|Polynomial| |#1|))) "\\spad{radicalSolve(rf)} finds the solutions expressed in terms of radicals of the equation \\spad{rf} = 0,{} where \\spad{rf} is a univariate rational function.") (((|List| (|Equation| (|Expression| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{radicalSolve(rf,x)} finds the solutions expressed in terms of radicals of the equation \\spad{rf} = 0 with respect to the symbol \\spad{x},{} where \\spad{rf} is a rational function."))) 
 NIL 
 NIL 
-(-1134 R) 
+(-1136 R) 
 ((|constructor| (NIL "This package finds the function func3 where func1 and func2 \\indented{1}{are given and\\space{2}func1 = func3(func2) .\\space{2}If there is no solution then} \\indented{1}{function func1 will be returned.} \\indented{1}{An example would be\\space{2}\\spad{func1:= 8*X**3+32*X**2-14*X ::EXPR INT} and} \\indented{1}{\\spad{func2:=2*X ::EXPR INT} convert them via univariate} \\indented{1}{to FRAC SUP EXPR INT and then the solution is \\spad{func3:=X**3+X**2-X}} \\indented{1}{of type FRAC SUP EXPR INT}")) (|unvectorise| (((|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Vector| (|Expression| |#1|)) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Integer|)) "\\spad{unvectorise(vect, var, n)} returns \\spad{vect(1) + vect(2)*var + ... + vect(n+1)*var**(n)} where \\spad{vect} is the vector of the coefficients of the polynomail ,{} \\spad{var} the new variable and \\spad{n} the degree.")) (|decomposeFunc| (((|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|)))) "\\spad{decomposeFunc(func1, func2, newvar)} returns a function func3 where \\spad{func1} = func3(\\spad{func2}) and expresses it in the new variable newvar. If there is no solution then \\spad{func1} will be returned."))) 
 NIL 
 NIL 
-(-1135 R) 
+(-1137 R) 
 ((|constructor| (NIL "This package tries to find solutions of equations of type Expression(\\spad{R}). This means expressions involving transcendental,{} exponential,{} logarithmic and nthRoot functions. After trying to transform different kernels to one kernel by applying several rules,{} it calls zerosOf for the SparseUnivariatePolynomial in the remaining kernel. For example the expression \\spad{sin(x)*cos(x)-2} will be transformed to \\indented{3}{\\spad{-2 tan(x/2)**4 -2 tan(x/2)**3 -4 tan(x/2)**2 +2 tan(x/2) -2}} by using the function normalize and then to \\indented{3}{\\spad{-2 tan(x)**2 + tan(x) -2}} with help of subsTan. This function tries to express the given function in terms of \\spad{tan(x/2)} to express in terms of \\spad{tan(x)} . Other examples are the expressions \\spad{sqrt(x+1)+sqrt(x+7)+1} or \\indented{1}{\\spad{sqrt(sin(x))+1} .}")) (|solve| (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Equation| (|Expression| |#1|))) (|List| (|Symbol|))) "\\spad{solve(leqs, lvar)} returns a list of solutions to the list of equations \\spad{leqs} with respect to the list of symbols lvar.") (((|List| (|Equation| (|Expression| |#1|))) (|Expression| |#1|) (|Symbol|)) "\\spad{solve(expr,x)} finds the solutions of the equation \\spad{expr} = 0 with respect to the symbol \\spad{x} where \\spad{expr} is a function of type Expression(\\spad{R}).") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Expression| |#1|)) (|Symbol|)) "\\spad{solve(eq,x)} finds the solutions of the equation \\spad{eq} where \\spad{eq} is an equation of functions of type Expression(\\spad{R}) with respect to the symbol \\spad{x}.") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Expression| |#1|))) "\\spad{solve(eq)} finds the solutions of the equation \\spad{eq} where \\spad{eq} is an equation of functions of type Expression(\\spad{R}) with respect to the unique symbol \\spad{x} appearing in \\spad{eq}.") (((|List| (|Equation| (|Expression| |#1|))) (|Expression| |#1|)) "\\spad{solve(expr)} finds the solutions of the equation \\spad{expr} = 0 where \\spad{expr} is a function of type Expression(\\spad{R}) with respect to the unique symbol \\spad{x} appearing in eq."))) 
 NIL 
 NIL 
-(-1136 S A) 
+(-1138 S A) 
 ((|constructor| (NIL "This package exports sorting algorithnms")) (|insertionSort!| ((|#2| |#2|) "\\spad{insertionSort! }\\undocumented") ((|#2| |#2| (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{insertionSort!(a,f)} \\undocumented")) (|bubbleSort!| ((|#2| |#2|) "\\spad{bubbleSort!(a)} \\undocumented") ((|#2| |#2| (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{bubbleSort!(a,f)} \\undocumented"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-853)))) 
-(-1137 R) 
+((|HasCategory| |#1| (QUOTE (-855)))) 
+(-1139 R) 
 ((|constructor| (NIL "The domain ThreeSpace is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them."))) 
 NIL 
 NIL 
-(-1138 R) 
+(-1140 R) 
 ((|constructor| (NIL "The category ThreeSpaceCategory is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(s)} returns the \\spadtype{ThreeSpace} \\spad{s} to Output format.")) (|subspace| (((|SubSpace| 3 |#1|) $) "\\spad{subspace(s)} returns the \\spadtype{SubSpace} which holds all the point information in the \\spadtype{ThreeSpace},{} \\spad{s}.")) (|check| (($ $) "\\spad{check(s)} returns lllpt,{} list of lists of lists of point information about the \\spadtype{ThreeSpace} \\spad{s}.")) (|objects| (((|Record| (|:| |points| (|NonNegativeInteger|)) (|:| |curves| (|NonNegativeInteger|)) (|:| |polygons| (|NonNegativeInteger|)) (|:| |constructs| (|NonNegativeInteger|))) $) "\\spad{objects(s)} returns the \\spadtype{ThreeSpace},{} \\spad{s},{} in the form of a 3D object record containing information on the number of points,{} curves,{} polygons and constructs comprising the \\spadtype{ThreeSpace}..")) (|lprop| (((|List| (|SubSpaceComponentProperty|)) $) "\\spad{lprop(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of subspace component properties,{} and if so,{} returns the list; An error is signaled otherwise.")) (|llprop| (((|List| (|List| (|SubSpaceComponentProperty|))) $) "\\spad{llprop(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of curves which are lists of the subspace component properties of the curves,{} and if so,{} returns the list of lists; An error is signaled otherwise.")) (|lllp| (((|List| (|List| (|List| (|Point| |#1|)))) $) "\\spad{lllp(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of components,{} which are lists of curves,{} which are lists of points,{} and if so,{} returns the list of lists of lists; An error is signaled otherwise.")) (|lllip| (((|List| (|List| (|List| (|NonNegativeInteger|)))) $) "\\spad{lllip(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of components,{} which are lists of curves,{} which are lists of indices to points,{} and if so,{} returns the list of lists of lists; An error is signaled otherwise.")) (|lp| (((|List| (|Point| |#1|)) $) "\\spad{lp(s)} returns the list of points component which the \\spadtype{ThreeSpace},{} \\spad{s},{} contains; these points are used by reference,{} \\spadignore{i.e.} the component holds indices referring to the points rather than the points themselves. This allows for sharing of the points.")) (|mesh?| (((|Boolean|) $) "\\spad{mesh?(s)} returns \\spad{true} if the \\spadtype{ThreeSpace} \\spad{s} is composed of one component,{} a mesh comprising a list of curves which are lists of points,{} or returns \\spad{false} if otherwise")) (|mesh| (((|List| (|List| (|Point| |#1|))) $) "\\spad{mesh(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single surface component defined by a list curves which contain lists of points,{} and if so,{} returns the list of lists of points; An error is signaled otherwise.") (($ (|List| (|List| (|Point| |#1|))) (|Boolean|) (|Boolean|)) "\\spad{mesh([[p0],[p1],...,[pn]], close1, close2)} creates a surface defined over a list of curves,{} \\spad{p0} through \\spad{pn},{} which are lists of points; the booleans \\spad{close1} and close2 indicate how the surface is to be closed: \\spad{close1} set to \\spad{true} means that each individual list (a curve) is to be closed (that is,{} the last point of the list is to be connected to the first point); close2 set to \\spad{true} means that the boundary at one end of the surface is to be connected to the boundary at the other end (the boundaries are defined as the first list of points (curve) and the last list of points (curve)); the \\spadtype{ThreeSpace} containing this surface is returned.") (($ (|List| (|List| (|Point| |#1|)))) "\\spad{mesh([[p0],[p1],...,[pn]])} creates a surface defined by a list of curves which are lists,{} \\spad{p0} through \\spad{pn},{} of points,{} and returns a \\spadtype{ThreeSpace} whose component is the surface.") (($ $ (|List| (|List| (|List| |#1|))) (|Boolean|) (|Boolean|)) "\\spad{mesh(s,[ [[r10]...,[r1m]], [[r20]...,[r2m]],..., [[rn0]...,[rnm]] ], close1, close2)} adds a surface component to the \\spadtype{ThreeSpace} \\spad{s},{} which is defined over a rectangular domain of size \\spad{WxH} where \\spad{W} is the number of lists of points from the domain \\spad{PointDomain(R)} and \\spad{H} is the number of elements in each of those lists; the booleans \\spad{close1} and close2 indicate how the surface is to be closed: if \\spad{close1} is \\spad{true} this means that each individual list (a curve) is to be closed (\\spadignore{i.e.} the last point of the list is to be connected to the first point); if close2 is \\spad{true},{} this means that the boundary at one end of the surface is to be connected to the boundary at the other end (the boundaries are defined as the first list of points (curve) and the last list of points (curve)).") (($ $ (|List| (|List| (|Point| |#1|))) (|Boolean|) (|Boolean|)) "\\spad{mesh(s,[[p0],[p1],...,[pn]], close1, close2)} adds a surface component to the \\spadtype{ThreeSpace},{} which is defined over a list of curves,{} in which each of these curves is a list of points. The boolean arguments \\spad{close1} and close2 indicate how the surface is to be closed. Argument \\spad{close1} equal \\spad{true} means that each individual list (a curve) is to be closed,{} \\spadignore{i.e.} the last point of the list is to be connected to the first point. Argument close2 equal \\spad{true} means that the boundary at one end of the surface is to be connected to the boundary at the other end,{} \\spadignore{i.e.} the boundaries are defined as the first list of points (curve) and the last list of points (curve).") (($ $ (|List| (|List| (|List| |#1|))) (|List| (|SubSpaceComponentProperty|)) (|SubSpaceComponentProperty|)) "\\spad{mesh(s,[ [[r10]...,[r1m]], [[r20]...,[r2m]],..., [[rn0]...,[rnm]] ], [props], prop)} adds a surface component to the \\spadtype{ThreeSpace} \\spad{s},{} which is defined over a rectangular domain of size \\spad{WxH} where \\spad{W} is the number of lists of points from the domain \\spad{PointDomain(R)} and \\spad{H} is the number of elements in each of those lists; lprops is the list of the subspace component properties for each curve list,{} and prop is the subspace component property by which the points are defined.") (($ $ (|List| (|List| (|Point| |#1|))) (|List| (|SubSpaceComponentProperty|)) (|SubSpaceComponentProperty|)) "\\spad{mesh(s,[[p0],[p1],...,[pn]],[props],prop)} adds a surface component,{} defined over a list curves which contains lists of points,{} to the \\spadtype{ThreeSpace} \\spad{s}; props is a list which contains the subspace component properties for each surface parameter,{} and \\spad{prop} is the subspace component property by which the points are defined.")) (|polygon?| (((|Boolean|) $) "\\spad{polygon?(s)} returns \\spad{true} if the \\spadtype{ThreeSpace} \\spad{s} contains a single polygon component,{} or \\spad{false} otherwise.")) (|polygon| (((|List| (|Point| |#1|)) $) "\\spad{polygon(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single polygon component defined by a list of points,{} and if so,{} returns the list of points; An error is signaled otherwise.") (($ (|List| (|Point| |#1|))) "\\spad{polygon([p0,p1,...,pn])} creates a polygon defined by a list of points,{} \\spad{p0} through \\spad{pn},{} and returns a \\spadtype{ThreeSpace} whose component is the polygon.") (($ $ (|List| (|List| |#1|))) "\\spad{polygon(s,[[r0],[r1],...,[rn]])} adds a polygon component defined by a list of points \\spad{r0} through \\spad{rn},{} which are lists of elements from the domain \\spad{PointDomain(m,R)} to the \\spadtype{ThreeSpace} \\spad{s},{} where \\spad{m} is the dimension of the points and \\spad{R} is the \\spadtype{Ring} over which the points are defined.") (($ $ (|List| (|Point| |#1|))) "\\spad{polygon(s,[p0,p1,...,pn])} adds a polygon component defined by a list of points,{} \\spad{p0} throught \\spad{pn},{} to the \\spadtype{ThreeSpace} \\spad{s}.")) (|closedCurve?| (((|Boolean|) $) "\\spad{closedCurve?(s)} returns \\spad{true} if the \\spadtype{ThreeSpace} \\spad{s} contains a single closed curve component,{} \\spadignore{i.e.} the first element of the curve is also the last element,{} or \\spad{false} otherwise.")) (|closedCurve| (((|List| (|Point| |#1|)) $) "\\spad{closedCurve(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single closed curve component defined by a list of points in which the first point is also the last point,{} all of which are from the domain \\spad{PointDomain(m,R)} and if so,{} returns the list of points. An error is signaled otherwise.") (($ (|List| (|Point| |#1|))) "\\spad{closedCurve(lp)} sets a list of points defined by the first element of \\spad{lp} through the last element of \\spad{lp} and back to the first elelment again and returns a \\spadtype{ThreeSpace} whose component is the closed curve defined by \\spad{lp}.") (($ $ (|List| (|List| |#1|))) "\\spad{closedCurve(s,[[lr0],[lr1],...,[lrn],[lr0]])} adds a closed curve component defined by a list of points \\spad{lr0} through \\spad{lrn},{} which are lists of elements from the domain \\spad{PointDomain(m,R)},{} where \\spad{R} is the \\spadtype{Ring} over which the point elements are defined and \\spad{m} is the dimension of the points,{} in which the last element of the list of points contains a copy of the first element list,{} \\spad{lr0}. The closed curve is added to the \\spadtype{ThreeSpace},{} \\spad{s}.") (($ $ (|List| (|Point| |#1|))) "\\spad{closedCurve(s,[p0,p1,...,pn,p0])} adds a closed curve component which is a list of points defined by the first element \\spad{p0} through the last element \\spad{pn} and back to the first element \\spad{p0} again,{} to the \\spadtype{ThreeSpace} \\spad{s}.")) (|curve?| (((|Boolean|) $) "\\spad{curve?(s)} queries whether the \\spadtype{ThreeSpace},{} \\spad{s},{} is a curve,{} \\spadignore{i.e.} has one component,{} a list of list of points,{} and returns \\spad{true} if it is,{} or \\spad{false} otherwise.")) (|curve| (((|List| (|Point| |#1|)) $) "\\spad{curve(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single curve defined by a list of points and if so,{} returns the curve,{} \\spadignore{i.e.} list of points. An error is signaled otherwise.") (($ (|List| (|Point| |#1|))) "\\spad{curve([p0,p1,p2,...,pn])} creates a space curve defined by the list of points \\spad{p0} through \\spad{pn},{} and returns the \\spadtype{ThreeSpace} whose component is the curve.") (($ $ (|List| (|List| |#1|))) "\\spad{curve(s,[[p0],[p1],...,[pn]])} adds a space curve which is a list of points \\spad{p0} through \\spad{pn} defined by lists of elements from the domain \\spad{PointDomain(m,R)},{} where \\spad{R} is the \\spadtype{Ring} over which the point elements are defined and \\spad{m} is the dimension of the points,{} to the \\spadtype{ThreeSpace} \\spad{s}.") (($ $ (|List| (|Point| |#1|))) "\\spad{curve(s,[p0,p1,...,pn])} adds a space curve component defined by a list of points \\spad{p0} through \\spad{pn},{} to the \\spadtype{ThreeSpace} \\spad{s}.")) (|point?| (((|Boolean|) $) "\\spad{point?(s)} queries whether the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single component which is a point and returns the boolean result.")) (|point| (((|Point| |#1|) $) "\\spad{point(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of only a single point and if so,{} returns the point. An error is signaled otherwise.") (($ (|Point| |#1|)) "\\spad{point(p)} returns a \\spadtype{ThreeSpace} object which is composed of one component,{} the point \\spad{p}.") (($ $ (|NonNegativeInteger|)) "\\spad{point(s,i)} adds a point component which is placed into a component list of the \\spadtype{ThreeSpace},{} \\spad{s},{} at the index given by \\spad{i}.") (($ $ (|List| |#1|)) "\\spad{point(s,[x,y,z])} adds a point component defined by a list of elements which are from the \\spad{PointDomain(R)} to the \\spadtype{ThreeSpace},{} \\spad{s},{} where \\spad{R} is the \\spadtype{Ring} over which the point elements are defined.") (($ $ (|Point| |#1|)) "\\spad{point(s,p)} adds a point component defined by the point,{} \\spad{p},{} specified as a list from \\spad{List(R)},{} to the \\spadtype{ThreeSpace},{} \\spad{s},{} where \\spad{R} is the \\spadtype{Ring} over which the point is defined.")) (|modifyPointData| (($ $ (|NonNegativeInteger|) (|Point| |#1|)) "\\spad{modifyPointData(s,i,p)} changes the point at the indexed location \\spad{i} in the \\spadtype{ThreeSpace},{} \\spad{s},{} to that of point \\spad{p}. This is useful for making changes to a point which has been transformed.")) (|enterPointData| (((|NonNegativeInteger|) $ (|List| (|Point| |#1|))) "\\spad{enterPointData(s,[p0,p1,...,pn])} adds a list of points from \\spad{p0} through \\spad{pn} to the \\spadtype{ThreeSpace},{} \\spad{s},{} and returns the index,{} to the starting point of the list.")) (|copy| (($ $) "\\spad{copy(s)} returns a new \\spadtype{ThreeSpace} that is an exact copy of \\spad{s}.")) (|composites| (((|List| $) $) "\\spad{composites(s)} takes the \\spadtype{ThreeSpace} \\spad{s},{} and creates a list containing a unique \\spadtype{ThreeSpace} for each single composite of \\spad{s}. If \\spad{s} has no composites defined (composites need to be explicitly created),{} the list returned is empty. Note that not all the components need to be part of a composite.")) (|components| (((|List| $) $) "\\spad{components(s)} takes the \\spadtype{ThreeSpace} \\spad{s},{} and creates a list containing a unique \\spadtype{ThreeSpace} for each single component of \\spad{s}. If \\spad{s} has no components defined,{} the list returned is empty.")) (|composite| (($ (|List| $)) "\\spad{composite([s1,s2,...,sn])} will create a new \\spadtype{ThreeSpace} that is a union of all the components from each \\spadtype{ThreeSpace} in the parameter list,{} grouped as a composite.")) (|merge| (($ $ $) "\\spad{merge(s1,s2)} will create a new \\spadtype{ThreeSpace} that has the components of \\spad{s1} and \\spad{s2}; Groupings of components into composites are maintained.") (($ (|List| $)) "\\spad{merge([s1,s2,...,sn])} will create a new \\spadtype{ThreeSpace} that has the components of all the ones in the list; Groupings of components into composites are maintained.")) (|numberOfComposites| (((|NonNegativeInteger|) $) "\\spad{numberOfComposites(s)} returns the number of supercomponents,{} or composites,{} in the \\spadtype{ThreeSpace},{} \\spad{s}; Composites are arbitrary groupings of otherwise distinct and unrelated components; A \\spadtype{ThreeSpace} need not have any composites defined at all and,{} outside of the requirement that no component can belong to more than one composite at a time,{} the definition and interpretation of composites are unrestricted.")) (|numberOfComponents| (((|NonNegativeInteger|) $) "\\spad{numberOfComponents(s)} returns the number of distinct object components in the indicated \\spadtype{ThreeSpace},{} \\spad{s},{} such as points,{} curves,{} polygons,{} and constructs.")) (|create3Space| (($ (|SubSpace| 3 |#1|)) "\\spad{create3Space(s)} creates a \\spadtype{ThreeSpace} object containing objects pre-defined within some \\spadtype{SubSpace} \\spad{s}.") (($) "\\spad{create3Space()} creates a \\spadtype{ThreeSpace} object capable of holding point,{} curve,{} mesh components and any combination."))) 
 NIL 
 NIL 
-(-1139) 
+(-1141) 
 ((|constructor| (NIL "This domain represents a kind of base domain \\indented{2}{for Spad syntax domain.\\space{2}It merely exists as a kind of} \\indented{2}{of abstract base in object-oriented programming language.} \\indented{2}{However,{} this is not an abstract class.}"))) 
 NIL 
 NIL 
-(-1140) 
+(-1142) 
 ((|constructor| (NIL "\\indented{1}{This package provides a simple Spad algebra parser.} Related Constructors: Syntax. See Also: Syntax.")) (|parse| (((|List| (|Syntax|)) (|String|)) "\\spad{parse(f)} parses the source file \\spad{f} (supposedly containing Spad algebras) and returns a List Syntax. The filename \\spad{f} is supposed to have the proper extension. Note that this function has the side effect of executing any system command contained in the file \\spad{f},{} even if it might not be meaningful."))) 
 NIL 
 NIL 
-(-1141) 
+(-1143) 
 ((|constructor| (NIL "This category describes the exported \\indented{2}{signatures of the SpadAst domain.}")) (|autoCoerce| (((|Integer|) $) "\\spad{autoCoerce(s)} returns the Integer view of \\spad{`s'}. Left at the discretion of the compiler.") (((|String|) $) "\\spad{autoCoerce(s)} returns the String view of \\spad{`s'}. Left at the discretion of the compiler.") (((|Identifier|) $) "\\spad{autoCoerce(s)} returns the Identifier view of \\spad{`s'}. Left at the discretion of the compiler.") (((|IsAst|) $) "\\spad{autoCoerce(s)} returns the IsAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|HasAst|) $) "\\spad{autoCoerce(s)} returns the HasAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|CaseAst|) $) "\\spad{autoCoerce(s)} returns the CaseAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|ColonAst|) $) "\\spad{autoCoerce(s)} returns the ColoonAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|SuchThatAst|) $) "\\spad{autoCoerce(s)} returns the SuchThatAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|LetAst|) $) "\\spad{autoCoerce(s)} returns the LetAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|SequenceAst|) $) "\\spad{autoCoerce(s)} returns the SequenceAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|SegmentAst|) $) "\\spad{autoCoerce(s)} returns the SegmentAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|RestrictAst|) $) "\\spad{autoCoerce(s)} returns the RestrictAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|PretendAst|) $) "\\spad{autoCoerce(s)} returns the PretendAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|CoerceAst|) $) "\\spad{autoCoerce(s)} returns the CoerceAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|ReturnAst|) $) "\\spad{autoCoerce(s)} returns the ReturnAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|ExitAst|) $) "\\spad{autoCoerce(s)} returns the ExitAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|ConstructAst|) $) "\\spad{autoCoerce(s)} returns the ConstructAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|CollectAst|) $) "\\spad{autoCoerce(s)} returns the CollectAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|StepAst|) $) "\\spad{autoCoerce(s)} returns the InAst view of \\spad{s}. Left at the discretion of the compiler.") (((|InAst|) $) "\\spad{autoCoerce(s)} returns the InAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|WhileAst|) $) "\\spad{autoCoerce(s)} returns the WhileAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|RepeatAst|) $) "\\spad{autoCoerce(s)} returns the RepeatAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|IfAst|) $) "\\spad{autoCoerce(s)} returns the IfAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|MappingAst|) $) "\\spad{autoCoerce(s)} returns the MappingAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|AttributeAst|) $) "\\spad{autoCoerce(s)} returns the AttributeAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|SignatureAst|) $) "\\spad{autoCoerce(s)} returns the SignatureAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|CapsuleAst|) $) "\\spad{autoCoerce(s)} returns the CapsuleAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|JoinAst|) $) "\\spad{autoCoerce(s)} returns the \\spadype{JoinAst} view of of the AST object \\spad{s}. Left at the discretion of the compiler.") (((|CategoryAst|) $) "\\spad{autoCoerce(s)} returns the CategoryAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|WhereAst|) $) "\\spad{autoCoerce(s)} returns the WhereAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|MacroAst|) $) "\\spad{autoCoerce(s)} returns the MacroAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|DefinitionAst|) $) "\\spad{autoCoerce(s)} returns the DefinitionAst view of \\spad{`s'}. Left at the discretion of the compiler.") (((|ImportAst|) $) "\\spad{autoCoerce(s)} returns the ImportAst view of \\spad{`s'}. Left at the discretion of the compiler.")) (|case| (((|Boolean|) $ (|[\|\|]| (|Integer|))) "\\spad{s case Integer} holds if \\spad{`s'} represents an integer literal.") (((|Boolean|) $ (|[\|\|]| (|String|))) "\\spad{s case String} holds if \\spad{`s'} represents a string literal.") (((|Boolean|) $ (|[\|\|]| (|Identifier|))) "\\spad{s case Identifier} holds if \\spad{`s'} represents an identifier.") (((|Boolean|) $ (|[\|\|]| (|IsAst|))) "\\spad{s case IsAst} holds if \\spad{`s'} represents an is-expression.") (((|Boolean|) $ (|[\|\|]| (|HasAst|))) "\\spad{s case HasAst} holds if \\spad{`s'} represents a has-expression.") (((|Boolean|) $ (|[\|\|]| (|CaseAst|))) "\\spad{s case CaseAst} holds if \\spad{`s'} represents a case-expression.") (((|Boolean|) $ (|[\|\|]| (|ColonAst|))) "\\spad{s case ColonAst} holds if \\spad{`s'} represents a colon-expression.") (((|Boolean|) $ (|[\|\|]| (|SuchThatAst|))) "\\spad{s case SuchThatAst} holds if \\spad{`s'} represents a qualified-expression.") (((|Boolean|) $ (|[\|\|]| (|LetAst|))) "\\spad{s case LetAst} holds if \\spad{`s'} represents an assignment-expression.") (((|Boolean|) $ (|[\|\|]| (|SequenceAst|))) "\\spad{s case SequenceAst} holds if \\spad{`s'} represents a sequence-of-statements.") (((|Boolean|) $ (|[\|\|]| (|SegmentAst|))) "\\spad{s case SegmentAst} holds if \\spad{`s'} represents a segment-expression.") (((|Boolean|) $ (|[\|\|]| (|RestrictAst|))) "\\spad{s case RestrictAst} holds if \\spad{`s'} represents a restrict-expression.") (((|Boolean|) $ (|[\|\|]| (|PretendAst|))) "\\spad{s case PretendAst} holds if \\spad{`s'} represents a pretend-expression.") (((|Boolean|) $ (|[\|\|]| (|CoerceAst|))) "\\spad{s case ReturnAst} holds if \\spad{`s'} represents a coerce-expression.") (((|Boolean|) $ (|[\|\|]| (|ReturnAst|))) "\\spad{s case ReturnAst} holds if \\spad{`s'} represents a return-statement.") (((|Boolean|) $ (|[\|\|]| (|ExitAst|))) "\\spad{s case ExitAst} holds if \\spad{`s'} represents an exit-expression.") (((|Boolean|) $ (|[\|\|]| (|ConstructAst|))) "\\spad{s case ConstructAst} holds if \\spad{`s'} represents a list-expression.") (((|Boolean|) $ (|[\|\|]| (|CollectAst|))) "\\spad{s case CollectAst} holds if \\spad{`s'} represents a list-comprehension.") (((|Boolean|) $ (|[\|\|]| (|StepAst|))) "\\spad{s case StepAst} holds if \\spad{s} represents an arithmetic progression iterator.") (((|Boolean|) $ (|[\|\|]| (|InAst|))) "\\spad{s case InAst} holds if \\spad{`s'} represents a in-iterator") (((|Boolean|) $ (|[\|\|]| (|WhileAst|))) "\\spad{s case WhileAst} holds if \\spad{`s'} represents a while-iterator") (((|Boolean|) $ (|[\|\|]| (|RepeatAst|))) "\\spad{s case RepeatAst} holds if \\spad{`s'} represents an repeat-loop.") (((|Boolean|) $ (|[\|\|]| (|IfAst|))) "\\spad{s case IfAst} holds if \\spad{`s'} represents an if-statement.") (((|Boolean|) $ (|[\|\|]| (|MappingAst|))) "\\spad{s case MappingAst} holds if \\spad{`s'} represents a mapping type.") (((|Boolean|) $ (|[\|\|]| (|AttributeAst|))) "\\spad{s case AttributeAst} holds if \\spad{`s'} represents an attribute.") (((|Boolean|) $ (|[\|\|]| (|SignatureAst|))) "\\spad{s case SignatureAst} holds if \\spad{`s'} represents a signature export.") (((|Boolean|) $ (|[\|\|]| (|CapsuleAst|))) "\\spad{s case CapsuleAst} holds if \\spad{`s'} represents a domain capsule.") (((|Boolean|) $ (|[\|\|]| (|JoinAst|))) "\\spad{s case JoinAst} holds is the syntax object \\spad{s} denotes the join of several categories.") (((|Boolean|) $ (|[\|\|]| (|CategoryAst|))) "\\spad{s case CategoryAst} holds if \\spad{`s'} represents an unnamed category.") (((|Boolean|) $ (|[\|\|]| (|WhereAst|))) "\\spad{s case WhereAst} holds if \\spad{`s'} represents an expression with local definitions.") (((|Boolean|) $ (|[\|\|]| (|MacroAst|))) "\\spad{s case MacroAst} holds if \\spad{`s'} represents a macro definition.") (((|Boolean|) $ (|[\|\|]| (|DefinitionAst|))) "\\spad{s case DefinitionAst} holds if \\spad{`s'} represents a definition.") (((|Boolean|) $ (|[\|\|]| (|ImportAst|))) "\\spad{s case ImportAst} holds if \\spad{`s'} represents an `import' statement."))) 
 NIL 
 NIL 
-(-1142) 
+(-1144) 
 ((|constructor| (NIL "SpecialOutputPackage allows FORTRAN,{} Tex and \\indented{2}{Script Formula Formatter output from programs.}")) (|outputAsTex| (((|Void|) (|List| (|OutputForm|))) "\\spad{outputAsTex(l)} sends (for each expression in the list \\spad{l}) output in Tex format to the destination as defined by \\spadsyscom{set output tex}.") (((|Void|) (|OutputForm|)) "\\spad{outputAsTex(o)} sends output \\spad{o} in Tex format to the destination defined by \\spadsyscom{set output tex}.")) (|outputAsScript| (((|Void|) (|List| (|OutputForm|))) "\\spad{outputAsScript(l)} sends (for each expression in the list \\spad{l}) output in Script Formula Formatter format to the destination defined. by \\spadsyscom{set output forumula}.") (((|Void|) (|OutputForm|)) "\\spad{outputAsScript(o)} sends output \\spad{o} in Script Formula Formatter format to the destination defined by \\spadsyscom{set output formula}.")) (|outputAsFortran| (((|Void|) (|List| (|OutputForm|))) "\\spad{outputAsFortran(l)} sends (for each expression in the list \\spad{l}) output in FORTRAN format to the destination defined by \\spadsyscom{set output fortran}.") (((|Void|) (|OutputForm|)) "\\spad{outputAsFortran(o)} sends output \\spad{o} in FORTRAN format.") (((|Void|) (|String|) (|OutputForm|)) "\\spad{outputAsFortran(v,o)} sends output \\spad{v} = \\spad{o} in FORTRAN format to the destination defined by \\spadsyscom{set output fortran}."))) 
 NIL 
 NIL 
-(-1143) 
+(-1145) 
 ((|constructor| (NIL "Category for the other special functions.")) (|airyBi| (($ $) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}.")) (|airyAi| (($ $) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}.")) (|besselK| (($ $ $) "\\spad{besselK(v,z)} is the modified Bessel function of the second kind.")) (|besselI| (($ $ $) "\\spad{besselI(v,z)} is the modified Bessel function of the first kind.")) (|besselY| (($ $ $) "\\spad{besselY(v,z)} is the Bessel function of the second kind.")) (|besselJ| (($ $ $) "\\spad{besselJ(v,z)} is the Bessel function of the first kind.")) (|polygamma| (($ $ $) "\\spad{polygamma(k,x)} is the \\spad{k-th} derivative of \\spad{digamma(x)},{} (often written \\spad{psi(k,x)} in the literature).")) (|digamma| (($ $) "\\spad{digamma(x)} is the logarithmic derivative of \\spad{Gamma(x)} (often written \\spad{psi(x)} in the literature).")) (|Beta| (($ $ $) "\\spad{Beta(x,y)} is \\spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.")) (|Gamma| (($ $ $) "\\spad{Gamma(a,x)} is the incomplete Gamma function.") (($ $) "\\spad{Gamma(x)} is the Euler Gamma function.")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x}."))) 
 NIL 
 NIL 
-(-1144 V C) 
+(-1146 V C) 
 ((|constructor| (NIL "This domain exports a modest implementation for the vertices of splitting trees. These vertices are called here splitting nodes. Every of these nodes store 3 informations. The first one is its value,{} that is the current expression to evaluate. The second one is its condition,{} that is the hypothesis under which the value has to be evaluated. The last one is its status,{} that is a boolean flag which is \\spad{true} iff the value is the result of its evaluation under its condition. Two splitting vertices are equal iff they have the sane values and the same conditions (so their status do not matter).")) (|subNode?| (((|Boolean|) $ $ (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{subNode?(\\spad{n1},{}\\spad{n2},{}o2)} returns \\spad{true} iff \\axiom{value(\\spad{n1}) = value(\\spad{n2})} and \\axiom{o2(condition(\\spad{n1}),{}condition(\\spad{n2}))}")) (|infLex?| (((|Boolean|) $ $ (|Mapping| (|Boolean|) |#1| |#1|) (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{infLex?(\\spad{n1},{}\\spad{n2},{}o1,{}o2)} returns \\spad{true} iff \\axiom{o1(value(\\spad{n1}),{}value(\\spad{n2}))} or \\axiom{value(\\spad{n1}) = value(\\spad{n2})} and \\axiom{o2(condition(\\spad{n1}),{}condition(\\spad{n2}))}.")) (|setEmpty!| (($ $) "\\axiom{setEmpty!(\\spad{n})} replaces \\spad{n} by \\axiom{empty()\\$\\%}.")) (|setStatus!| (($ $ (|Boolean|)) "\\axiom{setStatus!(\\spad{n},{}\\spad{b})} returns \\spad{n} whose status has been replaced by \\spad{b} if it is not empty,{} else an error is produced.")) (|setCondition!| (($ $ |#2|) "\\axiom{setCondition!(\\spad{n},{}\\spad{t})} returns \\spad{n} whose condition has been replaced by \\spad{t} if it is not empty,{} else an error is produced.")) (|setValue!| (($ $ |#1|) "\\axiom{setValue!(\\spad{n},{}\\spad{v})} returns \\spad{n} whose value has been replaced by \\spad{v} if it is not empty,{} else an error is produced.")) (|copy| (($ $) "\\axiom{copy(\\spad{n})} returns a copy of \\spad{n}.")) (|construct| (((|List| $) |#1| (|List| |#2|)) "\\axiom{construct(\\spad{v},{}\\spad{lt})} returns the same as \\axiom{[construct(\\spad{v},{}\\spad{t}) for \\spad{t} in \\spad{lt}]}") (((|List| $) (|List| (|Record| (|:| |val| |#1|) (|:| |tower| |#2|)))) "\\axiom{construct(\\spad{lvt})} returns the same as \\axiom{[construct(\\spad{vt}.val,{}\\spad{vt}.tower) for \\spad{vt} in \\spad{lvt}]}") (($ (|Record| (|:| |val| |#1|) (|:| |tower| |#2|))) "\\axiom{construct(\\spad{vt})} returns the same as \\axiom{construct(\\spad{vt}.val,{}\\spad{vt}.tower)}") (($ |#1| |#2|) "\\axiom{construct(\\spad{v},{}\\spad{t})} returns the same as \\axiom{construct(\\spad{v},{}\\spad{t},{}\\spad{false})}") (($ |#1| |#2| (|Boolean|)) "\\axiom{construct(\\spad{v},{}\\spad{t},{}\\spad{b})} returns the non-empty node with value \\spad{v},{} condition \\spad{t} and flag \\spad{b}")) (|status| (((|Boolean|) $) "\\axiom{status(\\spad{n})} returns the status of the node \\spad{n}.")) (|condition| ((|#2| $) "\\axiom{condition(\\spad{n})} returns the condition of the node \\spad{n}.")) (|value| ((|#1| $) "\\axiom{value(\\spad{n})} returns the value of the node \\spad{n}.")) (|empty?| (((|Boolean|) $) "\\axiom{empty?(\\spad{n})} returns \\spad{true} iff the node \\spad{n} is \\axiom{empty()\\$\\%}.")) (|empty| (($) "\\axiom{empty()} returns the same as \\axiom{[empty()\\$\\spad{V},{}empty()\\$\\spad{C},{}\\spad{false}]\\$\\%}"))) 
 NIL 
 NIL 
-(-1145 V C) 
+(-1147 V C) 
 ((|constructor| (NIL "This domain exports a modest implementation of splitting trees. Spliiting trees are needed when the evaluation of some quantity under some hypothesis requires to split the hypothesis into sub-cases. For instance by adding some new hypothesis on one hand and its negation on another hand. The computations are terminated is a splitting tree \\axiom{a} when \\axiom{status(value(a))} is \\axiom{\\spad{true}}. Thus,{} if for the splitting tree \\axiom{a} the flag \\axiom{status(value(a))} is \\axiom{\\spad{true}},{} then \\axiom{status(value(\\spad{d}))} is \\axiom{\\spad{true}} for any subtree \\axiom{\\spad{d}} of \\axiom{a}. This property of splitting trees is called the termination condition. If no vertex in a splitting tree \\axiom{a} is equal to another,{} \\axiom{a} is said to satisfy the no-duplicates condition. The splitting tree \\axiom{a} will satisfy this condition if nodes are added to \\axiom{a} by mean of \\axiom{splitNodeOf!} and if \\axiom{construct} is only used to create the root of \\axiom{a} with no children.")) (|splitNodeOf!| (($ $ $ (|List| (|SplittingNode| |#1| |#2|)) (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{splitNodeOf!(\\spad{l},{}a,{}\\spad{ls},{}sub?)} returns \\axiom{a} where the children list of \\axiom{\\spad{l}} has been set to \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in \\spad{ls} | not subNodeOf?(\\spad{s},{}a,{}sub?)]}. Thus,{} if \\axiom{\\spad{l}} is not a node of \\axiom{a},{} this latter splitting tree is unchanged.") (($ $ $ (|List| (|SplittingNode| |#1| |#2|))) "\\axiom{splitNodeOf!(\\spad{l},{}a,{}\\spad{ls})} returns \\axiom{a} where the children list of \\axiom{\\spad{l}} has been set to \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in \\spad{ls} | not nodeOf?(\\spad{s},{}a)]}. Thus,{} if \\axiom{\\spad{l}} is not a node of \\axiom{a},{} this latter splitting tree is unchanged.")) (|remove!| (($ (|SplittingNode| |#1| |#2|) $) "\\axiom{remove!(\\spad{s},{}a)} replaces a by remove(\\spad{s},{}a)")) (|remove| (($ (|SplittingNode| |#1| |#2|) $) "\\axiom{remove(\\spad{s},{}a)} returns the splitting tree obtained from a by removing every sub-tree \\axiom{\\spad{b}} such that \\axiom{value(\\spad{b})} and \\axiom{\\spad{s}} have the same value,{} condition and status.")) (|subNodeOf?| (((|Boolean|) (|SplittingNode| |#1| |#2|) $ (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{subNodeOf?(\\spad{s},{}a,{}sub?)} returns \\spad{true} iff for some node \\axiom{\\spad{n}} in \\axiom{a} we have \\axiom{\\spad{s} = \\spad{n}} or \\axiom{status(\\spad{n})} and \\axiom{subNode?(\\spad{s},{}\\spad{n},{}sub?)}.")) (|nodeOf?| (((|Boolean|) (|SplittingNode| |#1| |#2|) $) "\\axiom{nodeOf?(\\spad{s},{}a)} returns \\spad{true} iff some node of \\axiom{a} is equal to \\axiom{\\spad{s}}")) (|result| (((|List| (|Record| (|:| |val| |#1|) (|:| |tower| |#2|))) $) "\\axiom{result(a)} where \\axiom{\\spad{ls}} is the leaves list of \\axiom{a} returns \\axiom{[[value(\\spad{s}),{}condition(\\spad{s})]\\$\\spad{VT} for \\spad{s} in \\spad{ls}]} if the computations are terminated in \\axiom{a} else an error is produced.")) (|conditions| (((|List| |#2|) $) "\\axiom{conditions(a)} returns the list of the conditions of the leaves of a")) (|construct| (($ |#1| |#2| |#1| (|List| |#2|)) "\\axiom{construct(\\spad{v1},{}\\spad{t},{}\\spad{v2},{}\\spad{lt})} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with children list given by \\axiom{[[[\\spad{v},{}\\spad{t}]\\$\\spad{S}]\\$\\% for \\spad{s} in \\spad{ls}]}.") (($ |#1| |#2| (|List| (|SplittingNode| |#1| |#2|))) "\\axiom{construct(\\spad{v},{}\\spad{t},{}\\spad{ls})} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with children list given by \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in \\spad{ls}]}.") (($ |#1| |#2| (|List| $)) "\\axiom{construct(\\spad{v},{}\\spad{t},{}la)} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with \\axiom{la} as children list.") (($ (|SplittingNode| |#1| |#2|)) "\\axiom{construct(\\spad{s})} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{\\spad{s}} and no children. Thus,{} if the status of \\axiom{\\spad{s}} is \\spad{false},{} \\axiom{[\\spad{s}]} represents the starting point of the evaluation \\axiom{value(\\spad{s})} under the hypothesis \\axiom{condition(\\spad{s})}.")) (|updateStatus!| (($ $) "\\axiom{updateStatus!(a)} returns a where the status of the vertices are updated to satisfy the \"termination condition\".")) (|extractSplittingLeaf| (((|Union| $ "failed") $) "\\axiom{extractSplittingLeaf(a)} returns the left most leaf (as a tree) whose status is \\spad{false} if any,{} else \"failed\" is returned."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-1144 |#1| |#2|) (LIST (QUOTE -311) (LIST (QUOTE -1144) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1144 |#1| |#2|) (QUOTE (-1105)))) (|HasCategory| (-1144 |#1| |#2|) (QUOTE (-1105))) (-3962 (-12 (|HasCategory| (-1144 |#1| |#2|) (LIST (QUOTE -311) (LIST (QUOTE -1144) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1144 |#1| |#2|) (QUOTE (-1105)))) (|HasCategory| (-1144 |#1| |#2|) (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| (-1144 |#1| |#2|) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1146 |ndim| R) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-1146 |#1| |#2|) (LIST (QUOTE -312) (LIST (QUOTE -1146) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1146 |#1| |#2|) (QUOTE (-1107)))) (|HasCategory| (-1146 |#1| |#2|) (QUOTE (-1107))) (-3969 (-12 (|HasCategory| (-1146 |#1| |#2|) (LIST (QUOTE -312) (LIST (QUOTE -1146) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1146 |#1| |#2|) (QUOTE (-1107)))) (|HasCategory| (-1146 |#1| |#2|) (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| (-1146 |#1| |#2|) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1148 |ndim| R) 
 ((|constructor| (NIL "\\spadtype{SquareMatrix} is a matrix domain of square matrices,{} where the number of rows (= number of columns) is a parameter of the type.")) (|unitsKnown| ((|attribute|) "the invertible matrices are simply the matrices whose determinants are units in the Ring \\spad{R}.")) (|central| ((|attribute|) "the elements of the Ring \\spad{R},{} viewed as diagonal matrices,{} commute with all matrices and,{} indeed,{} are the only matrices which commute with all matrices.")) (|squareMatrix| (($ (|Matrix| |#2|)) "\\spad{squareMatrix(m)} converts a matrix of type \\spadtype{Matrix} to a matrix of type \\spadtype{SquareMatrix}.")) (|transpose| (($ $) "\\spad{transpose(m)} returns the transpose of the matrix \\spad{m}.")) (|new| (($ |#2|) "\\spad{new(c)} constructs a new \\spadtype{SquareMatrix} object of dimension \\spad{ndim} with initial entries equal to \\spad{c}."))) 
-((-4424 . T) (-4416 |has| |#2| (-6 (-4429 "*"))) (-4427 . T) (-4421 . T) (-4422 . T)) 
-((|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasAttribute| |#2| (QUOTE (-4429 "*"))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))))) (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (QUOTE (-366))) (-3962 (|HasAttribute| |#2| (QUOTE (-4429 "*"))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181))))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-173)))) 
-(-1147 S) 
+((-4431 . T) (-4423 |has| |#2| (-6 (-4436 "*"))) (-4434 . T) (-4428 . T) (-4429 . T)) 
+((|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasAttribute| |#2| (QUOTE (-4436 "*"))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (-12 (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#2| (QUOTE (-310))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (QUOTE (-367))) (-3969 (|HasAttribute| |#2| (QUOTE (-4436 "*"))) (|HasCategory| |#2| (QUOTE (-234))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183))))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-173)))) 
+(-1149 S) 
 ((|constructor| (NIL "A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.")) (|elt| (($ $ $) "\\spad{elt(s,t)} returns the concatenation of \\spad{s} and \\spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \\axiom{\"smoo\" \"shed\"} returns \\axiom{\"smooshed\"}.")) (|rightTrim| (($ $ (|CharacterClass|)) "\\spad{rightTrim(s,cc)} returns \\spad{s} with all trailing occurences of characters in \\spad{cc} deleted. For example,{} \\axiom{rightTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"(abc\"}.") (($ $ (|Character|)) "\\spad{rightTrim(s,c)} returns \\spad{s} with all trailing occurrences of \\spad{c} deleted. For example,{} \\axiom{rightTrim(\" abc \",{} char \" \")} returns \\axiom{\" abc\"}.")) (|leftTrim| (($ $ (|CharacterClass|)) "\\spad{leftTrim(s,cc)} returns \\spad{s} with all leading characters in \\spad{cc} deleted. For example,{} \\axiom{leftTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc)\"}.") (($ $ (|Character|)) "\\spad{leftTrim(s,c)} returns \\spad{s} with all leading characters \\spad{c} deleted. For example,{} \\axiom{leftTrim(\" abc \",{} char \" \")} returns \\axiom{\"abc \"}.")) (|trim| (($ $ (|CharacterClass|)) "\\spad{trim(s,cc)} returns \\spad{s} with all characters in \\spad{cc} deleted from right and left ends. For example,{} \\axiom{trim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc\"}.") (($ $ (|Character|)) "\\spad{trim(s,c)} returns \\spad{s} with all characters \\spad{c} deleted from right and left ends. For example,{} \\axiom{trim(\" abc \",{} char \" \")} returns \\axiom{\"abc\"}.")) (|split| (((|List| $) $ (|CharacterClass|)) "\\spad{split(s,cc)} returns a list of substrings delimited by characters in \\spad{cc}.") (((|List| $) $ (|Character|)) "\\spad{split(s,c)} returns a list of substrings delimited by character \\spad{c}.")) (|coerce| (($ (|Character|)) "\\spad{coerce(c)} returns \\spad{c} as a string \\spad{s} with the character \\spad{c}.")) (|position| (((|Integer|) (|CharacterClass|) $ (|Integer|)) "\\spad{position(cc,t,i)} returns the position \\axiom{\\spad{j} \\spad{>=} \\spad{i}} in \\spad{t} of the first character belonging to \\spad{cc}.") (((|Integer|) $ $ (|Integer|)) "\\spad{position(s,t,i)} returns the position \\spad{j} of the substring \\spad{s} in string \\spad{t},{} where \\axiom{\\spad{j} \\spad{>=} \\spad{i}} is required.")) (|replace| (($ $ (|UniversalSegment| (|Integer|)) $) "\\spad{replace(s,i..j,t)} replaces the substring \\axiom{\\spad{s}(\\spad{i}..\\spad{j})} of \\spad{s} by string \\spad{t}.")) (|match?| (((|Boolean|) $ $ (|Character|)) "\\spad{match?(s,t,c)} tests if \\spad{s} matches \\spad{t} except perhaps for multiple and consecutive occurrences of character \\spad{c}. Typically \\spad{c} is the blank character.")) (|match| (((|NonNegativeInteger|) $ $ (|Character|)) "\\spad{match(p,s,wc)} tests if pattern \\axiom{\\spad{p}} matches subject \\axiom{\\spad{s}} where \\axiom{\\spad{wc}} is a wild card character. If no match occurs,{} the index \\axiom{0} is returned; otheriwse,{} the value returned is the first index of the first character in the subject matching the subject (excluding that matched by an initial wild-card). For example,{} \\axiom{match(\"*to*\",{}\"yorktown\",{}\\spad{\"*\"})} returns \\axiom{5} indicating a successful match starting at index \\axiom{5} of \\axiom{\"yorktown\"}.")) (|substring?| (((|Boolean|) $ $ (|Integer|)) "\\spad{substring?(s,t,i)} tests if \\spad{s} is a substring of \\spad{t} beginning at index \\spad{i}. Note: \\axiom{substring?(\\spad{s},{}\\spad{t},{}0) = prefix?(\\spad{s},{}\\spad{t})}.")) (|suffix?| (((|Boolean|) $ $) "\\spad{suffix?(s,t)} tests if the string \\spad{s} is the final substring of \\spad{t}. Note: \\axiom{suffix?(\\spad{s},{}\\spad{t}) \\spad{==} reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.(\\spad{n} - \\spad{m} + \\spad{i}) for \\spad{i} in 0..maxIndex \\spad{s}])} where \\spad{m} and \\spad{n} denote the maxIndex of \\spad{s} and \\spad{t} respectively.")) (|prefix?| (((|Boolean|) $ $) "\\spad{prefix?(s,t)} tests if the string \\spad{s} is the initial substring of \\spad{t}. Note: \\axiom{prefix?(\\spad{s},{}\\spad{t}) \\spad{==} reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.\\spad{i} for \\spad{i} in 0..maxIndex \\spad{s}])}.")) (|upperCase!| (($ $) "\\spad{upperCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by upper case characters.")) (|upperCase| (($ $) "\\spad{upperCase(s)} returns the string with all characters in upper case.")) (|lowerCase!| (($ $) "\\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by lower case.")) (|lowerCase| (($ $) "\\spad{lowerCase(s)} returns the string with all characters in lower case."))) 
 NIL 
 NIL 
-(-1148) 
+(-1150) 
 ((|constructor| (NIL "A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.")) (|elt| (($ $ $) "\\spad{elt(s,t)} returns the concatenation of \\spad{s} and \\spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \\axiom{\"smoo\" \"shed\"} returns \\axiom{\"smooshed\"}.")) (|rightTrim| (($ $ (|CharacterClass|)) "\\spad{rightTrim(s,cc)} returns \\spad{s} with all trailing occurences of characters in \\spad{cc} deleted. For example,{} \\axiom{rightTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"(abc\"}.") (($ $ (|Character|)) "\\spad{rightTrim(s,c)} returns \\spad{s} with all trailing occurrences of \\spad{c} deleted. For example,{} \\axiom{rightTrim(\" abc \",{} char \" \")} returns \\axiom{\" abc\"}.")) (|leftTrim| (($ $ (|CharacterClass|)) "\\spad{leftTrim(s,cc)} returns \\spad{s} with all leading characters in \\spad{cc} deleted. For example,{} \\axiom{leftTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc)\"}.") (($ $ (|Character|)) "\\spad{leftTrim(s,c)} returns \\spad{s} with all leading characters \\spad{c} deleted. For example,{} \\axiom{leftTrim(\" abc \",{} char \" \")} returns \\axiom{\"abc \"}.")) (|trim| (($ $ (|CharacterClass|)) "\\spad{trim(s,cc)} returns \\spad{s} with all characters in \\spad{cc} deleted from right and left ends. For example,{} \\axiom{trim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc\"}.") (($ $ (|Character|)) "\\spad{trim(s,c)} returns \\spad{s} with all characters \\spad{c} deleted from right and left ends. For example,{} \\axiom{trim(\" abc \",{} char \" \")} returns \\axiom{\"abc\"}.")) (|split| (((|List| $) $ (|CharacterClass|)) "\\spad{split(s,cc)} returns a list of substrings delimited by characters in \\spad{cc}.") (((|List| $) $ (|Character|)) "\\spad{split(s,c)} returns a list of substrings delimited by character \\spad{c}.")) (|coerce| (($ (|Character|)) "\\spad{coerce(c)} returns \\spad{c} as a string \\spad{s} with the character \\spad{c}.")) (|position| (((|Integer|) (|CharacterClass|) $ (|Integer|)) "\\spad{position(cc,t,i)} returns the position \\axiom{\\spad{j} \\spad{>=} \\spad{i}} in \\spad{t} of the first character belonging to \\spad{cc}.") (((|Integer|) $ $ (|Integer|)) "\\spad{position(s,t,i)} returns the position \\spad{j} of the substring \\spad{s} in string \\spad{t},{} where \\axiom{\\spad{j} \\spad{>=} \\spad{i}} is required.")) (|replace| (($ $ (|UniversalSegment| (|Integer|)) $) "\\spad{replace(s,i..j,t)} replaces the substring \\axiom{\\spad{s}(\\spad{i}..\\spad{j})} of \\spad{s} by string \\spad{t}.")) (|match?| (((|Boolean|) $ $ (|Character|)) "\\spad{match?(s,t,c)} tests if \\spad{s} matches \\spad{t} except perhaps for multiple and consecutive occurrences of character \\spad{c}. Typically \\spad{c} is the blank character.")) (|match| (((|NonNegativeInteger|) $ $ (|Character|)) "\\spad{match(p,s,wc)} tests if pattern \\axiom{\\spad{p}} matches subject \\axiom{\\spad{s}} where \\axiom{\\spad{wc}} is a wild card character. If no match occurs,{} the index \\axiom{0} is returned; otheriwse,{} the value returned is the first index of the first character in the subject matching the subject (excluding that matched by an initial wild-card). For example,{} \\axiom{match(\"*to*\",{}\"yorktown\",{}\\spad{\"*\"})} returns \\axiom{5} indicating a successful match starting at index \\axiom{5} of \\axiom{\"yorktown\"}.")) (|substring?| (((|Boolean|) $ $ (|Integer|)) "\\spad{substring?(s,t,i)} tests if \\spad{s} is a substring of \\spad{t} beginning at index \\spad{i}. Note: \\axiom{substring?(\\spad{s},{}\\spad{t},{}0) = prefix?(\\spad{s},{}\\spad{t})}.")) (|suffix?| (((|Boolean|) $ $) "\\spad{suffix?(s,t)} tests if the string \\spad{s} is the final substring of \\spad{t}. Note: \\axiom{suffix?(\\spad{s},{}\\spad{t}) \\spad{==} reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.(\\spad{n} - \\spad{m} + \\spad{i}) for \\spad{i} in 0..maxIndex \\spad{s}])} where \\spad{m} and \\spad{n} denote the maxIndex of \\spad{s} and \\spad{t} respectively.")) (|prefix?| (((|Boolean|) $ $) "\\spad{prefix?(s,t)} tests if the string \\spad{s} is the initial substring of \\spad{t}. Note: \\axiom{prefix?(\\spad{s},{}\\spad{t}) \\spad{==} reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.\\spad{i} for \\spad{i} in 0..maxIndex \\spad{s}])}.")) (|upperCase!| (($ $) "\\spad{upperCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by upper case characters.")) (|upperCase| (($ $) "\\spad{upperCase(s)} returns the string with all characters in upper case.")) (|lowerCase!| (($ $) "\\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by lower case.")) (|lowerCase| (($ $) "\\spad{lowerCase(s)} returns the string with all characters in lower case."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-1149 R E V P TS) 
+(-1151 R E V P TS) 
 ((|constructor| (NIL "A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are provided: in the sense of Zariski closure (like in Kalkbrener\\spad{'s} algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard- Moreno methods). This algorithm is valid for nay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \\spad{TS} and the packages \\spad{QCMPPK(R,E,V,P,TS)} and \\spad{RSETGCD(R,E,V,P,TS)}. The same way it does not care about the way univariate polynomial gcds (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these gcds need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \\axiomType{\\spad{TS}}. Thus,{} the operations of this package are not documented.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) 
 NIL 
 NIL 
-(-1150 R E V P) 
+(-1152 R E V P) 
 ((|constructor| (NIL "This domain provides an implementation of square-free regular chains. Moreover,{} the operation \\axiomOpFrom{zeroSetSplit}{SquareFreeRegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.} \\indented{2}{Version: 2}")) (|preprocess| (((|Record| (|:| |val| (|List| |#4|)) (|:| |towers| (|List| $))) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{pre_process(\\spad{lp},{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|internalZeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalZeroSetSplit(\\spad{lp},{}\\spad{b1},{}\\spad{b2},{}\\spad{b3})} is an internal subroutine,{} exported only for developement.")) (|zeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(\\spad{lp},{}\\spad{b1},{}\\spad{b2}.\\spad{b3},{}\\spad{b4})} is an internal subroutine,{} exported only for developement.") (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(\\spad{lp},{}clos?,{}info?)} has the same specifications as \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory} from \\spadtype{RegularTriangularSetCategory} Moreover,{} if \\axiom{clos?} then solves in the sense of the Zariski closure else solves in the sense of the regular zeros. If \\axiom{info?} then do print messages during the computations.")) (|internalAugment| (((|List| $) |#4| $ (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalAugment(\\spad{p},{}\\spad{ts},{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement."))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#4| (LIST (QUOTE -311) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#3| (QUOTE (-371))) (|HasCategory| |#4| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1151 S) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#4| (LIST (QUOTE -312) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-372))) (|HasCategory| |#4| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1153 S) 
 ((|constructor| (NIL "Linked List implementation of a Stack")) (|stack| (($ (|List| |#1|)) "\\spad{stack([x,y,...,z])} creates a stack with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last element \\spad{z}."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1152 A S) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1154 A S) 
 ((|constructor| (NIL "A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \\spadtype{Stream}. From streams,{} a number of infinite structures such power series can be built. A stream aggregate may also be infinite since it may be cyclic. For example,{} see \\spadtype{DecimalExpansion}.")) (|possiblyInfinite?| (((|Boolean|) $) "\\spad{possiblyInfinite?(s)} tests if the stream \\spad{s} could possibly have an infinite number of elements. Note: for many datatypes,{} \\axiom{possiblyInfinite?(\\spad{s}) = not explictlyFinite?(\\spad{s})}.")) (|explicitlyFinite?| (((|Boolean|) $) "\\spad{explicitlyFinite?(s)} tests if the stream has a finite number of elements,{} and \\spad{false} otherwise. Note: for many datatypes,{} \\axiom{explicitlyFinite?(\\spad{s}) = not possiblyInfinite?(\\spad{s})}."))) 
 NIL 
 NIL 
-(-1153 S) 
+(-1155 S) 
 ((|constructor| (NIL "A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \\spadtype{Stream}. From streams,{} a number of infinite structures such power series can be built. A stream aggregate may also be infinite since it may be cyclic. For example,{} see \\spadtype{DecimalExpansion}.")) (|possiblyInfinite?| (((|Boolean|) $) "\\spad{possiblyInfinite?(s)} tests if the stream \\spad{s} could possibly have an infinite number of elements. Note: for many datatypes,{} \\axiom{possiblyInfinite?(\\spad{s}) = not explictlyFinite?(\\spad{s})}.")) (|explicitlyFinite?| (((|Boolean|) $) "\\spad{explicitlyFinite?(s)} tests if the stream has a finite number of elements,{} and \\spad{false} otherwise. Note: for many datatypes,{} \\axiom{explicitlyFinite?(\\spad{s}) = not possiblyInfinite?(\\spad{s})}."))) 
 NIL 
 NIL 
-(-1154 |Key| |Ent| |dent|) 
+(-1156 |Key| |Ent| |dent|) 
 ((|constructor| (NIL "A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key."))) 
-((-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-853))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) 
-(-1155) 
+((-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-855))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) 
+(-1157) 
 ((|constructor| (NIL "A class of objects which can be 'stepped through'. Repeated applications of \\spadfun{nextItem} is guaranteed never to return duplicate items and only return \"failed\" after exhausting all elements of the domain. This assumes that the sequence starts with \\spad{init()}. For infinite domains,{} repeated application of \\spadfun{nextItem} is not required to reach all possible domain elements starting from any initial element. \\blankline Conditional attributes: \\indented{2}{infinite\\tab{15}repeated \\spad{nextItem}\\spad{'s} are never \"failed\".}")) (|nextItem| (((|Union| $ "failed") $) "\\spad{nextItem(x)} returns the next item,{} or \"failed\" if domain is exhausted.")) (|init| (($) "\\spad{init()} chooses an initial object for stepping."))) 
 NIL 
 NIL 
-(-1156) 
+(-1158) 
 ((|constructor| (NIL "This domain represents an arithmetic progression iterator syntax.")) (|step| (((|SpadAst|) $) "\\spad{step(i)} returns the Spad AST denoting the step of the arithmetic progression represented by the iterator \\spad{i}.")) (|upperBound| (((|Maybe| (|SpadAst|)) $) "If the set of values assumed by the iteration variable is bounded from above,{} \\spad{upperBound(i)} returns the upper bound. Otherwise,{} its returns \\spad{nothing}.")) (|lowerBound| (((|SpadAst|) $) "\\spad{lowerBound(i)} returns the lower bound on the values assumed by the iteration variable.")) (|iterationVar| (((|Identifier|) $) "\\spad{iterationVar(i)} returns the name of the iterating variable of the arithmetic progression iterator \\spad{i}."))) 
 NIL 
 NIL 
-(-1157 |Coef|) 
+(-1159 |Coef|) 
 ((|constructor| (NIL "This package computes infinite products of Taylor series over an integral domain of characteristic 0. Here Taylor series are represented by streams of Taylor coefficients.")) (|generalInfiniteProduct| (((|Stream| |#1|) (|Stream| |#1|) (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) 
 NIL 
 NIL 
-(-1158 S) 
+(-1160 S) 
 ((|constructor| (NIL "A stream is an implementation of an infinite sequence using a list of terms that have been computed and a function closure to compute additional terms when needed.")) (|filterUntil| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterUntil(p,s)} returns \\spad{[x0,x1,...,x(n)]} where \\spad{s = [x0,x1,x2,..]} and \\spad{n} is the smallest index such that \\spad{p(xn) = true}.")) (|filterWhile| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterWhile(p,s)} returns \\spad{[x0,x1,...,x(n-1)]} where \\spad{s = [x0,x1,x2,..]} and \\spad{n} is the smallest index such that \\spad{p(xn) = false}.")) (|generate| (($ (|Mapping| |#1| |#1|) |#1|) "\\spad{generate(f,x)} creates an infinite stream whose first element is \\spad{x} and whose \\spad{n}th element (\\spad{n > 1}) is \\spad{f} applied to the previous element. Note: \\spad{generate(f,x) = [x,f(x),f(f(x)),...]}.") (($ (|Mapping| |#1|)) "\\spad{generate(f)} creates an infinite stream all of whose elements are equal to \\spad{f()}. Note: \\spad{generate(f) = [f(),f(),f(),...]}.")) (|setrest!| (($ $ (|Integer|) $) "\\spad{setrest!(x,n,y)} sets rest(\\spad{x},{}\\spad{n}) to \\spad{y}. The function will expand cycles if necessary.")) (|showAll?| (((|Boolean|)) "\\spad{showAll?()} returns \\spad{true} if all computed entries of streams will be displayed.")) (|showAllElements| (((|OutputForm|) $) "\\spad{showAllElements(s)} creates an output form which displays all computed elements.")) (|output| (((|Void|) (|Integer|) $) "\\spad{output(n,st)} computes and displays the first \\spad{n} entries of \\spad{st}.")) (|cons| (($ |#1| $) "\\spad{cons(a,s)} returns a stream whose \\spad{first} is \\spad{a} and whose \\spad{rest} is \\spad{s}. Note: \\spad{cons(a,s) = concat(a,s)}.")) (|delay| (($ (|Mapping| $)) "\\spad{delay(f)} creates a stream with a lazy evaluation defined by function \\spad{f}. Caution: This function can only be called in compiled code.")) (|findCycle| (((|Record| (|:| |cycle?| (|Boolean|)) (|:| |prefix| (|NonNegativeInteger|)) (|:| |period| (|NonNegativeInteger|))) (|NonNegativeInteger|) $) "\\spad{findCycle(n,st)} determines if \\spad{st} is periodic within \\spad{n}.")) (|repeating?| (((|Boolean|) (|List| |#1|) $) "\\spad{repeating?(l,s)} returns \\spad{true} if a stream \\spad{s} is periodic with period \\spad{l},{} and \\spad{false} otherwise.")) (|repeating| (($ (|List| |#1|)) "\\spad{repeating(l)} is a repeating stream whose period is the list \\spad{l}.")) (|shallowlyMutable| ((|attribute|) "one may destructively alter a stream by assigning new values to its entries."))) 
-((-4428 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1159 S) 
+((-4435 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1161 S) 
 ((|constructor| (NIL "Functions defined on streams with entries in one set.")) (|concat| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{concat(u)} returns the left-to-right concatentation of the streams in \\spad{u}. Note: \\spad{concat(u) = reduce(concat,u)}."))) 
 NIL 
 NIL 
-(-1160 A B) 
+(-1162 A B) 
 ((|constructor| (NIL "Functions defined on streams with entries in two sets.")) (|reduce| ((|#2| |#2| (|Mapping| |#2| |#1| |#2|) (|Stream| |#1|)) "\\spad{reduce(b,f,u)},{} where \\spad{u} is a finite stream \\spad{[x0,x1,...,xn]},{} returns the value \\spad{r(n)} computed as follows: \\spad{r0 = f(x0,b), r1 = f(x1,r0),..., r(n) = f(xn,r(n-1))}.")) (|scan| (((|Stream| |#2|) |#2| (|Mapping| |#2| |#1| |#2|) (|Stream| |#1|)) "\\spad{scan(b,h,[x0,x1,x2,...])} returns \\spad{[y0,y1,y2,...]},{} where \\spad{y0 = h(x0,b)},{} \\spad{y1 = h(x1,y0)},{}\\spad{...} \\spad{yn = h(xn,y(n-1))}.")) (|map| (((|Stream| |#2|) (|Mapping| |#2| |#1|) (|Stream| |#1|)) "\\spad{map(f,s)} returns a stream whose elements are the function \\spad{f} applied to the corresponding elements of \\spad{s}. Note: \\spad{map(f,[x0,x1,x2,...]) = [f(x0),f(x1),f(x2),..]}."))) 
 NIL 
 NIL 
-(-1161 A B C) 
+(-1163 A B C) 
 ((|constructor| (NIL "Functions defined on streams with entries in three sets.")) (|map| (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|Stream| |#1|) (|Stream| |#2|)) "\\spad{map(f,st1,st2)} returns the stream whose elements are the function \\spad{f} applied to the corresponding elements of \\spad{st1} and \\spad{st2}. Note: \\spad{map(f,[x0,x1,x2,..],[y0,y1,y2,..]) = [f(x0,y0),f(x1,y1),..]}."))) 
 NIL 
 NIL 
-(-1162) 
+(-1164) 
 ((|constructor| (NIL "A category for string-like objects")) (|string| (($ (|Integer|)) "\\spad{string(i)} returns the decimal representation of \\spad{i} in a string"))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-1163) 
+(-1165) 
 NIL 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| (-144) (QUOTE (-853))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144))))) (-12 (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144)))))) (|HasCategory| (-144) (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-144) (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| (-144) (QUOTE (-1105))) (|HasCategory| (-144) (LIST (QUOTE -311) (QUOTE (-144)))))) 
-(-1164 |Entry|) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| (-144) (QUOTE (-855))) (|HasCategory| (-144) (LIST (QUOTE -312) (QUOTE (-144))))) (-12 (|HasCategory| (-144) (QUOTE (-1107))) (|HasCategory| (-144) (LIST (QUOTE -312) (QUOTE (-144)))))) (|HasCategory| (-144) (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-144) (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| (-144) (QUOTE (-1107))) (|HasCategory| (-144) (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| (-144) (QUOTE (-1107))) (|HasCategory| (-144) (LIST (QUOTE -312) (QUOTE (-144)))))) 
+(-1166 |Entry|) 
 ((|constructor| (NIL "This domain provides tables where the keys are strings. A specialized hash function for strings is used."))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (QUOTE (-1163))) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (QUOTE (-1105)))) (-3962 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (QUOTE (-1105))) (|HasCategory| (-1163) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 (-1163)) (|:| -2256 |#1|)) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1165 A) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (QUOTE (-1165))) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (QUOTE (-1107)))) (-3969 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (QUOTE (-1107))) (|HasCategory| (-1165) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 (-1165)) (|:| -2263 |#1|)) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1167 A) 
 ((|constructor| (NIL "StreamTaylorSeriesOperations implements Taylor series arithmetic,{} where a Taylor series is represented by a stream of its coefficients.")) (|power| (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{power(a,f)} returns the power series \\spad{f} raised to the power \\spad{a}.")) (|lazyGintegrate| (((|Stream| |#1|) (|Mapping| |#1| (|Integer|)) |#1| (|Mapping| (|Stream| |#1|))) "\\spad{lazyGintegrate(f,r,g)} is used for fixed point computations.")) (|mapdiv| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{mapdiv([a0,a1,..],[b0,b1,..])} returns \\spad{[a0/b0,a1/b1,..]}.")) (|powern| (((|Stream| |#1|) (|Fraction| (|Integer|)) (|Stream| |#1|)) "\\spad{powern(r,f)} raises power series \\spad{f} to the power \\spad{r}.")) (|nlde| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{nlde(u)} solves a first order non-linear differential equation described by \\spad{u} of the form \\spad{[[b<0,0>,b<0,1>,...],[b<1,0>,b<1,1>,.],...]}. the differential equation has the form \\spad{y' = sum(i=0 to infinity,j=0 to infinity,b<i,j>*(x**i)*(y**j))}.")) (|lazyIntegrate| (((|Stream| |#1|) |#1| (|Mapping| (|Stream| |#1|))) "\\spad{lazyIntegrate(r,f)} is a local function used for fixed point computations.")) (|integrate| (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{integrate(r,a)} returns the integral of the power series \\spad{a} with respect to the power series variableintegration where \\spad{r} denotes the constant of integration. Thus \\spad{integrate(a,[a0,a1,a2,...]) = [a,a0,a1/2,a2/3,...]}.")) (|invmultisect| (((|Stream| |#1|) (|Integer|) (|Integer|) (|Stream| |#1|)) "\\spad{invmultisect(a,b,st)} substitutes \\spad{x**((a+b)*n)} for \\spad{x**n} and multiplies by \\spad{x**b}.")) (|multisect| (((|Stream| |#1|) (|Integer|) (|Integer|) (|Stream| |#1|)) "\\spad{multisect(a,b,st)} selects the coefficients of \\spad{x**((a+b)*n+a)},{} and changes them to \\spad{x**n}.")) (|generalLambert| (((|Stream| |#1|) (|Stream| |#1|) (|Integer|) (|Integer|)) "\\spad{generalLambert(f(x),a,d)} returns \\spad{f(x**a) + f(x**(a + d)) + f(x**(a + 2 d)) + ...}. \\spad{f(x)} should have zero constant coefficient and \\spad{a} and \\spad{d} should be positive.")) (|evenlambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{evenlambert(st)} computes \\spad{f(x**2) + f(x**4) + f(x**6) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f(x)} is a power series with constant coefficient 1,{} then \\spad{prod(f(x**(2*n)),n=1..infinity) = exp(evenlambert(log(f(x))))}.")) (|oddlambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{oddlambert(st)} computes \\spad{f(x) + f(x**3) + f(x**5) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f}(\\spad{x}) is a power series with constant coefficient 1 then \\spad{prod(f(x**(2*n-1)),n=1..infinity) = exp(oddlambert(log(f(x))))}.")) (|lambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{lambert(st)} computes \\spad{f(x) + f(x**2) + f(x**3) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f(x)} is a power series with constant coefficient 1 then \\spad{prod(f(x**n),n = 1..infinity) = exp(lambert(log(f(x))))}.")) (|addiag| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{addiag(x)} performs diagonal addition of a stream of streams. if \\spad{x} = \\spad{[[a<0,0>,a<0,1>,..],[a<1,0>,a<1,1>,..],[a<2,0>,a<2,1>,..],..]} and \\spad{addiag(x) = [b<0,b<1>,...], then b<k> = sum(i+j=k,a<i,j>)}.")) (|revert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{revert(a)} computes the inverse of a power series \\spad{a} with respect to composition. the series should have constant coefficient 0 and first order coefficient should be invertible.")) (|lagrange| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{lagrange(g)} produces the power series for \\spad{f} where \\spad{f} is implicitly defined as \\spad{f(z) = z*g(f(z))}.")) (|compose| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{compose(a,b)} composes the power series \\spad{a} with the power series \\spad{b}.")) (|eval| (((|Stream| |#1|) (|Stream| |#1|) |#1|) "\\spad{eval(a,r)} returns a stream of partial sums of the power series \\spad{a} evaluated at the power series variable equal to \\spad{r}.")) (|coerce| (((|Stream| |#1|) |#1|) "\\spad{coerce(r)} converts a ring element \\spad{r} to a stream with one element.")) (|gderiv| (((|Stream| |#1|) (|Mapping| |#1| (|Integer|)) (|Stream| |#1|)) "\\spad{gderiv(f,[a0,a1,a2,..])} returns \\spad{[f(0)*a0,f(1)*a1,f(2)*a2,..]}.")) (|deriv| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{deriv(a)} returns the derivative of the power series with respect to the power series variable. Thus \\spad{deriv([a0,a1,a2,...])} returns \\spad{[a1,2 a2,3 a3,...]}.")) (|mapmult| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{mapmult([a0,a1,..],[b0,b1,..])} returns \\spad{[a0*b0,a1*b1,..]}.")) (|int| (((|Stream| |#1|) |#1|) "\\spad{int(r)} returns [\\spad{r},{}\\spad{r+1},{}\\spad{r+2},{}...],{} where \\spad{r} is a ring element.")) (|oddintegers| (((|Stream| (|Integer|)) (|Integer|)) "\\spad{oddintegers(n)} returns \\spad{[n,n+2,n+4,...]}.")) (|integers| (((|Stream| (|Integer|)) (|Integer|)) "\\spad{integers(n)} returns \\spad{[n,n+1,n+2,...]}.")) (|monom| (((|Stream| |#1|) |#1| (|Integer|)) "\\spad{monom(deg,coef)} is a monomial of degree \\spad{deg} with coefficient \\spad{coef}.")) (|recip| (((|Union| (|Stream| |#1|) "failed") (|Stream| |#1|)) "\\spad{recip(a)} returns the power series reciprocal of \\spad{a},{} or \"failed\" if not possible.")) (/ (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a / b} returns the power series quotient of \\spad{a} by \\spad{b}. An error message is returned if \\spad{b} is not invertible. This function is used in fixed point computations.")) (|exquo| (((|Union| (|Stream| |#1|) "failed") (|Stream| |#1|) (|Stream| |#1|)) "\\spad{exquo(a,b)} returns the power series quotient of \\spad{a} by \\spad{b},{} if the quotient exists,{} and \"failed\" otherwise")) (* (((|Stream| |#1|) (|Stream| |#1|) |#1|) "\\spad{a * r} returns the power series scalar multiplication of \\spad{a} by \\spad{r:} \\spad{[a0,a1,...] * r = [a0 * r,a1 * r,...]}") (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{r * a} returns the power series scalar multiplication of \\spad{r} by \\spad{a}: \\spad{r * [a0,a1,...] = [r * a0,r * a1,...]}") (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a * b} returns the power series (Cauchy) product of \\spad{a} and \\spad{b:} \\spad{[a0,a1,...] * [b0,b1,...] = [c0,c1,...]} where \\spad{ck = sum(i + j = k,ai * bk)}.")) (- (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{- a} returns the power series negative of \\spad{a}: \\spad{- [a0,a1,...] = [- a0,- a1,...]}") (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a - b} returns the power series difference of \\spad{a} and \\spad{b}: \\spad{[a0,a1,..] - [b0,b1,..] = [a0 - b0,a1 - b1,..]}")) (+ (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a + b} returns the power series sum of \\spad{a} and \\spad{b}: \\spad{[a0,a1,..] + [b0,b1,..] = [a0 + b0,a1 + b1,..]}"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550)))))) 
-(-1166 |Coef|) 
+((|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551)))))) 
+(-1168 |Coef|) 
 ((|constructor| (NIL "StreamTranscendentalFunctions implements transcendental functions on Taylor series,{} where a Taylor series is represented by a stream of its coefficients.")) (|acsch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsch(st)} computes the inverse hyperbolic cosecant of a power series \\spad{st}.")) (|asech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asech(st)} computes the inverse hyperbolic secant of a power series \\spad{st}.")) (|acoth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acoth(st)} computes the inverse hyperbolic cotangent of a power series \\spad{st}.")) (|atanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atanh(st)} computes the inverse hyperbolic tangent of a power series \\spad{st}.")) (|acosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acosh(st)} computes the inverse hyperbolic cosine of a power series \\spad{st}.")) (|asinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asinh(st)} computes the inverse hyperbolic sine of a power series \\spad{st}.")) (|csch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csch(st)} computes the hyperbolic cosecant of a power series \\spad{st}.")) (|sech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sech(st)} computes the hyperbolic secant of a power series \\spad{st}.")) (|coth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{coth(st)} computes the hyperbolic cotangent of a power series \\spad{st}.")) (|tanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tanh(st)} computes the hyperbolic tangent of a power series \\spad{st}.")) (|cosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cosh(st)} computes the hyperbolic cosine of a power series \\spad{st}.")) (|sinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sinh(st)} computes the hyperbolic sine of a power series \\spad{st}.")) (|sinhcosh| (((|Record| (|:| |sinh| (|Stream| |#1|)) (|:| |cosh| (|Stream| |#1|))) (|Stream| |#1|)) "\\spad{sinhcosh(st)} returns a record containing the hyperbolic sine and cosine of a power series \\spad{st}.")) (|acsc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsc(st)} computes arccosecant of a power series \\spad{st}.")) (|asec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asec(st)} computes arcsecant of a power series \\spad{st}.")) (|acot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acot(st)} computes arccotangent of a power series \\spad{st}.")) (|atan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atan(st)} computes arctangent of a power series \\spad{st}.")) (|acos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acos(st)} computes arccosine of a power series \\spad{st}.")) (|asin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asin(st)} computes arcsine of a power series \\spad{st}.")) (|csc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csc(st)} computes cosecant of a power series \\spad{st}.")) (|sec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sec(st)} computes secant of a power series \\spad{st}.")) (|cot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cot(st)} computes cotangent of a power series \\spad{st}.")) (|tan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tan(st)} computes tangent of a power series \\spad{st}.")) (|cos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cos(st)} computes cosine of a power series \\spad{st}.")) (|sin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sin(st)} computes sine of a power series \\spad{st}.")) (|sincos| (((|Record| (|:| |sin| (|Stream| |#1|)) (|:| |cos| (|Stream| |#1|))) (|Stream| |#1|)) "\\spad{sincos(st)} returns a record containing the sine and cosine of a power series \\spad{st}.")) (** (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{st1 ** st2} computes the power of a power series \\spad{st1} by another power series \\spad{st2}.")) (|log| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{log(st)} computes the log of a power series.")) (|exp| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{exp(st)} computes the exponential of a power series \\spad{st}."))) 
 NIL 
 NIL 
-(-1167 |Coef|) 
+(-1169 |Coef|) 
 ((|constructor| (NIL "StreamTranscendentalFunctionsNonCommutative implements transcendental functions on Taylor series over a non-commutative ring,{} where a Taylor series is represented by a stream of its coefficients.")) (|acsch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsch(st)} computes the inverse hyperbolic cosecant of a power series \\spad{st}.")) (|asech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asech(st)} computes the inverse hyperbolic secant of a power series \\spad{st}.")) (|acoth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acoth(st)} computes the inverse hyperbolic cotangent of a power series \\spad{st}.")) (|atanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atanh(st)} computes the inverse hyperbolic tangent of a power series \\spad{st}.")) (|acosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acosh(st)} computes the inverse hyperbolic cosine of a power series \\spad{st}.")) (|asinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asinh(st)} computes the inverse hyperbolic sine of a power series \\spad{st}.")) (|csch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csch(st)} computes the hyperbolic cosecant of a power series \\spad{st}.")) (|sech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sech(st)} computes the hyperbolic secant of a power series \\spad{st}.")) (|coth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{coth(st)} computes the hyperbolic cotangent of a power series \\spad{st}.")) (|tanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tanh(st)} computes the hyperbolic tangent of a power series \\spad{st}.")) (|cosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cosh(st)} computes the hyperbolic cosine of a power series \\spad{st}.")) (|sinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sinh(st)} computes the hyperbolic sine of a power series \\spad{st}.")) (|acsc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsc(st)} computes arccosecant of a power series \\spad{st}.")) (|asec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asec(st)} computes arcsecant of a power series \\spad{st}.")) (|acot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acot(st)} computes arccotangent of a power series \\spad{st}.")) (|atan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atan(st)} computes arctangent of a power series \\spad{st}.")) (|acos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acos(st)} computes arccosine of a power series \\spad{st}.")) (|asin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asin(st)} computes arcsine of a power series \\spad{st}.")) (|csc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csc(st)} computes cosecant of a power series \\spad{st}.")) (|sec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sec(st)} computes secant of a power series \\spad{st}.")) (|cot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cot(st)} computes cotangent of a power series \\spad{st}.")) (|tan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tan(st)} computes tangent of a power series \\spad{st}.")) (|cos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cos(st)} computes cosine of a power series \\spad{st}.")) (|sin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sin(st)} computes sine of a power series \\spad{st}.")) (** (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{st1 ** st2} computes the power of a power series \\spad{st1} by another power series \\spad{st2}.")) (|log| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{log(st)} computes the log of a power series.")) (|exp| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{exp(st)} computes the exponential of a power series \\spad{st}."))) 
 NIL 
 NIL 
-(-1168 R UP) 
+(-1170 R UP) 
 ((|constructor| (NIL "This package computes the subresultants of two polynomials which is needed for the `Lazard Rioboo' enhancement to Tragers integrations formula For efficiency reasons this has been rewritten to call Lionel Ducos package which is currently the best one. \\blankline")) (|primitivePart| ((|#2| |#2| |#1|) "\\spad{primitivePart(p, q)} reduces the coefficient of \\spad{p} modulo \\spad{q},{} takes the primitive part of the result,{} and ensures that the leading coefficient of that result is monic.")) (|subresultantVector| (((|PrimitiveArray| |#2|) |#2| |#2|) "\\spad{subresultantVector(p, q)} returns \\spad{[p0,...,pn]} where \\spad{pi} is the \\spad{i}-th subresultant of \\spad{p} and \\spad{q}. In particular,{} \\spad{p0 = resultant(p, q)}."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-309)))) 
-(-1169 |n| R) 
+((|HasCategory| |#1| (QUOTE (-310)))) 
+(-1171 |n| R) 
 ((|constructor| (NIL "This domain \\undocumented")) (|pointData| (((|List| (|Point| |#2|)) $) "\\spad{pointData(s)} returns the list of points from the point data field of the 3 dimensional subspace \\spad{s}.")) (|parent| (($ $) "\\spad{parent(s)} returns the subspace which is the parent of the indicated 3 dimensional subspace \\spad{s}. If \\spad{s} is the top level subspace an error message is returned.")) (|level| (((|NonNegativeInteger|) $) "\\spad{level(s)} returns a non negative integer which is the current level field of the indicated 3 dimensional subspace \\spad{s}.")) (|extractProperty| (((|SubSpaceComponentProperty|) $) "\\spad{extractProperty(s)} returns the property of domain \\spadtype{SubSpaceComponentProperty} of the indicated 3 dimensional subspace \\spad{s}.")) (|extractClosed| (((|Boolean|) $) "\\spad{extractClosed(s)} returns the \\spadtype{Boolean} value of the closed property for the indicated 3 dimensional subspace \\spad{s}. If the property is closed,{} \\spad{True} is returned,{} otherwise \\spad{False} is returned.")) (|extractIndex| (((|NonNegativeInteger|) $) "\\spad{extractIndex(s)} returns a non negative integer which is the current index of the 3 dimensional subspace \\spad{s}.")) (|extractPoint| (((|Point| |#2|) $) "\\spad{extractPoint(s)} returns the point which is given by the current index location into the point data field of the 3 dimensional subspace \\spad{s}.")) (|traverse| (($ $ (|List| (|NonNegativeInteger|))) "\\spad{traverse(s,li)} follows the branch list of the 3 dimensional subspace,{} \\spad{s},{} along the path dictated by the list of non negative integers,{} \\spad{li},{} which points to the component which has been traversed to. The subspace,{} \\spad{s},{} is returned,{} where \\spad{s} is now the subspace pointed to by \\spad{li}.")) (|defineProperty| (($ $ (|List| (|NonNegativeInteger|)) (|SubSpaceComponentProperty|)) "\\spad{defineProperty(s,li,p)} defines the component property in the 3 dimensional subspace,{} \\spad{s},{} to be that of \\spad{p},{} where \\spad{p} is of the domain \\spadtype{SubSpaceComponentProperty}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component whose property is being defined. The subspace,{} \\spad{s},{} is returned with the component property definition.")) (|closeComponent| (($ $ (|List| (|NonNegativeInteger|)) (|Boolean|)) "\\spad{closeComponent(s,li,b)} sets the property of the component in the 3 dimensional subspace,{} \\spad{s},{} to be closed if \\spad{b} is \\spad{true},{} or open if \\spad{b} is \\spad{false}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component whose closed property is to be set. The subspace,{} \\spad{s},{} is returned with the component property modification.")) (|modifyPoint| (($ $ (|NonNegativeInteger|) (|Point| |#2|)) "\\spad{modifyPoint(s,ind,p)} modifies the point referenced by the index location,{} \\spad{ind},{} by replacing it with the point,{} \\spad{p} in the 3 dimensional subspace,{} \\spad{s}. An error message occurs if \\spad{s} is empty,{} otherwise the subspace \\spad{s} is returned with the point modification.") (($ $ (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{modifyPoint(s,li,i)} replaces an existing point in the 3 dimensional subspace,{} \\spad{s},{} with the 4 dimensional point indicated by the index location,{} \\spad{i}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the existing point is to be modified. An error message occurs if \\spad{s} is empty,{} otherwise the subspace \\spad{s} is returned with the point modification.") (($ $ (|List| (|NonNegativeInteger|)) (|Point| |#2|)) "\\spad{modifyPoint(s,li,p)} replaces an existing point in the 3 dimensional subspace,{} \\spad{s},{} with the 4 dimensional point,{} \\spad{p}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the existing point is to be modified. An error message occurs if \\spad{s} is empty,{} otherwise the subspace \\spad{s} is returned with the point modification.")) (|addPointLast| (($ $ $ (|Point| |#2|) (|NonNegativeInteger|)) "\\spad{addPointLast(s,s2,li,p)} adds the 4 dimensional point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s}. \\spad{s2} point to the end of the subspace \\spad{s}. \\spad{n} is the path in the \\spad{s2} component. The subspace \\spad{s} is returned with the additional point.")) (|addPoint2| (($ $ (|Point| |#2|)) "\\spad{addPoint2(s,p)} adds the 4 dimensional point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s}. The subspace \\spad{s} is returned with the additional point.")) (|addPoint| (((|NonNegativeInteger|) $ (|Point| |#2|)) "\\spad{addPoint(s,p)} adds the point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s},{} and returns the new total number of points in \\spad{s}.") (($ $ (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{addPoint(s,li,i)} adds the 4 dimensional point indicated by the index location,{} \\spad{i},{} to the 3 dimensional subspace,{} \\spad{s}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the point is to be added. It\\spad{'s} length should range from 0 to \\spad{n - 1} where \\spad{n} is the dimension of the subspace. If the length is \\spad{n - 1},{} then a specific lowest level component is being referenced. If it is less than \\spad{n - 1},{} then some higher level component (0 indicates top level component) is being referenced and a component of that level with the desired point is created. The subspace \\spad{s} is returned with the additional point.") (($ $ (|List| (|NonNegativeInteger|)) (|Point| |#2|)) "\\spad{addPoint(s,li,p)} adds the 4 dimensional point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the point is to be added. It\\spad{'s} length should range from 0 to \\spad{n - 1} where \\spad{n} is the dimension of the subspace. If the length is \\spad{n - 1},{} then a specific lowest level component is being referenced. If it is less than \\spad{n - 1},{} then some higher level component (0 indicates top level component) is being referenced and a component of that level with the desired point is created. The subspace \\spad{s} is returned with the additional point.")) (|separate| (((|List| $) $) "\\spad{separate(s)} makes each of the components of the \\spadtype{SubSpace},{} \\spad{s},{} into a list of separate and distinct subspaces and returns the list.")) (|merge| (($ (|List| $)) "\\spad{merge(ls)} a list of subspaces,{} \\spad{ls},{} into one subspace.") (($ $ $) "\\spad{merge(s1,s2)} the subspaces \\spad{s1} and \\spad{s2} into a single subspace.")) (|deepCopy| (($ $) "\\spad{deepCopy(x)} \\undocumented")) (|shallowCopy| (($ $) "\\spad{shallowCopy(x)} \\undocumented")) (|numberOfChildren| (((|NonNegativeInteger|) $) "\\spad{numberOfChildren(x)} \\undocumented")) (|children| (((|List| $) $) "\\spad{children(x)} \\undocumented")) (|child| (($ $ (|NonNegativeInteger|)) "\\spad{child(x,n)} \\undocumented")) (|birth| (($ $) "\\spad{birth(x)} \\undocumented")) (|subspace| (($) "\\spad{subspace()} \\undocumented")) (|new| (($) "\\spad{new()} \\undocumented")) (|internal?| (((|Boolean|) $) "\\spad{internal?(x)} \\undocumented")) (|root?| (((|Boolean|) $) "\\spad{root?(x)} \\undocumented")) (|leaf?| (((|Boolean|) $) "\\spad{leaf?(x)} \\undocumented"))) 
 NIL 
 NIL 
-(-1170 S1 S2) 
+(-1172 S1 S2) 
 ((|constructor| (NIL "This domain implements \"such that\" forms")) (|rhs| ((|#2| $) "\\spad{rhs(f)} returns the right side of \\spad{f}")) (|lhs| ((|#1| $) "\\spad{lhs(f)} returns the left side of \\spad{f}")) (|construct| (($ |#1| |#2|) "\\spad{construct(s,t)} makes a form \\spad{s:t}"))) 
 NIL 
 NIL 
-(-1171) 
+(-1173) 
 ((|constructor| (NIL "This domain represents the filter iterator syntax.")) (|predicate| (((|SpadAst|) $) "\\spad{predicate(e)} returns the syntax object for the predicate in the filter iterator syntax `e'."))) 
 NIL 
 NIL 
-(-1172 |Coef| |var| |cen|) 
+(-1174 |Coef| |var| |cen|) 
 ((|constructor| (NIL "Sparse Laurent series in one variable \\indented{2}{\\spadtype{SparseUnivariateLaurentSeries} is a domain representing Laurent} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{SparseUnivariateLaurentSeries(Integer,x,3)} represents Laurent} \\indented{2}{series in \\spad{(x - 3)} with integer coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a Laurent series."))) 
-(((-4429 "*") -3962 (-3258 (|has| |#1| (-366)) (|has| (-1179 |#1| |#2| |#3|) (-823))) (|has| |#1| (-173)) (-3258 (|has| |#1| (-366)) (|has| (-1179 |#1| |#2| |#3|) (-914)))) (-4420 -3962 (-3258 (|has| |#1| (-366)) (|has| (-1179 |#1| |#2| |#3|) (-823))) (|has| |#1| (-561)) (-3258 (|has| |#1| (-366)) (|has| (-1179 |#1| |#2| |#3|) (-914)))) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1179 |#1| |#2| |#3|) (QUOTE (-823)))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1179 |#1| |#2| |#3|) (QUOTE (-914)))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1179 |#1| |#2| |#3|) (LIST (QUOTE -617) (QUOTE (-539))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1179 |#1| |#2| |#3|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1179 |#1| |#2| |#3|) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1179 |#1| |#2| |#3|) (LIST (QUOTE -288) (LIST (QUOTE -1179) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|)) (LIST (QUOTE -1179) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1179 |#1| |#2| |#3|) (LIST (QUOTE -311) (LIST (QUOTE -1179) (|devaluate| |#1|) (|devaluate| 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-(-1173 R -3498) 
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+(-1175 R -3505) 
 ((|constructor| (NIL "computes sums of top-level expressions.")) (|sum| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{sum(f(n), n = a..b)} returns \\spad{f}(a) + \\spad{f}(a+1) + ... + \\spad{f}(\\spad{b}).") ((|#2| |#2| (|Symbol|)) "\\spad{sum(a(n), n)} returns A(\\spad{n}) such that A(\\spad{n+1}) - A(\\spad{n}) = a(\\spad{n})."))) 
 NIL 
 NIL 
-(-1174 R) 
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 ((|constructor| (NIL "Computes sums of rational functions.")) (|sum| (((|Union| (|Fraction| (|Polynomial| |#1|)) (|Expression| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|Fraction| (|Polynomial| |#1|)))) "\\spad{sum(f(n), n = a..b)} returns \\spad{f(a) + f(a+1) + ... f(b)}.") (((|Fraction| (|Polynomial| |#1|)) (|Polynomial| |#1|) (|SegmentBinding| (|Polynomial| |#1|))) "\\spad{sum(f(n), n = a..b)} returns \\spad{f(a) + f(a+1) + ... f(b)}.") (((|Union| (|Fraction| (|Polynomial| |#1|)) (|Expression| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{sum(a(n), n)} returns \\spad{A} which is the indefinite sum of \\spad{a} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{A(n+1) - A(n) = a(n)}.") (((|Fraction| (|Polynomial| |#1|)) (|Polynomial| |#1|) (|Symbol|)) "\\spad{sum(a(n), n)} returns \\spad{A} which is the indefinite sum of \\spad{a} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{A(n+1) - A(n) = a(n)}."))) 
 NIL 
 NIL 
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 ((|constructor| (NIL "This domain represents univariate polynomials over arbitrary (not necessarily commutative) coefficient rings. The variable is unspecified so that the variable displays as \\spad{?} on output. If it is necessary to specify the variable name,{} use type \\spadtype{UnivariatePolynomial}. The representation is sparse in the sense that only non-zero terms are represented.")) (|fmecg| (($ $ (|NonNegativeInteger|) |#1| $) "\\spad{fmecg(p1,e,r,p2)} finds \\spad{X} : \\spad{p1} - \\spad{r} * X**e * \\spad{p2}")) (|outputForm| (((|OutputForm|) $ (|OutputForm|)) "\\spad{outputForm(p,var)} converts the SparseUnivariatePolynomial \\spad{p} to an output form (see \\spadtype{OutputForm}) printed as a polynomial in the output form variable."))) 
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-((|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-1086) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-1086) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-1086) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-1086) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-1086) (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#1| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-914)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-456))) (|HasCategory| |#1| (QUOTE (-914)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-1155))) (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#1| (QUOTE (-234))) (|HasAttribute| |#1| (QUOTE -4425)) (|HasCategory| |#1| (QUOTE (-456))) (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
-(-1176 R S) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4430 |has| |#1| (-367)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-1088) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-1088) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-1088) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-1088) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-1088) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#1| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-457))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-1157))) (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#1| (QUOTE (-234))) (|HasAttribute| |#1| (QUOTE -4432)) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#1| (QUOTE (-145))))) 
+(-1178 R S) 
 ((|constructor| (NIL "This package lifts a mapping from coefficient rings \\spad{R} to \\spad{S} to a mapping from sparse univariate polynomial over \\spad{R} to a sparse univariate polynomial over \\spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.")) (|map| (((|SparseUnivariatePolynomial| |#2|) (|Mapping| |#2| |#1|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{map(func, poly)} creates a new polynomial by applying \\spad{func} to every non-zero coefficient of the polynomial poly."))) 
 NIL 
 NIL 
-(-1177 E OV R P) 
+(-1179 E OV R P) 
 ((|constructor| (NIL "\\indented{1}{SupFractionFactorize} contains the factor function for univariate polynomials over the quotient field of a ring \\spad{S} such that the package MultivariateFactorize works for \\spad{S}")) (|squareFree| (((|Factored| (|SparseUnivariatePolynomial| (|Fraction| |#4|))) (|SparseUnivariatePolynomial| (|Fraction| |#4|))) "\\spad{squareFree(p)} returns the square-free factorization of the univariate polynomial \\spad{p} with coefficients which are fractions of polynomials over \\spad{R}. Each factor has no repeated roots and the factors are pairwise relatively prime.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| (|Fraction| |#4|))) (|SparseUnivariatePolynomial| (|Fraction| |#4|))) "\\spad{factor(p)} factors the univariate polynomial \\spad{p} with coefficients which are fractions of polynomials over \\spad{R}."))) 
 NIL 
 NIL 
-(-1178 |Coef| |var| |cen|) 
+(-1180 |Coef| |var| |cen|) 
 ((|constructor| (NIL "Sparse Puiseux series in one variable \\indented{2}{\\spadtype{SparseUnivariatePuiseuxSeries} is a domain representing Puiseux} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{SparseUnivariatePuiseuxSeries(Integer,x,3)} represents Puiseux} \\indented{2}{series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|)))) (|HasCategory| (-411 (-550)) (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-964))) (|HasCategory| |#1| (QUOTE (-1206))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4246) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1181))))) (|HasSignature| |#1| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#1|))))))) 
-(-1179 |Coef| |var| |cen|) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|)))) (|HasCategory| (-412 (-551)) (QUOTE (-1118))) (|HasCategory| |#1| (QUOTE (-367))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasSignature| |#1| (LIST (QUOTE -4387) (LIST (|devaluate| |#1|) (QUOTE (-1183)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1208))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasSignature| |#1| (LIST (QUOTE -4253) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1183))))) (|HasSignature| |#1| (LIST (QUOTE -3494) (LIST (LIST (QUOTE -646) (QUOTE (-1183))) (|devaluate| |#1|))))))) 
+(-1181 |Coef| |var| |cen|) 
 ((|constructor| (NIL "Sparse Taylor series in one variable \\indented{2}{\\spadtype{SparseUnivariateTaylorSeries} is a domain representing Taylor} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spadtype{SparseUnivariateTaylorSeries}(Integer,{}\\spad{x},{}3) represents Taylor} \\indented{2}{series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x),x)} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} computes the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-774)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-774)) (|devaluate| |#1|)))) (|HasCategory| (-774) (QUOTE (-1116))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-774))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-774))))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-964))) (|HasCategory| |#1| (QUOTE (-1206))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4246) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1181))))) (|HasSignature| |#1| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#1|))))))) 
-(-1180) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-776)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-776)) (|devaluate| |#1|)))) (|HasCategory| (-776) (QUOTE (-1118))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-776))))) (|HasSignature| |#1| (LIST (QUOTE -4387) (LIST (|devaluate| |#1|) (QUOTE (-1183)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-776))))) (|HasCategory| |#1| (QUOTE (-367))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1208))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasSignature| |#1| (LIST (QUOTE -4253) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1183))))) (|HasSignature| |#1| (LIST (QUOTE -3494) (LIST (LIST (QUOTE -646) (QUOTE (-1183))) (|devaluate| |#1|))))))) 
+(-1182) 
 ((|constructor| (NIL "This domain builds representations of boolean expressions for use with the \\axiomType{FortranCode} domain.")) (NOT (($ $) "\\spad{NOT(x)} returns the \\axiomType{Switch} expression representing \\spad{\\~~x}.") (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{NOT(x)} returns the \\axiomType{Switch} expression representing \\spad{\\~~x}.")) (AND (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{AND(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x and y}.")) (EQ (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{EQ(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x = y}.")) (OR (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{OR(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x or y}.")) (GE (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{GE(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x>=y}.")) (LE (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{LE(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x<=y}.")) (GT (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{GT(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x>y}.")) (LT (($ (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $)) (|Union| (|:| I (|Expression| (|Integer|))) (|:| F (|Expression| (|Float|))) (|:| CF (|Expression| (|Complex| (|Float|)))) (|:| |switch| $))) "\\spad{LT(x,y)} returns the \\axiomType{Switch} expression representing \\spad{x<y}.")) (|coerce| (($ (|Symbol|)) "\\spad{coerce(s)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-1181) 
+(-1183) 
 ((|constructor| (NIL "Basic and scripted symbols.")) (|sample| (($) "\\spad{sample()} returns a sample of \\%")) (|list| (((|List| $) $) "\\spad{list(sy)} takes a scripted symbol and produces a list of the name followed by the scripts.")) (|string| (((|String|) $) "\\spad{string(s)} converts the symbol \\spad{s} to a string. Error: if the symbol is subscripted.")) (|elt| (($ $ (|List| (|OutputForm|))) "\\spad{elt(s,[a1,...,an])} or \\spad{s}([a1,{}...,{}an]) returns \\spad{s} subscripted by \\spad{[a1,...,an]}.")) (|argscript| (($ $ (|List| (|OutputForm|))) "\\spad{argscript(s, [a1,...,an])} returns \\spad{s} arg-scripted by \\spad{[a1,...,an]}.")) (|superscript| (($ $ (|List| (|OutputForm|))) "\\spad{superscript(s, [a1,...,an])} returns \\spad{s} superscripted by \\spad{[a1,...,an]}.")) (|subscript| (($ $ (|List| (|OutputForm|))) "\\spad{subscript(s, [a1,...,an])} returns \\spad{s} subscripted by \\spad{[a1,...,an]}.")) (|script| (($ $ (|Record| (|:| |sub| (|List| (|OutputForm|))) (|:| |sup| (|List| (|OutputForm|))) (|:| |presup| (|List| (|OutputForm|))) (|:| |presub| (|List| (|OutputForm|))) (|:| |args| (|List| (|OutputForm|))))) "\\spad{script(s, [a,b,c,d,e])} returns \\spad{s} with subscripts a,{} superscripts \\spad{b},{} pre-superscripts \\spad{c},{} pre-subscripts \\spad{d},{} and argument-scripts \\spad{e}.") (($ $ (|List| (|List| (|OutputForm|)))) "\\spad{script(s, [a,b,c,d,e])} returns \\spad{s} with subscripts a,{} superscripts \\spad{b},{} pre-superscripts \\spad{c},{} pre-subscripts \\spad{d},{} and argument-scripts \\spad{e}. Omitted components are taken to be empty. For example,{} \\spad{script(s, [a,b,c])} is equivalent to \\spad{script(s,[a,b,c,[],[]])}.")) (|scripts| (((|Record| (|:| |sub| (|List| (|OutputForm|))) (|:| |sup| (|List| (|OutputForm|))) (|:| |presup| (|List| (|OutputForm|))) (|:| |presub| (|List| (|OutputForm|))) (|:| |args| (|List| (|OutputForm|)))) $) "\\spad{scripts(s)} returns all the scripts of \\spad{s}.")) (|scripted?| (((|Boolean|) $) "\\spad{scripted?(s)} is \\spad{true} if \\spad{s} has been given any scripts.")) (|name| (($ $) "\\spad{name(s)} returns \\spad{s} without its scripts.")) (|resetNew| (((|Void|)) "\\spad{resetNew()} resets the internals counters that new() and new(\\spad{s}) use to return distinct symbols every time.")) (|new| (($ $) "\\spad{new(s)} returns a new symbol whose name starts with \\%\\spad{s}.") (($) "\\spad{new()} returns a new symbol whose name starts with \\%."))) 
 NIL 
 NIL 
-(-1182 R) 
+(-1184 R) 
 ((|constructor| (NIL "Computes all the symmetric functions in \\spad{n} variables.")) (|symFunc| (((|Vector| |#1|) |#1| (|PositiveInteger|)) "\\spad{symFunc(r, n)} returns the vector of the elementary symmetric functions in \\spad{[r,r,...,r]} \\spad{n} times.") (((|Vector| |#1|) (|List| |#1|)) "\\spad{symFunc([r1,...,rn])} returns the vector of the elementary symmetric functions in the \\spad{ri's}: \\spad{[r1 + ... + rn, r1 r2 + ... + r(n-1) rn, ..., r1 r2 ... rn]}."))) 
 NIL 
 NIL 
-(-1183 R) 
+(-1185 R) 
 ((|constructor| (NIL "This domain implements symmetric polynomial"))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-6 -4425)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-3962 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-456))) (-12 (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| (-975) (QUOTE (-131)))) (|HasAttribute| |#1| (QUOTE -4425))) 
-(-1184) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-6 -4432)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-3969 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-457))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| (-977) (QUOTE (-131)))) (|HasAttribute| |#1| (QUOTE -4432))) 
+(-1186) 
 ((|constructor| (NIL "Creates and manipulates one global symbol table for FORTRAN code generation,{} containing details of types,{} dimensions,{} and argument lists.")) (|symbolTableOf| (((|SymbolTable|) (|Symbol|) $) "\\spad{symbolTableOf(f,tab)} returns the symbol table of \\spad{f}")) (|argumentListOf| (((|List| (|Symbol|)) (|Symbol|) $) "\\spad{argumentListOf(f,tab)} returns the argument list of \\spad{f}")) (|returnTypeOf| (((|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1="void")) (|Symbol|) $) "\\spad{returnTypeOf(f,tab)} returns the type of the object returned by \\spad{f}")) (|empty| (($) "\\spad{empty()} creates a new,{} empty symbol table.")) (|printTypes| (((|Void|) (|Symbol|)) "\\spad{printTypes(tab)} produces FORTRAN type declarations from \\spad{tab},{} on the current FORTRAN output stream")) (|printHeader| (((|Void|)) "\\spad{printHeader()} produces the FORTRAN header for the current subprogram in the global symbol table on the current FORTRAN output stream.") (((|Void|) (|Symbol|)) "\\spad{printHeader(f)} produces the FORTRAN header for subprogram \\spad{f} in the global symbol table on the current FORTRAN output stream.") (((|Void|) (|Symbol|) $) "\\spad{printHeader(f,tab)} produces the FORTRAN header for subprogram \\spad{f} in symbol table \\spad{tab} on the current FORTRAN output stream.")) (|returnType!| (((|Void|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#))) "\\spad{returnType!(t)} declares that the return type of he current subprogram in the global symbol table is \\spad{t}.") (((|Void|) (|Symbol|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#))) "\\spad{returnType!(f,t)} declares that the return type of subprogram \\spad{f} in the global symbol table is \\spad{t}.") (((|Void|) (|Symbol|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#)) $) "\\spad{returnType!(f,t,tab)} declares that the return type of subprogram \\spad{f} in symbol table \\spad{tab} is \\spad{t}.")) (|argumentList!| (((|Void|) (|List| (|Symbol|))) "\\spad{argumentList!(l)} declares that the argument list for the current subprogram in the global symbol table is \\spad{l}.") (((|Void|) (|Symbol|) (|List| (|Symbol|))) "\\spad{argumentList!(f,l)} declares that the argument list for subprogram \\spad{f} in the global symbol table is \\spad{l}.") (((|Void|) (|Symbol|) (|List| (|Symbol|)) $) "\\spad{argumentList!(f,l,tab)} declares that the argument list for subprogram \\spad{f} in symbol table \\spad{tab} is \\spad{l}.")) (|endSubProgram| (((|Symbol|)) "\\spad{endSubProgram()} asserts that we are no longer processing the current subprogram.")) (|currentSubProgram| (((|Symbol|)) "\\spad{currentSubProgram()} returns the name of the current subprogram being processed")) (|newSubProgram| (((|Void|) (|Symbol|)) "\\spad{newSubProgram(f)} asserts that from now on type declarations are part of subprogram \\spad{f}.")) (|declare!| (((|FortranType|) (|Symbol|) (|FortranType|) (|Symbol|)) "\\spad{declare!(u,t,asp)} declares the parameter \\spad{u} to have type \\spad{t} in \\spad{asp}.") (((|FortranType|) (|Symbol|) (|FortranType|)) "\\spad{declare!(u,t)} declares the parameter \\spad{u} to have type \\spad{t} in the current level of the symbol table.") (((|FortranType|) (|List| (|Symbol|)) (|FortranType|) (|Symbol|) $) "\\spad{declare!(u,t,asp,tab)} declares the parameters \\spad{u} of subprogram \\spad{asp} to have type \\spad{t} in symbol table \\spad{tab}.") (((|FortranType|) (|Symbol|) (|FortranType|) (|Symbol|) $) "\\spad{declare!(u,t,asp,tab)} declares the parameter \\spad{u} of subprogram \\spad{asp} to have type \\spad{t} in symbol table \\spad{tab}.")) (|clearTheSymbolTable| (((|Void|) (|Symbol|)) "\\spad{clearTheSymbolTable(x)} removes the symbol \\spad{x} from the table") (((|Void|)) "\\spad{clearTheSymbolTable()} clears the current symbol table.")) (|showTheSymbolTable| (($) "\\spad{showTheSymbolTable()} returns the current symbol table."))) 
 NIL 
 NIL 
-(-1185) 
+(-1187) 
 ((|constructor| (NIL "Create and manipulate a symbol table for generated FORTRAN code")) (|symbolTable| (($ (|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| (|FortranType|))))) "\\spad{symbolTable(l)} creates a symbol table from the elements of \\spad{l}.")) (|printTypes| (((|Void|) $) "\\spad{printTypes(tab)} produces FORTRAN type declarations from \\spad{tab},{} on the current FORTRAN output stream")) (|newTypeLists| (((|SExpression|) $) "\\spad{newTypeLists(x)} \\undocumented")) (|typeLists| (((|List| (|List| (|Union| (|:| |name| (|Symbol|)) (|:| |bounds| (|List| (|Union| (|:| S (|Symbol|)) (|:| P (|Polynomial| (|Integer|))))))))) $) "\\spad{typeLists(tab)} returns a list of lists of types of objects in \\spad{tab}")) (|externalList| (((|List| (|Symbol|)) $) "\\spad{externalList(tab)} returns a list of all the external symbols in \\spad{tab}")) (|typeList| (((|List| (|Union| (|:| |name| (|Symbol|)) (|:| |bounds| (|List| (|Union| (|:| S (|Symbol|)) (|:| P (|Polynomial| (|Integer|)))))))) (|FortranScalarType|) $) "\\spad{typeList(t,tab)} returns a list of all the objects of type \\spad{t} in \\spad{tab}")) (|parametersOf| (((|List| (|Symbol|)) $) "\\spad{parametersOf(tab)} returns a list of all the symbols declared in \\spad{tab}")) (|fortranTypeOf| (((|FortranType|) (|Symbol|) $) "\\spad{fortranTypeOf(u,tab)} returns the type of \\spad{u} in \\spad{tab}")) (|declare!| (((|FortranType|) (|Symbol|) (|FortranType|) $) "\\spad{declare!(u,t,tab)} creates a new entry in \\spad{tab},{} declaring \\spad{u} to be of type \\spad{t}") (((|FortranType|) (|List| (|Symbol|)) (|FortranType|) $) "\\spad{declare!(l,t,tab)} creates new entrys in \\spad{tab},{} declaring each of \\spad{l} to be of type \\spad{t}")) (|empty| (($) "\\spad{empty()} returns a new,{} empty symbol table")) (|coerce| (((|Table| (|Symbol|) (|FortranType|)) $) "\\spad{coerce(x)} returns a table view of \\spad{x}"))) 
 NIL 
 NIL 
-(-1186) 
+(-1188) 
 ((|constructor| (NIL "\\indented{1}{This domain provides a simple domain,{} general enough for} \\indented{2}{building complete representation of Spad programs as objects} \\indented{2}{of a term algebra built from ground terms of type integers,{} foats,{}} \\indented{2}{identifiers,{} and strings.} \\indented{2}{This domain differs from InputForm in that it represents} \\indented{2}{any entity in a Spad program,{} not just expressions.\\space{2}Furthermore,{}} \\indented{2}{while InputForm may contain atoms like vectors and other Lisp} \\indented{2}{objects,{} the Syntax domain is supposed to contain only that} \\indented{2}{initial algebra build from the primitives listed above.} Related Constructors: \\indented{2}{Integer,{} DoubleFloat,{} Identifier,{} String,{} SExpression.} See Also: SExpression,{} InputForm. The equality supported by this domain is structural.")) (|case| (((|Boolean|) $ (|[\|\|]| (|String|))) "\\spad{x case String} is \\spad{true} if \\spad{`x'} really is a String") (((|Boolean|) $ (|[\|\|]| (|Identifier|))) "\\spad{x case Identifier} is \\spad{true} if \\spad{`x'} really is an Identifier") (((|Boolean|) $ (|[\|\|]| (|DoubleFloat|))) "\\spad{x case DoubleFloat} is \\spad{true} if \\spad{`x'} really is a DoubleFloat") (((|Boolean|) $ (|[\|\|]| (|Integer|))) "\\spad{x case Integer} is \\spad{true} if \\spad{`x'} really is an Integer")) (|compound?| (((|Boolean|) $) "\\spad{compound? x} is \\spad{true} when \\spad{`x'} is not an atomic syntax.")) (|getOperands| (((|List| $) $) "\\spad{getOperands(x)} returns the list of operands to the operator in \\spad{`x'}.")) (|getOperator| (((|Union| (|Integer|) (|DoubleFloat|) (|Identifier|) (|String|) $) $) "\\spad{getOperator(x)} returns the operator,{} or tag,{} of the syntax \\spad{`x'}. The value returned is itself a syntax if \\spad{`x'} really is an application of a function symbol as opposed to being an atomic ground term.")) (|nil?| (((|Boolean|) $) "\\spad{nil?(s)} is \\spad{true} when \\spad{`s'} is a syntax for the constant nil.")) (|buildSyntax| (($ $ (|List| $)) "\\spad{buildSyntax(op, [a1, ..., an])} builds a syntax object for \\spad{op}(a1,{}...,{}an).") (($ (|Identifier|) (|List| $)) "\\spad{buildSyntax(op, [a1, ..., an])} builds a syntax object for \\spad{op}(a1,{}...,{}an).")) (|autoCoerce| (((|String|) $) "\\spad{autoCoerce(s)} forcibly extracts a string value from the syntax \\spad{`s'}; no check performed. To be called only at the discretion of the compiler.") (((|Identifier|) $) "\\spad{autoCoerce(s)} forcibly extracts an identifier from the Syntax domain \\spad{`s'}; no check performed. To be called only at at the discretion of the compiler.") (((|DoubleFloat|) $) "\\spad{autoCoerce(s)} forcibly extracts a float value from the syntax \\spad{`s'}; no check performed. To be called only at the discretion of the compiler") (((|Integer|) $) "\\spad{autoCoerce(s)} forcibly extracts an integer value from the syntax \\spad{`s'}; no check performed. To be called only at the discretion of the compiler.")) (|coerce| (((|String|) $) "\\spad{coerce(s)} extracts a string value from the syntax \\spad{`s'}.") (((|Identifier|) $) "\\spad{coerce(s)} extracts an identifier from the syntax \\spad{`s'}.") (((|DoubleFloat|) $) "\\spad{coerce(s)} extracts a float value from the syntax \\spad{`s'}.") (((|Integer|) $) "\\spad{coerce(s)} extracts and integer value from the syntax \\spad{`s'}")) (|convert| (($ (|SExpression|)) "\\spad{convert(s)} converts an \\spad{s}-expression to Syntax. Note,{} when \\spad{`s'} is not an atom,{} it is expected that it designates a proper list,{} \\spadignore{e.g.} a sequence of cons cells ending with nil.") (((|SExpression|) $) "\\spad{convert(s)} returns the \\spad{s}-expression representation of a syntax."))) 
 NIL 
 NIL 
-(-1187 N) 
+(-1189 N) 
 ((|constructor| (NIL "This domain implements sized (signed) integer datatypes parameterized by the precision (or width) of the underlying representation. The intent is that they map directly to the hosting hardware natural integer datatypes. Consequently,{} natural values for \\spad{N} are: 8,{} 16,{} 32,{} 64,{} etc. These datatypes are mostly useful for system programming tasks,{} \\spadignore{i.e.} interfacting with the hosting operating system,{} reading/writing external binary format files.")) (|sample| (($) "\\spad{sample} gives a sample datum of this type."))) 
 NIL 
 NIL 
-(-1188 N) 
+(-1190 N) 
 ((|constructor| (NIL "This domain implements sized (unsigned) integer datatypes parameterized by the precision (or width) of the underlying representation. The intent is that they map directly to the hosting hardware natural integer datatypes. Consequently,{} natural values for \\spad{N} are: 8,{} 16,{} 32,{} 64,{} etc. These datatypes are mostly useful for system programming tasks,{} \\spadignore{i.e.} interfacting with the hosting operating system,{} reading/writing external binary format files.")) (|sample| (($) "\\spad{sample} gives a sample datum of type Byte.")) (|bitior| (($ $ $) "bitor(\\spad{x},{}\\spad{y}) returns the bitwise `inclusive or' of \\spad{`x'} and \\spad{`y'}.")) (|bitand| (($ $ $) "\\spad{bitand(x,y)} returns the bitwise `and' of \\spad{`x'} and \\spad{`y'}."))) 
 NIL 
 NIL 
-(-1189) 
+(-1191) 
 ((|constructor| (NIL "This domain is a datatype system-level pointer values."))) 
 NIL 
 NIL 
-(-1190 R) 
+(-1192 R) 
 ((|triangularSystems| (((|List| (|List| (|Polynomial| |#1|))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{triangularSystems(lf,lv)} solves the system of equations defined by \\spad{lf} with respect to the list of symbols \\spad{lv}; the system of equations is obtaining by equating to zero the list of rational functions \\spad{lf}. The output is a list of solutions where each solution is expressed as a \"reduced\" triangular system of polynomials.")) (|solve| (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{solve(eq)} finds the solutions of the equation \\spad{eq} with respect to the unique variable appearing in \\spad{eq}.") (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Fraction| (|Polynomial| |#1|))) "\\spad{solve(p)} finds the solution of a rational function \\spad{p} = 0 with respect to the unique variable appearing in \\spad{p}.") (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{solve(eq,v)} finds the solutions of the equation \\spad{eq} with respect to the variable \\spad{v}.") (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{solve(p,v)} solves the equation \\spad{p=0},{} where \\spad{p} is a rational function with respect to the variable \\spad{v}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) "\\spad{solve(le)} finds the solutions of the list \\spad{le} of equations of rational functions with respect to all symbols appearing in \\spad{le}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{solve(lp)} finds the solutions of the list \\spad{lp} of rational functions with respect to all symbols appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|List| (|Symbol|))) "\\spad{solve(le,lv)} finds the solutions of the list \\spad{le} of equations of rational functions with respect to the list of symbols \\spad{lv}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{solve(lp,lv)} finds the solutions of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}."))) 
 NIL 
 NIL 
-(-1191) 
+(-1193) 
 ((|constructor| (NIL "The package \\spadtype{System} provides information about the runtime system and its characteristics.")) (|loadNativeModule| (((|Void|) (|String|)) "\\spad{loadNativeModule(path)} loads the native modile designated by \\spadvar{\\spad{path}}.")) (|nativeModuleExtension| (((|String|)) "\\spad{nativeModuleExtension} is a string representation of a filename extension for native modules.")) (|hostByteOrder| (((|ByteOrder|)) "\\sapd{hostByteOrder}")) (|hostPlatform| (((|String|)) "\\spad{hostPlatform} is a string `triplet' description of the platform hosting the running OpenAxiom system.")) (|rootDirectory| (((|String|)) "\\spad{rootDirectory()} returns the pathname of the root directory for the running OpenAxiom system."))) 
 NIL 
 NIL 
-(-1192 S) 
+(-1194 S) 
 ((|constructor| (NIL "TableauBumpers implements the Schenstead-Knuth correspondence between sequences and pairs of Young tableaux. The 2 Young tableaux are represented as a single tableau with pairs as components.")) (|mr| (((|Record| (|:| |f1| (|List| |#1|)) (|:| |f2| (|List| (|List| (|List| |#1|)))) (|:| |f3| (|List| (|List| |#1|))) (|:| |f4| (|List| (|List| (|List| |#1|))))) (|List| (|List| (|List| |#1|)))) "\\spad{mr(t)} is an auxiliary function which finds the position of the maximum element of a tableau \\spad{t} which is in the lowest row,{} producing a record of results")) (|maxrow| (((|Record| (|:| |f1| (|List| |#1|)) (|:| |f2| (|List| (|List| (|List| |#1|)))) (|:| |f3| (|List| (|List| |#1|))) (|:| |f4| (|List| (|List| (|List| |#1|))))) (|List| |#1|) (|List| (|List| (|List| |#1|))) (|List| (|List| |#1|)) (|List| (|List| (|List| |#1|))) (|List| (|List| (|List| |#1|))) (|List| (|List| (|List| |#1|)))) "\\spad{maxrow(a,b,c,d,e)} is an auxiliary function for \\spad{mr}")) (|inverse| (((|List| |#1|) (|List| |#1|)) "\\spad{inverse(ls)} forms the inverse of a sequence \\spad{ls}")) (|slex| (((|List| (|List| |#1|)) (|List| |#1|)) "\\spad{slex(ls)} sorts the argument sequence \\spad{ls},{} then zips (see \\spadfunFrom{map}{ListFunctions3}) the original argument sequence with the sorted result to a list of pairs")) (|lex| (((|List| (|List| |#1|)) (|List| (|List| |#1|))) "\\spad{lex(ls)} sorts a list of pairs to lexicographic order")) (|tab| (((|Tableau| (|List| |#1|)) (|List| |#1|)) "\\spad{tab(ls)} creates a tableau from \\spad{ls} by first creating a list of pairs using \\spadfunFrom{slex}{TableauBumpers},{} then creating a tableau using \\spadfunFrom{tab1}{TableauBumpers}.")) (|tab1| (((|List| (|List| (|List| |#1|))) (|List| (|List| |#1|))) "\\spad{tab1(lp)} creates a tableau from a list of pairs \\spad{lp}")) (|bat| (((|List| (|List| |#1|)) (|Tableau| (|List| |#1|))) "\\spad{bat(ls)} unbumps a tableau \\spad{ls}")) (|bat1| (((|List| (|List| |#1|)) (|List| (|List| (|List| |#1|)))) "\\spad{bat1(llp)} unbumps a tableau \\spad{llp}. Operation bat1 is the inverse of tab1.")) (|untab| (((|List| (|List| |#1|)) (|List| (|List| |#1|)) (|List| (|List| (|List| |#1|)))) "\\spad{untab(lp,llp)} is an auxiliary function which unbumps a tableau \\spad{llp},{} using \\spad{lp} to accumulate pairs")) (|bumptab1| (((|List| (|List| (|List| |#1|))) (|List| |#1|) (|List| (|List| (|List| |#1|)))) "\\spad{bumptab1(pr,t)} bumps a tableau \\spad{t} with a pair \\spad{pr} using comparison function \\spadfun{<},{} returning a new tableau")) (|bumptab| (((|List| (|List| (|List| |#1|))) (|Mapping| (|Boolean|) |#1| |#1|) (|List| |#1|) (|List| (|List| (|List| |#1|)))) "\\spad{bumptab(cf,pr,t)} bumps a tableau \\spad{t} with a pair \\spad{pr} using comparison function \\spad{cf},{} returning a new tableau")) (|bumprow| (((|Record| (|:| |fs| (|Boolean|)) (|:| |sd| (|List| |#1|)) (|:| |td| (|List| (|List| |#1|)))) (|Mapping| (|Boolean|) |#1| |#1|) (|List| |#1|) (|List| (|List| |#1|))) "\\spad{bumprow(cf,pr,r)} is an auxiliary function which bumps a row \\spad{r} with a pair \\spad{pr} using comparison function \\spad{cf},{} and returns a record"))) 
 NIL 
 NIL 
-(-1193 |Key| |Entry|) 
+(-1195 |Key| |Entry|) 
 ((|constructor| (NIL "This is the general purpose table type. The keys are hashed to look up the entries. This creates a \\spadtype{HashTable} if equal for the Key domain is consistent with Lisp EQUAL otherwise an \\spadtype{AssociationList}"))) 
-((-4427 . T) (-4428 . T)) 
-((-12 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -311) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4294) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2256) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -617) (QUOTE (-539)))) (-12 (|HasCategory| |#2| (QUOTE (-1105))) (|HasCategory| |#2| (LIST (QUOTE -311) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#2| (QUOTE (-1105))) (-3962 (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#2| (LIST (QUOTE -616) (QUOTE (-866)))) (|HasCategory| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1194 S) 
+((-4434 . T) (-4435 . T)) 
+((-12 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -312) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -4301) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2263) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -619) (QUOTE (-540)))) (-12 (|HasCategory| |#2| (QUOTE (-1107))) (|HasCategory| |#2| (LIST (QUOTE -312) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#2| (QUOTE (-1107))) (-3969 (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -618) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1196 S) 
 ((|constructor| (NIL "\\indented{1}{The tableau domain is for printing Young tableaux,{} and} coercions to and from List List \\spad{S} where \\spad{S} is a set.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(t)} converts a tableau \\spad{t} to an output form.")) (|listOfLists| (((|List| (|List| |#1|)) $) "\\spad{listOfLists t} converts a tableau \\spad{t} to a list of lists.")) (|tableau| (($ (|List| (|List| |#1|))) "\\spad{tableau(ll)} converts a list of lists \\spad{ll} to a tableau."))) 
 NIL 
 NIL 
-(-1195 R) 
+(-1197 R) 
 ((|constructor| (NIL "Expands tangents of sums and scalar products.")) (|tanNa| ((|#1| |#1| (|Integer|)) "\\spad{tanNa(a, n)} returns \\spad{f(a)} such that if \\spad{a = tan(u)} then \\spad{f(a) = tan(n * u)}.")) (|tanAn| (((|SparseUnivariatePolynomial| |#1|) |#1| (|PositiveInteger|)) "\\spad{tanAn(a, n)} returns \\spad{P(x)} such that if \\spad{a = tan(u)} then \\spad{P(tan(u/n)) = 0}.")) (|tanSum| ((|#1| (|List| |#1|)) "\\spad{tanSum([a1,...,an])} returns \\spad{f(a1,...,an)} such that if \\spad{ai = tan(ui)} then \\spad{f(a1,...,an) = tan(u1 + ... + un)}."))) 
 NIL 
 NIL 
-(-1196 S |Key| |Entry|) 
+(-1198 S |Key| |Entry|) 
 ((|constructor| (NIL "A table aggregate is a model of a table,{} \\spadignore{i.e.} a discrete many-to-one mapping from keys to entries.")) (|map| (($ (|Mapping| |#3| |#3| |#3|) $ $) "\\spad{map(fn,t1,t2)} creates a new table \\spad{t} from given tables \\spad{t1} and \\spad{t2} with elements \\spad{fn}(\\spad{x},{}\\spad{y}) where \\spad{x} and \\spad{y} are corresponding elements from \\spad{t1} and \\spad{t2} respectively.")) (|table| (($ (|List| (|Record| (|:| |key| |#2|) (|:| |entry| |#3|)))) "\\spad{table([x,y,...,z])} creates a table consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{table()}\\$\\spad{T} creates an empty table of type \\spad{T}.")) (|setelt| ((|#3| $ |#2| |#3|) "\\spad{setelt(t,k,e)} (also written \\axiom{\\spad{t}.\\spad{k} \\spad{:=} \\spad{e}}) is equivalent to \\axiom{(insert([\\spad{k},{}\\spad{e}],{}\\spad{t}); \\spad{e})}."))) 
 NIL 
 NIL 
-(-1197 |Key| |Entry|) 
+(-1199 |Key| |Entry|) 
 ((|constructor| (NIL "A table aggregate is a model of a table,{} \\spadignore{i.e.} a discrete many-to-one mapping from keys to entries.")) (|map| (($ (|Mapping| |#2| |#2| |#2|) $ $) "\\spad{map(fn,t1,t2)} creates a new table \\spad{t} from given tables \\spad{t1} and \\spad{t2} with elements \\spad{fn}(\\spad{x},{}\\spad{y}) where \\spad{x} and \\spad{y} are corresponding elements from \\spad{t1} and \\spad{t2} respectively.")) (|table| (($ (|List| (|Record| (|:| |key| |#1|) (|:| |entry| |#2|)))) "\\spad{table([x,y,...,z])} creates a table consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{table()}\\$\\spad{T} creates an empty table of type \\spad{T}.")) (|setelt| ((|#2| $ |#1| |#2|) "\\spad{setelt(t,k,e)} (also written \\axiom{\\spad{t}.\\spad{k} \\spad{:=} \\spad{e}}) is equivalent to \\axiom{(insert([\\spad{k},{}\\spad{e}],{}\\spad{t}); \\spad{e})}."))) 
-((-4428 . T)) 
+((-4435 . T)) 
 NIL 
-(-1198 |Key| |Entry|) 
+(-1200 |Key| |Entry|) 
 ((|constructor| (NIL "\\axiom{TabulatedComputationPackage(Key ,{}Entry)} provides some modest support for dealing with operations with type \\axiom{Key \\spad{->} Entry}. The result of such operations can be stored and retrieved with this package by using a hash-table. The user does not need to worry about the management of this hash-table. However,{} onnly one hash-table is built by calling \\axiom{TabulatedComputationPackage(Key ,{}Entry)}.")) (|insert!| (((|Void|) |#1| |#2|) "\\axiom{insert!(\\spad{x},{}\\spad{y})} stores the item whose key is \\axiom{\\spad{x}} and whose entry is \\axiom{\\spad{y}}.")) (|extractIfCan| (((|Union| |#2| "failed") |#1|) "\\axiom{extractIfCan(\\spad{x})} searches the item whose key is \\axiom{\\spad{x}}.")) (|makingStats?| (((|Boolean|)) "\\axiom{makingStats?()} returns \\spad{true} iff the statisitics process is running.")) (|printingInfo?| (((|Boolean|)) "\\axiom{printingInfo?()} returns \\spad{true} iff messages are printed when manipulating items from the hash-table.")) (|usingTable?| (((|Boolean|)) "\\axiom{usingTable?()} returns \\spad{true} iff the hash-table is used")) (|clearTable!| (((|Void|)) "\\axiom{clearTable!()} clears the hash-table and assumes that it will no longer be used.")) (|printStats!| (((|Void|)) "\\axiom{printStats!()} prints the statistics.")) (|startStats!| (((|Void|) (|String|)) "\\axiom{startStats!(\\spad{x})} initializes the statisitics process and sets the comments to display when statistics are printed")) (|printInfo!| (((|Void|) (|String|) (|String|)) "\\axiom{printInfo!(\\spad{x},{}\\spad{y})} initializes the mesages to be printed when manipulating items from the hash-table. If a key is retrieved then \\axiom{\\spad{x}} is displayed. If an item is stored then \\axiom{\\spad{y}} is displayed.")) (|initTable!| (((|Void|)) "\\axiom{initTable!()} initializes the hash-table."))) 
 NIL 
 NIL 
-(-1199) 
+(-1201) 
 ((|constructor| (NIL "This package provides functions for template manipulation")) (|stripCommentsAndBlanks| (((|String|) (|String|)) "\\spad{stripCommentsAndBlanks(s)} treats \\spad{s} as a piece of AXIOM input,{} and removes comments,{} and leading and trailing blanks.")) (|interpretString| (((|Any|) (|String|)) "\\spad{interpretString(s)} treats a string as a piece of AXIOM input,{} by parsing and interpreting it."))) 
 NIL 
 NIL 
-(-1200) 
+(-1202) 
 ((|constructor| (NIL "\\spadtype{TexFormat} provides a coercion from \\spadtype{OutputForm} to \\TeX{} format. The particular dialect of \\TeX{} used is \\LaTeX{}. The basic object consists of three parts: a prologue,{} a tex part and an epilogue. The functions \\spadfun{prologue},{} \\spadfun{tex} and \\spadfun{epilogue} extract these parts,{} respectively. The main guts of the expression go into the tex part. The other parts can be set (\\spadfun{setPrologue!},{} \\spadfun{setEpilogue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain \\spad{``}\\verb+\\spad{\\[}+\\spad{''} and \\spad{``}\\verb+\\spad{\\]}+\\spad{''},{} respectively,{} so that the TeX section will be printed in LaTeX display math mode.")) (|setPrologue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setPrologue!(t,strings)} sets the prologue section of a TeX form \\spad{t} to \\spad{strings}.")) (|setTex!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setTex!(t,strings)} sets the TeX section of a TeX form \\spad{t} to \\spad{strings}.")) (|setEpilogue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setEpilogue!(t,strings)} sets the epilogue section of a TeX form \\spad{t} to \\spad{strings}.")) (|prologue| (((|List| (|String|)) $) "\\spad{prologue(t)} extracts the prologue section of a TeX form \\spad{t}.")) (|new| (($) "\\spad{new()} create a new,{} empty object. Use \\spadfun{setPrologue!},{} \\spadfun{setTex!} and \\spadfun{setEpilogue!} to set the various components of this object.")) (|tex| (((|List| (|String|)) $) "\\spad{tex(t)} extracts the TeX section of a TeX form \\spad{t}.")) (|epilogue| (((|List| (|String|)) $) "\\spad{epilogue(t)} extracts the epilogue section of a TeX form \\spad{t}.")) (|display| (((|Void|) $) "\\spad{display(t)} outputs the TeX formatted code \\spad{t} so that each line has length less than or equal to the value set by the system command \\spadsyscom{set output length}.") (((|Void|) $ (|Integer|)) "\\spad{display(t,width)} outputs the TeX formatted code \\spad{t} so that each line has length less than or equal to \\spadvar{\\spad{width}}.")) (|convert| (($ (|OutputForm|) (|Integer|) (|OutputForm|)) "\\spad{convert(o,step,type)} changes \\spad{o} in standard output format to TeX format and also adds the given \\spad{step} number and \\spad{type}. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \\spad{step} numbers.") (($ (|OutputForm|) (|Integer|)) "\\spad{convert(o,step)} changes \\spad{o} in standard output format to TeX format and also adds the given \\spad{step} number. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \\spad{step} numbers."))) 
 NIL 
 NIL 
-(-1201 S) 
+(-1203 S) 
 ((|constructor| (NIL "\\spadtype{TexFormat1} provides a utility coercion for changing to TeX format anything that has a coercion to the standard output format.")) (|coerce| (((|TexFormat|) |#1|) "\\spad{coerce(s)} provides a direct coercion from a domain \\spad{S} to TeX format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to TeX format."))) 
 NIL 
 NIL 
-(-1202) 
+(-1204) 
 ((|constructor| (NIL "This domain provides an implementation of text files. Text is stored in these files using the native character set of the computer.")) (|endOfFile?| (((|Boolean|) $) "\\spad{endOfFile?(f)} tests whether the file \\spad{f} is positioned after the end of all text. If the file is open for output,{} then this test is always \\spad{true}.")) (|readIfCan!| (((|Union| (|String|) "failed") $) "\\spad{readIfCan!(f)} returns a string of the contents of a line from file \\spad{f},{} if possible. If \\spad{f} is not readable or if it is positioned at the end of file,{} then \\spad{\"failed\"} is returned.")) (|readLineIfCan!| (((|Union| (|String|) "failed") $) "\\spad{readLineIfCan!(f)} returns a string of the contents of a line from file \\spad{f},{} if possible. If \\spad{f} is not readable or if it is positioned at the end of file,{} then \\spad{\"failed\"} is returned.")) (|readLine!| (((|String|) $) "\\spad{readLine!(f)} returns a string of the contents of a line from the file \\spad{f}.")) (|writeLine!| (((|String|) $) "\\spad{writeLine!(f)} finishes the current line in the file \\spad{f}. An empty string is returned. The call \\spad{writeLine!(f)} is equivalent to \\spad{writeLine!(f,\"\")}.") (((|String|) $ (|String|)) "\\spad{writeLine!(f,s)} writes the contents of the string \\spad{s} and finishes the current line in the file \\spad{f}. The value of \\spad{s} is returned."))) 
 NIL 
 NIL 
-(-1203 R) 
+(-1205 R) 
 ((|constructor| (NIL "Tools for the sign finding utilities.")) (|direction| (((|Integer|) (|String|)) "\\spad{direction(s)} \\undocumented")) (|nonQsign| (((|Union| (|Integer|) "failed") |#1|) "\\spad{nonQsign(r)} \\undocumented")) (|sign| (((|Union| (|Integer|) "failed") |#1|) "\\spad{sign(r)} \\undocumented"))) 
 NIL 
 NIL 
-(-1204) 
+(-1206) 
 ((|constructor| (NIL "This package exports a function for making a \\spadtype{ThreeSpace}")) (|createThreeSpace| (((|ThreeSpace| (|DoubleFloat|))) "\\spad{createThreeSpace()} creates a \\spadtype{ThreeSpace(DoubleFloat)} object capable of holding point,{} curve,{} mesh components and any combination."))) 
 NIL 
 NIL 
-(-1205 S) 
+(-1207 S) 
 ((|constructor| (NIL "Category for the transcendental elementary functions.")) (|pi| (($) "\\spad{pi()} returns the constant \\spad{pi}."))) 
 NIL 
 NIL 
-(-1206) 
+(-1208) 
 ((|constructor| (NIL "Category for the transcendental elementary functions.")) (|pi| (($) "\\spad{pi()} returns the constant \\spad{pi}."))) 
 NIL 
 NIL 
-(-1207 S) 
+(-1209 S) 
 ((|constructor| (NIL "\\spadtype{Tree(S)} is a basic domains of tree structures. Each tree is either empty or else is a {\\it node} consisting of a value and a list of (sub)trees.")) (|cyclicParents| (((|List| $) $) "\\spad{cyclicParents(t)} returns a list of cycles that are parents of \\spad{t}.")) (|cyclicEqual?| (((|Boolean|) $ $) "\\spad{cyclicEqual?(t1, t2)} tests of two cyclic trees have the same structure.")) (|cyclicEntries| (((|List| $) $) "\\spad{cyclicEntries(t)} returns a list of top-level cycles in tree \\spad{t}.")) (|cyclicCopy| (($ $) "\\spad{cyclicCopy(l)} makes a copy of a (possibly) cyclic tree \\spad{l}.")) (|cyclic?| (((|Boolean|) $) "\\spad{cyclic?(t)} tests if \\spad{t} is a cyclic tree.")) (|tree| (($ |#1|) "\\spad{tree(nd)} creates a tree with value \\spad{nd},{} and no children") (($ (|List| |#1|)) "\\spad{tree(ls)} creates a tree from a list of elements of \\spad{s}.") (($ |#1| (|List| $)) "\\spad{tree(nd,ls)} creates a tree with value \\spad{nd},{} and children \\spad{ls}."))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1105))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1208 S) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1107))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1210 S) 
 ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) 
 NIL 
 NIL 
-(-1209) 
+(-1211) 
 ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) 
 NIL 
 NIL 
-(-1210 R -3498) 
+(-1212 R -3505) 
 ((|constructor| (NIL "\\spadtype{TrigonometricManipulations} provides transformations from trigonometric functions to complex exponentials and logarithms,{} and back.")) (|complexForm| (((|Complex| |#2|) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f, imag f]}.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| ((|#2| |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| ((|#2| |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f, x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f, x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) 
 NIL 
 NIL 
-(-1211 R |Row| |Col| M) 
+(-1213 R |Row| |Col| M) 
 ((|constructor| (NIL "This package provides functions that compute \"fraction-free\" inverses of upper and lower triangular matrices over a integral domain. By \"fraction-free inverses\" we mean the following: given a matrix \\spad{B} with entries in \\spad{R} and an element \\spad{d} of \\spad{R} such that \\spad{d} * inv(\\spad{B}) also has entries in \\spad{R},{} we return \\spad{d} * inv(\\spad{B}). Thus,{} it is not necessary to pass to the quotient field in any of our computations.")) (|LowTriBddDenomInv| ((|#4| |#4| |#1|) "\\spad{LowTriBddDenomInv(B,d)} returns \\spad{M},{} where \\spad{B} is a non-singular lower triangular matrix and \\spad{d} is an element of \\spad{R} such that \\spad{M = d * inv(B)} has entries in \\spad{R}.")) (|UpTriBddDenomInv| ((|#4| |#4| |#1|) "\\spad{UpTriBddDenomInv(B,d)} returns \\spad{M},{} where \\spad{B} is a non-singular upper triangular matrix and \\spad{d} is an element of \\spad{R} such that \\spad{M = d * inv(B)} has entries in \\spad{R}."))) 
 NIL 
 NIL 
-(-1212 R -3498) 
+(-1214 R -3505) 
 ((|constructor| (NIL "TranscendentalManipulations provides functions to simplify and expand expressions involving transcendental operators.")) (|expandTrigProducts| ((|#2| |#2|) "\\spad{expandTrigProducts(e)} replaces \\axiom{sin(\\spad{x})*sin(\\spad{y})} by \\spad{(cos(x-y)-cos(x+y))/2},{} \\axiom{cos(\\spad{x})*cos(\\spad{y})} by \\spad{(cos(x-y)+cos(x+y))/2},{} and \\axiom{sin(\\spad{x})*cos(\\spad{y})} by \\spad{(sin(x-y)+sin(x+y))/2}. Note that this operation uses the pattern matcher and so is relatively expensive. To avoid getting into an infinite loop the transformations are applied at most ten times.")) (|removeSinhSq| ((|#2| |#2|) "\\spad{removeSinhSq(f)} converts every \\spad{sinh(u)**2} appearing in \\spad{f} into \\spad{1 - cosh(x)**2},{} and also reduces higher powers of \\spad{sinh(u)} with that formula.")) (|removeCoshSq| ((|#2| |#2|) "\\spad{removeCoshSq(f)} converts every \\spad{cosh(u)**2} appearing in \\spad{f} into \\spad{1 - sinh(x)**2},{} and also reduces higher powers of \\spad{cosh(u)} with that formula.")) (|removeSinSq| ((|#2| |#2|) "\\spad{removeSinSq(f)} converts every \\spad{sin(u)**2} appearing in \\spad{f} into \\spad{1 - cos(x)**2},{} and also reduces higher powers of \\spad{sin(u)} with that formula.")) (|removeCosSq| ((|#2| |#2|) "\\spad{removeCosSq(f)} converts every \\spad{cos(u)**2} appearing in \\spad{f} into \\spad{1 - sin(x)**2},{} and also reduces higher powers of \\spad{cos(u)} with that formula.")) (|coth2tanh| ((|#2| |#2|) "\\spad{coth2tanh(f)} converts every \\spad{coth(u)} appearing in \\spad{f} into \\spad{1/tanh(u)}.")) (|cot2tan| ((|#2| |#2|) "\\spad{cot2tan(f)} converts every \\spad{cot(u)} appearing in \\spad{f} into \\spad{1/tan(u)}.")) (|tanh2coth| ((|#2| |#2|) "\\spad{tanh2coth(f)} converts every \\spad{tanh(u)} appearing in \\spad{f} into \\spad{1/coth(u)}.")) (|tan2cot| ((|#2| |#2|) "\\spad{tan2cot(f)} converts every \\spad{tan(u)} appearing in \\spad{f} into \\spad{1/cot(u)}.")) (|tanh2trigh| ((|#2| |#2|) "\\spad{tanh2trigh(f)} converts every \\spad{tanh(u)} appearing in \\spad{f} into \\spad{sinh(u)/cosh(u)}.")) (|tan2trig| ((|#2| |#2|) "\\spad{tan2trig(f)} converts every \\spad{tan(u)} appearing in \\spad{f} into \\spad{sin(u)/cos(u)}.")) (|sinh2csch| ((|#2| |#2|) "\\spad{sinh2csch(f)} converts every \\spad{sinh(u)} appearing in \\spad{f} into \\spad{1/csch(u)}.")) (|sin2csc| ((|#2| |#2|) "\\spad{sin2csc(f)} converts every \\spad{sin(u)} appearing in \\spad{f} into \\spad{1/csc(u)}.")) (|sech2cosh| ((|#2| |#2|) "\\spad{sech2cosh(f)} converts every \\spad{sech(u)} appearing in \\spad{f} into \\spad{1/cosh(u)}.")) (|sec2cos| ((|#2| |#2|) "\\spad{sec2cos(f)} converts every \\spad{sec(u)} appearing in \\spad{f} into \\spad{1/cos(u)}.")) (|csch2sinh| ((|#2| |#2|) "\\spad{csch2sinh(f)} converts every \\spad{csch(u)} appearing in \\spad{f} into \\spad{1/sinh(u)}.")) (|csc2sin| ((|#2| |#2|) "\\spad{csc2sin(f)} converts every \\spad{csc(u)} appearing in \\spad{f} into \\spad{1/sin(u)}.")) (|coth2trigh| ((|#2| |#2|) "\\spad{coth2trigh(f)} converts every \\spad{coth(u)} appearing in \\spad{f} into \\spad{cosh(u)/sinh(u)}.")) (|cot2trig| ((|#2| |#2|) "\\spad{cot2trig(f)} converts every \\spad{cot(u)} appearing in \\spad{f} into \\spad{cos(u)/sin(u)}.")) (|cosh2sech| ((|#2| |#2|) "\\spad{cosh2sech(f)} converts every \\spad{cosh(u)} appearing in \\spad{f} into \\spad{1/sech(u)}.")) (|cos2sec| ((|#2| |#2|) "\\spad{cos2sec(f)} converts every \\spad{cos(u)} appearing in \\spad{f} into \\spad{1/sec(u)}.")) (|expandLog| ((|#2| |#2|) "\\spad{expandLog(f)} converts every \\spad{log(a/b)} appearing in \\spad{f} into \\spad{log(a) - log(b)},{} and every \\spad{log(a*b)} into \\spad{log(a) + log(b)}..")) (|expandPower| ((|#2| |#2|) "\\spad{expandPower(f)} converts every power \\spad{(a/b)**c} appearing in \\spad{f} into \\spad{a**c * b**(-c)}.")) (|simplifyLog| ((|#2| |#2|) "\\spad{simplifyLog(f)} converts every \\spad{log(a) - log(b)} appearing in \\spad{f} into \\spad{log(a/b)},{} every \\spad{log(a) + log(b)} into \\spad{log(a*b)} and every \\spad{n*log(a)} into \\spad{log(a^n)}.")) (|simplifyExp| ((|#2| |#2|) "\\spad{simplifyExp(f)} converts every product \\spad{exp(a)*exp(b)} appearing in \\spad{f} into \\spad{exp(a+b)}.")) (|htrigs| ((|#2| |#2|) "\\spad{htrigs(f)} converts all the exponentials in \\spad{f} into hyperbolic sines and cosines.")) (|simplify| ((|#2| |#2|) "\\spad{simplify(f)} performs the following simplifications on \\spad{f:}\\begin{items} \\item 1. rewrites trigs and hyperbolic trigs in terms of \\spad{sin} ,{}\\spad{cos},{} \\spad{sinh},{} \\spad{cosh}. \\item 2. rewrites \\spad{sin**2} and \\spad{sinh**2} in terms of \\spad{cos} and \\spad{cosh},{} \\item 3. rewrites \\spad{exp(a)*exp(b)} as \\spad{exp(a+b)}. \\item 4. rewrites \\spad{(a**(1/n))**m * (a**(1/s))**t} as a single power of a single radical of \\spad{a}. \\end{items}")) (|expand| ((|#2| |#2|) "\\spad{expand(f)} performs the following expansions on \\spad{f:}\\begin{items} \\item 1. logs of products are expanded into sums of logs,{} \\item 2. trigonometric and hyperbolic trigonometric functions of sums are expanded into sums of products of trigonometric and hyperbolic trigonometric functions. \\item 3. formal powers of the form \\spad{(a/b)**c} are expanded into \\spad{a**c * b**(-c)}. \\end{items}"))) 
 NIL 
-((-12 (|HasCategory| |#1| (LIST (QUOTE -617) (LIST (QUOTE -894) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -890) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -890) (|devaluate| |#1|))))) 
-(-1213 |Coef|) 
+((-12 (|HasCategory| |#1| (LIST (QUOTE -619) (LIST (QUOTE -896) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -892) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -892) (|devaluate| |#1|))))) 
+(-1215 |Coef|) 
 ((|constructor| (NIL "\\spadtype{TaylorSeries} is a general multivariate Taylor series domain over the ring Coef and with variables of type Symbol.")) (|fintegrate| (($ (|Mapping| $) (|Symbol|) |#1|) "\\spad{fintegrate(f,v,c)} is the integral of \\spad{f()} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.} \\indented{1}{The evaluation of \\spad{f()} is delayed.}")) (|integrate| (($ $ (|Symbol|) |#1|) "\\spad{integrate(s,v,c)} is the integral of \\spad{s} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.}")) (|coerce| (($ (|Polynomial| |#1|)) "\\spad{coerce(s)} regroups terms of \\spad{s} by total degree \\indented{1}{and forms a series.}") (($ (|Symbol|)) "\\spad{coerce(s)} converts a variable to a Taylor series")) (|coefficient| (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{coefficient(s, n)} gives the terms of total degree \\spad{n}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-1214 S R E V P) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-147))) (|HasCategory| |#1| (QUOTE (-145))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-1216 S R E V P) 
 ((|constructor| (NIL "The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \\axiom{\\spad{R}} be an integral domain and \\axiom{\\spad{V}} a finite ordered set of variables,{} say \\axiom{\\spad{X1} < \\spad{X2} < ... < \\spad{Xn}}. A set \\axiom{\\spad{S}} of polynomials in \\axiom{\\spad{R}[\\spad{X1},{}\\spad{X2},{}...,{}\\spad{Xn}]} is triangular if no elements of \\axiom{\\spad{S}} lies in \\axiom{\\spad{R}},{} and if two distinct elements of \\axiom{\\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a non-constant polynomial \\axiom{\\spad{Q}} if the degree of \\axiom{\\spad{P}} in the main variable of \\axiom{\\spad{Q}} is less than the main degree of \\axiom{\\spad{Q}}. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a triangular set \\axiom{\\spad{T}} if it is reduced \\spad{w}.\\spad{r}.\\spad{t}. every polynomial of \\axiom{\\spad{T}}. \\newline References : \\indented{1}{[1] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)}")) (|coHeight| (((|NonNegativeInteger|) $) "\\axiom{coHeight(\\spad{ts})} returns \\axiom{size()\\spad{\\$}\\spad{V}} minus \\axiom{\\spad{\\#}\\spad{ts}}.")) (|extend| (($ $ |#5|) "\\axiom{extend(\\spad{ts},{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{\\spad{ts}},{} according to the properties of triangular sets of the current category If the required properties do not hold an error is returned.")) (|extendIfCan| (((|Union| $ "failed") $ |#5|) "\\axiom{extendIfCan(\\spad{ts},{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{\\spad{ts}},{} according to the properties of triangular sets of the current domain. If the required properties do not hold then \"failed\" is returned. This operation encodes in some sense the properties of the triangular sets of the current category. Is is used to implement the \\axiom{construct} operation to guarantee that every triangular set build from a list of polynomials has the required properties.")) (|select| (((|Union| |#5| "failed") $ |#4|) "\\axiom{select(\\spad{ts},{}\\spad{v})} returns the polynomial of \\axiom{\\spad{ts}} with \\axiom{\\spad{v}} as main variable,{} if any.")) (|algebraic?| (((|Boolean|) |#4| $) "\\axiom{algebraic?(\\spad{v},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{\\spad{ts}}.")) (|algebraicVariables| (((|List| |#4|) $) "\\axiom{algebraicVariables(\\spad{ts})} returns the decreasingly sorted list of the main variables of the polynomials of \\axiom{\\spad{ts}}.")) (|rest| (((|Union| $ "failed") $) "\\axiom{rest(\\spad{ts})} returns the polynomials of \\axiom{\\spad{ts}} with smaller main variable than \\axiom{mvar(\\spad{ts})} if \\axiom{\\spad{ts}} is not empty,{} otherwise returns \"failed\"")) (|last| (((|Union| |#5| "failed") $) "\\axiom{last(\\spad{ts})} returns the polynomial of \\axiom{\\spad{ts}} with smallest main variable if \\axiom{\\spad{ts}} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|first| (((|Union| |#5| "failed") $) "\\axiom{first(\\spad{ts})} returns the polynomial of \\axiom{\\spad{ts}} with greatest main variable if \\axiom{\\spad{ts}} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|zeroSetSplitIntoTriangularSystems| (((|List| (|Record| (|:| |close| $) (|:| |open| (|List| |#5|)))) (|List| |#5|)) "\\axiom{zeroSetSplitIntoTriangularSystems(\\spad{lp})} returns a list of triangular systems \\axiom{[[\\spad{ts1},{}\\spad{qs1}],{}...,{}[\\spad{tsn},{}\\spad{qsn}]]} such that the zero set of \\axiom{\\spad{lp}} is the union of the closures of the \\axiom{W_i} where \\axiom{W_i} consists of the zeros of \\axiom{\\spad{ts}} which do not cancel any polynomial in \\axiom{qsi}.")) (|zeroSetSplit| (((|List| $) (|List| |#5|)) "\\axiom{zeroSetSplit(\\spad{lp})} returns a list \\axiom{\\spad{lts}} of triangular sets such that the zero set of \\axiom{\\spad{lp}} is the union of the closures of the regular zero sets of the members of \\axiom{\\spad{lts}}.")) (|reduceByQuasiMonic| ((|#5| |#5| $) "\\axiom{reduceByQuasiMonic(\\spad{p},{}\\spad{ts})} returns the same as \\axiom{remainder(\\spad{p},{}collectQuasiMonic(\\spad{ts})).polnum}.")) (|collectQuasiMonic| (($ $) "\\axiom{collectQuasiMonic(\\spad{ts})} returns the subset of \\axiom{\\spad{ts}} consisting of the polynomials with initial in \\axiom{\\spad{R}}.")) (|removeZero| ((|#5| |#5| $) "\\axiom{removeZero(\\spad{p},{}\\spad{ts})} returns \\axiom{0} if \\axiom{\\spad{p}} reduces to \\axiom{0} by pseudo-division \\spad{w}.\\spad{r}.\\spad{t} \\axiom{\\spad{ts}} otherwise returns a polynomial \\axiom{\\spad{q}} computed from \\axiom{\\spad{p}} by removing any coefficient in \\axiom{\\spad{p}} reducing to \\axiom{0}.")) (|initiallyReduce| ((|#5| |#5| $) "\\axiom{initiallyReduce(\\spad{p},{}\\spad{ts})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}\\spad{ts})} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}.")) (|headReduce| ((|#5| |#5| $) "\\axiom{headReduce(\\spad{p},{}\\spad{ts})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduce?(\\spad{r},{}\\spad{ts})} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}.")) (|stronglyReduce| ((|#5| |#5| $) "\\axiom{stronglyReduce(\\spad{p},{}\\spad{ts})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{stronglyReduced?(\\spad{r},{}\\spad{ts})} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}.")) (|rewriteSetWithReduction| (((|List| |#5|) (|List| |#5|) $ (|Mapping| |#5| |#5| |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{rewriteSetWithReduction(\\spad{lp},{}\\spad{ts},{}redOp,{}redOp?)} returns a list \\axiom{\\spad{lq}} of polynomials such that \\axiom{[reduce(\\spad{p},{}\\spad{ts},{}redOp,{}redOp?) for \\spad{p} in \\spad{lp}]} and \\axiom{\\spad{lp}} have the same zeros inside the regular zero set of \\axiom{\\spad{ts}}. Moreover,{} for every polynomial \\axiom{\\spad{q}} in \\axiom{\\spad{lq}} and every polynomial \\axiom{\\spad{t}} in \\axiom{\\spad{ts}} \\axiom{redOp?(\\spad{q},{}\\spad{t})} holds and there exists a polynomial \\axiom{\\spad{p}} in the ideal generated by \\axiom{\\spad{lp}} and a product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = \\spad{f*q} + redOp(\\spad{p},{}\\spad{q})}.")) (|reduce| ((|#5| |#5| $ (|Mapping| |#5| |#5| |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{reduce(\\spad{p},{}\\spad{ts},{}redOp,{}redOp?)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{redOp?(\\spad{r},{}\\spad{p})} holds for every \\axiom{\\spad{p}} of \\axiom{\\spad{ts}} and there exists some product \\axiom{\\spad{h}} of the initials of the members of \\axiom{\\spad{ts}} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = \\spad{f*q} + redOp(\\spad{p},{}\\spad{q})}.")) (|autoReduced?| (((|Boolean|) $ (|Mapping| (|Boolean|) |#5| (|List| |#5|))) "\\axiom{autoReduced?(\\spad{ts},{}redOp?)} returns \\spad{true} iff every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to every other in the sense of \\axiom{redOp?}")) (|initiallyReduced?| (((|Boolean|) $) "\\spad{initiallyReduced?(ts)} returns \\spad{true} iff for every element \\axiom{\\spad{p}} of \\axiom{\\spad{ts}} \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the other elements of \\axiom{\\spad{ts}} with the same main variable.") (((|Boolean|) |#5| $) "\\axiom{initiallyReduced?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the elements of \\axiom{\\spad{ts}} with the same main variable.")) (|headReduced?| (((|Boolean|) $) "\\spad{headReduced?(ts)} returns \\spad{true} iff the head of every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{\\spad{ts}}.") (((|Boolean|) |#5| $) "\\axiom{headReduced?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff the head of \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ts}}.")) (|stronglyReduced?| (((|Boolean|) $) "\\axiom{stronglyReduced?(\\spad{ts})} returns \\spad{true} iff every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{\\spad{ts}}.") (((|Boolean|) |#5| $) "\\axiom{stronglyReduced?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ts}}.")) (|reduced?| (((|Boolean|) |#5| $ (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{reduced?(\\spad{p},{}\\spad{ts},{}redOp?)} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. in the sense of the operation \\axiom{redOp?},{} that is if for every \\axiom{\\spad{t}} in \\axiom{\\spad{ts}} \\axiom{redOp?(\\spad{p},{}\\spad{t})} holds.")) (|normalized?| (((|Boolean|) $) "\\axiom{normalized?(\\spad{ts})} returns \\spad{true} iff for every axiom{\\spad{p}} in axiom{\\spad{ts}} we have \\axiom{normalized?(\\spad{p},{}us)} where \\axiom{us} is \\axiom{collectUnder(\\spad{ts},{}mvar(\\spad{p}))}.") (((|Boolean|) |#5| $) "\\axiom{normalized?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variables of the polynomials of \\axiom{\\spad{ts}}")) (|quasiComponent| (((|Record| (|:| |close| (|List| |#5|)) (|:| |open| (|List| |#5|))) $) "\\axiom{quasiComponent(\\spad{ts})} returns \\axiom{[\\spad{lp},{}\\spad{lq}]} where \\axiom{\\spad{lp}} is the list of the members of \\axiom{\\spad{ts}} and \\axiom{\\spad{lq}}is \\axiom{initials(\\spad{ts})}.")) (|degree| (((|NonNegativeInteger|) $) "\\axiom{degree(\\spad{ts})} returns the product of main degrees of the members of \\axiom{\\spad{ts}}.")) (|initials| (((|List| |#5|) $) "\\axiom{initials(\\spad{ts})} returns the list of the non-constant initials of the members of \\axiom{\\spad{ts}}.")) (|basicSet| (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#5|))) "failed") (|List| |#5|) (|Mapping| (|Boolean|) |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{basicSet(\\spad{ps},{}pred?,{}redOp?)} returns the same as \\axiom{basicSet(\\spad{qs},{}redOp?)} where \\axiom{\\spad{qs}} consists of the polynomials of \\axiom{\\spad{ps}} satisfying property \\axiom{pred?}.") (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#5|))) "failed") (|List| |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{basicSet(\\spad{ps},{}redOp?)} returns \\axiom{[\\spad{bs},{}\\spad{ts}]} where \\axiom{concat(\\spad{bs},{}\\spad{ts})} is \\axiom{\\spad{ps}} and \\axiom{\\spad{bs}} is a basic set in Wu Wen Tsun sense of \\axiom{\\spad{ps}} \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?},{} if no non-zero constant polynomial lie in \\axiom{\\spad{ps}},{} otherwise \\axiom{\"failed\"} is returned.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(\\spad{ts1},{}\\spad{ts2})} returns \\spad{true} iff \\axiom{\\spad{ts2}} has higher rank than \\axiom{\\spad{ts1}} in Wu Wen Tsun sense."))) 
 NIL 
-((|HasCategory| |#4| (QUOTE (-371)))) 
-(-1215 R E V P) 
+((|HasCategory| |#4| (QUOTE (-372)))) 
+(-1217 R E V P) 
 ((|constructor| (NIL "The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \\axiom{\\spad{R}} be an integral domain and \\axiom{\\spad{V}} a finite ordered set of variables,{} say \\axiom{\\spad{X1} < \\spad{X2} < ... < \\spad{Xn}}. A set \\axiom{\\spad{S}} of polynomials in \\axiom{\\spad{R}[\\spad{X1},{}\\spad{X2},{}...,{}\\spad{Xn}]} is triangular if no elements of \\axiom{\\spad{S}} lies in \\axiom{\\spad{R}},{} and if two distinct elements of \\axiom{\\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a non-constant polynomial \\axiom{\\spad{Q}} if the degree of \\axiom{\\spad{P}} in the main variable of \\axiom{\\spad{Q}} is less than the main degree of \\axiom{\\spad{Q}}. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a triangular set \\axiom{\\spad{T}} if it is reduced \\spad{w}.\\spad{r}.\\spad{t}. every polynomial of \\axiom{\\spad{T}}. \\newline References : \\indented{1}{[1] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)}")) (|coHeight| (((|NonNegativeInteger|) $) "\\axiom{coHeight(\\spad{ts})} returns \\axiom{size()\\spad{\\$}\\spad{V}} minus \\axiom{\\spad{\\#}\\spad{ts}}.")) (|extend| (($ $ |#4|) "\\axiom{extend(\\spad{ts},{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{\\spad{ts}},{} according to the properties of triangular sets of the current category If the required properties do not hold an error is returned.")) (|extendIfCan| (((|Union| $ "failed") $ |#4|) "\\axiom{extendIfCan(\\spad{ts},{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{\\spad{ts}},{} according to the properties of triangular sets of the current domain. If the required properties do not hold then \"failed\" is returned. This operation encodes in some sense the properties of the triangular sets of the current category. Is is used to implement the \\axiom{construct} operation to guarantee that every triangular set build from a list of polynomials has the required properties.")) (|select| (((|Union| |#4| "failed") $ |#3|) "\\axiom{select(\\spad{ts},{}\\spad{v})} returns the polynomial of \\axiom{\\spad{ts}} with \\axiom{\\spad{v}} as main variable,{} if any.")) (|algebraic?| (((|Boolean|) |#3| $) "\\axiom{algebraic?(\\spad{v},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{\\spad{ts}}.")) (|algebraicVariables| (((|List| |#3|) $) "\\axiom{algebraicVariables(\\spad{ts})} returns the decreasingly sorted list of the main variables of the polynomials of \\axiom{\\spad{ts}}.")) (|rest| (((|Union| $ "failed") $) "\\axiom{rest(\\spad{ts})} returns the polynomials of \\axiom{\\spad{ts}} with smaller main variable than \\axiom{mvar(\\spad{ts})} if \\axiom{\\spad{ts}} is not empty,{} otherwise returns \"failed\"")) (|last| (((|Union| |#4| "failed") $) "\\axiom{last(\\spad{ts})} returns the polynomial of \\axiom{\\spad{ts}} with smallest main variable if \\axiom{\\spad{ts}} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|first| (((|Union| |#4| "failed") $) "\\axiom{first(\\spad{ts})} returns the polynomial of \\axiom{\\spad{ts}} with greatest main variable if \\axiom{\\spad{ts}} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|zeroSetSplitIntoTriangularSystems| (((|List| (|Record| (|:| |close| $) (|:| |open| (|List| |#4|)))) (|List| |#4|)) "\\axiom{zeroSetSplitIntoTriangularSystems(\\spad{lp})} returns a list of triangular systems \\axiom{[[\\spad{ts1},{}\\spad{qs1}],{}...,{}[\\spad{tsn},{}\\spad{qsn}]]} such that the zero set of \\axiom{\\spad{lp}} is the union of the closures of the \\axiom{W_i} where \\axiom{W_i} consists of the zeros of \\axiom{\\spad{ts}} which do not cancel any polynomial in \\axiom{qsi}.")) (|zeroSetSplit| (((|List| $) (|List| |#4|)) "\\axiom{zeroSetSplit(\\spad{lp})} returns a list \\axiom{\\spad{lts}} of triangular sets such that the zero set of \\axiom{\\spad{lp}} is the union of the closures of the regular zero sets of the members of \\axiom{\\spad{lts}}.")) (|reduceByQuasiMonic| ((|#4| |#4| $) "\\axiom{reduceByQuasiMonic(\\spad{p},{}\\spad{ts})} returns the same as \\axiom{remainder(\\spad{p},{}collectQuasiMonic(\\spad{ts})).polnum}.")) (|collectQuasiMonic| (($ $) "\\axiom{collectQuasiMonic(\\spad{ts})} returns the subset of \\axiom{\\spad{ts}} consisting of the polynomials with initial in \\axiom{\\spad{R}}.")) (|removeZero| ((|#4| |#4| $) "\\axiom{removeZero(\\spad{p},{}\\spad{ts})} returns \\axiom{0} if \\axiom{\\spad{p}} reduces to \\axiom{0} by pseudo-division \\spad{w}.\\spad{r}.\\spad{t} \\axiom{\\spad{ts}} otherwise returns a polynomial \\axiom{\\spad{q}} computed from \\axiom{\\spad{p}} by removing any coefficient in \\axiom{\\spad{p}} reducing to \\axiom{0}.")) (|initiallyReduce| ((|#4| |#4| $) "\\axiom{initiallyReduce(\\spad{p},{}\\spad{ts})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}\\spad{ts})} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}.")) (|headReduce| ((|#4| |#4| $) "\\axiom{headReduce(\\spad{p},{}\\spad{ts})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduce?(\\spad{r},{}\\spad{ts})} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}.")) (|stronglyReduce| ((|#4| |#4| $) "\\axiom{stronglyReduce(\\spad{p},{}\\spad{ts})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{stronglyReduced?(\\spad{r},{}\\spad{ts})} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}.")) (|rewriteSetWithReduction| (((|List| |#4|) (|List| |#4|) $ (|Mapping| |#4| |#4| |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{rewriteSetWithReduction(\\spad{lp},{}\\spad{ts},{}redOp,{}redOp?)} returns a list \\axiom{\\spad{lq}} of polynomials such that \\axiom{[reduce(\\spad{p},{}\\spad{ts},{}redOp,{}redOp?) for \\spad{p} in \\spad{lp}]} and \\axiom{\\spad{lp}} have the same zeros inside the regular zero set of \\axiom{\\spad{ts}}. Moreover,{} for every polynomial \\axiom{\\spad{q}} in \\axiom{\\spad{lq}} and every polynomial \\axiom{\\spad{t}} in \\axiom{\\spad{ts}} \\axiom{redOp?(\\spad{q},{}\\spad{t})} holds and there exists a polynomial \\axiom{\\spad{p}} in the ideal generated by \\axiom{\\spad{lp}} and a product \\axiom{\\spad{h}} of \\axiom{initials(\\spad{ts})} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = \\spad{f*q} + redOp(\\spad{p},{}\\spad{q})}.")) (|reduce| ((|#4| |#4| $ (|Mapping| |#4| |#4| |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{reduce(\\spad{p},{}\\spad{ts},{}redOp,{}redOp?)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{redOp?(\\spad{r},{}\\spad{p})} holds for every \\axiom{\\spad{p}} of \\axiom{\\spad{ts}} and there exists some product \\axiom{\\spad{h}} of the initials of the members of \\axiom{\\spad{ts}} such that \\axiom{\\spad{h*p} - \\spad{r}} lies in the ideal generated by \\axiom{\\spad{ts}}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = \\spad{f*q} + redOp(\\spad{p},{}\\spad{q})}.")) (|autoReduced?| (((|Boolean|) $ (|Mapping| (|Boolean|) |#4| (|List| |#4|))) "\\axiom{autoReduced?(\\spad{ts},{}redOp?)} returns \\spad{true} iff every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to every other in the sense of \\axiom{redOp?}")) (|initiallyReduced?| (((|Boolean|) $) "\\spad{initiallyReduced?(ts)} returns \\spad{true} iff for every element \\axiom{\\spad{p}} of \\axiom{\\spad{ts}} \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the other elements of \\axiom{\\spad{ts}} with the same main variable.") (((|Boolean|) |#4| $) "\\axiom{initiallyReduced?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the elements of \\axiom{\\spad{ts}} with the same main variable.")) (|headReduced?| (((|Boolean|) $) "\\spad{headReduced?(ts)} returns \\spad{true} iff the head of every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{\\spad{ts}}.") (((|Boolean|) |#4| $) "\\axiom{headReduced?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff the head of \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ts}}.")) (|stronglyReduced?| (((|Boolean|) $) "\\axiom{stronglyReduced?(\\spad{ts})} returns \\spad{true} iff every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{\\spad{ts}}.") (((|Boolean|) |#4| $) "\\axiom{stronglyReduced?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{ts}}.")) (|reduced?| (((|Boolean|) |#4| $ (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{reduced?(\\spad{p},{}\\spad{ts},{}redOp?)} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. in the sense of the operation \\axiom{redOp?},{} that is if for every \\axiom{\\spad{t}} in \\axiom{\\spad{ts}} \\axiom{redOp?(\\spad{p},{}\\spad{t})} holds.")) (|normalized?| (((|Boolean|) $) "\\axiom{normalized?(\\spad{ts})} returns \\spad{true} iff for every axiom{\\spad{p}} in axiom{\\spad{ts}} we have \\axiom{normalized?(\\spad{p},{}us)} where \\axiom{us} is \\axiom{collectUnder(\\spad{ts},{}mvar(\\spad{p}))}.") (((|Boolean|) |#4| $) "\\axiom{normalized?(\\spad{p},{}\\spad{ts})} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variables of the polynomials of \\axiom{\\spad{ts}}")) (|quasiComponent| (((|Record| (|:| |close| (|List| |#4|)) (|:| |open| (|List| |#4|))) $) "\\axiom{quasiComponent(\\spad{ts})} returns \\axiom{[\\spad{lp},{}\\spad{lq}]} where \\axiom{\\spad{lp}} is the list of the members of \\axiom{\\spad{ts}} and \\axiom{\\spad{lq}}is \\axiom{initials(\\spad{ts})}.")) (|degree| (((|NonNegativeInteger|) $) "\\axiom{degree(\\spad{ts})} returns the product of main degrees of the members of \\axiom{\\spad{ts}}.")) (|initials| (((|List| |#4|) $) "\\axiom{initials(\\spad{ts})} returns the list of the non-constant initials of the members of \\axiom{\\spad{ts}}.")) (|basicSet| (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#4|))) "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{basicSet(\\spad{ps},{}pred?,{}redOp?)} returns the same as \\axiom{basicSet(\\spad{qs},{}redOp?)} where \\axiom{\\spad{qs}} consists of the polynomials of \\axiom{\\spad{ps}} satisfying property \\axiom{pred?}.") (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#4|))) "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{basicSet(\\spad{ps},{}redOp?)} returns \\axiom{[\\spad{bs},{}\\spad{ts}]} where \\axiom{concat(\\spad{bs},{}\\spad{ts})} is \\axiom{\\spad{ps}} and \\axiom{\\spad{bs}} is a basic set in Wu Wen Tsun sense of \\axiom{\\spad{ps}} \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?},{} if no non-zero constant polynomial lie in \\axiom{\\spad{ps}},{} otherwise \\axiom{\"failed\"} is returned.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(\\spad{ts1},{}\\spad{ts2})} returns \\spad{true} iff \\axiom{\\spad{ts2}} has higher rank than \\axiom{\\spad{ts1}} in Wu Wen Tsun sense."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-1216 |Curve|) 
+(-1218 |Curve|) 
 ((|constructor| (NIL "\\indented{2}{Package for constructing tubes around 3-dimensional parametric curves.} Domain of tubes around 3-dimensional parametric curves.")) (|tube| (($ |#1| (|List| (|List| (|Point| (|DoubleFloat|)))) (|Boolean|)) "\\spad{tube(c,ll,b)} creates a tube of the domain \\spadtype{TubePlot} from a space curve \\spad{c} of the category \\spadtype{PlottableSpaceCurveCategory},{} a list of lists of points (loops) \\spad{ll} and a boolean \\spad{b} which if \\spad{true} indicates a closed tube,{} or if \\spad{false} an open tube.")) (|setClosed| (((|Boolean|) $ (|Boolean|)) "\\spad{setClosed(t,b)} declares the given tube plot \\spad{t} to be closed if \\spad{b} is \\spad{true},{} or if \\spad{b} is \\spad{false},{} \\spad{t} is set to be open.")) (|open?| (((|Boolean|) $) "\\spad{open?(t)} tests whether the given tube plot \\spad{t} is open.")) (|closed?| (((|Boolean|) $) "\\spad{closed?(t)} tests whether the given tube plot \\spad{t} is closed.")) (|listLoops| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{listLoops(t)} returns the list of lists of points,{} or the 'loops',{} of the given tube plot \\spad{t}.")) (|getCurve| ((|#1| $) "\\spad{getCurve(t)} returns the \\spadtype{PlottableSpaceCurveCategory} representing the parametric curve of the given tube plot \\spad{t}."))) 
 NIL 
 NIL 
-(-1217) 
+(-1219) 
 ((|constructor| (NIL "Tools for constructing tubes around 3-dimensional parametric curves.")) (|loopPoints| (((|List| (|Point| (|DoubleFloat|))) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|List| (|List| (|DoubleFloat|)))) "\\spad{loopPoints(p,n,b,r,lls)} creates and returns a list of points which form the loop with radius \\spad{r},{} around the center point indicated by the point \\spad{p},{} with the principal normal vector of the space curve at point \\spad{p} given by the point(vector) \\spad{n},{} and the binormal vector given by the point(vector) \\spad{b},{} and a list of lists,{} \\spad{lls},{} which is the \\spadfun{cosSinInfo} of the number of points defining the loop.")) (|cosSinInfo| (((|List| (|List| (|DoubleFloat|))) (|Integer|)) "\\spad{cosSinInfo(n)} returns the list of lists of values for \\spad{n},{} in the form: \\spad{[[cos(n - 1) a,sin(n - 1) a],...,[cos 2 a,sin 2 a],[cos a,sin a]]} where \\spad{a = 2 pi/n}. Note: \\spad{n} should be greater than 2.")) (|unitVector| (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{unitVector(p)} creates the unit vector of the point \\spad{p} and returns the result as a point. Note: \\spad{unitVector(p) = p/|p|}.")) (|cross| (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{cross(p,q)} computes the cross product of the two points \\spad{p} and \\spad{q} using only the first three coordinates,{} and keeping the color of the first point \\spad{p}. The result is returned as a point.")) (|dot| (((|DoubleFloat|) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{dot(p,q)} computes the dot product of the two points \\spad{p} and \\spad{q} using only the first three coordinates,{} and returns the resulting \\spadtype{DoubleFloat}.")) (- (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{p - q} computes and returns a point whose coordinates are the differences of the coordinates of two points \\spad{p} and \\spad{q},{} using the color,{} or fourth coordinate,{} of the first point \\spad{p} as the color also of the point \\spad{q}.")) (+ (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{p + q} computes and returns a point whose coordinates are the sums of the coordinates of the two points \\spad{p} and \\spad{q},{} using the color,{} or fourth coordinate,{} of the first point \\spad{p} as the color also of the point \\spad{q}.")) (* (((|Point| (|DoubleFloat|)) (|DoubleFloat|) (|Point| (|DoubleFloat|))) "\\spad{s * p} returns a point whose coordinates are the scalar multiple of the point \\spad{p} by the scalar \\spad{s},{} preserving the color,{} or fourth coordinate,{} of \\spad{p}.")) (|point| (((|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{point(x1,x2,x3,c)} creates and returns a point from the three specified coordinates \\spad{x1},{} \\spad{x2},{} \\spad{x3},{} and also a fourth coordinate,{} \\spad{c},{} which is generally used to specify the color of the point."))) 
 NIL 
 NIL 
-(-1218 S) 
+(-1220 S) 
 ((|constructor| (NIL "\\indented{1}{This domain is used to interface with the interpreter\\spad{'s} notion} of comma-delimited sequences of values.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length(x)} returns the number of elements in tuple \\spad{x}")) (|select| ((|#1| $ (|NonNegativeInteger|)) "\\spad{select(x,n)} returns the \\spad{n}-th element of tuple \\spad{x}. tuples are 0-based"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1219 -3498) 
+((|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1221 -3505) 
 ((|constructor| (NIL "A basic package for the factorization of bivariate polynomials over a finite field. The functions here represent the base step for the multivariate factorizer.")) (|twoFactor| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|)) (|Integer|)) "\\spad{twoFactor(p,n)} returns the factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}. Also,{} \\spad{p} is assumed primitive and square-free and \\spad{n} is the degree of the inner variable of \\spad{p} (maximum of the degrees of the coefficients of \\spad{p}).")) (|generalSqFr| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) "\\spad{generalSqFr(p)} returns the square-free factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}.")) (|generalTwoFactor| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) "\\spad{generalTwoFactor(p)} returns the factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}."))) 
 NIL 
 NIL 
-(-1220) 
+(-1222) 
 ((|constructor| (NIL "The fundamental Type."))) 
 NIL 
 NIL 
-(-1221) 
+(-1223) 
 ((|constructor| (NIL "This domain represents a type AST."))) 
 NIL 
 NIL 
-(-1222 S) 
+(-1224 S) 
 ((|constructor| (NIL "Provides functions to force a partial ordering on any set.")) (|more?| (((|Boolean|) |#1| |#1|) "\\spad{more?(a, b)} compares \\spad{a} and \\spad{b} in the partial ordering induced by setOrder,{} and uses the ordering on \\spad{S} if \\spad{a} and \\spad{b} are not comparable in the partial ordering.")) (|userOrdered?| (((|Boolean|)) "\\spad{userOrdered?()} tests if the partial ordering induced by \\spadfunFrom{setOrder}{UserDefinedPartialOrdering} is not empty.")) (|largest| ((|#1| (|List| |#1|)) "\\spad{largest l} returns the largest element of \\spad{l} where the partial ordering induced by setOrder is completed into a total one by the ordering on \\spad{S}.") ((|#1| (|List| |#1|) (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{largest(l, fn)} returns the largest element of \\spad{l} where the partial ordering induced by setOrder is completed into a total one by \\spad{fn}.")) (|less?| (((|Boolean|) |#1| |#1| (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{less?(a, b, fn)} compares \\spad{a} and \\spad{b} in the partial ordering induced by setOrder,{} and returns \\spad{fn(a, b)} if \\spad{a} and \\spad{b} are not comparable in that ordering.") (((|Union| (|Boolean|) "failed") |#1| |#1|) "\\spad{less?(a, b)} compares \\spad{a} and \\spad{b} in the partial ordering induced by setOrder.")) (|getOrder| (((|Record| (|:| |low| (|List| |#1|)) (|:| |high| (|List| |#1|)))) "\\spad{getOrder()} returns \\spad{[[b1,...,bm], [a1,...,an]]} such that the partial ordering on \\spad{S} was given by \\spad{setOrder([b1,...,bm],[a1,...,an])}.")) (|setOrder| (((|Void|) (|List| |#1|) (|List| |#1|)) "\\spad{setOrder([b1,...,bm], [a1,...,an])} defines a partial ordering on \\spad{S} given \\spad{by:} \\indented{3}{(1)\\space{2}\\spad{b1 < b2 < ... < bm < a1 < a2 < ... < an}.} \\indented{3}{(2)\\space{2}\\spad{bj < c < ai}\\space{2}for \\spad{c} not among the \\spad{ai}\\spad{'s} and \\spad{bj}\\spad{'s}.} \\indented{3}{(3)\\space{2}undefined on \\spad{(c,d)} if neither is among the \\spad{ai}\\spad{'s},{}\\spad{bj}\\spad{'s}.}") (((|Void|) (|List| |#1|)) "\\spad{setOrder([a1,...,an])} defines a partial ordering on \\spad{S} given \\spad{by:} \\indented{3}{(1)\\space{2}\\spad{a1 < a2 < ... < an}.} \\indented{3}{(2)\\space{2}\\spad{b < ai\\space{3}for i = 1..n} and \\spad{b} not among the \\spad{ai}\\spad{'s}.} \\indented{3}{(3)\\space{2}undefined on \\spad{(b, c)} if neither is among the \\spad{ai}\\spad{'s}.}"))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-853)))) 
-(-1223) 
+((|HasCategory| |#1| (QUOTE (-855)))) 
+(-1225) 
 ((|constructor| (NIL "This packages provides functions to allow the user to select the ordering on the variables and operators for displaying polynomials,{} fractions and expressions. The ordering affects the display only and not the computations.")) (|resetVariableOrder| (((|Void|)) "\\spad{resetVariableOrder()} cancels any previous use of setVariableOrder and returns to the default system ordering.")) (|getVariableOrder| (((|Record| (|:| |high| (|List| (|Symbol|))) (|:| |low| (|List| (|Symbol|))))) "\\spad{getVariableOrder()} returns \\spad{[[b1,...,bm], [a1,...,an]]} such that the ordering on the variables was given by \\spad{setVariableOrder([b1,...,bm], [a1,...,an])}.")) (|setVariableOrder| (((|Void|) (|List| (|Symbol|)) (|List| (|Symbol|))) "\\spad{setVariableOrder([b1,...,bm], [a1,...,an])} defines an ordering on the variables given by \\spad{b1 > b2 > ... > bm >} other variables \\spad{> a1 > a2 > ... > an}.") (((|Void|) (|List| (|Symbol|))) "\\spad{setVariableOrder([a1,...,an])} defines an ordering on the variables given by \\spad{a1 > a2 > ... > an > other variables}."))) 
 NIL 
 NIL 
-(-1224 S) 
+(-1226 S) 
 ((|constructor| (NIL "A constructive unique factorization domain,{} \\spadignore{i.e.} where we can constructively factor members into a product of a finite number of irreducible elements.")) (|factor| (((|Factored| $) $) "\\spad{factor(x)} returns the factorization of \\spad{x} into irreducibles.")) (|squareFreePart| (($ $) "\\spad{squareFreePart(x)} returns a product of prime factors of \\spad{x} each taken with multiplicity one.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns the square-free factorization of \\spad{x} \\spadignore{i.e.} such that the factors are pairwise relatively prime and each has multiple prime factors.")) (|prime?| (((|Boolean|) $) "\\spad{prime?(x)} tests if \\spad{x} can never be written as the product of two non-units of the ring,{} \\spadignore{i.e.} \\spad{x} is an irreducible element."))) 
 NIL 
 NIL 
-(-1225) 
+(-1227) 
 ((|constructor| (NIL "A constructive unique factorization domain,{} \\spadignore{i.e.} where we can constructively factor members into a product of a finite number of irreducible elements.")) (|factor| (((|Factored| $) $) "\\spad{factor(x)} returns the factorization of \\spad{x} into irreducibles.")) (|squareFreePart| (($ $) "\\spad{squareFreePart(x)} returns a product of prime factors of \\spad{x} each taken with multiplicity one.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns the square-free factorization of \\spad{x} \\spadignore{i.e.} such that the factors are pairwise relatively prime and each has multiple prime factors.")) (|prime?| (((|Boolean|) $) "\\spad{prime?(x)} tests if \\spad{x} can never be written as the product of two non-units of the ring,{} \\spadignore{i.e.} \\spad{x} is an irreducible element."))) 
-((-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1226) 
+(-1228) 
 ((|constructor| (NIL "This domain is a datatype for (unsigned) integer values of precision 16 bits."))) 
 NIL 
 NIL 
-(-1227) 
+(-1229) 
 ((|constructor| (NIL "This domain is a datatype for (unsigned) integer values of precision 32 bits."))) 
 NIL 
 NIL 
-(-1228) 
+(-1230) 
 ((|constructor| (NIL "This domain is a datatype for (unsigned) integer values of precision 64 bits."))) 
 NIL 
 NIL 
-(-1229) 
+(-1231) 
 ((|constructor| (NIL "This domain is a datatype for (unsigned) integer values of precision 8 bits."))) 
 NIL 
 NIL 
-(-1230 |Coef| |var| |cen|) 
+(-1232 |Coef| |var| |cen|) 
 ((|constructor| (NIL "Dense Laurent series in one variable \\indented{2}{\\spadtype{UnivariateLaurentSeries} is a domain representing Laurent} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{UnivariateLaurentSeries(Integer,x,3)} represents Laurent series in} \\indented{2}{\\spad{(x - 3)} with integer coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a Laurent series."))) 
-(((-4429 "*") -3962 (-3258 (|has| |#1| (-366)) (|has| (-1260 |#1| |#2| |#3|) (-823))) (|has| |#1| (-173)) (-3258 (|has| |#1| (-366)) (|has| (-1260 |#1| |#2| |#3|) (-914)))) (-4420 -3962 (-3258 (|has| |#1| (-366)) (|has| (-1260 |#1| |#2| |#3|) (-823))) (|has| |#1| (-561)) (-3258 (|has| |#1| (-366)) (|has| (-1260 |#1| |#2| |#3|) (-914)))) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
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|#2| |#3|) (LIST (QUOTE -518) (QUOTE (-1181)) (LIST (QUOTE -1260) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (LIST (QUOTE -642) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (LIST (QUOTE -1042) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (LIST (QUOTE -1042) (QUOTE (-1181))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (QUOTE (-823)))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (QUOTE (-853)))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (QUOTE (-1024)))) 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(|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-550)) (|devaluate| |#1|))))) (|HasCategory| (-550) (QUOTE (-1116))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-366))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (QUOTE (-914)))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (LIST (QUOTE -1042) (QUOTE (-1181))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (LIST (QUOTE -617) (QUOTE (-539))))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (QUOTE (-1024)))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (QUOTE (-823)))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| (-1260 |#1| |#2| |#3|) (QUOTE (-823)))) 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-(-1231 |Coef1| |Coef2| |var1| |var2| |cen1| |cen2|) 
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+(-1233 |Coef1| |Coef2| |var1| |var2| |cen1| |cen2|) 
 ((|constructor| (NIL "Mapping package for univariate Laurent series \\indented{2}{This package allows one to apply a function to the coefficients of} \\indented{2}{a univariate Laurent series.}")) (|map| (((|UnivariateLaurentSeries| |#2| |#4| |#6|) (|Mapping| |#2| |#1|) (|UnivariateLaurentSeries| |#1| |#3| |#5|)) "\\spad{map(f,g(x))} applies the map \\spad{f} to the coefficients of the Laurent series \\spad{g(x)}."))) 
 NIL 
 NIL 
-(-1232 |Coef|) 
+(-1234 |Coef|) 
 ((|constructor| (NIL "\\spadtype{UnivariateLaurentSeriesCategory} is the category of Laurent series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 1. We may integrate a series when we can divide coefficients by integers.")) (|rationalFunction| (((|Fraction| (|Polynomial| |#1|)) $ (|Integer|) (|Integer|)) "\\spad{rationalFunction(f,k1,k2)} returns a rational function consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Fraction| (|Polynomial| |#1|)) $ (|Integer|)) "\\spad{rationalFunction(f,k)} returns a rational function consisting of the sum of all terms of \\spad{f} of degree \\spad{<=} \\spad{k}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(f,sum(n = n0..infinity,a[n] * x**n)) = sum(n = 0..infinity,f(n) * a[n] * x**n)}. This function is used when Puiseux series are represented by a Laurent series and an exponent.")) (|series| (($ (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents."))) 
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 NIL 
-(-1233 S |Coef| UTS) 
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 ((|constructor| (NIL "This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair \\spad{[n,f(x)]},{} where \\spad{n} is an arbitrary integer and \\spad{f(x)} is a Taylor series. This pair represents the Laurent series \\spad{x**n * f(x)}.")) (|taylorIfCan| (((|Union| |#3| "failed") $) "\\spad{taylorIfCan(f(x))} converts the Laurent series \\spad{f(x)} to a Taylor series,{} if possible. If this is not possible,{} \"failed\" is returned.")) (|taylor| ((|#3| $) "\\spad{taylor(f(x))} converts the Laurent series \\spad{f}(\\spad{x}) to a Taylor series,{} if possible. Error: if this is not possible.")) (|removeZeroes| (($ (|Integer|) $) "\\spad{removeZeroes(n,f(x))} removes up to \\spad{n} leading zeroes from the Laurent series \\spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable.") (($ $) "\\spad{removeZeroes(f(x))} removes leading zeroes from the representation of the Laurent series \\spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable. Note: \\spad{removeZeroes(f)} removes all leading zeroes from \\spad{f}")) (|taylorRep| ((|#3| $) "\\spad{taylorRep(f(x))} returns \\spad{g(x)},{} where \\spad{f = x**n * g(x)} is represented by \\spad{[n,g(x)]}.")) (|degree| (((|Integer|) $) "\\spad{degree(f(x))} returns the degree of the lowest order term of \\spad{f(x)},{} which may have zero as a coefficient.")) (|laurent| (($ (|Integer|) |#3|) "\\spad{laurent(n,f(x))} returns \\spad{x**n * f(x)}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-366)))) 
-(-1234 |Coef| UTS) 
+((|HasCategory| |#2| (QUOTE (-367)))) 
+(-1236 |Coef| UTS) 
 ((|constructor| (NIL "This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair \\spad{[n,f(x)]},{} where \\spad{n} is an arbitrary integer and \\spad{f(x)} is a Taylor series. This pair represents the Laurent series \\spad{x**n * f(x)}.")) (|taylorIfCan| (((|Union| |#2| "failed") $) "\\spad{taylorIfCan(f(x))} converts the Laurent series \\spad{f(x)} to a Taylor series,{} if possible. If this is not possible,{} \"failed\" is returned.")) (|taylor| ((|#2| $) "\\spad{taylor(f(x))} converts the Laurent series \\spad{f}(\\spad{x}) to a Taylor series,{} if possible. Error: if this is not possible.")) (|removeZeroes| (($ (|Integer|) $) "\\spad{removeZeroes(n,f(x))} removes up to \\spad{n} leading zeroes from the Laurent series \\spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable.") (($ $) "\\spad{removeZeroes(f(x))} removes leading zeroes from the representation of the Laurent series \\spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable. Note: \\spad{removeZeroes(f)} removes all leading zeroes from \\spad{f}")) (|taylorRep| ((|#2| $) "\\spad{taylorRep(f(x))} returns \\spad{g(x)},{} where \\spad{f = x**n * g(x)} is represented by \\spad{[n,g(x)]}.")) (|degree| (((|Integer|) $) "\\spad{degree(f(x))} returns the degree of the lowest order term of \\spad{f(x)},{} which may have zero as a coefficient.")) (|laurent| (($ (|Integer|) |#2|) "\\spad{laurent(n,f(x))} returns \\spad{x**n * f(x)}."))) 
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 NIL 
-(-1235 |Coef| UTS) 
+(-1237 |Coef| UTS) 
 ((|constructor| (NIL "This package enables one to construct a univariate Laurent series domain from a univariate Taylor series domain. Univariate Laurent series are represented by a pair \\spad{[n,f(x)]},{} where \\spad{n} is an arbitrary integer and \\spad{f(x)} is a Taylor series. This pair represents the Laurent series \\spad{x**n * f(x)}."))) 
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 ((|constructor| (NIL "Package for the factorization of univariate polynomials with integer coefficients. The factorization is done by \"lifting\" (HENSEL) the factorization over a finite field.")) (|henselFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|)) "\\spad{henselFact(m,flag)} returns the factorization of \\spad{m},{} FinalFact is a Record \\spad{s}.\\spad{t}. FinalFact.contp=content \\spad{m},{} FinalFact.factors=List of irreducible factors of \\spad{m} with exponent ,{} if \\spad{flag} =true the polynomial is assumed square free.")) (|factorSquareFree| (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(m)} returns the factorization of \\spad{m} square free polynomial")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(m)} returns the factorization of \\spad{m}"))) 
 NIL 
 NIL 
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 ((|constructor| (NIL "This domain provides segments which may be half open. That is,{} ranges of the form \\spad{a..} or \\spad{a..b}.")) (|hasHi| (((|Boolean|) $) "\\spad{hasHi(s)} tests whether the segment \\spad{s} has an upper bound.")) (|coerce| (($ (|Segment| |#1|)) "\\spad{coerce(x)} allows \\spadtype{Segment} values to be used as \\%.")) (|segment| (($ |#1|) "\\spad{segment(l)} is an alternate way to construct the segment \\spad{l..}.")) (SEGMENT (($ |#1|) "\\spad{l..} produces a half open segment,{} that is,{} one with no upper bound."))) 
 NIL 
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 ((|constructor| (NIL "This package provides operations for mapping functions onto segments.")) (|map| (((|Stream| |#2|) (|Mapping| |#2| |#1|) (|UniversalSegment| |#1|)) "\\spad{map(f,s)} expands the segment \\spad{s},{} applying \\spad{f} to each value.") (((|UniversalSegment| |#2|) (|Mapping| |#2| |#1|) (|UniversalSegment| |#1|)) "\\spad{map(f,seg)} returns the new segment obtained by applying \\spad{f} to the endpoints of \\spad{seg}."))) 
 NIL 
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 ((|constructor| (NIL "This domain represents univariate polynomials in some symbol over arbitrary (not necessarily commutative) coefficient rings. The representation is sparse in the sense that only non-zero terms are represented.")) (|fmecg| (($ $ (|NonNegativeInteger|) |#2| $) "\\spad{fmecg(p1,e,r,p2)} finds \\spad{X} : \\spad{p1} - \\spad{r} * X**e * \\spad{p2}"))) 
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-((|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-561)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-381)))) (|HasCategory| (-1086) (LIST (QUOTE -890) (QUOTE (-381))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -890) (QUOTE (-550)))) (|HasCategory| (-1086) (LIST (QUOTE -890) (QUOTE (-550))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381))))) (|HasCategory| (-1086) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-381)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550))))) (|HasCategory| (-1086) (LIST (QUOTE -617) (LIST (QUOTE -894) (QUOTE (-550)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| (-1086) (LIST (QUOTE -617) (QUOTE (-539))))) (|HasCategory| |#2| (LIST (QUOTE -642) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (QUOTE (-550)))) (-3962 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| |#2| (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (-3962 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-914)))) (-3962 (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-914)))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-1155))) (|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasAttribute| |#2| (QUOTE -4425)) (|HasCategory| |#2| (QUOTE (-456))) (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (-3962 (-12 (|HasCategory| |#2| (QUOTE (-914))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
-(-1240 |x| R |y| S) 
+(((-4436 "*") |has| |#2| (-173)) (-4427 |has| |#2| (-562)) (-4430 |has| |#2| (-367)) (-4432 |has| |#2| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-173))) (-3969 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-382)))) (|HasCategory| (-1088) (LIST (QUOTE -892) (QUOTE (-382))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -892) (QUOTE (-551)))) (|HasCategory| (-1088) (LIST (QUOTE -892) (QUOTE (-551))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382))))) (|HasCategory| (-1088) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-382)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551))))) (|HasCategory| (-1088) (LIST (QUOTE -619) (LIST (QUOTE -896) (QUOTE (-551)))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| (-1088) (LIST (QUOTE -619) (QUOTE (-540))))) (|HasCategory| |#2| (LIST (QUOTE -644) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-147))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (QUOTE (-551)))) (-3969 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| |#2| (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (-3969 (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-3969 (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-3969 (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-1157))) (|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasCategory| |#2| (QUOTE (-234))) (|HasAttribute| |#2| (QUOTE -4432)) (|HasCategory| |#2| (QUOTE (-457))) (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (-3969 (-12 (|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| $ (QUOTE (-145)))) (|HasCategory| |#2| (QUOTE (-145))))) 
+(-1242 |x| R |y| S) 
 ((|constructor| (NIL "This package lifts a mapping from coefficient rings \\spad{R} to \\spad{S} to a mapping from \\spadtype{UnivariatePolynomial}(\\spad{x},{}\\spad{R}) to \\spadtype{UnivariatePolynomial}(\\spad{y},{}\\spad{S}). Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.")) (|map| (((|UnivariatePolynomial| |#3| |#4|) (|Mapping| |#4| |#2|) (|UnivariatePolynomial| |#1| |#2|)) "\\spad{map(func, poly)} creates a new polynomial by applying \\spad{func} to every non-zero coefficient of the polynomial poly."))) 
 NIL 
 NIL 
-(-1241 R Q UP) 
+(-1243 R Q UP) 
 ((|constructor| (NIL "UnivariatePolynomialCommonDenominator provides functions to compute the common denominator of the coefficients of univariate polynomials over the quotient field of a \\spad{gcd} domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#3|) "\\spad{splitDenominator(q)} returns \\spad{[p, d]} such that \\spad{q = p/d} and \\spad{d} is a common denominator for the coefficients of \\spad{q}.")) (|clearDenominator| ((|#3| |#3|) "\\spad{clearDenominator(q)} returns \\spad{p} such that \\spad{q = p/d} where \\spad{d} is a common denominator for the coefficients of \\spad{q}.")) (|commonDenominator| ((|#1| |#3|) "\\spad{commonDenominator(q)} returns a common denominator \\spad{d} for the coefficients of \\spad{q}."))) 
 NIL 
 NIL 
-(-1242 R UP) 
+(-1244 R UP) 
 ((|constructor| (NIL "UnivariatePolynomialDecompositionPackage implements functional decomposition of univariate polynomial with coefficients in an \\spad{IntegralDomain} of \\spad{CharacteristicZero}.")) (|monicCompleteDecompose| (((|List| |#2|) |#2|) "\\spad{monicCompleteDecompose(f)} returns a list of factors of \\spad{f} for the functional decomposition ([ \\spad{f1},{} ...,{} \\spad{fn} ] means \\spad{f} = \\spad{f1} \\spad{o} ... \\spad{o} \\spad{fn}).")) (|monicDecomposeIfCan| (((|Union| (|Record| (|:| |left| |#2|) (|:| |right| |#2|)) "failed") |#2|) "\\spad{monicDecomposeIfCan(f)} returns a functional decomposition of the monic polynomial \\spad{f} of \"failed\" if it has not found any.")) (|leftFactorIfCan| (((|Union| |#2| "failed") |#2| |#2|) "\\spad{leftFactorIfCan(f,h)} returns the left factor (\\spad{g} in \\spad{f} = \\spad{g} \\spad{o} \\spad{h}) of the functional decomposition of the polynomial \\spad{f} with given \\spad{h} or \\spad{\"failed\"} if \\spad{g} does not exist.")) (|rightFactorIfCan| (((|Union| |#2| "failed") |#2| (|NonNegativeInteger|) |#1|) "\\spad{rightFactorIfCan(f,d,c)} returns a candidate to be the right factor (\\spad{h} in \\spad{f} = \\spad{g} \\spad{o} \\spad{h}) of degree \\spad{d} with leading coefficient \\spad{c} of a functional decomposition of the polynomial \\spad{f} or \\spad{\"failed\"} if no such candidate.")) (|monicRightFactorIfCan| (((|Union| |#2| "failed") |#2| (|NonNegativeInteger|)) "\\spad{monicRightFactorIfCan(f,d)} returns a candidate to be the monic right factor (\\spad{h} in \\spad{f} = \\spad{g} \\spad{o} \\spad{h}) of degree \\spad{d} of a functional decomposition of the polynomial \\spad{f} or \\spad{\"failed\"} if no such candidate."))) 
 NIL 
 NIL 
-(-1243 R UP) 
+(-1245 R UP) 
 ((|constructor| (NIL "UnivariatePolynomialDivisionPackage provides a division for non monic univarite polynomials with coefficients in an \\spad{IntegralDomain}.")) (|divideIfCan| (((|Union| (|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) "failed") |#2| |#2|) "\\spad{divideIfCan(f,g)} returns quotient and remainder of the division of \\spad{f} by \\spad{g} or \"failed\" if it has not succeeded."))) 
 NIL 
 NIL 
-(-1244 R U) 
+(-1246 R U) 
 ((|constructor| (NIL "This package implements Karatsuba\\spad{'s} trick for multiplying (large) univariate polynomials. It could be improved with a version doing the work on place and also with a special case for squares. We've done this in Basicmath,{} but we believe that this out of the scope of AXIOM.")) (|karatsuba| ((|#2| |#2| |#2| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{karatsuba(a,b,l,k)} returns \\spad{a*b} by applying Karatsuba\\spad{'s} trick provided that both \\spad{a} and \\spad{b} have at least \\spad{l} terms and \\spad{k > 0} holds and by calling \\spad{noKaratsuba} otherwise. The other multiplications are performed by recursive calls with the same third argument and \\spad{k-1} as fourth argument.")) (|karatsubaOnce| ((|#2| |#2| |#2|) "\\spad{karatsuba(a,b)} returns \\spad{a*b} by applying Karatsuba\\spad{'s} trick once. The other multiplications are performed by calling \\spad{*} from \\spad{U}.")) (|noKaratsuba| ((|#2| |#2| |#2|) "\\spad{noKaratsuba(a,b)} returns \\spad{a*b} without using Karatsuba\\spad{'s} trick at all."))) 
 NIL 
 NIL 
-(-1245 S R) 
+(-1247 S R) 
 ((|constructor| (NIL "The category of univariate polynomials over a ring \\spad{R}. No particular model is assumed - implementations can be either sparse or dense.")) (|integrate| (($ $) "\\spad{integrate(p)} integrates the univariate polynomial \\spad{p} with respect to its distinguished variable.")) (|additiveValuation| ((|attribute|) "euclideanSize(a*b) = euclideanSize(a) + euclideanSize(\\spad{b})")) (|separate| (((|Record| (|:| |primePart| $) (|:| |commonPart| $)) $ $) "\\spad{separate(p, q)} returns \\spad{[a, b]} such that polynomial \\spad{p = a b} and \\spad{a} is relatively prime to \\spad{q}.")) (|pseudoDivide| (((|Record| (|:| |coef| |#2|) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{pseudoDivide(p,q)} returns \\spad{[c, q, r]},{} when \\spad{p' := p*lc(q)**(deg p - deg q + 1) = c * p} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|pseudoQuotient| (($ $ $) "\\spad{pseudoQuotient(p,q)} returns \\spad{r},{} the quotient when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|composite| (((|Union| (|Fraction| $) "failed") (|Fraction| $) $) "\\spad{composite(f, q)} returns \\spad{h} if \\spad{f} = \\spad{h}(\\spad{q}),{} and \"failed\" is no such \\spad{h} exists.") (((|Union| $ "failed") $ $) "\\spad{composite(p, q)} returns \\spad{h} if \\spad{p = h(q)},{} and \"failed\" no such \\spad{h} exists.")) (|subResultantGcd| (($ $ $) "\\spad{subResultantGcd(p,q)} computes the \\spad{gcd} of the polynomials \\spad{p} and \\spad{q} using the SubResultant \\spad{GCD} algorithm.")) (|order| (((|NonNegativeInteger|) $ $) "\\spad{order(p, q)} returns the largest \\spad{n} such that \\spad{q**n} divides polynomial \\spad{p} \\spadignore{i.e.} the order of \\spad{p(x)} at \\spad{q(x)=0}.")) (|elt| ((|#2| (|Fraction| $) |#2|) "\\spad{elt(a,r)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by the constant \\spad{r}.") (((|Fraction| $) (|Fraction| $) (|Fraction| $)) "\\spad{elt(a,b)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by \\spad{b}.")) (|resultant| ((|#2| $ $) "\\spad{resultant(p,q)} returns the resultant of the polynomials \\spad{p} and \\spad{q}.")) (|discriminant| ((|#2| $) "\\spad{discriminant(p)} returns the discriminant of the polynomial \\spad{p}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|) $) "\\spad{differentiate(p, d, x')} extends the \\spad{R}-derivation \\spad{d} to an extension \\spad{D} in \\spad{R[x]} where \\spad{Dx} is given by \\spad{x'},{} and returns \\spad{Dp}.")) (|pseudoRemainder| (($ $ $) "\\spad{pseudoRemainder(p,q)} = \\spad{r},{} for polynomials \\spad{p} and \\spad{q},{} returns the remainder when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|shiftLeft| (($ $ (|NonNegativeInteger|)) "\\spad{shiftLeft(p,n)} returns \\spad{p * monomial(1,n)}")) (|shiftRight| (($ $ (|NonNegativeInteger|)) "\\spad{shiftRight(p,n)} returns \\spad{monicDivide(p,monomial(1,n)).quotient}")) (|karatsubaDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ (|NonNegativeInteger|)) "\\spad{karatsubaDivide(p,n)} returns the same as \\spad{monicDivide(p,monomial(1,n))}")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicDivide(p,q)} divide the polynomial \\spad{p} by the monic polynomial \\spad{q},{} returning the pair \\spad{[quotient, remainder]}. Error: if \\spad{q} isn\\spad{'t} monic.")) (|divideExponents| (((|Union| $ "failed") $ (|NonNegativeInteger|)) "\\spad{divideExponents(p,n)} returns a new polynomial resulting from dividing all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n},{} or \"failed\" if some exponent is not exactly divisible by \\spad{n}.")) (|multiplyExponents| (($ $ (|NonNegativeInteger|)) "\\spad{multiplyExponents(p,n)} returns a new polynomial resulting from multiplying all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n}.")) (|unmakeSUP| (($ (|SparseUnivariatePolynomial| |#2|)) "\\spad{unmakeSUP(sup)} converts \\spad{sup} of type \\spadtype{SparseUnivariatePolynomial(R)} to be a member of the given type. Note: converse of makeSUP.")) (|makeSUP| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{makeSUP(p)} converts the polynomial \\spad{p} to be of type SparseUnivariatePolynomial over the same coefficients.")) (|vectorise| (((|Vector| |#2|) $ (|NonNegativeInteger|)) "\\spad{vectorise(p, n)} returns \\spad{[a0,...,a(n-1)]} where \\spad{p = a0 + a1*x + ... + a(n-1)*x**(n-1)} + higher order terms. The degree of polynomial \\spad{p} can be different from \\spad{n-1}."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-366))) (|HasCategory| |#2| (QUOTE (-456))) (|HasCategory| |#2| (QUOTE (-561))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-1155)))) 
-(-1246 R) 
+((|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (QUOTE (-367))) (|HasCategory| |#2| (QUOTE (-457))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (QUOTE (-1157)))) 
+(-1248 R) 
 ((|constructor| (NIL "The category of univariate polynomials over a ring \\spad{R}. No particular model is assumed - implementations can be either sparse or dense.")) (|integrate| (($ $) "\\spad{integrate(p)} integrates the univariate polynomial \\spad{p} with respect to its distinguished variable.")) (|additiveValuation| ((|attribute|) "euclideanSize(a*b) = euclideanSize(a) + euclideanSize(\\spad{b})")) (|separate| (((|Record| (|:| |primePart| $) (|:| |commonPart| $)) $ $) "\\spad{separate(p, q)} returns \\spad{[a, b]} such that polynomial \\spad{p = a b} and \\spad{a} is relatively prime to \\spad{q}.")) (|pseudoDivide| (((|Record| (|:| |coef| |#1|) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{pseudoDivide(p,q)} returns \\spad{[c, q, r]},{} when \\spad{p' := p*lc(q)**(deg p - deg q + 1) = c * p} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|pseudoQuotient| (($ $ $) "\\spad{pseudoQuotient(p,q)} returns \\spad{r},{} the quotient when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|composite| (((|Union| (|Fraction| $) "failed") (|Fraction| $) $) "\\spad{composite(f, q)} returns \\spad{h} if \\spad{f} = \\spad{h}(\\spad{q}),{} and \"failed\" is no such \\spad{h} exists.") (((|Union| $ "failed") $ $) "\\spad{composite(p, q)} returns \\spad{h} if \\spad{p = h(q)},{} and \"failed\" no such \\spad{h} exists.")) (|subResultantGcd| (($ $ $) "\\spad{subResultantGcd(p,q)} computes the \\spad{gcd} of the polynomials \\spad{p} and \\spad{q} using the SubResultant \\spad{GCD} algorithm.")) (|order| (((|NonNegativeInteger|) $ $) "\\spad{order(p, q)} returns the largest \\spad{n} such that \\spad{q**n} divides polynomial \\spad{p} \\spadignore{i.e.} the order of \\spad{p(x)} at \\spad{q(x)=0}.")) (|elt| ((|#1| (|Fraction| $) |#1|) "\\spad{elt(a,r)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by the constant \\spad{r}.") (((|Fraction| $) (|Fraction| $) (|Fraction| $)) "\\spad{elt(a,b)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by \\spad{b}.")) (|resultant| ((|#1| $ $) "\\spad{resultant(p,q)} returns the resultant of the polynomials \\spad{p} and \\spad{q}.")) (|discriminant| ((|#1| $) "\\spad{discriminant(p)} returns the discriminant of the polynomial \\spad{p}.")) (|differentiate| (($ $ (|Mapping| |#1| |#1|) $) "\\spad{differentiate(p, d, x')} extends the \\spad{R}-derivation \\spad{d} to an extension \\spad{D} in \\spad{R[x]} where \\spad{Dx} is given by \\spad{x'},{} and returns \\spad{Dp}.")) (|pseudoRemainder| (($ $ $) "\\spad{pseudoRemainder(p,q)} = \\spad{r},{} for polynomials \\spad{p} and \\spad{q},{} returns the remainder when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|shiftLeft| (($ $ (|NonNegativeInteger|)) "\\spad{shiftLeft(p,n)} returns \\spad{p * monomial(1,n)}")) (|shiftRight| (($ $ (|NonNegativeInteger|)) "\\spad{shiftRight(p,n)} returns \\spad{monicDivide(p,monomial(1,n)).quotient}")) (|karatsubaDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ (|NonNegativeInteger|)) "\\spad{karatsubaDivide(p,n)} returns the same as \\spad{monicDivide(p,monomial(1,n))}")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicDivide(p,q)} divide the polynomial \\spad{p} by the monic polynomial \\spad{q},{} returning the pair \\spad{[quotient, remainder]}. Error: if \\spad{q} isn\\spad{'t} monic.")) (|divideExponents| (((|Union| $ "failed") $ (|NonNegativeInteger|)) "\\spad{divideExponents(p,n)} returns a new polynomial resulting from dividing all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n},{} or \"failed\" if some exponent is not exactly divisible by \\spad{n}.")) (|multiplyExponents| (($ $ (|NonNegativeInteger|)) "\\spad{multiplyExponents(p,n)} returns a new polynomial resulting from multiplying all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n}.")) (|unmakeSUP| (($ (|SparseUnivariatePolynomial| |#1|)) "\\spad{unmakeSUP(sup)} converts \\spad{sup} of type \\spadtype{SparseUnivariatePolynomial(R)} to be a member of the given type. Note: converse of makeSUP.")) (|makeSUP| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{makeSUP(p)} converts the polynomial \\spad{p} to be of type SparseUnivariatePolynomial over the same coefficients.")) (|vectorise| (((|Vector| |#1|) $ (|NonNegativeInteger|)) "\\spad{vectorise(p, n)} returns \\spad{[a0,...,a(n-1)]} where \\spad{p = a0 + a1*x + ... + a(n-1)*x**(n-1)} + higher order terms. The degree of polynomial \\spad{p} can be different from \\spad{n-1}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4423 |has| |#1| (-366)) (-4425 |has| |#1| (-6 -4425)) (-4422 . T) (-4421 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4430 |has| |#1| (-367)) (-4432 |has| |#1| (-6 -4432)) (-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
-(-1247 R PR S PS) 
+(-1249 R PR S PS) 
 ((|constructor| (NIL "Mapping from polynomials over \\spad{R} to polynomials over \\spad{S} given a map from \\spad{R} to \\spad{S} assumed to send zero to zero.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f, p)} takes a function \\spad{f} from \\spad{R} to \\spad{S},{} and applies it to each (non-zero) coefficient of a polynomial \\spad{p} over \\spad{R},{} getting a new polynomial over \\spad{S}. Note: since the map is not applied to zero elements,{} it may map zero to zero."))) 
 NIL 
 NIL 
-(-1248 S |Coef| |Expon|) 
+(-1250 S |Coef| |Expon|) 
 ((|constructor| (NIL "\\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.")) (|eval| (((|Stream| |#2|) $ |#2|) "\\spad{eval(f,a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.")) (|extend| (($ $ |#3|) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree \\spad{<=} \\spad{n} to be computed.")) (|approximate| ((|#2| $ |#3|) "\\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.")) (|truncate| (($ $ |#3| |#3|) "\\spad{truncate(f,k1,k2)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (($ $ |#3|) "\\spad{truncate(f,k)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| ((|#3| $ |#3|) "\\spad{order(f,n) = min(m,n)},{} where \\spad{m} is the degree of the lowest order non-zero term in \\spad{f}.") ((|#3| $) "\\spad{order(f)} is the degree of the lowest order non-zero term in \\spad{f}. This will result in an infinite loop if \\spad{f} has no non-zero terms.")) (|multiplyExponents| (($ $ (|PositiveInteger|)) "\\spad{multiplyExponents(f,n)} multiplies all exponents of the power series \\spad{f} by the positive integer \\spad{n}.")) (|center| ((|#2| $) "\\spad{center(f)} returns the point about which the series \\spad{f} is expanded.")) (|variable| (((|Symbol|) $) "\\spad{variable(f)} returns the (unique) power series variable of the power series \\spad{f}.")) (|elt| ((|#2| $ |#3|) "\\spad{elt(f(x),r)} returns the coefficient of the term of degree \\spad{r} in \\spad{f(x)}. This is the same as the function \\spadfun{coefficient}.")) (|terms| (((|Stream| (|Record| (|:| |k| |#3|) (|:| |c| |#2|))) $) "\\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#2| (LIST (QUOTE *) (LIST (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#2|)))) (|HasCategory| |#3| (QUOTE (-1116))) (|HasSignature| |#2| (LIST (QUOTE **) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (|devaluate| |#3|)))) (|HasSignature| |#2| (LIST (QUOTE -4380) (LIST (|devaluate| |#2|) (QUOTE (-1181)))))) 
-(-1249 |Coef| |Expon|) 
+((|HasCategory| |#2| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#2| (LIST (QUOTE *) (LIST (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#2|)))) (|HasCategory| |#3| (QUOTE (-1118))) (|HasSignature| |#2| (LIST (QUOTE **) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (|devaluate| |#3|)))) (|HasSignature| |#2| (LIST (QUOTE -4387) (LIST (|devaluate| |#2|) (QUOTE (-1183)))))) 
+(-1251 |Coef| |Expon|) 
 ((|constructor| (NIL "\\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.")) (|eval| (((|Stream| |#1|) $ |#1|) "\\spad{eval(f,a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.")) (|extend| (($ $ |#2|) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree \\spad{<=} \\spad{n} to be computed.")) (|approximate| ((|#1| $ |#2|) "\\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.")) (|truncate| (($ $ |#2| |#2|) "\\spad{truncate(f,k1,k2)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (($ $ |#2|) "\\spad{truncate(f,k)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| ((|#2| $ |#2|) "\\spad{order(f,n) = min(m,n)},{} where \\spad{m} is the degree of the lowest order non-zero term in \\spad{f}.") ((|#2| $) "\\spad{order(f)} is the degree of the lowest order non-zero term in \\spad{f}. This will result in an infinite loop if \\spad{f} has no non-zero terms.")) (|multiplyExponents| (($ $ (|PositiveInteger|)) "\\spad{multiplyExponents(f,n)} multiplies all exponents of the power series \\spad{f} by the positive integer \\spad{n}.")) (|center| ((|#1| $) "\\spad{center(f)} returns the point about which the series \\spad{f} is expanded.")) (|variable| (((|Symbol|) $) "\\spad{variable(f)} returns the (unique) power series variable of the power series \\spad{f}.")) (|elt| ((|#1| $ |#2|) "\\spad{elt(f(x),r)} returns the coefficient of the term of degree \\spad{r} in \\spad{f(x)}. This is the same as the function \\spadfun{coefficient}.")) (|terms| (((|Stream| (|Record| (|:| |k| |#2|) (|:| |c| |#1|))) $) "\\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1250 RC P) 
+(-1252 RC P) 
 ((|constructor| (NIL "This package provides for square-free decomposition of univariate polynomials over arbitrary rings,{} \\spadignore{i.e.} a partial factorization such that each factor is a product of irreducibles with multiplicity one and the factors are pairwise relatively prime. If the ring has characteristic zero,{} the result is guaranteed to satisfy this condition. If the ring is an infinite ring of finite characteristic,{} then it may not be possible to decide when polynomials contain factors which are \\spad{p}th powers. In this case,{} the flag associated with that polynomial is set to \"nil\" (meaning that that polynomials are not guaranteed to be square-free).")) (|BumInSepFFE| (((|Record| (|:| |flg| (|Union| #1="nil" #2="sqfr" #3="irred" #4="prime")) (|:| |fctr| |#2|) (|:| |xpnt| (|Integer|))) (|Record| (|:| |flg| (|Union| #1# #2# #3# #4#)) (|:| |fctr| |#2|) (|:| |xpnt| (|Integer|)))) "\\spad{BumInSepFFE(f)} is a local function,{} exported only because it has multiple conditional definitions.")) (|squareFreePart| ((|#2| |#2|) "\\spad{squareFreePart(p)} returns a polynomial which has the same irreducible factors as the univariate polynomial \\spad{p},{} but each factor has multiplicity one.")) (|squareFree| (((|Factored| |#2|) |#2|) "\\spad{squareFree(p)} computes the square-free factorization of the univariate polynomial \\spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime.")) (|gcd| (($ $ $) "\\spad{gcd(p,q)} computes the greatest-common-divisor of \\spad{p} and \\spad{q}."))) 
 NIL 
 NIL 
-(-1251 |Coef| |var| |cen|) 
+(-1253 |Coef| |var| |cen|) 
 ((|constructor| (NIL "Dense Puiseux series in one variable \\indented{2}{\\spadtype{UnivariatePuiseuxSeries} is a domain representing Puiseux} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{UnivariatePuiseuxSeries(Integer,x,3)} represents Puiseux series in} \\indented{2}{\\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} returns the derivative of \\spad{f(x)} with respect to \\spad{x}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|)))) (|HasCategory| (-411 (-550)) (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-964))) (|HasCategory| |#1| (QUOTE (-1206))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4246) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1181))))) (|HasSignature| |#1| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#1|))))))) 
-(-1252 |Coef1| |Coef2| |var1| |var2| |cen1| |cen2|) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-173))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551))) (|devaluate| |#1|)))) (|HasCategory| (-412 (-551)) (QUOTE (-1118))) (|HasCategory| |#1| (QUOTE (-367))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-3969 (|HasCategory| |#1| (QUOTE (-367))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasSignature| |#1| (LIST (QUOTE -4387) (LIST (|devaluate| |#1|) (QUOTE (-1183)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -412) (QUOTE (-551)))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1208))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasSignature| |#1| (LIST (QUOTE -4253) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1183))))) (|HasSignature| |#1| (LIST (QUOTE -3494) (LIST (LIST (QUOTE -646) (QUOTE (-1183))) (|devaluate| |#1|))))))) 
+(-1254 |Coef1| |Coef2| |var1| |var2| |cen1| |cen2|) 
 ((|constructor| (NIL "Mapping package for univariate Puiseux series. This package allows one to apply a function to the coefficients of a univariate Puiseux series.")) (|map| (((|UnivariatePuiseuxSeries| |#2| |#4| |#6|) (|Mapping| |#2| |#1|) (|UnivariatePuiseuxSeries| |#1| |#3| |#5|)) "\\spad{map(f,g(x))} applies the map \\spad{f} to the coefficients of the Puiseux series \\spad{g(x)}."))) 
 NIL 
 NIL 
-(-1253 |Coef|) 
+(-1255 |Coef|) 
 ((|constructor| (NIL "\\spadtype{UnivariatePuiseuxSeriesCategory} is the category of Puiseux series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),var)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{var}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 1. We may integrate a series when we can divide coefficients by rational numbers.")) (|multiplyExponents| (($ $ (|Fraction| (|Integer|))) "\\spad{multiplyExponents(f,r)} multiplies all exponents of the power series \\spad{f} by the positive rational number \\spad{r}.")) (|series| (($ (|NonNegativeInteger|) (|Stream| (|Record| (|:| |k| (|Fraction| (|Integer|))) (|:| |c| |#1|)))) "\\spad{series(n,st)} creates a series from a common denomiator and a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents and \\spad{n} should be a common denominator for the exponents in the stream of terms."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1254 S |Coef| ULS) 
+(-1256 S |Coef| ULS) 
 ((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x^r)}.")) (|laurentIfCan| (((|Union| |#3| "failed") $) "\\spad{laurentIfCan(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. If this is not possible,{} \"failed\" is returned.")) (|laurent| ((|#3| $) "\\spad{laurent(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. Error: if this is not possible.")) (|degree| (((|Fraction| (|Integer|)) $) "\\spad{degree(f(x))} returns the degree of the leading term of the Puiseux series \\spad{f(x)},{} which may have zero as a coefficient.")) (|laurentRep| ((|#3| $) "\\spad{laurentRep(f(x))} returns \\spad{g(x)} where the Puiseux series \\spad{f(x) = g(x^r)} is represented by \\spad{[r,g(x)]}.")) (|rationalPower| (((|Fraction| (|Integer|)) $) "\\spad{rationalPower(f(x))} returns \\spad{r} where the Puiseux series \\spad{f(x) = g(x^r)}.")) (|puiseux| (($ (|Fraction| (|Integer|)) |#3|) "\\spad{puiseux(r,f(x))} returns \\spad{f(x^r)}."))) 
 NIL 
 NIL 
-(-1255 |Coef| ULS) 
+(-1257 |Coef| ULS) 
 ((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x^r)}.")) (|laurentIfCan| (((|Union| |#2| "failed") $) "\\spad{laurentIfCan(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. If this is not possible,{} \"failed\" is returned.")) (|laurent| ((|#2| $) "\\spad{laurent(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. Error: if this is not possible.")) (|degree| (((|Fraction| (|Integer|)) $) "\\spad{degree(f(x))} returns the degree of the leading term of the Puiseux series \\spad{f(x)},{} which may have zero as a coefficient.")) (|laurentRep| ((|#2| $) "\\spad{laurentRep(f(x))} returns \\spad{g(x)} where the Puiseux series \\spad{f(x) = g(x^r)} is represented by \\spad{[r,g(x)]}.")) (|rationalPower| (((|Fraction| (|Integer|)) $) "\\spad{rationalPower(f(x))} returns \\spad{r} where the Puiseux series \\spad{f(x) = g(x^r)}.")) (|puiseux| (($ (|Fraction| (|Integer|)) |#2|) "\\spad{puiseux(r,f(x))} returns \\spad{f(x^r)}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1256 |Coef| ULS) 
+(-1258 |Coef| ULS) 
 ((|constructor| (NIL "This package enables one to construct a univariate Puiseux series domain from a univariate Laurent series domain. Univariate Puiseux series are represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x^r)}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4425 |has| |#1| (-366)) (-4419 |has| |#1| (-366)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#1| (QUOTE (-173))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550))) (|devaluate| |#1|)))) (|HasCategory| (-411 (-550)) (QUOTE (-1116))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-3962 (|HasCategory| |#1| (QUOTE (-366))) (|HasCategory| |#1| (QUOTE (-561)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -411) (QUOTE (-550)))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-964))) (|HasCategory| |#1| (QUOTE (-1206))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4246) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1181))))) (|HasSignature| |#1| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#1|)))))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550)))))) 
-(-1257 R FE |var| |cen|) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4432 |has| |#1| (-367)) (-4426 |has| |#1| (-367)) (-4428 . T) (-4429 . T) (-4431 . T)) 
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+(-1259 R FE |var| |cen|) 
 ((|constructor| (NIL "UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent functions with essential singularities. Objects in this domain are sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series. Thus,{} the elements of this domain are sums of expressions of the form \\spad{g(x) * exp(f(x))},{} where \\spad{g}(\\spad{x}) is a univariate Puiseux series and \\spad{f}(\\spad{x}) is a univariate Puiseux series with no terms of non-negative degree.")) (|dominantTerm| (((|Union| (|Record| (|:| |%term| (|Record| (|:| |%coef| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expon| (|ExponentialOfUnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expTerms| (|List| (|Record| (|:| |k| (|Fraction| (|Integer|))) (|:| |c| |#2|)))))) (|:| |%type| (|String|))) "failed") $) "\\spad{dominantTerm(f(var))} returns the term that dominates the limiting behavior of \\spad{f(var)} as \\spad{var -> cen+} together with a \\spadtype{String} which briefly describes that behavior. The value of the \\spadtype{String} will be \\spad{\"zero\"} (resp. \\spad{\"infinity\"}) if the term tends to zero (resp. infinity) exponentially and will \\spad{\"series\"} if the term is a Puiseux series.")) (|limitPlus| (((|Union| (|OrderedCompletion| |#2|) "failed") $) "\\spad{limitPlus(f(var))} returns \\spad{limit(var -> cen+,f(var))}."))) 
-(((-4429 "*") |has| (-1251 |#2| |#3| |#4|) (-173)) (-4420 |has| (-1251 |#2| |#3| |#4|) (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| (-1251 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-1251 |#2| |#3| |#4|) (QUOTE (-145))) (|HasCategory| (-1251 |#2| |#3| |#4|) (QUOTE (-147))) (|HasCategory| (-1251 |#2| |#3| |#4|) (QUOTE (-173))) (-3962 (|HasCategory| (-1251 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-1251 |#2| |#3| |#4|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550)))))) (|HasCategory| (-1251 |#2| |#3| |#4|) (LIST (QUOTE -1042) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| (-1251 |#2| |#3| |#4|) (LIST (QUOTE -1042) (QUOTE (-550)))) (|HasCategory| (-1251 |#2| |#3| |#4|) (QUOTE (-366))) (|HasCategory| (-1251 |#2| |#3| |#4|) (QUOTE (-456))) (|HasCategory| (-1251 |#2| |#3| |#4|) (QUOTE (-561)))) 
-(-1258 A S) 
+(((-4436 "*") |has| (-1253 |#2| |#3| |#4|) (-173)) (-4427 |has| (-1253 |#2| |#3| |#4|) (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| (-1253 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-1253 |#2| |#3| |#4|) (QUOTE (-145))) (|HasCategory| (-1253 |#2| |#3| |#4|) (QUOTE (-147))) (|HasCategory| (-1253 |#2| |#3| |#4|) (QUOTE (-173))) (-3969 (|HasCategory| (-1253 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-1253 |#2| |#3| |#4|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551)))))) (|HasCategory| (-1253 |#2| |#3| |#4|) (LIST (QUOTE -1044) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| (-1253 |#2| |#3| |#4|) (LIST (QUOTE -1044) (QUOTE (-551)))) (|HasCategory| (-1253 |#2| |#3| |#4|) (QUOTE (-367))) (|HasCategory| (-1253 |#2| |#3| |#4|) (QUOTE (-457))) (|HasCategory| (-1253 |#2| |#3| |#4|) (QUOTE (-562)))) 
+(-1260 A S) 
 ((|constructor| (NIL "A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \\spadfun{value} of the list designating the head,{} or \\spadfun{first},{} of the list,{} and the child designating the tail,{} or \\spadfun{rest},{} of the list. A node with no child then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.")) (|split!| (($ $ (|Integer|)) "\\spad{split!(u,n)} splits \\spad{u} into two aggregates: \\axiom{\\spad{v} = rest(\\spad{u},{}\\spad{n})} and \\axiom{\\spad{w} = first(\\spad{u},{}\\spad{n})},{} returning \\axiom{\\spad{v}}. Note: afterwards \\axiom{rest(\\spad{u},{}\\spad{n})} returns \\axiom{empty()}.")) (|setlast!| ((|#2| $ |#2|) "\\spad{setlast!(u,x)} destructively changes the last element of \\spad{u} to \\spad{x}.")) (|setrest!| (($ $ $) "\\spad{setrest!(u,v)} destructively changes the rest of \\spad{u} to \\spad{v}.")) (|setelt| ((|#2| $ "last" |#2|) "\\spad{setelt(u,\"last\",x)} (also written: \\axiom{\\spad{u}.last \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setlast!(\\spad{u},{}\\spad{v})}.") (($ $ "rest" $) "\\spad{setelt(u,\"rest\",v)} (also written: \\axiom{\\spad{u}.rest \\spad{:=} \\spad{v}}) is equivalent to \\axiom{setrest!(\\spad{u},{}\\spad{v})}.") ((|#2| $ "first" |#2|) "\\spad{setelt(u,\"first\",x)} (also written: \\axiom{\\spad{u}.first \\spad{:=} \\spad{x}}) is equivalent to \\axiom{setfirst!(\\spad{u},{}\\spad{x})}.")) (|setfirst!| ((|#2| $ |#2|) "\\spad{setfirst!(u,x)} destructively changes the first element of a to \\spad{x}.")) (|cycleSplit!| (($ $) "\\spad{cycleSplit!(u)} splits the aggregate by dropping off the cycle. The value returned is the cycle entry,{} or nil if none exists. For example,{} if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} is the cyclic list where \\spad{v} is the head of the cycle,{} \\axiom{cycleSplit!(\\spad{w})} will drop \\spad{v} off \\spad{w} thus destructively changing \\spad{w} to \\spad{u},{} and returning \\spad{v}.")) (|concat!| (($ $ |#2|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}. Note: \\axiom{concat!(a,{}\\spad{x}) = setlast!(a,{}[\\spad{x}])}.") (($ $ $) "\\spad{concat!(u,v)} destructively concatenates \\spad{v} to the end of \\spad{u}. Note: \\axiom{concat!(\\spad{u},{}\\spad{v}) = setlast!(\\spad{u},{}\\spad{v})}.")) (|cycleTail| (($ $) "\\spad{cycleTail(u)} returns the last node in the cycle,{} or empty if none exists.")) (|cycleLength| (((|NonNegativeInteger|) $) "\\spad{cycleLength(u)} returns the length of a top-level cycle contained in aggregate \\spad{u},{} or 0 is \\spad{u} has no such cycle.")) (|cycleEntry| (($ $) "\\spad{cycleEntry(u)} returns the head of a top-level cycle contained in aggregate \\spad{u},{} or \\axiom{empty()} if none exists.")) (|third| ((|#2| $) "\\spad{third(u)} returns the third element of \\spad{u}. Note: \\axiom{third(\\spad{u}) = first(rest(rest(\\spad{u})))}.")) (|second| ((|#2| $) "\\spad{second(u)} returns the second element of \\spad{u}. Note: \\axiom{second(\\spad{u}) = first(rest(\\spad{u}))}.")) (|tail| (($ $) "\\spad{tail(u)} returns the last node of \\spad{u}. Note: if \\spad{u} is \\axiom{shallowlyMutable},{} \\axiom{setrest(tail(\\spad{u}),{}\\spad{v}) = concat(\\spad{u},{}\\spad{v})}.")) (|last| (($ $ (|NonNegativeInteger|)) "\\spad{last(u,n)} returns a copy of the last \\spad{n} (\\axiom{\\spad{n} \\spad{>=} 0}) nodes of \\spad{u}. Note: \\axiom{last(\\spad{u},{}\\spad{n})} is a list of \\spad{n} elements.") ((|#2| $) "\\spad{last(u)} resturn the last element of \\spad{u}. Note: for lists,{} \\axiom{last(\\spad{u}) = \\spad{u} . (maxIndex \\spad{u}) = \\spad{u} . (\\# \\spad{u} - 1)}.")) (|rest| (($ $ (|NonNegativeInteger|)) "\\spad{rest(u,n)} returns the \\axiom{\\spad{n}}th (\\spad{n} \\spad{>=} 0) node of \\spad{u}. Note: \\axiom{rest(\\spad{u},{}0) = \\spad{u}}.") (($ $) "\\spad{rest(u)} returns an aggregate consisting of all but the first element of \\spad{u} (equivalently,{} the next node of \\spad{u}).")) (|elt| ((|#2| $ "last") "\\spad{elt(u,\"last\")} (also written: \\axiom{\\spad{u} . last}) is equivalent to last \\spad{u}.") (($ $ "rest") "\\spad{elt(\\%,\"rest\")} (also written: \\axiom{\\spad{u}.rest}) is equivalent to \\axiom{rest \\spad{u}}.") ((|#2| $ "first") "\\spad{elt(u,\"first\")} (also written: \\axiom{\\spad{u} . first}) is equivalent to first \\spad{u}.")) (|first| (($ $ (|NonNegativeInteger|)) "\\spad{first(u,n)} returns a copy of the first \\spad{n} (\\axiom{\\spad{n} \\spad{>=} 0}) elements of \\spad{u}.") ((|#2| $) "\\spad{first(u)} returns the first element of \\spad{u} (equivalently,{} the value at the current node).")) (|concat| (($ |#2| $) "\\spad{concat(x,u)} returns aggregate consisting of \\spad{x} followed by the elements of \\spad{u}. Note: if \\axiom{\\spad{v} = concat(\\spad{x},{}\\spad{u})} then \\axiom{\\spad{x} = first \\spad{v}} and \\axiom{\\spad{u} = rest \\spad{v}}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate \\spad{w} consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: \\axiom{\\spad{v} = rest(\\spad{w},{}\\#a)}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4428))) 
-(-1259 S) 
+((|HasAttribute| |#1| (QUOTE -4435))) 
+(-1261 S) 
 ((|constructor| (NIL "A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \\spadfun{value} of the list designating the head,{} or \\spadfun{first},{} of the list,{} and the child designating the tail,{} or \\spadfun{rest},{} of the list. A node with no child then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.")) (|split!| (($ $ (|Integer|)) "\\spad{split!(u,n)} splits \\spad{u} into two aggregates: \\axiom{\\spad{v} = rest(\\spad{u},{}\\spad{n})} and \\axiom{\\spad{w} = first(\\spad{u},{}\\spad{n})},{} returning \\axiom{\\spad{v}}. Note: afterwards \\axiom{rest(\\spad{u},{}\\spad{n})} returns \\axiom{empty()}.")) (|setlast!| ((|#1| $ |#1|) "\\spad{setlast!(u,x)} destructively changes the last element of \\spad{u} to \\spad{x}.")) (|setrest!| (($ $ $) "\\spad{setrest!(u,v)} destructively changes the rest of \\spad{u} to \\spad{v}.")) (|setelt| ((|#1| $ "last" |#1|) "\\spad{setelt(u,\"last\",x)} (also written: \\axiom{\\spad{u}.last \\spad{:=} \\spad{b}}) is equivalent to \\axiom{setlast!(\\spad{u},{}\\spad{v})}.") (($ $ "rest" $) "\\spad{setelt(u,\"rest\",v)} (also written: \\axiom{\\spad{u}.rest \\spad{:=} \\spad{v}}) is equivalent to \\axiom{setrest!(\\spad{u},{}\\spad{v})}.") ((|#1| $ "first" |#1|) "\\spad{setelt(u,\"first\",x)} (also written: \\axiom{\\spad{u}.first \\spad{:=} \\spad{x}}) is equivalent to \\axiom{setfirst!(\\spad{u},{}\\spad{x})}.")) (|setfirst!| ((|#1| $ |#1|) "\\spad{setfirst!(u,x)} destructively changes the first element of a to \\spad{x}.")) (|cycleSplit!| (($ $) "\\spad{cycleSplit!(u)} splits the aggregate by dropping off the cycle. The value returned is the cycle entry,{} or nil if none exists. For example,{} if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} is the cyclic list where \\spad{v} is the head of the cycle,{} \\axiom{cycleSplit!(\\spad{w})} will drop \\spad{v} off \\spad{w} thus destructively changing \\spad{w} to \\spad{u},{} and returning \\spad{v}.")) (|concat!| (($ $ |#1|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}. Note: \\axiom{concat!(a,{}\\spad{x}) = setlast!(a,{}[\\spad{x}])}.") (($ $ $) "\\spad{concat!(u,v)} destructively concatenates \\spad{v} to the end of \\spad{u}. Note: \\axiom{concat!(\\spad{u},{}\\spad{v}) = setlast!(\\spad{u},{}\\spad{v})}.")) (|cycleTail| (($ $) "\\spad{cycleTail(u)} returns the last node in the cycle,{} or empty if none exists.")) (|cycleLength| (((|NonNegativeInteger|) $) "\\spad{cycleLength(u)} returns the length of a top-level cycle contained in aggregate \\spad{u},{} or 0 is \\spad{u} has no such cycle.")) (|cycleEntry| (($ $) "\\spad{cycleEntry(u)} returns the head of a top-level cycle contained in aggregate \\spad{u},{} or \\axiom{empty()} if none exists.")) (|third| ((|#1| $) "\\spad{third(u)} returns the third element of \\spad{u}. Note: \\axiom{third(\\spad{u}) = first(rest(rest(\\spad{u})))}.")) (|second| ((|#1| $) "\\spad{second(u)} returns the second element of \\spad{u}. Note: \\axiom{second(\\spad{u}) = first(rest(\\spad{u}))}.")) (|tail| (($ $) "\\spad{tail(u)} returns the last node of \\spad{u}. Note: if \\spad{u} is \\axiom{shallowlyMutable},{} \\axiom{setrest(tail(\\spad{u}),{}\\spad{v}) = concat(\\spad{u},{}\\spad{v})}.")) (|last| (($ $ (|NonNegativeInteger|)) "\\spad{last(u,n)} returns a copy of the last \\spad{n} (\\axiom{\\spad{n} \\spad{>=} 0}) nodes of \\spad{u}. Note: \\axiom{last(\\spad{u},{}\\spad{n})} is a list of \\spad{n} elements.") ((|#1| $) "\\spad{last(u)} resturn the last element of \\spad{u}. Note: for lists,{} \\axiom{last(\\spad{u}) = \\spad{u} . (maxIndex \\spad{u}) = \\spad{u} . (\\# \\spad{u} - 1)}.")) (|rest| (($ $ (|NonNegativeInteger|)) "\\spad{rest(u,n)} returns the \\axiom{\\spad{n}}th (\\spad{n} \\spad{>=} 0) node of \\spad{u}. Note: \\axiom{rest(\\spad{u},{}0) = \\spad{u}}.") (($ $) "\\spad{rest(u)} returns an aggregate consisting of all but the first element of \\spad{u} (equivalently,{} the next node of \\spad{u}).")) (|elt| ((|#1| $ "last") "\\spad{elt(u,\"last\")} (also written: \\axiom{\\spad{u} . last}) is equivalent to last \\spad{u}.") (($ $ "rest") "\\spad{elt(\\%,\"rest\")} (also written: \\axiom{\\spad{u}.rest}) is equivalent to \\axiom{rest \\spad{u}}.") ((|#1| $ "first") "\\spad{elt(u,\"first\")} (also written: \\axiom{\\spad{u} . first}) is equivalent to first \\spad{u}.")) (|first| (($ $ (|NonNegativeInteger|)) "\\spad{first(u,n)} returns a copy of the first \\spad{n} (\\axiom{\\spad{n} \\spad{>=} 0}) elements of \\spad{u}.") ((|#1| $) "\\spad{first(u)} returns the first element of \\spad{u} (equivalently,{} the value at the current node).")) (|concat| (($ |#1| $) "\\spad{concat(x,u)} returns aggregate consisting of \\spad{x} followed by the elements of \\spad{u}. Note: if \\axiom{\\spad{v} = concat(\\spad{x},{}\\spad{u})} then \\axiom{\\spad{x} = first \\spad{v}} and \\axiom{\\spad{u} = rest \\spad{v}}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate \\spad{w} consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: \\axiom{\\spad{v} = rest(\\spad{w},{}\\#a)}."))) 
 NIL 
 NIL 
-(-1260 |Coef| |var| |cen|) 
+(-1262 |Coef| |var| |cen|) 
 ((|constructor| (NIL "Dense Taylor series in one variable \\spadtype{UnivariateTaylorSeries} is a domain representing Taylor series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring,{} the power series variable,{} and the center of the power series expansion. For example,{} \\spadtype{UnivariateTaylorSeries}(Integer,{}\\spad{x},{}3) represents Taylor series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x),x)} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|invmultisect| (($ (|Integer|) (|Integer|) $) "\\spad{invmultisect(a,b,f(x))} substitutes \\spad{x^((a+b)*n)} \\indented{1}{for \\spad{x^n} and multiples by \\spad{x^b}.}")) (|multisect| (($ (|Integer|) (|Integer|) $) "\\spad{multisect(a,b,f(x))} selects the coefficients of \\indented{1}{\\spad{x^((a+b)*n+a)},{} and changes this monomial to \\spad{x^n}.}")) (|revert| (($ $) "\\spad{revert(f(x))} returns a Taylor series \\spad{g(x)} such that \\spad{f(g(x)) = g(f(x)) = x}. Series \\spad{f(x)} should have constant coefficient 0 and invertible 1st order coefficient.")) (|generalLambert| (($ $ (|Integer|) (|Integer|)) "\\spad{generalLambert(f(x),a,d)} returns \\spad{f(x^a) + f(x^(a + d)) + \\indented{1}{f(x^(a + 2 d)) + ... }. \\spad{f(x)} should have zero constant} \\indented{1}{coefficient and \\spad{a} and \\spad{d} should be positive.}")) (|evenlambert| (($ $) "\\spad{evenlambert(f(x))} returns \\spad{f(x^2) + f(x^4) + f(x^6) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n))) = exp(log(evenlambert(f(x))))}.}")) (|oddlambert| (($ $) "\\spad{oddlambert(f(x))} returns \\spad{f(x) + f(x^3) + f(x^5) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n-1)))=exp(log(oddlambert(f(x))))}.}")) (|lambert| (($ $) "\\spad{lambert(f(x))} returns \\spad{f(x) + f(x^2) + f(x^3) + ...}. \\indented{1}{This function is used for computing infinite products.} \\indented{1}{\\spad{f(x)} should have zero constant coefficient.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n = 1..infinity,f(x^n)) = exp(log(lambert(f(x))))}.}")) (|lagrange| (($ $) "\\spad{lagrange(g(x))} produces the Taylor series for \\spad{f(x)} \\indented{1}{where \\spad{f(x)} is implicitly defined as \\spad{f(x) = x*g(f(x))}.}")) (|differentiate| (($ $ (|Variable| |#2|)) "\\spad{differentiate(f(x),x)} computes the derivative of \\spad{f(x)} with respect to \\spad{x}.")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (QUOTE (-561))) (-3962 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-561)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -904) (QUOTE (-1181)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-774)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-774)) (|devaluate| |#1|)))) (|HasCategory| (-774) (QUOTE (-1116))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-774))))) (|HasSignature| |#1| (LIST (QUOTE -4380) (LIST (|devaluate| |#1|) (QUOTE (-1181)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-774))))) (|HasCategory| |#1| (QUOTE (-366))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-964))) (|HasCategory| |#1| (QUOTE (-1206))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-550))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasSignature| |#1| (LIST (QUOTE -4246) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1181))))) (|HasSignature| |#1| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#1|))))))) 
-(-1261 |Coef1| |Coef2| UTS1 UTS2) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (QUOTE (-562))) (-3969 (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-145))) (|HasCategory| |#1| (QUOTE (-147))) (-12 (|HasCategory| |#1| (LIST (QUOTE -906) (QUOTE (-1183)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-776)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-776)) (|devaluate| |#1|)))) (|HasCategory| (-776) (QUOTE (-1118))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-776))))) (|HasSignature| |#1| (LIST (QUOTE -4387) (LIST (|devaluate| |#1|) (QUOTE (-1183)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-776))))) (|HasCategory| |#1| (QUOTE (-367))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1208))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-551))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasSignature| |#1| (LIST (QUOTE -4253) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1183))))) (|HasSignature| |#1| (LIST (QUOTE -3494) (LIST (LIST (QUOTE -646) (QUOTE (-1183))) (|devaluate| |#1|))))))) 
+(-1263 |Coef1| |Coef2| UTS1 UTS2) 
 ((|constructor| (NIL "Mapping package for univariate Taylor series. \\indented{2}{This package allows one to apply a function to the coefficients of} \\indented{2}{a univariate Taylor series.}")) (|map| ((|#4| (|Mapping| |#2| |#1|) |#3|) "\\spad{map(f,g(x))} applies the map \\spad{f} to the coefficients of \\indented{1}{the Taylor series \\spad{g(x)}.}"))) 
 NIL 
 NIL 
-(-1262 S |Coef|) 
+(-1264 S |Coef|) 
 ((|constructor| (NIL "\\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (** (($ $ |#2|) "\\spad{f(x) ** a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.")) (|polynomial| (((|Polynomial| |#2|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#2|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|multiplyCoefficients| (($ (|Mapping| |#2| (|Integer|)) $) "\\spad{multiplyCoefficients(f,sum(n = 0..infinity,a[n] * x**n))} returns \\spad{sum(n = 0..infinity,f(n) * a[n] * x**n)}. This function is used when Laurent series are represented by a Taylor series and an order.")) (|quoByVar| (($ $) "\\spad{quoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.")) (|coefficients| (((|Stream| |#2|) $) "\\spad{coefficients(a0 + a1 x + a2 x**2 + ...)} returns a stream of coefficients: \\spad{[a0,a1,a2,...]}. The entries of the stream may be zero.")) (|series| (($ (|Stream| |#2|)) "\\spad{series([a0,a1,a2,...])} is the Taylor series \\spad{a0 + a1 x + a2 x**2 + ...}.") (($ (|Stream| (|Record| (|:| |k| (|NonNegativeInteger|)) (|:| |c| |#2|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents."))) 
 NIL 
-((|HasCategory| |#2| (LIST (QUOTE -29) (QUOTE (-550)))) (|HasCategory| |#2| (QUOTE (-964))) (|HasCategory| |#2| (QUOTE (-1206))) (|HasSignature| |#2| (LIST (QUOTE -3487) (LIST (LIST (QUOTE -644) (QUOTE (-1181))) (|devaluate| |#2|)))) (|HasSignature| |#2| (LIST (QUOTE -4246) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1181))))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasCategory| |#2| (QUOTE (-366)))) 
-(-1263 |Coef|) 
+((|HasCategory| |#2| (LIST (QUOTE -29) (QUOTE (-551)))) (|HasCategory| |#2| (QUOTE (-966))) (|HasCategory| |#2| (QUOTE (-1208))) (|HasSignature| |#2| (LIST (QUOTE -3494) (LIST (LIST (QUOTE -646) (QUOTE (-1183))) (|devaluate| |#2|)))) (|HasSignature| |#2| (LIST (QUOTE -4253) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1183))))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasCategory| |#2| (QUOTE (-367)))) 
+(-1265 |Coef|) 
 ((|constructor| (NIL "\\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (** (($ $ |#1|) "\\spad{f(x) ** a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.")) (|polynomial| (((|Polynomial| |#1|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(f,sum(n = 0..infinity,a[n] * x**n))} returns \\spad{sum(n = 0..infinity,f(n) * a[n] * x**n)}. This function is used when Laurent series are represented by a Taylor series and an order.")) (|quoByVar| (($ $) "\\spad{quoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.")) (|coefficients| (((|Stream| |#1|) $) "\\spad{coefficients(a0 + a1 x + a2 x**2 + ...)} returns a stream of coefficients: \\spad{[a0,a1,a2,...]}. The entries of the stream may be zero.")) (|series| (($ (|Stream| |#1|)) "\\spad{series([a0,a1,a2,...])} is the Taylor series \\spad{a0 + a1 x + a2 x**2 + ...}.") (($ (|Stream| (|Record| (|:| |k| (|NonNegativeInteger|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents."))) 
-(((-4429 "*") |has| |#1| (-173)) (-4420 |has| |#1| (-561)) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") |has| |#1| (-173)) (-4427 |has| |#1| (-562)) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1264 |Coef| UTS) 
+(-1266 |Coef| UTS) 
 ((|constructor| (NIL "\\indented{1}{This package provides Taylor series solutions to regular} linear or non-linear ordinary differential equations of arbitrary order.")) (|mpsode| (((|List| |#2|) (|List| |#1|) (|List| (|Mapping| |#2| (|List| |#2|)))) "\\spad{mpsode(r,f)} solves the system of differential equations \\spad{dy[i]/dx =f[i] [x,y[1],y[2],...,y[n]]},{} \\spad{y[i](a) = r[i]} for \\spad{i} in 1..\\spad{n}.")) (|ode| ((|#2| (|Mapping| |#2| (|List| |#2|)) (|List| |#1|)) "\\spad{ode(f,cl)} is the solution to \\spad{y<n>=f(y,y',..,y<n-1>)} such that \\spad{y<i>(a) = cl.i} for \\spad{i} in 1..\\spad{n}.")) (|ode2| ((|#2| (|Mapping| |#2| |#2| |#2|) |#1| |#1|) "\\spad{ode2(f,c0,c1)} is the solution to \\spad{y'' = f(y,y')} such that \\spad{y(a) = c0} and \\spad{y'(a) = c1}.")) (|ode1| ((|#2| (|Mapping| |#2| |#2|) |#1|) "\\spad{ode1(f,c)} is the solution to \\spad{y' = f(y)} such that \\spad{y(a) = c}.")) (|fixedPointExquo| ((|#2| |#2| |#2|) "\\spad{fixedPointExquo(f,g)} computes the exact quotient of \\spad{f} and \\spad{g} using a fixed point computation.")) (|stFuncN| (((|Mapping| (|Stream| |#1|) (|List| (|Stream| |#1|))) (|Mapping| |#2| (|List| |#2|))) "\\spad{stFuncN(f)} is a local function xported due to compiler problem. This function is of no interest to the top-level user.")) (|stFunc2| (((|Mapping| (|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) (|Mapping| |#2| |#2| |#2|)) "\\spad{stFunc2(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user.")) (|stFunc1| (((|Mapping| (|Stream| |#1|) (|Stream| |#1|)) (|Mapping| |#2| |#2|)) "\\spad{stFunc1(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user."))) 
 NIL 
 NIL 
-(-1265 -3498 UP L UTS) 
+(-1267 -3505 UP L UTS) 
 ((|constructor| (NIL "\\spad{RUTSodetools} provides tools to interface with the series \\indented{1}{ODE solver when presented with linear ODEs.}")) (RF2UTS ((|#4| (|Fraction| |#2|)) "\\spad{RF2UTS(f)} converts \\spad{f} to a Taylor series.")) (LODO2FUN (((|Mapping| |#4| (|List| |#4|)) |#3|) "\\spad{LODO2FUN(op)} returns the function to pass to the series ODE solver in order to solve \\spad{op y = 0}.")) (UTS2UP ((|#2| |#4| (|NonNegativeInteger|)) "\\spad{UTS2UP(s, n)} converts the first \\spad{n} terms of \\spad{s} to a univariate polynomial.")) (UP2UTS ((|#4| |#2|) "\\spad{UP2UTS(p)} converts \\spad{p} to a Taylor series."))) 
 NIL 
-((|HasCategory| |#1| (QUOTE (-561)))) 
-(-1266) 
+((|HasCategory| |#1| (QUOTE (-562)))) 
+(-1268) 
 ((|constructor| (NIL "The category of domains that act like unions. UnionType,{} like Type or Category,{} acts mostly as a take that communicates `union-like' intended semantics to the compiler. A domain \\spad{D} that satifies UnionType should provide definitions for `case' operators,{} with corresponding `autoCoerce' operators."))) 
 NIL 
 NIL 
-(-1267 |sym|) 
+(-1269 |sym|) 
 ((|constructor| (NIL "This domain implements variables")) (|variable| (((|Symbol|)) "\\spad{variable()} returns the symbol")) (|coerce| (((|Symbol|) $) "\\spad{coerce(x)} returns the symbol"))) 
 NIL 
 NIL 
-(-1268 S R) 
+(-1270 S R) 
 ((|constructor| (NIL "\\spadtype{VectorCategory} represents the type of vector like objects,{} \\spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.")) (|magnitude| ((|#2| $) "\\spad{magnitude(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the length")) (|length| ((|#2| $) "\\spad{length(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the magnitude")) (|cross| (($ $ $) "vectorProduct(\\spad{u},{}\\spad{v}) constructs the cross product of \\spad{u} and \\spad{v}. Error: if \\spad{u} and \\spad{v} are not of length 3.")) (|outerProduct| (((|Matrix| |#2|) $ $) "\\spad{outerProduct(u,v)} constructs the matrix whose (\\spad{i},{}\\spad{j})\\spad{'}th element is \\spad{u}(\\spad{i})\\spad{*v}(\\spad{j}).")) (|dot| ((|#2| $ $) "\\spad{dot(x,y)} computes the inner product of the two vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.")) (* (($ $ |#2|) "\\spad{y * r} multiplies each component of the vector \\spad{y} by the element \\spad{r}.") (($ |#2| $) "\\spad{r * y} multiplies the element \\spad{r} times each component of the vector \\spad{y}.") (($ (|Integer|) $) "\\spad{n * y} multiplies each component of the vector \\spad{y} by the integer \\spad{n}.")) (- (($ $ $) "\\spad{x - y} returns the component-wise difference of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.") (($ $) "\\spad{-x} negates all components of the vector \\spad{x}.")) (|zero| (($ (|NonNegativeInteger|)) "\\spad{zero(n)} creates a zero vector of length \\spad{n}.")) (+ (($ $ $) "\\spad{x + y} returns the component-wise sum of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-1006))) (|HasCategory| |#2| (QUOTE (-1053))) (|HasCategory| |#2| (QUOTE (-729))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-25)))) 
-(-1269 R) 
+((|HasCategory| |#2| (QUOTE (-1008))) (|HasCategory| |#2| (QUOTE (-1055))) (|HasCategory| |#2| (QUOTE (-731))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-25)))) 
+(-1271 R) 
 ((|constructor| (NIL "\\spadtype{VectorCategory} represents the type of vector like objects,{} \\spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.")) (|magnitude| ((|#1| $) "\\spad{magnitude(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the length")) (|length| ((|#1| $) "\\spad{length(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the magnitude")) (|cross| (($ $ $) "vectorProduct(\\spad{u},{}\\spad{v}) constructs the cross product of \\spad{u} and \\spad{v}. Error: if \\spad{u} and \\spad{v} are not of length 3.")) (|outerProduct| (((|Matrix| |#1|) $ $) "\\spad{outerProduct(u,v)} constructs the matrix whose (\\spad{i},{}\\spad{j})\\spad{'}th element is \\spad{u}(\\spad{i})\\spad{*v}(\\spad{j}).")) (|dot| ((|#1| $ $) "\\spad{dot(x,y)} computes the inner product of the two vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.")) (* (($ $ |#1|) "\\spad{y * r} multiplies each component of the vector \\spad{y} by the element \\spad{r}.") (($ |#1| $) "\\spad{r * y} multiplies the element \\spad{r} times each component of the vector \\spad{y}.") (($ (|Integer|) $) "\\spad{n * y} multiplies each component of the vector \\spad{y} by the integer \\spad{n}.")) (- (($ $ $) "\\spad{x - y} returns the component-wise difference of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.") (($ $) "\\spad{-x} negates all components of the vector \\spad{x}.")) (|zero| (($ (|NonNegativeInteger|)) "\\spad{zero(n)} creates a zero vector of length \\spad{n}.")) (+ (($ $ $) "\\spad{x + y} returns the component-wise sum of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length."))) 
-((-4428 . T) (-4427 . T)) 
+((-4435 . T) (-4434 . T)) 
 NIL 
-(-1270 R) 
+(-1272 R) 
 ((|constructor| (NIL "This type represents vector like objects with varying lengths and indexed by a finite segment of integers starting at 1.")) (|vector| (($ (|List| |#1|)) "\\spad{vector(l)} converts the list \\spad{l} to a vector."))) 
-((-4428 . T) (-4427 . T)) 
-((-3962 (-12 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) (-3962 (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866))))) (|HasCategory| |#1| (LIST (QUOTE -617) (QUOTE (-539)))) (-3962 (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105)))) (|HasCategory| |#1| (QUOTE (-853))) (|HasCategory| (-550) (QUOTE (-853))) (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-729))) (|HasCategory| |#1| (QUOTE (-1053))) (-12 (|HasCategory| |#1| (QUOTE (-1006))) (|HasCategory| |#1| (QUOTE (-1053)))) (|HasCategory| |#1| (LIST (QUOTE -616) (QUOTE (-866)))) (-12 (|HasCategory| |#1| (QUOTE (-1105))) (|HasCategory| |#1| (LIST (QUOTE -311) (|devaluate| |#1|))))) 
-(-1271 A B) 
+((-4435 . T) (-4434 . T)) 
+((-3969 (-12 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) (-3969 (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-540)))) (-3969 (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107)))) (|HasCategory| |#1| (QUOTE (-855))) (|HasCategory| (-551) (QUOTE (-855))) (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-731))) (|HasCategory| |#1| (QUOTE (-1055))) (-12 (|HasCategory| |#1| (QUOTE (-1008))) (|HasCategory| |#1| (QUOTE (-1055)))) (|HasCategory| |#1| (LIST (QUOTE -618) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1107))) (|HasCategory| |#1| (LIST (QUOTE -312) (|devaluate| |#1|))))) 
+(-1273 A B) 
 ((|constructor| (NIL "\\indented{2}{This package provides operations which all take as arguments} vectors of elements of some type \\spad{A} and functions from \\spad{A} to another of type \\spad{B}. The operations all iterate over their vector argument and either return a value of type \\spad{B} or a vector over \\spad{B}.")) (|map| (((|Union| (|Vector| |#2|) "failed") (|Mapping| (|Union| |#2| "failed") |#1|) (|Vector| |#1|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values or \\spad{\"failed\"}.") (((|Vector| |#2|) (|Mapping| |#2| |#1|) (|Vector| |#1|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|Vector| |#1|) |#2|) "\\spad{reduce(func,vec,ident)} combines the elements in \\spad{vec} using the binary function \\spad{func}. Argument \\spad{ident} is returned if \\spad{vec} is empty.")) (|scan| (((|Vector| |#2|) (|Mapping| |#2| |#1| |#2|) (|Vector| |#1|) |#2|) "\\spad{scan(func,vec,ident)} creates a new vector whose elements are the result of applying reduce to the binary function \\spad{func},{} increasing initial subsequences of the vector \\spad{vec},{} and the element \\spad{ident}."))) 
 NIL 
 NIL 
-(-1272) 
+(-1274) 
 ((|constructor| (NIL "ViewportPackage provides functions for creating GraphImages and TwoDimensionalViewports from lists of lists of points.")) (|coerce| (((|TwoDimensionalViewport|) (|GraphImage|)) "\\spad{coerce(gi)} converts the indicated \\spadtype{GraphImage},{} \\spad{gi},{} into the \\spadtype{TwoDimensionalViewport} form.")) (|drawCurves| (((|TwoDimensionalViewport|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|DrawOption|))) "\\spad{drawCurves([[p0],[p1],...,[pn]],[options])} creates a \\spadtype{TwoDimensionalViewport} from the list of lists of points,{} \\spad{p0} throught \\spad{pn},{} using the options specified in the list \\spad{options}.") (((|TwoDimensionalViewport|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|Palette|) (|Palette|) (|PositiveInteger|) (|List| (|DrawOption|))) "\\spad{drawCurves([[p0],[p1],...,[pn]],ptColor,lineColor,ptSize,[options])} creates a \\spadtype{TwoDimensionalViewport} from the list of lists of points,{} \\spad{p0} throught \\spad{pn},{} using the options specified in the list \\spad{options}. The point color is specified by \\spad{ptColor},{} the line color is specified by \\spad{lineColor},{} and the point size is specified by \\spad{ptSize}.")) (|graphCurves| (((|GraphImage|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|DrawOption|))) "\\spad{graphCurves([[p0],[p1],...,[pn]],[options])} creates a \\spadtype{GraphImage} from the list of lists of points,{} \\spad{p0} throught \\spad{pn},{} using the options specified in the list \\spad{options}.") (((|GraphImage|) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{graphCurves([[p0],[p1],...,[pn]])} creates a \\spadtype{GraphImage} from the list of lists of points indicated by \\spad{p0} through \\spad{pn}.") (((|GraphImage|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|Palette|) (|Palette|) (|PositiveInteger|) (|List| (|DrawOption|))) "\\spad{graphCurves([[p0],[p1],...,[pn]],ptColor,lineColor,ptSize,[options])} creates a \\spadtype{GraphImage} from the list of lists of points,{} \\spad{p0} throught \\spad{pn},{} using the options specified in the list \\spad{options}. The graph point color is specified by \\spad{ptColor},{} the graph line color is specified by \\spad{lineColor},{} and the size of the points is specified by \\spad{ptSize}."))) 
 NIL 
 NIL 
-(-1273) 
+(-1275) 
 ((|constructor| (NIL "TwoDimensionalViewport creates viewports to display graphs.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(v)} returns the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport} as output of the domain \\spadtype{OutputForm}.")) (|key| (((|Integer|) $) "\\spad{key(v)} returns the process ID number of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport}.")) (|reset| (((|Void|) $) "\\spad{reset(v)} sets the current state of the graph characteristics of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} back to their initial settings.")) (|write| (((|String|) $ (|String|) (|List| (|String|))) "\\spad{write(v,s,lf)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v} and the optional file types indicated by the list \\spad{lf}.") (((|String|) $ (|String|) (|String|)) "\\spad{write(v,s,f)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v} and an optional file type \\spad{f}.") (((|String|) $ (|String|)) "\\spad{write(v,s)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v}.")) (|resize| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{resize(v,w,h)} displays the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with a width of \\spad{w} and a height of \\spad{h},{} keeping the upper left-hand corner position unchanged.")) (|update| (((|Void|) $ (|GraphImage|) (|PositiveInteger|)) "\\spad{update(v,gr,n)} drops the graph \\spad{gr} in slot \\spad{n} of viewport \\spad{v}. The graph \\spad{gr} must have been transmitted already and acquired an integer key.")) (|move| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{move(v,x,y)} displays the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the upper left-hand corner of the viewport window at the screen coordinate position \\spad{x},{} \\spad{y}.")) (|show| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{show(v,n,s)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the graph if \\spad{s} is \"off\".")) (|translate| (((|Void|) $ (|PositiveInteger|) (|Float|) (|Float|)) "\\spad{translate(v,n,dx,dy)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} translated by \\spad{dx} in the \\spad{x}-coordinate direction from the center of the viewport,{} and by \\spad{dy} in the \\spad{y}-coordinate direction from the center. Setting \\spad{dx} and \\spad{dy} to \\spad{0} places the center of the graph at the center of the viewport.")) (|scale| (((|Void|) $ (|PositiveInteger|) (|Float|) (|Float|)) "\\spad{scale(v,n,sx,sy)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} scaled by the factor \\spad{sx} in the \\spad{x}-coordinate direction and by the factor \\spad{sy} in the \\spad{y}-coordinate direction.")) (|dimensions| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{dimensions(v,x,y,width,height)} sets the position of the upper left-hand corner of the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} to the window coordinate \\spad{x},{} \\spad{y},{} and sets the dimensions of the window to that of \\spad{width},{} \\spad{height}. The new dimensions are not displayed until the function \\spadfun{makeViewport2D} is executed again for \\spad{v}.")) (|close| (((|Void|) $) "\\spad{close(v)} closes the viewport window of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and terminates the corresponding process ID.")) (|controlPanel| (((|Void|) $ (|String|)) "\\spad{controlPanel(v,s)} displays the control panel of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or hides the control panel if \\spad{s} is \"off\".")) (|connect| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{connect(v,n,s)} displays the lines connecting the graph points in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the lines if \\spad{s} is \"off\".")) (|region| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{region(v,n,s)} displays the bounding box of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the bounding box if \\spad{s} is \"off\".")) (|points| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{points(v,n,s)} displays the points of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the points if \\spad{s} is \"off\".")) (|units| (((|Void|) $ (|PositiveInteger|) (|Palette|)) "\\spad{units(v,n,c)} displays the units of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the units color set to the given palette color \\spad{c}.") (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{units(v,n,s)} displays the units of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the units if \\spad{s} is \"off\".")) (|axes| (((|Void|) $ (|PositiveInteger|) (|Palette|)) "\\spad{axes(v,n,c)} displays the axes of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the axes color set to the given palette color \\spad{c}.") (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{axes(v,n,s)} displays the axes of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the axes if \\spad{s} is \"off\".")) (|getGraph| (((|GraphImage|) $ (|PositiveInteger|)) "\\spad{getGraph(v,n)} returns the graph which is of the domain \\spadtype{GraphImage} which is located in graph field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of the domain \\spadtype{TwoDimensionalViewport}.")) (|putGraph| (((|Void|) $ (|GraphImage|) (|PositiveInteger|)) "\\spad{putGraph(v,gi,n)} sets the graph field indicated by \\spad{n},{} of the indicated two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} to be the graph,{} \\spad{gi} of domain \\spadtype{GraphImage}. The contents of viewport,{} \\spad{v},{} will contain \\spad{gi} when the function \\spadfun{makeViewport2D} is called to create the an updated viewport \\spad{v}.")) (|title| (((|Void|) $ (|String|)) "\\spad{title(v,s)} changes the title which is shown in the two-dimensional viewport window,{} \\spad{v} of domain \\spadtype{TwoDimensionalViewport}.")) (|graphs| (((|Vector| (|Union| (|GraphImage|) "undefined")) $) "\\spad{graphs(v)} returns a vector,{} or list,{} which is a union of all the graphs,{} of the domain \\spadtype{GraphImage},{} which are allocated for the two-dimensional viewport,{} \\spad{v},{} of domain \\spadtype{TwoDimensionalViewport}. Those graphs which have no data are labeled \"undefined\",{} otherwise their contents are shown.")) (|graphStates| (((|Vector| (|Record| (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)) (|:| |points| (|Integer|)) (|:| |connect| (|Integer|)) (|:| |spline| (|Integer|)) (|:| |axes| (|Integer|)) (|:| |axesColor| (|Palette|)) (|:| |units| (|Integer|)) (|:| |unitsColor| (|Palette|)) (|:| |showing| (|Integer|)))) $) "\\spad{graphStates(v)} returns and shows a listing of a record containing the current state of the characteristics of each of the ten graph records in the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport}.")) (|graphState| (((|Void|) $ (|PositiveInteger|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Palette|) (|Integer|) (|Palette|) (|Integer|)) "\\spad{graphState(v,num,sX,sY,dX,dY,pts,lns,box,axes,axesC,un,unC,cP)} sets the state of the characteristics for the graph indicated by \\spad{num} in the given two-dimensional viewport \\spad{v},{} of domain \\spadtype{TwoDimensionalViewport},{} to the values given as parameters. The scaling of the graph in the \\spad{x} and \\spad{y} component directions is set to be \\spad{sX} and \\spad{sY}; the window translation in the \\spad{x} and \\spad{y} component directions is set to be \\spad{dX} and \\spad{dY}; The graph points,{} lines,{} bounding \\spad{box},{} \\spad{axes},{} or units will be shown in the viewport if their given parameters \\spad{pts},{} \\spad{lns},{} \\spad{box},{} \\spad{axes} or \\spad{un} are set to be \\spad{1},{} but will not be shown if they are set to \\spad{0}. The color of the \\spad{axes} and the color of the units are indicated by the palette colors \\spad{axesC} and \\spad{unC} respectively. To display the control panel when the viewport window is displayed,{} set \\spad{cP} to \\spad{1},{} otherwise set it to \\spad{0}.")) (|options| (($ $ (|List| (|DrawOption|))) "\\spad{options(v,lopt)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and returns \\spad{v} with it\\spad{'s} draw options modified to be those which are indicated in the given list,{} \\spad{lopt} of domain \\spadtype{DrawOption}.") (((|List| (|DrawOption|)) $) "\\spad{options(v)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and returns a list containing the draw options from the domain \\spadtype{DrawOption} for \\spad{v}.")) (|makeViewport2D| (($ (|GraphImage|) (|List| (|DrawOption|))) "\\spad{makeViewport2D(gi,lopt)} creates and displays a viewport window of the domain \\spadtype{TwoDimensionalViewport} whose graph field is assigned to be the given graph,{} \\spad{gi},{} of domain \\spadtype{GraphImage},{} and whose options field is set to be the list of options,{} \\spad{lopt} of domain \\spadtype{DrawOption}.") (($ $) "\\spad{makeViewport2D(v)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and displays a viewport window on the screen which contains the contents of \\spad{v}.")) (|viewport2D| (($) "\\spad{viewport2D()} returns an undefined two-dimensional viewport of the domain \\spadtype{TwoDimensionalViewport} whose contents are empty.")) (|getPickedPoints| (((|List| (|Point| (|DoubleFloat|))) $) "\\spad{getPickedPoints(x)} returns a list of small floats for the points the user interactively picked on the viewport for full integration into the system,{} some design issues need to be addressed: \\spadignore{e.g.} how to go through the GraphImage interface,{} how to default to graphs,{} etc."))) 
 NIL 
 NIL 
-(-1274) 
+(-1276) 
 ((|key| (((|Integer|) $) "\\spad{key(v)} returns the process ID number of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|close| (((|Void|) $) "\\spad{close(v)} closes the viewport window of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and terminates the corresponding process ID.")) (|write| (((|String|) $ (|String|) (|List| (|String|))) "\\spad{write(v,s,lf)} takes the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data file for \\spad{v} and the optional file types indicated by the list \\spad{lf}.") (((|String|) $ (|String|) (|String|)) "\\spad{write(v,s,f)} takes the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data file for \\spad{v} and an optional file type \\spad{f}.") (((|String|) $ (|String|)) "\\spad{write(v,s)} takes the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data file for \\spad{v}.")) (|colorDef| (((|Void|) $ (|Color|) (|Color|)) "\\spad{colorDef(v,c1,c2)} sets the range of colors along the colormap so that the lower end of the colormap is defined by \\spad{c1} and the top end of the colormap is defined by \\spad{c2},{} for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|reset| (((|Void|) $) "\\spad{reset(v)} sets the current state of the graph characteristics of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} back to their initial settings.")) (|intensity| (((|Void|) $ (|Float|)) "\\spad{intensity(v,i)} sets the intensity of the light source to \\spad{i},{} for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|lighting| (((|Void|) $ (|Float|) (|Float|) (|Float|)) "\\spad{lighting(v,x,y,z)} sets the position of the light source to the coordinates \\spad{x},{} \\spad{y},{} and \\spad{z} and displays the graph for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|clipSurface| (((|Void|) $ (|String|)) "\\spad{clipSurface(v,s)} displays the graph with the specified clipping region removed if \\spad{s} is \"on\",{} or displays the graph without clipping implemented if \\spad{s} is \"off\",{} for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|showClipRegion| (((|Void|) $ (|String|)) "\\spad{showClipRegion(v,s)} displays the clipping region of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the region if \\spad{s} is \"off\".")) (|showRegion| (((|Void|) $ (|String|)) "\\spad{showRegion(v,s)} displays the bounding box of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the box if \\spad{s} is \"off\".")) (|hitherPlane| (((|Void|) $ (|Float|)) "\\spad{hitherPlane(v,h)} sets the hither clipping plane of the graph to \\spad{h},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|eyeDistance| (((|Void|) $ (|Float|)) "\\spad{eyeDistance(v,d)} sets the distance of the observer from the center of the graph to \\spad{d},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|perspective| (((|Void|) $ (|String|)) "\\spad{perspective(v,s)} displays the graph in perspective if \\spad{s} is \"on\",{} or does not display perspective if \\spad{s} is \"off\" for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|translate| (((|Void|) $ (|Float|) (|Float|)) "\\spad{translate(v,dx,dy)} sets the horizontal viewport offset to \\spad{dx} and the vertical viewport offset to \\spad{dy},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|zoom| (((|Void|) $ (|Float|) (|Float|) (|Float|)) "\\spad{zoom(v,sx,sy,sz)} sets the graph scaling factors for the \\spad{x}-coordinate axis to \\spad{sx},{} the \\spad{y}-coordinate axis to \\spad{sy} and the \\spad{z}-coordinate axis to \\spad{sz} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.") (((|Void|) $ (|Float|)) "\\spad{zoom(v,s)} sets the graph scaling factor to \\spad{s},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|rotate| (((|Void|) $ (|Integer|) (|Integer|)) "\\spad{rotate(v,th,phi)} rotates the graph to the longitudinal view angle \\spad{th} degrees and the latitudinal view angle \\spad{phi} degrees for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new rotation position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.") (((|Void|) $ (|Float|) (|Float|)) "\\spad{rotate(v,th,phi)} rotates the graph to the longitudinal view angle \\spad{th} radians and the latitudinal view angle \\spad{phi} radians for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|drawStyle| (((|Void|) $ (|String|)) "\\spad{drawStyle(v,s)} displays the surface for the given three-dimensional viewport \\spad{v} which is of domain \\spadtype{ThreeDimensionalViewport} in the style of drawing indicated by \\spad{s}. If \\spad{s} is not a valid drawing style the style is wireframe by default. Possible styles are \\spad{\"shade\"},{} \\spad{\"solid\"} or \\spad{\"opaque\"},{} \\spad{\"smooth\"},{} and \\spad{\"wireMesh\"}.")) (|outlineRender| (((|Void|) $ (|String|)) "\\spad{outlineRender(v,s)} displays the polygon outline showing either triangularized surface or a quadrilateral surface outline depending on the whether the \\spadfun{diagonals} function has been set,{} for the given three-dimensional viewport \\spad{v} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the polygon outline if \\spad{s} is \"off\".")) (|diagonals| (((|Void|) $ (|String|)) "\\spad{diagonals(v,s)} displays the diagonals of the polygon outline showing a triangularized surface instead of a quadrilateral surface outline,{} for the given three-dimensional viewport \\spad{v} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the diagonals if \\spad{s} is \"off\".")) (|axes| (((|Void|) $ (|String|)) "\\spad{axes(v,s)} displays the axes of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the axes if \\spad{s} is \"off\".")) (|controlPanel| (((|Void|) $ (|String|)) "\\spad{controlPanel(v,s)} displays the control panel of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or hides the control panel if \\spad{s} is \"off\".")) (|viewpoint| (((|Void|) $ (|Float|) (|Float|) (|Float|)) "\\spad{viewpoint(v,rotx,roty,rotz)} sets the rotation about the \\spad{x}-axis to be \\spad{rotx} radians,{} sets the rotation about the \\spad{y}-axis to be \\spad{roty} radians,{} and sets the rotation about the \\spad{z}-axis to be \\spad{rotz} radians,{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport} and displays \\spad{v} with the new view position.") (((|Void|) $ (|Float|) (|Float|)) "\\spad{viewpoint(v,th,phi)} sets the longitudinal view angle to \\spad{th} radians and the latitudinal view angle to \\spad{phi} radians for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.") (((|Void|) $ (|Integer|) (|Integer|) (|Float|) (|Float|) (|Float|)) "\\spad{viewpoint(v,th,phi,s,dx,dy)} sets the longitudinal view angle to \\spad{th} degrees,{} the latitudinal view angle to \\spad{phi} degrees,{} the scale factor to \\spad{s},{} the horizontal viewport offset to \\spad{dx},{} and the vertical viewport offset to \\spad{dy} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.") (((|Void|) $ (|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)))) "\\spad{viewpoint(v,viewpt)} sets the viewpoint for the viewport. The viewport record consists of the latitudal and longitudal angles,{} the zoom factor,{} the \\spad{X},{} \\spad{Y},{} and \\spad{Z} scales,{} and the \\spad{X} and \\spad{Y} displacements.") (((|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|))) $) "\\spad{viewpoint(v)} returns the current viewpoint setting of the given viewport,{} \\spad{v}. This function is useful in the situation where the user has created a viewport,{} proceeded to interact with it via the control panel and desires to save the values of the viewpoint as the default settings for another viewport to be created using the system.") (((|Void|) $ (|Float|) (|Float|) (|Float|) (|Float|) (|Float|)) "\\spad{viewpoint(v,th,phi,s,dx,dy)} sets the longitudinal view angle to \\spad{th} radians,{} the latitudinal view angle to \\spad{phi} radians,{} the scale factor to \\spad{s},{} the horizontal viewport offset to \\spad{dx},{} and the vertical viewport offset to \\spad{dy} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.")) (|dimensions| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{dimensions(v,x,y,width,height)} sets the position of the upper left-hand corner of the three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} to the window coordinate \\spad{x},{} \\spad{y},{} and sets the dimensions of the window to that of \\spad{width},{} \\spad{height}. The new dimensions are not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.")) (|title| (((|Void|) $ (|String|)) "\\spad{title(v,s)} changes the title which is shown in the three-dimensional viewport window,{} \\spad{v} of domain \\spadtype{ThreeDimensionalViewport}.")) (|resize| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{resize(v,w,h)} displays the three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} with a width of \\spad{w} and a height of \\spad{h},{} keeping the upper left-hand corner position unchanged.")) (|move| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{move(v,x,y)} displays the three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} with the upper left-hand corner of the viewport window at the screen coordinate position \\spad{x},{} \\spad{y}.")) (|options| (($ $ (|List| (|DrawOption|))) "\\spad{options(v,lopt)} takes the viewport,{} \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport} and sets the draw options being used by \\spad{v} to those indicated in the list,{} \\spad{lopt},{} which is a list of options from the domain \\spad{DrawOption}.") (((|List| (|DrawOption|)) $) "\\spad{options(v)} takes the viewport,{} \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport} and returns a list of all the draw options from the domain \\spad{DrawOption} which are being used by \\spad{v}.")) (|modifyPointData| (((|Void|) $ (|NonNegativeInteger|) (|Point| (|DoubleFloat|))) "\\spad{modifyPointData(v,ind,pt)} takes the viewport,{} \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport},{} and places the data point,{} \\spad{pt} into the list of points database of \\spad{v} at the index location given by \\spad{ind}.")) (|subspace| (($ $ (|ThreeSpace| (|DoubleFloat|))) "\\spad{subspace(v,sp)} places the contents of the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport},{} in the subspace \\spad{sp},{} which is of the domain \\spad{ThreeSpace}.") (((|ThreeSpace| (|DoubleFloat|)) $) "\\spad{subspace(v)} returns the contents of the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport},{} as a subspace of the domain \\spad{ThreeSpace}.")) (|makeViewport3D| (($ (|ThreeSpace| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{makeViewport3D(sp,lopt)} takes the given space,{} \\spad{sp} which is of the domain \\spadtype{ThreeSpace} and displays a viewport window on the screen which contains the contents of \\spad{sp},{} and whose draw options are indicated by the list \\spad{lopt},{} which is a list of options from the domain \\spad{DrawOption}.") (($ (|ThreeSpace| (|DoubleFloat|)) (|String|)) "\\spad{makeViewport3D(sp,s)} takes the given space,{} \\spad{sp} which is of the domain \\spadtype{ThreeSpace} and displays a viewport window on the screen which contains the contents of \\spad{sp},{} and whose title is given by \\spad{s}.") (($ $) "\\spad{makeViewport3D(v)} takes the given three-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{ThreeDimensionalViewport} and displays a viewport window on the screen which contains the contents of \\spad{v}.")) (|viewport3D| (($) "\\spad{viewport3D()} returns an undefined three-dimensional viewport of the domain \\spadtype{ThreeDimensionalViewport} whose contents are empty.")) (|viewDeltaYDefault| (((|Float|) (|Float|)) "\\spad{viewDeltaYDefault(dy)} sets the current default vertical offset from the center of the viewport window to be \\spad{dy} and returns \\spad{dy}.") (((|Float|)) "\\spad{viewDeltaYDefault()} returns the current default vertical offset from the center of the viewport window.")) (|viewDeltaXDefault| (((|Float|) (|Float|)) "\\spad{viewDeltaXDefault(dx)} sets the current default horizontal offset from the center of the viewport window to be \\spad{dx} and returns \\spad{dx}.") (((|Float|)) "\\spad{viewDeltaXDefault()} returns the current default horizontal offset from the center of the viewport window.")) (|viewZoomDefault| (((|Float|) (|Float|)) "\\spad{viewZoomDefault(s)} sets the current default graph scaling value to \\spad{s} and returns \\spad{s}.") (((|Float|)) "\\spad{viewZoomDefault()} returns the current default graph scaling value.")) (|viewPhiDefault| (((|Float|) (|Float|)) "\\spad{viewPhiDefault(p)} sets the current default latitudinal view angle in radians to the value \\spad{p} and returns \\spad{p}.") (((|Float|)) "\\spad{viewPhiDefault()} returns the current default latitudinal view angle in radians.")) (|viewThetaDefault| (((|Float|) (|Float|)) "\\spad{viewThetaDefault(t)} sets the current default longitudinal view angle in radians to the value \\spad{t} and returns \\spad{t}.") (((|Float|)) "\\spad{viewThetaDefault()} returns the current default longitudinal view angle in radians."))) 
 NIL 
 NIL 
-(-1275) 
+(-1277) 
 ((|constructor| (NIL "ViewportDefaultsPackage describes default and user definable values for graphics")) (|tubeRadiusDefault| (((|DoubleFloat|)) "\\spad{tubeRadiusDefault()} returns the radius used for a 3D tube plot.") (((|DoubleFloat|) (|Float|)) "\\spad{tubeRadiusDefault(r)} sets the default radius for a 3D tube plot to \\spad{r}.")) (|tubePointsDefault| (((|PositiveInteger|)) "\\spad{tubePointsDefault()} returns the number of points to be used when creating the circle to be used in creating a 3D tube plot.") (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{tubePointsDefault(i)} sets the number of points to use when creating the circle to be used in creating a 3D tube plot to \\spad{i}.")) (|var2StepsDefault| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{var2StepsDefault(i)} sets the number of steps to take when creating a 3D mesh in the direction of the first defined free variable to \\spad{i} (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).") (((|PositiveInteger|)) "\\spad{var2StepsDefault()} is the current setting for the number of steps to take when creating a 3D mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).")) (|var1StepsDefault| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{var1StepsDefault(i)} sets the number of steps to take when creating a 3D mesh in the direction of the first defined free variable to \\spad{i} (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).") (((|PositiveInteger|)) "\\spad{var1StepsDefault()} is the current setting for the number of steps to take when creating a 3D mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).")) (|viewWriteAvailable| (((|List| (|String|))) "\\spad{viewWriteAvailable()} returns a list of available methods for writing,{} such as BITMAP,{} POSTSCRIPT,{} etc.")) (|viewWriteDefault| (((|List| (|String|)) (|List| (|String|))) "\\spad{viewWriteDefault(l)} sets the default list of things to write in a viewport data file to the strings in \\spad{l}; a viewAlone file is always genereated.") (((|List| (|String|))) "\\spad{viewWriteDefault()} returns the list of things to write in a viewport data file; a viewAlone file is always generated.")) (|viewDefaults| (((|Void|)) "\\spad{viewDefaults()} resets all the default graphics settings.")) (|viewSizeDefault| (((|List| (|PositiveInteger|)) (|List| (|PositiveInteger|))) "\\spad{viewSizeDefault([w,h])} sets the default viewport width to \\spad{w} and height to \\spad{h}.") (((|List| (|PositiveInteger|))) "\\spad{viewSizeDefault()} returns the default viewport width and height.")) (|viewPosDefault| (((|List| (|NonNegativeInteger|)) (|List| (|NonNegativeInteger|))) "\\spad{viewPosDefault([x,y])} sets the default \\spad{X} and \\spad{Y} position of a viewport window unless overriden explicityly,{} newly created viewports will have th \\spad{X} and \\spad{Y} coordinates \\spad{x},{} \\spad{y}.") (((|List| (|NonNegativeInteger|))) "\\spad{viewPosDefault()} returns the default \\spad{X} and \\spad{Y} position of a viewport window unless overriden explicityly,{} newly created viewports will have this \\spad{X} and \\spad{Y} coordinate.")) (|pointSizeDefault| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{pointSizeDefault(i)} sets the default size of the points in a 2D viewport to \\spad{i}.") (((|PositiveInteger|)) "\\spad{pointSizeDefault()} returns the default size of the points in a 2D viewport.")) (|unitsColorDefault| (((|Palette|) (|Palette|)) "\\spad{unitsColorDefault(p)} sets the default color of the unit ticks in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{unitsColorDefault()} returns the default color of the unit ticks in a 2D viewport.")) (|axesColorDefault| (((|Palette|) (|Palette|)) "\\spad{axesColorDefault(p)} sets the default color of the axes in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{axesColorDefault()} returns the default color of the axes in a 2D viewport.")) (|lineColorDefault| (((|Palette|) (|Palette|)) "\\spad{lineColorDefault(p)} sets the default color of lines connecting points in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{lineColorDefault()} returns the default color of lines connecting points in a 2D viewport.")) (|pointColorDefault| (((|Palette|) (|Palette|)) "\\spad{pointColorDefault(p)} sets the default color of points in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{pointColorDefault()} returns the default color of points in a 2D viewport."))) 
 NIL 
 NIL 
-(-1276) 
+(-1278) 
 ((|constructor| (NIL "This type is used when no value is needed,{} \\spadignore{e.g.} in the \\spad{then} part of a one armed \\spad{if}. All values can be coerced to type Void. Once a value has been coerced to Void,{} it cannot be recovered.")) (|void| (($) "\\spad{void()} produces a void object."))) 
 NIL 
 NIL 
-(-1277 A S) 
+(-1279 A S) 
 ((|constructor| (NIL "Vector Spaces (not necessarily finite dimensional) over a field.")) (|dimension| (((|CardinalNumber|)) "\\spad{dimension()} returns the dimensionality of the vector space.")) (/ (($ $ |#2|) "\\spad{x/y} divides the vector \\spad{x} by the scalar \\spad{y}."))) 
 NIL 
 NIL 
-(-1278 S) 
+(-1280 S) 
 ((|constructor| (NIL "Vector Spaces (not necessarily finite dimensional) over a field.")) (|dimension| (((|CardinalNumber|)) "\\spad{dimension()} returns the dimensionality of the vector space.")) (/ (($ $ |#1|) "\\spad{x/y} divides the vector \\spad{x} by the scalar \\spad{y}."))) 
-((-4422 . T) (-4421 . T)) 
+((-4429 . T) (-4428 . T)) 
 NIL 
-(-1279 R) 
+(-1281 R) 
 ((|constructor| (NIL "This package implements the Weierstrass preparation theorem \\spad{f} or multivariate power series. weierstrass(\\spad{v},{}\\spad{p}) where \\spad{v} is a variable,{} and \\spad{p} is a TaylorSeries(\\spad{R}) in which the terms of lowest degree \\spad{s} must include c*v**s where \\spad{c} is a constant,{}\\spad{s>0},{} is a list of TaylorSeries coefficients A[\\spad{i}] of the equivalent polynomial A = A[0] + A[1]\\spad{*v} + A[2]*v**2 + ... + A[\\spad{s}-1]*v**(\\spad{s}-1) + v**s such that p=A*B ,{} \\spad{B} being a TaylorSeries of minimum degree 0")) (|qqq| (((|Mapping| (|Stream| (|TaylorSeries| |#1|)) (|Stream| (|TaylorSeries| |#1|))) (|NonNegativeInteger|) (|TaylorSeries| |#1|) (|Stream| (|TaylorSeries| |#1|))) "\\spad{qqq(n,s,st)} is used internally.")) (|weierstrass| (((|List| (|TaylorSeries| |#1|)) (|Symbol|) (|TaylorSeries| |#1|)) "\\spad{weierstrass(v,ts)} where \\spad{v} is a variable and \\spad{ts} is \\indented{1}{a TaylorSeries,{} impements the Weierstrass Preparation} \\indented{1}{Theorem. The result is a list of TaylorSeries that} \\indented{1}{are the coefficients of the equivalent series.}")) (|clikeUniv| (((|Mapping| (|SparseUnivariatePolynomial| (|Polynomial| |#1|)) (|Polynomial| |#1|)) (|Symbol|)) "\\spad{clikeUniv(v)} is used internally.")) (|sts2stst| (((|Stream| (|Stream| (|Polynomial| |#1|))) (|Symbol|) (|Stream| (|Polynomial| |#1|))) "\\spad{sts2stst(v,s)} is used internally.")) (|cfirst| (((|Mapping| (|Stream| (|Polynomial| |#1|)) (|Stream| (|Polynomial| |#1|))) (|NonNegativeInteger|)) "\\spad{cfirst n} is used internally.")) (|crest| (((|Mapping| (|Stream| (|Polynomial| |#1|)) (|Stream| (|Polynomial| |#1|))) (|NonNegativeInteger|)) "\\spad{crest n} is used internally."))) 
 NIL 
 NIL 
-(-1280 K R UP -3498) 
+(-1282 K R UP -3505) 
 ((|constructor| (NIL "In this package \\spad{K} is a finite field,{} \\spad{R} is a ring of univariate polynomials over \\spad{K},{} and \\spad{F} is a framed algebra over \\spad{R}. The package provides a function to compute the integral closure of \\spad{R} in the quotient field of \\spad{F} as well as a function to compute a \"local integral basis\" at a specific prime.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) |#2|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{R} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}."))) 
 NIL 
 NIL 
-(-1281) 
+(-1283) 
 ((|constructor| (NIL "This domain represents the syntax of a `where' expression.")) (|qualifier| (((|SpadAst|) $) "\\spad{qualifier(e)} returns the qualifier of the expression `e'.")) (|mainExpression| (((|SpadAst|) $) "\\spad{mainExpression(e)} returns the main expression of the `where' expression `e'."))) 
 NIL 
 NIL 
-(-1282) 
+(-1284) 
 ((|constructor| (NIL "This domain represents the `while' iterator syntax.")) (|condition| (((|SpadAst|) $) "\\spad{condition(i)} returns the condition of the while iterator `i'."))) 
 NIL 
 NIL 
-(-1283 R |VarSet| E P |vl| |wl| |wtlevel|) 
+(-1285 R |VarSet| E P |vl| |wl| |wtlevel|) 
 ((|constructor| (NIL "This domain represents truncated weighted polynomials over a general (not necessarily commutative) polynomial type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.")) (|changeWeightLevel| (((|Void|) (|NonNegativeInteger|)) "\\spad{changeWeightLevel(n)} changes the weight level to the new value given: \\spad{NB:} previously calculated terms are not affected")) (/ (((|Union| $ "failed") $ $) "\\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \\spad{R} is a Field)"))) 
-((-4422 |has| |#1| (-173)) (-4421 |has| |#1| (-173)) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366)))) 
-(-1284 R E V P) 
+((-4429 |has| |#1| (-173)) (-4428 |has| |#1| (-173)) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367)))) 
+(-1286 R E V P) 
 ((|constructor| (NIL "A domain constructor of the category \\axiomType{GeneralTriangularSet}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. The \\axiomOpFrom{construct}{WuWenTsunTriangularSet} operation does not check the previous requirement. Triangular sets are stored as sorted lists \\spad{w}.\\spad{r}.\\spad{t}. the main variables of their members. Furthermore,{} this domain exports operations dealing with the characteristic set method of Wu Wen Tsun and some optimizations mainly proposed by Dong Ming Wang.\\newline References : \\indented{1}{[1] \\spad{W}. \\spad{T}. WU \"A Zero Structure Theorem for polynomial equations solving\"} \\indented{6}{\\spad{MM} Research Preprints,{} 1987.} \\indented{1}{[2] \\spad{D}. \\spad{M}. WANG \"An implementation of the characteristic set method in Maple\"} \\indented{6}{Proc. DISCO'92. Bath,{} England.}")) (|characteristicSerie| (((|List| $) (|List| |#4|)) "\\axiom{characteristicSerie(\\spad{ps})} returns the same as \\axiom{characteristicSerie(\\spad{ps},{}initiallyReduced?,{}initiallyReduce)}.") (((|List| $) (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{characteristicSerie(\\spad{ps},{}redOp?,{}redOp)} returns a list \\axiom{\\spad{lts}} of triangular sets such that the zero set of \\axiom{\\spad{ps}} is the union of the regular zero sets of the members of \\axiom{\\spad{lts}}. This is made by the Ritt and Wu Wen Tsun process applying the operation \\axiom{characteristicSet(\\spad{ps},{}redOp?,{}redOp)} to compute characteristic sets in Wu Wen Tsun sense.")) (|characteristicSet| (((|Union| $ "failed") (|List| |#4|)) "\\axiom{characteristicSet(\\spad{ps})} returns the same as \\axiom{characteristicSet(\\spad{ps},{}initiallyReduced?,{}initiallyReduce)}.") (((|Union| $ "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{characteristicSet(\\spad{ps},{}redOp?,{}redOp)} returns a non-contradictory characteristic set of \\axiom{\\spad{ps}} in Wu Wen Tsun sense \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?} (using \\axiom{redOp} to reduce polynomials \\spad{w}.\\spad{r}.\\spad{t} a \\axiom{redOp?} basic set),{} if no non-zero constant polynomial appear during those reductions,{} else \\axiom{\"failed\"} is returned. The operations \\axiom{redOp} and \\axiom{redOp?} must satisfy the following conditions: \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} holds for every polynomials \\axiom{\\spad{p},{}\\spad{q}} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that we have \\axiom{init(\\spad{q})^e*p = \\spad{f*q} + redOp(\\spad{p},{}\\spad{q})}.")) (|medialSet| (((|Union| $ "failed") (|List| |#4|)) "\\axiom{medial(\\spad{ps})} returns the same as \\axiom{medialSet(\\spad{ps},{}initiallyReduced?,{}initiallyReduce)}.") (((|Union| $ "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{medialSet(\\spad{ps},{}redOp?,{}redOp)} returns \\axiom{\\spad{bs}} a basic set (in Wu Wen Tsun sense \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?}) of some set generating the same ideal as \\axiom{\\spad{ps}} (with rank not higher than any basic set of \\axiom{\\spad{ps}}),{} if no non-zero constant polynomials appear during the computatioms,{} else \\axiom{\"failed\"} is returned. In the former case,{} \\axiom{\\spad{bs}} has to be understood as a candidate for being a characteristic set of \\axiom{\\spad{ps}}. In the original algorithm,{} \\axiom{\\spad{bs}} is simply a basic set of \\axiom{\\spad{ps}}."))) 
-((-4428 . T) (-4427 . T)) 
-((-12 (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#4| (LIST (QUOTE -311) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -617) (QUOTE (-539)))) (|HasCategory| |#4| (QUOTE (-1105))) (|HasCategory| |#1| (QUOTE (-561))) (|HasCategory| |#3| (QUOTE (-371))) (|HasCategory| |#4| (LIST (QUOTE -616) (QUOTE (-866))))) 
-(-1285 R) 
+((-4435 . T) (-4434 . T)) 
+((-12 (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#4| (LIST (QUOTE -312) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-540)))) (|HasCategory| |#4| (QUOTE (-1107))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-372))) (|HasCategory| |#4| (LIST (QUOTE -618) (QUOTE (-868))))) 
+(-1287 R) 
 ((|constructor| (NIL "This is the category of algebras over non-commutative rings. It is used by constructors of non-commutative algebras such as: \\indented{4}{\\spadtype{XPolynomialRing}.} \\indented{4}{\\spadtype{XFreeAlgebra}} Author: Michel Petitot (petitot@lifl.\\spad{fr})"))) 
-((-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1286 |vl| R) 
+(-1288 |vl| R) 
 ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables do not commute. The coefficient ring may be non-commutative too. However,{} coefficients and variables commute."))) 
-((-4424 . T) (-4420 |has| |#2| (-6 -4420)) (-4422 . T) (-4421 . T)) 
-((|HasCategory| |#2| (QUOTE (-173))) (|HasAttribute| |#2| (QUOTE -4420))) 
-(-1287 R |VarSet| XPOLY) 
+((-4431 . T) (-4427 |has| |#2| (-6 -4427)) (-4429 . T) (-4428 . T)) 
+((|HasCategory| |#2| (QUOTE (-173))) (|HasAttribute| |#2| (QUOTE -4427))) 
+(-1289 R |VarSet| XPOLY) 
 ((|constructor| (NIL "This package provides computations of logarithms and exponentials for polynomials in non-commutative variables. \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr}).")) (|Hausdorff| ((|#3| |#3| |#3| (|NonNegativeInteger|)) "\\axiom{Hausdorff(a,{}\\spad{b},{}\\spad{n})} returns log(exp(a)*exp(\\spad{b})) truncated at order \\axiom{\\spad{n}}.")) (|log| ((|#3| |#3| (|NonNegativeInteger|)) "\\axiom{log(\\spad{p},{} \\spad{n})} returns the logarithm of \\axiom{\\spad{p}} truncated at order \\axiom{\\spad{n}}.")) (|exp| ((|#3| |#3| (|NonNegativeInteger|)) "\\axiom{exp(\\spad{p},{} \\spad{n})} returns the exponential of \\axiom{\\spad{p}} truncated at order \\axiom{\\spad{n}}."))) 
 NIL 
 NIL 
-(-1288 S -3498) 
+(-1290 S -3505) 
 ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|NonNegativeInteger|)) "\\spad{Frobenius(a,s)} returns \\spad{a**(q**s)} where \\spad{q} is the size()\\$\\spad{F}.") (($ $) "\\spad{Frobenius(a)} returns \\spad{a ** q} where \\spad{q} is the \\spad{size()\\$F}.")) (|transcendenceDegree| (((|NonNegativeInteger|)) "\\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.")) (|extensionDegree| (((|OnePointCompletion| (|PositiveInteger|))) "\\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \\spad{infinity} if it is not.")) (|degree| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{degree(a)} returns the degree of minimal polynomial of an element \\spad{a} if \\spad{a} is algebraic with respect to the ground field \\spad{F},{} and \\spad{infinity} otherwise.")) (|inGroundField?| (((|Boolean|) $) "\\spad{inGroundField?(a)} tests whether an element \\spad{a} is already in the ground field \\spad{F}.")) (|transcendent?| (((|Boolean|) $) "\\spad{transcendent?(a)} tests whether an element \\spad{a} is transcendent with respect to the ground field \\spad{F}.")) (|algebraic?| (((|Boolean|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) 
 NIL 
-((|HasCategory| |#2| (QUOTE (-371))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147)))) 
-(-1289 -3498) 
+((|HasCategory| |#2| (QUOTE (-372))) (|HasCategory| |#2| (QUOTE (-145))) (|HasCategory| |#2| (QUOTE (-147)))) 
+(-1291 -3505) 
 ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|NonNegativeInteger|)) "\\spad{Frobenius(a,s)} returns \\spad{a**(q**s)} where \\spad{q} is the size()\\$\\spad{F}.") (($ $) "\\spad{Frobenius(a)} returns \\spad{a ** q} where \\spad{q} is the \\spad{size()\\$F}.")) (|transcendenceDegree| (((|NonNegativeInteger|)) "\\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.")) (|extensionDegree| (((|OnePointCompletion| (|PositiveInteger|))) "\\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \\spad{infinity} if it is not.")) (|degree| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{degree(a)} returns the degree of minimal polynomial of an element \\spad{a} if \\spad{a} is algebraic with respect to the ground field \\spad{F},{} and \\spad{infinity} otherwise.")) (|inGroundField?| (((|Boolean|) $) "\\spad{inGroundField?(a)} tests whether an element \\spad{a} is already in the ground field \\spad{F}.")) (|transcendent?| (((|Boolean|) $) "\\spad{transcendent?(a)} tests whether an element \\spad{a} is transcendent with respect to the ground field \\spad{F}.")) (|algebraic?| (((|Boolean|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) 
-((-4419 . T) (-4425 . T) (-4420 . T) ((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+((-4426 . T) (-4432 . T) (-4427 . T) ((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
-(-1290 |vl| R) 
+(-1292 |vl| R) 
 ((|constructor| (NIL "This category specifies opeations for polynomials and formal series with non-commutative variables.")) (|varList| (((|List| |#1|) $) "\\spad{varList(x)} returns the list of variables which appear in \\spad{x}.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(fn,x)} returns \\spad{Sum(fn(r_i) w_i)} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|sh| (($ $ (|NonNegativeInteger|)) "\\spad{sh(x,n)} returns the shuffle power of \\spad{x} to the \\spad{n}.") (($ $ $) "\\spad{sh(x,y)} returns the shuffle-product of \\spad{x} by \\spad{y}. This multiplication is associative and commutative.")) (|quasiRegular| (($ $) "\\spad{quasiRegular(x)} return \\spad{x} minus its constant term.")) (|quasiRegular?| (((|Boolean|) $) "\\spad{quasiRegular?(x)} return \\spad{true} if \\spad{constant(x)} is zero.")) (|constant| ((|#2| $) "\\spad{constant(x)} returns the constant term of \\spad{x}.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(x)} returns \\spad{true} if \\spad{x} is constant.")) (|coerce| (($ |#1|) "\\spad{coerce(v)} returns \\spad{v}.")) (|mirror| (($ $) "\\spad{mirror(x)} returns \\spad{Sum(r_i mirror(w_i))} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} returns \\spad{true} if \\spad{x} is a monomial")) (|monom| (($ (|OrderedFreeMonoid| |#1|) |#2|) "\\spad{monom(w,r)} returns the product of the word \\spad{w} by the coefficient \\spad{r}.")) (|rquo| (($ $ $) "\\spad{rquo(x,y)} returns the right simplification of \\spad{x} by \\spad{y}.") (($ $ (|OrderedFreeMonoid| |#1|)) "\\spad{rquo(x,w)} returns the right simplification of \\spad{x} by \\spad{w}.") (($ $ |#1|) "\\spad{rquo(x,v)} returns the right simplification of \\spad{x} by the variable \\spad{v}.")) (|lquo| (($ $ $) "\\spad{lquo(x,y)} returns the left simplification of \\spad{x} by \\spad{y}.") (($ $ (|OrderedFreeMonoid| |#1|)) "\\spad{lquo(x,w)} returns the left simplification of \\spad{x} by the word \\spad{w}.") (($ $ |#1|) "\\spad{lquo(x,v)} returns the left simplification of \\spad{x} by the variable \\spad{v}.")) (|coef| ((|#2| $ $) "\\spad{coef(x,y)} returns scalar product of \\spad{x} by \\spad{y},{} the set of words being regarded as an orthogonal basis.") ((|#2| $ (|OrderedFreeMonoid| |#1|)) "\\spad{coef(x,w)} returns the coefficient of the word \\spad{w} in \\spad{x}.")) (|mindegTerm| (((|Record| (|:| |k| (|OrderedFreeMonoid| |#1|)) (|:| |c| |#2|)) $) "\\spad{mindegTerm(x)} returns the term whose word is \\spad{mindeg(x)}.")) (|mindeg| (((|OrderedFreeMonoid| |#1|) $) "\\spad{mindeg(x)} returns the little word which appears in \\spad{x}. Error if \\spad{x=0}.")) (* (($ $ |#2|) "\\spad{x * r} returns the product of \\spad{x} by \\spad{r}. Usefull if \\spad{R} is a non-commutative Ring.") (($ |#1| $) "\\spad{v * x} returns the product of a variable \\spad{x} by \\spad{x}."))) 
-((-4420 |has| |#2| (-6 -4420)) (-4422 . T) (-4421 . T) (-4424 . T)) 
+((-4427 |has| |#2| (-6 -4427)) (-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
-(-1291 |VarSet| R) 
+(-1293 |VarSet| R) 
 ((|constructor| (NIL "This domain constructor implements polynomials in non-commutative variables written in the Poincare-Birkhoff-Witt basis from the Lyndon basis. These polynomials can be used to compute Baker-Campbell-Hausdorff relations. \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr}).")) (|log| (($ $ (|NonNegativeInteger|)) "\\axiom{log(\\spad{p},{}\\spad{n})} returns the logarithm of \\axiom{\\spad{p}} (truncated up to order \\axiom{\\spad{n}}).")) (|exp| (($ $ (|NonNegativeInteger|)) "\\axiom{exp(\\spad{p},{}\\spad{n})} returns the exponential of \\axiom{\\spad{p}} (truncated up to order \\axiom{\\spad{n}}).")) (|product| (($ $ $ (|NonNegativeInteger|)) "\\axiom{product(a,{}\\spad{b},{}\\spad{n})} returns \\axiom{a*b} (truncated up to order \\axiom{\\spad{n}}).")) (|LiePolyIfCan| (((|Union| (|LiePolynomial| |#1| |#2|) "failed") $) "\\axiom{LiePolyIfCan(\\spad{p})} return \\axiom{\\spad{p}} if \\axiom{\\spad{p}} is a Lie polynomial.")) (|coerce| (((|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{p})} returns \\axiom{\\spad{p}} as a recursive polynomial.") (((|XDistributedPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{p})} returns \\axiom{\\spad{p}} as a distributed polynomial.") (($ (|LiePolynomial| |#1| |#2|)) "\\axiom{coerce(\\spad{p})} returns \\axiom{\\spad{p}}."))) 
-((-4420 |has| |#2| (-6 -4420)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -720) (LIST (QUOTE -411) (QUOTE (-550))))) (|HasAttribute| |#2| (QUOTE -4420))) 
-(-1292 R) 
+((-4427 |has| |#2| (-6 -4427)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#2| (QUOTE (-173))) (|HasCategory| |#2| (LIST (QUOTE -722) (LIST (QUOTE -412) (QUOTE (-551))))) (|HasAttribute| |#2| (QUOTE -4427))) 
+(-1294 R) 
 ((|constructor| (NIL "\\indented{2}{This type supports multivariate polynomials} whose set of variables is \\spadtype{Symbol}. The representation is recursive. The coefficient ring may be non-commutative and the variables do not commute. However,{} coefficients and variables commute."))) 
-((-4420 |has| |#1| (-6 -4420)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasAttribute| |#1| (QUOTE -4420))) 
-(-1293 |vl| R) 
+((-4427 |has| |#1| (-6 -4427)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasAttribute| |#1| (QUOTE -4427))) 
+(-1295 |vl| R) 
 ((|constructor| (NIL "The Category of polynomial rings with non-commutative variables. The coefficient ring may be non-commutative too. However coefficients commute with vaiables.")) (|trunc| (($ $ (|NonNegativeInteger|)) "\\spad{trunc(p,n)} returns the polynomial \\spad{p} truncated at order \\spad{n}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} returns the degree of \\spad{p}. \\indented{1}{Note that the degree of a word is its length.}")) (|maxdeg| (((|OrderedFreeMonoid| |#1|) $) "\\spad{maxdeg(p)} returns the greatest leading word in the support of \\spad{p}."))) 
-((-4420 |has| |#2| (-6 -4420)) (-4422 . T) (-4421 . T) (-4424 . T)) 
+((-4427 |has| |#2| (-6 -4427)) (-4429 . T) (-4428 . T) (-4431 . T)) 
 NIL 
-(-1294 R E) 
+(-1296 R E) 
 ((|constructor| (NIL "This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and words belonging to an arbitrary \\spadtype{OrderedMonoid}. This type is used,{} for instance,{} by the \\spadtype{XDistributedPolynomial} domain constructor where the Monoid is free.")) (|canonicalUnitNormal| ((|attribute|) "canonicalUnitNormal guarantees that the function unitCanonical returns the same representative for all associates of any particular element.")) (/ (($ $ |#1|) "\\spad{p/r} returns \\spad{p*(1/r)}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,x)} returns \\spad{Sum(fn(r_i) w_i)} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|quasiRegular| (($ $) "\\spad{quasiRegular(x)} return \\spad{x} minus its constant term.")) (|quasiRegular?| (((|Boolean|) $) "\\spad{quasiRegular?(x)} return \\spad{true} if \\spad{constant(p)} is zero.")) (|constant| ((|#1| $) "\\spad{constant(p)} return the constant term of \\spad{p}.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(p)} tests whether the polynomial \\spad{p} belongs to the coefficient ring.")) (|coef| ((|#1| $ |#2|) "\\spad{coef(p,e)} extracts the coefficient of the monomial \\spad{e}. Returns zero if \\spad{e} is not present.")) (|reductum| (($ $) "\\spad{reductum(p)} returns \\spad{p} minus its leading term. An error is produced if \\spad{p} is zero.")) (|mindeg| ((|#2| $) "\\spad{mindeg(p)} returns the smallest word occurring in the polynomial \\spad{p} with a non-zero coefficient. An error is produced if \\spad{p} is zero.")) (|maxdeg| ((|#2| $) "\\spad{maxdeg(p)} returns the greatest word occurring in the polynomial \\spad{p} with a non-zero coefficient. An error is produced if \\spad{p} is zero.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# p} returns the number of terms in \\spad{p}.")) (* (($ $ |#1|) "\\spad{p*r} returns the product of \\spad{p} by \\spad{r}."))) 
-((-4424 . T) (-4425 |has| |#1| (-6 -4425)) (-4420 |has| |#1| (-6 -4420)) (-4422 . T) (-4421 . T)) 
-((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-366))) (|HasAttribute| |#1| (QUOTE -4424)) (|HasAttribute| |#1| (QUOTE -4425)) (|HasAttribute| |#1| (QUOTE -4420))) 
-(-1295 |VarSet| R) 
+((-4431 . T) (-4432 |has| |#1| (-6 -4432)) (-4427 |has| |#1| (-6 -4427)) (-4429 . T) (-4428 . T)) 
+((|HasCategory| |#1| (QUOTE (-173))) (|HasCategory| |#1| (QUOTE (-367))) (|HasAttribute| |#1| (QUOTE -4431)) (|HasAttribute| |#1| (QUOTE -4432)) (|HasAttribute| |#1| (QUOTE -4427))) 
+(-1297 |VarSet| R) 
 ((|constructor| (NIL "\\indented{2}{This type supports multivariate polynomials} whose variables do not commute. The representation is recursive. The coefficient ring may be non-commutative. Coefficients and variables commute.")) (|RemainderList| (((|List| (|Record| (|:| |k| |#1|) (|:| |c| $))) $) "\\spad{RemainderList(p)} returns the regular part of \\spad{p} as a list of terms.")) (|unexpand| (($ (|XDistributedPolynomial| |#1| |#2|)) "\\spad{unexpand(p)} returns \\spad{p} in recursive form.")) (|expand| (((|XDistributedPolynomial| |#1| |#2|) $) "\\spad{expand(p)} returns \\spad{p} in distributed form."))) 
-((-4420 |has| |#2| (-6 -4420)) (-4422 . T) (-4421 . T) (-4424 . T)) 
-((|HasCategory| |#2| (QUOTE (-173))) (|HasAttribute| |#2| (QUOTE -4420))) 
-(-1296 A) 
+((-4427 |has| |#2| (-6 -4427)) (-4429 . T) (-4428 . T) (-4431 . T)) 
+((|HasCategory| |#2| (QUOTE (-173))) (|HasAttribute| |#2| (QUOTE -4427))) 
+(-1298 A) 
 ((|constructor| (NIL "This package implements fixed-point computations on streams.")) (Y (((|List| (|Stream| |#1|)) (|Mapping| (|List| (|Stream| |#1|)) (|List| (|Stream| |#1|))) (|Integer|)) "\\spad{Y(g,n)} computes a fixed point of the function \\spad{g},{} where \\spad{g} takes a list of \\spad{n} streams and returns a list of \\spad{n} streams.") (((|Stream| |#1|) (|Mapping| (|Stream| |#1|) (|Stream| |#1|))) "\\spad{Y(f)} computes a fixed point of the function \\spad{f}."))) 
 NIL 
 NIL 
-(-1297 R |ls| |ls2|) 
+(-1299 R |ls| |ls2|) 
 ((|constructor| (NIL "A package for computing symbolically the complex and real roots of zero-dimensional algebraic systems over the integer or rational numbers. Complex roots are given by means of univariate representations of irreducible regular chains. Real roots are given by means of tuples of coordinates lying in the \\spadtype{RealClosure} of the coefficient ring. This constructor takes three arguments. The first one \\spad{R} is the coefficient ring. The second one \\spad{ls} is the list of variables involved in the systems to solve. The third one must be \\spad{concat(ls,s)} where \\spad{s} is an additional symbol used for the univariate representations. WARNING: The third argument is not checked. All operations are based on triangular decompositions. The default is to compute these decompositions directly from the input system by using the \\spadtype{RegularChain} domain constructor. The lexTriangular algorithm can also be used for computing these decompositions (see the \\spadtype{LexTriangularPackage} package constructor). For that purpose,{} the operations \\axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage},{} \\axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage} and \\axiomOpFrom{positiveSolve}{ZeroDimensionalSolvePackage} admit an optional argument. \\newline Author: Marc Moreno Maza.")) (|convert| (((|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|))) (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#3|)) (|OrderedVariableList| |#3|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|)))) "\\spad{convert(st)} returns the members of \\spad{st}.") (((|SparseUnivariatePolynomial| (|RealClosure| (|Fraction| |#1|))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{convert(u)} converts \\spad{u}.") (((|Polynomial| (|RealClosure| (|Fraction| |#1|))) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|))) "\\spad{convert(q)} converts \\spad{q}.") (((|Polynomial| (|RealClosure| (|Fraction| |#1|))) (|Polynomial| |#1|)) "\\spad{convert(p)} converts \\spad{p}.") (((|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|)) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) "\\spad{convert(q)} converts \\spad{q}.")) (|squareFree| (((|List| (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#3|)) (|OrderedVariableList| |#3|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|)))) (|RegularChain| |#1| |#2|)) "\\spad{squareFree(ts)} returns the square-free factorization of \\spad{ts}. Moreover,{} each factor is a Lazard triangular set and the decomposition is a Kalkbrener split of \\spad{ts},{} which is enough here for the matter of solving zero-dimensional algebraic systems. WARNING: \\spad{ts} is not checked to be zero-dimensional.")) (|positiveSolve| (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|))) "\\spad{positiveSolve(lp)} returns the same as \\spad{positiveSolve(lp,false,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{positiveSolve(lp)} returns the same as \\spad{positiveSolve(lp,info?,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{positiveSolve(lp,info?,lextri?)} returns the set of the points in the variety associated with \\spad{lp} whose coordinates are (real) strictly positive. Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during decomposition into regular chains. If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\\spad{lp},{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}. WARNING: For each set of coordinates given by \\spad{positiveSolve(lp,info?,lextri?)} the ordering of the indeterminates is reversed \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ls}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|RegularChain| |#1| |#2|)) "\\spad{positiveSolve(ts)} returns the points of the regular set of \\spad{ts} with (real) strictly positive coordinates.")) (|realSolve| (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|))) "\\spad{realSolve(lp)} returns the same as \\spad{realSolve(ts,false,false,false)}") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{realSolve(ts,info?)} returns the same as \\spad{realSolve(ts,info?,false,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{realSolve(ts,info?,check?)} returns the same as \\spad{realSolve(ts,info?,check?,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|) (|Boolean|)) "\\spad{realSolve(ts,info?,check?,lextri?)} returns the set of the points in the variety associated with \\spad{lp} whose coordinates are all real. Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during decomposition into regular chains. If \\spad{check?} is \\spad{true} then the result is checked. If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\\spad{lp},{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}. WARNING: For each set of coordinates given by \\spad{realSolve(ts,info?,check?,lextri?)} the ordering of the indeterminates is reversed \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ls}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|RegularChain| |#1| |#2|)) "\\spad{realSolve(ts)} returns the set of the points in the regular zero set of \\spad{ts} whose coordinates are all real. WARNING: For each set of coordinates given by \\spad{realSolve(ts)} the ordering of the indeterminates is reversed \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ls}.")) (|univariateSolve| (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{univariateSolve(lp)} returns the same as \\spad{univariateSolve(lp,false,false,false)}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{univariateSolve(lp,info?)} returns the same as \\spad{univariateSolve(lp,info?,false,false)}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{univariateSolve(lp,info?,check?)} returns the same as \\spad{univariateSolve(lp,info?,check?,false)}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|) (|Boolean|)) "\\spad{univariateSolve(lp,info?,check?,lextri?)} returns a univariate representation of the variety associated with \\spad{lp}. Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during the decomposition into regular chains. If \\spad{check?} is \\spad{true} then the result is checked. See \\axiomOpFrom{rur}{RationalUnivariateRepresentationPackage}(\\spad{lp},{}\\spad{true}). If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\\spad{lp},{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|RegularChain| |#1| |#2|)) "\\spad{univariateSolve(ts)} returns a univariate representation of \\spad{ts}. See \\axiomOpFrom{rur}{RationalUnivariateRepresentationPackage}(\\spad{lp},{}\\spad{true}).")) (|triangSolve| (((|List| (|RegularChain| |#1| |#2|)) (|List| (|Polynomial| |#1|))) "\\spad{triangSolve(lp)} returns the same as \\spad{triangSolve(lp,false,false)}") (((|List| (|RegularChain| |#1| |#2|)) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{triangSolve(lp,info?)} returns the same as \\spad{triangSolve(lp,false)}") (((|List| (|RegularChain| |#1| |#2|)) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{triangSolve(lp,info?,lextri?)} decomposes the variety associated with \\axiom{\\spad{lp}} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{\\spad{lp}} needs to generate a zero-dimensional ideal. If \\axiom{\\spad{lp}} is not zero-dimensional then the result is only a decomposition of its zero-set in the sense of the closure (\\spad{w}.\\spad{r}.\\spad{t}. Zarisky topology). Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during the computations. See \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory}(\\spad{lp},{}\\spad{true},{}\\spad{info?}). If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\\spad{lp},{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}."))) 
 NIL 
 NIL 
-(-1298 R) 
+(-1300 R) 
 ((|constructor| (NIL "Test for linear dependence over the integers.")) (|solveLinearlyOverQ| (((|Union| (|Vector| (|Fraction| (|Integer|))) "failed") (|Vector| |#1|) |#1|) "\\spad{solveLinearlyOverQ([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such rational numbers \\spad{ci}\\spad{'s} exist.")) (|linearDependenceOverZ| (((|Union| (|Vector| (|Integer|)) "failed") (|Vector| |#1|)) "\\spad{linearlyDependenceOverZ([v1,...,vn])} returns \\spad{[c1,...,cn]} if \\spad{c1*v1 + ... + cn*vn = 0} and not all the \\spad{ci}\\spad{'s} are 0,{} \"failed\" if the \\spad{vi}\\spad{'s} are linearly independent over the integers.")) (|linearlyDependentOverZ?| (((|Boolean|) (|Vector| |#1|)) "\\spad{linearlyDependentOverZ?([v1,...,vn])} returns \\spad{true} if the \\spad{vi}\\spad{'s} are linearly dependent over the integers,{} \\spad{false} otherwise."))) 
 NIL 
 NIL 
-(-1299 |p|) 
+(-1301 |p|) 
 ((|constructor| (NIL "IntegerMod(\\spad{n}) creates the ring of integers reduced modulo the integer \\spad{n}."))) 
-(((-4429 "*") . T) (-4421 . T) (-4422 . T) (-4424 . T)) 
+(((-4436 "*") . T) (-4428 . T) (-4429 . T) (-4431 . T)) 
 NIL 
 NIL 
 NIL 
@@ -5144,4 +5152,4 @@ NIL
 NIL 
 NIL 
 NIL 
-((-3 NIL 2263889 2263894 2263899 2263904) (-2 NIL 2263869 2263874 2263879 2263884) (-1 NIL 2263849 2263854 2263859 2263864) (0 NIL 2263829 2263834 2263839 2263844) (-1299 "ZMOD.spad" 2263638 2263651 2263767 2263824) (-1298 "ZLINDEP.spad" 2262704 2262715 2263628 2263633) (-1297 "ZDSOLVE.spad" 2252649 2252671 2262694 2262699) (-1296 "YSTREAM.spad" 2252144 2252155 2252639 2252644) (-1295 "XRPOLY.spad" 2251364 2251384 2252000 2252069) (-1294 "XPR.spad" 2249159 2249172 2251082 2251181) (-1293 "XPOLYC.spad" 2248478 2248494 2249085 2249154) (-1292 "XPOLY.spad" 2248033 2248044 2248334 2248403) (-1291 "XPBWPOLY.spad" 2246470 2246490 2247813 2247882) (-1290 "XFALG.spad" 2243518 2243534 2246396 2246465) (-1289 "XF.spad" 2241981 2241996 2243420 2243513) (-1288 "XF.spad" 2240424 2240441 2241865 2241870) (-1287 "XEXPPKG.spad" 2239675 2239701 2240414 2240419) (-1286 "XDPOLY.spad" 2239289 2239305 2239531 2239600) (-1285 "XALG.spad" 2238949 2238960 2239245 2239284) (-1284 "WUTSET.spad" 2234788 2234805 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2142098 2142598 2142603) (-1246 "UPOLYC.spad" 2137119 2137130 2141921 2142074) (-1245 "UPOLYC.spad" 2132051 2132064 2136855 2136860) (-1244 "UPMP.spad" 2130951 2130964 2132041 2132046) (-1243 "UPDIVP.spad" 2130516 2130530 2130941 2130946) (-1242 "UPDECOMP.spad" 2128761 2128775 2130506 2130511) (-1241 "UPCDEN.spad" 2127970 2127986 2128751 2128756) (-1240 "UP2.spad" 2127334 2127355 2127960 2127965) (-1239 "UP.spad" 2124533 2124548 2124920 2125073) (-1238 "UNISEG2.spad" 2124030 2124043 2124489 2124494) (-1237 "UNISEG.spad" 2123383 2123394 2123949 2123954) (-1236 "UNIFACT.spad" 2122486 2122498 2123373 2123378) (-1235 "ULSCONS.spad" 2114882 2114902 2115252 2115401) (-1234 "ULSCCAT.spad" 2112619 2112639 2114728 2114877) (-1233 "ULSCCAT.spad" 2110464 2110486 2112575 2112580) (-1232 "ULSCAT.spad" 2108696 2108712 2110310 2110459) (-1231 "ULS2.spad" 2108210 2108263 2108686 2108691) (-1230 "ULS.spad" 2098768 2098796 2099855 2100284) (-1229 "UINT8.spad" 2098645 2098654 2098758 2098763) (-1228 "UINT64.spad" 2098521 2098530 2098635 2098640) (-1227 "UINT32.spad" 2098397 2098406 2098511 2098516) (-1226 "UINT16.spad" 2098273 2098282 2098387 2098392) (-1225 "UFD.spad" 2097338 2097347 2098199 2098268) (-1224 "UFD.spad" 2096465 2096476 2097328 2097333) (-1223 "UDVO.spad" 2095346 2095355 2096455 2096460) (-1222 "UDPO.spad" 2092839 2092850 2095302 2095307) (-1221 "TYPEAST.spad" 2092758 2092767 2092829 2092834) (-1220 "TYPE.spad" 2092690 2092699 2092748 2092753) (-1219 "TWOFACT.spad" 2091342 2091357 2092680 2092685) (-1218 "TUPLE.spad" 2090828 2090839 2091241 2091246) (-1217 "TUBETOOL.spad" 2087695 2087704 2090818 2090823) (-1216 "TUBE.spad" 2086342 2086359 2087685 2087690) (-1215 "TSETCAT.spad" 2073469 2073486 2086310 2086337) (-1214 "TSETCAT.spad" 2060582 2060601 2073425 2073430) (-1213 "TS.spad" 2059181 2059197 2060147 2060244) (-1212 "TRMANIP.spad" 2053547 2053564 2058887 2058892) (-1211 "TRIMAT.spad" 2052510 2052535 2053537 2053542) (-1210 "TRIGMNIP.spad" 2051037 2051054 2052500 2052505) (-1209 "TRIGCAT.spad" 2050549 2050558 2051027 2051032) (-1208 "TRIGCAT.spad" 2050059 2050070 2050539 2050544) (-1207 "TREE.spad" 2048634 2048645 2049666 2049693) (-1206 "TRANFUN.spad" 2048473 2048482 2048624 2048629) (-1205 "TRANFUN.spad" 2048310 2048321 2048463 2048468) (-1204 "TOPSP.spad" 2047984 2047993 2048300 2048305) (-1203 "TOOLSIGN.spad" 2047647 2047658 2047974 2047979) (-1202 "TEXTFILE.spad" 2046208 2046217 2047637 2047642) (-1201 "TEX1.spad" 2045764 2045775 2046198 2046203) (-1200 "TEX.spad" 2042910 2042919 2045754 2045759) (-1199 "TEMUTL.spad" 2042465 2042474 2042900 2042905) (-1198 "TBCMPPK.spad" 2040558 2040581 2042455 2042460) (-1197 "TBAGG.spad" 2039608 2039631 2040538 2040553) (-1196 "TBAGG.spad" 2038666 2038691 2039598 2039603) (-1195 "TANEXP.spad" 2038074 2038085 2038656 2038661) (-1194 "TABLEAU.spad" 2037555 2037566 2038064 2038069) (-1193 "TABLE.spad" 2035966 2035989 2036236 2036263) (-1192 "TABLBUMP.spad" 2032769 2032780 2035956 2035961) (-1191 "SYSTEM.spad" 2031997 2032006 2032759 2032764) (-1190 "SYSSOLP.spad" 2029480 2029491 2031987 2031992) (-1189 "SYSPTR.spad" 2029379 2029388 2029470 2029475) (-1188 "SYSNNI.spad" 2028561 2028572 2029369 2029374) (-1187 "SYSINT.spad" 2027965 2027976 2028551 2028556) (-1186 "SYNTAX.spad" 2024171 2024180 2027955 2027960) (-1185 "SYMTAB.spad" 2022239 2022248 2024161 2024166) (-1184 "SYMS.spad" 2018268 2018277 2022229 2022234) (-1183 "SYMPOLY.spad" 2017275 2017286 2017357 2017484) (-1182 "SYMFUNC.spad" 2016776 2016787 2017265 2017270) (-1181 "SYMBOL.spad" 2014279 2014288 2016766 2016771) (-1180 "SWITCH.spad" 2011050 2011059 2014269 2014274) (-1179 "SUTS.spad" 2007955 2007983 2009517 2009614) (-1178 "SUPXS.spad" 2005096 2005124 2006087 2006236) (-1177 "SUPFRACF.spad" 2004201 2004219 2005086 2005091) (-1176 "SUP2.spad" 2003593 2003606 2004191 2004196) (-1175 "SUP.spad" 2000406 2000417 2001179 2001332) (-1174 "SUMRF.spad" 1999380 1999391 2000396 2000401) (-1173 "SUMFS.spad" 1999017 1999034 1999370 1999375) (-1172 "SULS.spad" 1989562 1989590 1990662 1991091) (-1171 "SUCHTAST.spad" 1989331 1989340 1989552 1989557) (-1170 "SUCH.spad" 1989013 1989028 1989321 1989326) (-1169 "SUBSPACE.spad" 1981128 1981143 1989003 1989008) (-1168 "SUBRESP.spad" 1980298 1980312 1981084 1981089) (-1167 "STTFNC.spad" 1976766 1976782 1980288 1980293) (-1166 "STTF.spad" 1972865 1972881 1976756 1976761) (-1165 "STTAYLOR.spad" 1965500 1965511 1972746 1972751) (-1164 "STRTBL.spad" 1964005 1964022 1964154 1964181) (-1163 "STRING.spad" 1963414 1963423 1963428 1963455) (-1162 "STRICAT.spad" 1963202 1963211 1963382 1963409) (-1161 "STREAM3.spad" 1962775 1962790 1963192 1963197) (-1160 "STREAM2.spad" 1961903 1961916 1962765 1962770) (-1159 "STREAM1.spad" 1961609 1961620 1961893 1961898) (-1158 "STREAM.spad" 1958527 1958538 1961134 1961149) (-1157 "STINPROD.spad" 1957463 1957479 1958517 1958522) (-1156 "STEPAST.spad" 1956697 1956706 1957453 1957458) (-1155 "STEP.spad" 1955898 1955907 1956687 1956692) (-1154 "STBL.spad" 1954424 1954452 1954591 1954606) (-1153 "STAGG.spad" 1953499 1953510 1954414 1954419) (-1152 "STAGG.spad" 1952572 1952585 1953489 1953494) (-1151 "STACK.spad" 1951929 1951940 1952179 1952206) (-1150 "SREGSET.spad" 1949633 1949650 1951575 1951602) (-1149 "SRDCMPK.spad" 1948194 1948214 1949623 1949628) (-1148 "SRAGG.spad" 1943337 1943346 1948162 1948189) (-1147 "SRAGG.spad" 1938500 1938511 1943327 1943332) (-1146 "SQMATRIX.spad" 1936116 1936134 1937032 1937119) (-1145 "SPLTREE.spad" 1930668 1930681 1935552 1935579) (-1144 "SPLNODE.spad" 1927256 1927269 1930658 1930663) (-1143 "SPFCAT.spad" 1926065 1926074 1927246 1927251) (-1142 "SPECOUT.spad" 1924617 1924626 1926055 1926060) (-1141 "SPADXPT.spad" 1916212 1916221 1924607 1924612) (-1140 "spad-parser.spad" 1915677 1915686 1916202 1916207) (-1139 "SPADAST.spad" 1915378 1915387 1915667 1915672) (-1138 "SPACEC.spad" 1899577 1899588 1915368 1915373) (-1137 "SPACE3.spad" 1899353 1899364 1899567 1899572) (-1136 "SORTPAK.spad" 1898902 1898915 1899309 1899314) (-1135 "SOLVETRA.spad" 1896665 1896676 1898892 1898897) (-1134 "SOLVESER.spad" 1895193 1895204 1896655 1896660) (-1133 "SOLVERAD.spad" 1891219 1891230 1895183 1895188) (-1132 "SOLVEFOR.spad" 1889681 1889699 1891209 1891214) (-1131 "SNTSCAT.spad" 1889281 1889298 1889649 1889676) (-1130 "SMTS.spad" 1887553 1887579 1888846 1888943) (-1129 "SMP.spad" 1885028 1885048 1885418 1885545) (-1128 "SMITH.spad" 1883873 1883898 1885018 1885023) (-1127 "SMATCAT.spad" 1881983 1882013 1883817 1883868) (-1126 "SMATCAT.spad" 1880025 1880057 1881861 1881866) (-1125 "SKAGG.spad" 1878988 1878999 1879993 1880020) (-1124 "SINT.spad" 1877820 1877829 1878854 1878983) (-1123 "SIMPAN.spad" 1877548 1877557 1877810 1877815) (-1122 "SIGNRF.spad" 1876673 1876684 1877538 1877543) (-1121 "SIGNEF.spad" 1875959 1875976 1876663 1876668) (-1120 "SIGAST.spad" 1875344 1875353 1875949 1875954) (-1119 "SIG.spad" 1874674 1874683 1875334 1875339) (-1118 "SHP.spad" 1872602 1872617 1874630 1874635) (-1117 "SHDP.spad" 1862313 1862340 1862822 1862953) (-1116 "SGROUP.spad" 1861921 1861930 1862303 1862308) (-1115 "SGROUP.spad" 1861527 1861538 1861911 1861916) (-1114 "SGCF.spad" 1854690 1854699 1861517 1861522) (-1113 "SFRTCAT.spad" 1853620 1853637 1854658 1854685) (-1112 "SFRGCD.spad" 1852683 1852703 1853610 1853615) (-1111 "SFQCMPK.spad" 1847320 1847340 1852673 1852678) (-1110 "SFORT.spad" 1846759 1846773 1847310 1847315) (-1109 "SEXOF.spad" 1846602 1846642 1846749 1846754) (-1108 "SEXCAT.spad" 1844203 1844243 1846592 1846597) (-1107 "SEX.spad" 1844095 1844104 1844193 1844198) (-1106 "SETMN.spad" 1842547 1842564 1844085 1844090) (-1105 "SETCAT.spad" 1841869 1841878 1842537 1842542) (-1104 "SETCAT.spad" 1841189 1841200 1841859 1841864) (-1103 "SETAGG.spad" 1837738 1837749 1841169 1841184) (-1102 "SETAGG.spad" 1834295 1834308 1837728 1837733) (-1101 "SET.spad" 1832619 1832630 1833716 1833755) (-1100 "SEQAST.spad" 1832322 1832331 1832609 1832614) (-1099 "SEGXCAT.spad" 1831478 1831491 1832312 1832317) (-1098 "SEGCAT.spad" 1830403 1830414 1831468 1831473) (-1097 "SEGBIND2.spad" 1830101 1830114 1830393 1830398) (-1096 "SEGBIND.spad" 1829859 1829870 1830048 1830053) (-1095 "SEGAST.spad" 1829573 1829582 1829849 1829854) (-1094 "SEG2.spad" 1829008 1829021 1829529 1829534) (-1093 "SEG.spad" 1828821 1828832 1828927 1828932) (-1092 "SDVAR.spad" 1828097 1828108 1828811 1828816) (-1091 "SDPOL.spad" 1825523 1825534 1825814 1825941) (-1090 "SCPKG.spad" 1823612 1823623 1825513 1825518) (-1089 "SCOPE.spad" 1822765 1822774 1823602 1823607) (-1088 "SCACHE.spad" 1821461 1821472 1822755 1822760) (-1087 "SASTCAT.spad" 1821370 1821379 1821451 1821456) (-1086 "SAOS.spad" 1821242 1821251 1821360 1821365) (-1085 "SAERFFC.spad" 1820955 1820975 1821232 1821237) (-1084 "SAEFACT.spad" 1820656 1820676 1820945 1820950) (-1083 "SAE.spad" 1818831 1818847 1819442 1819577) (-1082 "RURPK.spad" 1816490 1816506 1818821 1818826) (-1081 "RULESET.spad" 1815943 1815967 1816480 1816485) (-1080 "RULECOLD.spad" 1815795 1815808 1815933 1815938) (-1079 "RULE.spad" 1814035 1814059 1815785 1815790) (-1078 "RTVALUE.spad" 1813770 1813779 1814025 1814030) (-1077 "RSTRCAST.spad" 1813487 1813496 1813760 1813765) (-1076 "RSETGCD.spad" 1809865 1809885 1813477 1813482) (-1075 "RSETCAT.spad" 1799801 1799818 1809833 1809860) (-1074 "RSETCAT.spad" 1789757 1789776 1799791 1799796) (-1073 "RSDCMPK.spad" 1788209 1788229 1789747 1789752) (-1072 "RRCC.spad" 1786593 1786623 1788199 1788204) (-1071 "RRCC.spad" 1784975 1785007 1786583 1786588) (-1070 "RPTAST.spad" 1784677 1784686 1784965 1784970) (-1069 "RPOLCAT.spad" 1764037 1764052 1784545 1784672) (-1068 "RPOLCAT.spad" 1743111 1743128 1763621 1763626) (-1067 "ROUTINE.spad" 1738994 1739003 1741758 1741785) (-1066 "ROMAN.spad" 1738322 1738331 1738860 1738989) (-1065 "ROIRC.spad" 1737402 1737434 1738312 1738317) (-1064 "RNS.spad" 1736305 1736314 1737304 1737397) (-1063 "RNS.spad" 1735294 1735305 1736295 1736300) (-1062 "RNGBIND.spad" 1734454 1734468 1735249 1735254) (-1061 "RNG.spad" 1734189 1734198 1734444 1734449) (-1060 "RMODULE.spad" 1733954 1733965 1734179 1734184) (-1059 "RMCAT2.spad" 1733374 1733431 1733944 1733949) (-1058 "RMATRIX.spad" 1732198 1732217 1732541 1732580) (-1057 "RMATCAT.spad" 1727777 1727808 1732154 1732193) (-1056 "RMATCAT.spad" 1723246 1723279 1727625 1727630) (-1055 "RLINSET.spad" 1722640 1722651 1723236 1723241) (-1054 "RINTERP.spad" 1722528 1722548 1722630 1722635) (-1053 "RING.spad" 1721998 1722007 1722508 1722523) (-1052 "RING.spad" 1721476 1721487 1721988 1721993) (-1051 "RIDIST.spad" 1720868 1720877 1721466 1721471) (-1050 "RGCHAIN.spad" 1719451 1719467 1720353 1720380) (-1049 "RGBCSPC.spad" 1719232 1719244 1719441 1719446) (-1048 "RGBCMDL.spad" 1718762 1718774 1719222 1719227) (-1047 "RFFACTOR.spad" 1718224 1718235 1718752 1718757) (-1046 "RFFACT.spad" 1717959 1717971 1718214 1718219) (-1045 "RFDIST.spad" 1716955 1716964 1717949 1717954) (-1044 "RF.spad" 1714597 1714608 1716945 1716950) (-1043 "RETSOL.spad" 1714016 1714029 1714587 1714592) (-1042 "RETRACT.spad" 1713444 1713455 1714006 1714011) (-1041 "RETRACT.spad" 1712870 1712883 1713434 1713439) (-1040 "RETAST.spad" 1712682 1712691 1712860 1712865) (-1039 "RESULT.spad" 1710742 1710751 1711329 1711356) (-1038 "RESRING.spad" 1710089 1710136 1710680 1710737) (-1037 "RESLATC.spad" 1709413 1709424 1710079 1710084) (-1036 "REPSQ.spad" 1709144 1709155 1709403 1709408) (-1035 "REPDB.spad" 1708851 1708862 1709134 1709139) (-1034 "REP2.spad" 1698509 1698520 1708693 1708698) (-1033 "REP1.spad" 1692705 1692716 1698459 1698464) (-1032 "REP.spad" 1690259 1690268 1692695 1692700) (-1031 "REGSET.spad" 1688056 1688073 1689905 1689932) (-1030 "REF.spad" 1687391 1687402 1688011 1688016) (-1029 "REDORDER.spad" 1686597 1686614 1687381 1687386) (-1028 "RECLOS.spad" 1685380 1685400 1686084 1686177) (-1027 "REALSOLV.spad" 1684520 1684529 1685370 1685375) (-1026 "REAL0Q.spad" 1681818 1681833 1684510 1684515) (-1025 "REAL0.spad" 1678662 1678677 1681808 1681813) (-1024 "REAL.spad" 1678534 1678543 1678652 1678657) (-1023 "RDUCEAST.spad" 1678255 1678264 1678524 1678529) (-1022 "RDIV.spad" 1677910 1677935 1678245 1678250) (-1021 "RDIST.spad" 1677477 1677488 1677900 1677905) (-1020 "RDETRS.spad" 1676341 1676359 1677467 1677472) (-1019 "RDETR.spad" 1674480 1674498 1676331 1676336) (-1018 "RDEEFS.spad" 1673579 1673596 1674470 1674475) (-1017 "RDEEF.spad" 1672589 1672606 1673569 1673574) (-1016 "RCFIELD.spad" 1669775 1669784 1672491 1672584) (-1015 "RCFIELD.spad" 1667047 1667058 1669765 1669770) (-1014 "RCAGG.spad" 1664975 1664986 1667037 1667042) (-1013 "RCAGG.spad" 1662830 1662843 1664894 1664899) (-1012 "RATRET.spad" 1662190 1662201 1662820 1662825) (-1011 "RATFACT.spad" 1661882 1661894 1662180 1662185) (-1010 "RANDSRC.spad" 1661201 1661210 1661872 1661877) (-1009 "RADUTIL.spad" 1660957 1660966 1661191 1661196) (-1008 "RADIX.spad" 1657878 1657892 1659424 1659517) (-1007 "RADFF.spad" 1656291 1656328 1656410 1656566) (-1006 "RADCAT.spad" 1655886 1655895 1656281 1656286) (-1005 "RADCAT.spad" 1655479 1655490 1655876 1655881) (-1004 "QUEUE.spad" 1654827 1654838 1655086 1655113) (-1003 "QUATCT2.spad" 1654447 1654466 1654817 1654822) (-1002 "QUATCAT.spad" 1652617 1652628 1654377 1654442) (-1001 "QUATCAT.spad" 1650538 1650551 1652300 1652305) (-1000 "QUAT.spad" 1649119 1649130 1649462 1649527) (-999 "QUAGG.spad" 1647947 1647957 1649087 1649114) (-998 "QQUTAST.spad" 1647716 1647724 1647937 1647942) (-997 "QFORM.spad" 1647181 1647195 1647706 1647711) (-996 "QFCAT2.spad" 1646874 1646890 1647171 1647176) (-995 "QFCAT.spad" 1645577 1645587 1646776 1646869) (-994 "QFCAT.spad" 1643871 1643883 1645072 1645077) (-993 "QEQUAT.spad" 1643430 1643438 1643861 1643866) (-992 "QCMPACK.spad" 1638177 1638196 1643420 1643425) (-991 "QALGSET2.spad" 1636173 1636191 1638167 1638172) (-990 "QALGSET.spad" 1632254 1632286 1636087 1636092) (-989 "PWFFINTB.spad" 1629670 1629691 1632244 1632249) (-988 "PUSHVAR.spad" 1629009 1629028 1629660 1629665) (-987 "PTRANFN.spad" 1625137 1625147 1628999 1629004) (-986 "PTPACK.spad" 1622225 1622235 1625127 1625132) (-985 "PTFUNC2.spad" 1622048 1622062 1622215 1622220) (-984 "PTCAT.spad" 1621303 1621313 1622016 1622043) (-983 "PSQFR.spad" 1620610 1620634 1621293 1621298) (-982 "PSEUDLIN.spad" 1619496 1619506 1620600 1620605) (-981 "PSETPK.spad" 1604929 1604945 1619374 1619379) (-980 "PSETCAT.spad" 1598849 1598872 1604909 1604924) (-979 "PSETCAT.spad" 1592743 1592768 1598805 1598810) (-978 "PSCURVE.spad" 1591726 1591734 1592733 1592738) (-977 "PSCAT.spad" 1590509 1590538 1591624 1591721) (-976 "PSCAT.spad" 1589382 1589413 1590499 1590504) (-975 "PRTITION.spad" 1588343 1588351 1589372 1589377) (-974 "PRTDAST.spad" 1588062 1588070 1588333 1588338) (-973 "PRS.spad" 1577624 1577641 1588018 1588023) (-972 "PRQAGG.spad" 1577059 1577069 1577592 1577619) (-971 "PROPLOG.spad" 1576358 1576366 1577049 1577054) (-970 "PROPFRML.spad" 1574926 1574937 1576348 1576353) (-969 "PROPERTY.spad" 1574414 1574422 1574916 1574921) (-968 "PRODUCT.spad" 1572096 1572108 1572380 1572435) (-967 "PRINT.spad" 1571848 1571856 1572086 1572091) (-966 "PRIMES.spad" 1570101 1570111 1571838 1571843) (-965 "PRIMELT.spad" 1568182 1568196 1570091 1570096) (-964 "PRIMCAT.spad" 1567809 1567817 1568172 1568177) (-963 "PRIMARR2.spad" 1566576 1566588 1567799 1567804) (-962 "PRIMARR.spad" 1565581 1565591 1565759 1565786) (-961 "PREASSOC.spad" 1564963 1564975 1565571 1565576) (-960 "PR.spad" 1563355 1563367 1564054 1564181) (-959 "PPCURVE.spad" 1562492 1562500 1563345 1563350) (-958 "PORTNUM.spad" 1562267 1562275 1562482 1562487) (-957 "POLYROOT.spad" 1561116 1561138 1562223 1562228) (-956 "POLYLIFT.spad" 1560381 1560404 1561106 1561111) (-955 "POLYCATQ.spad" 1558499 1558521 1560371 1560376) (-954 "POLYCAT.spad" 1551969 1551990 1558367 1558494) (-953 "POLYCAT.spad" 1544777 1544800 1551177 1551182) (-952 "POLY2UP.spad" 1544229 1544243 1544767 1544772) (-951 "POLY2.spad" 1543826 1543838 1544219 1544224) (-950 "POLY.spad" 1541161 1541171 1541676 1541803) (-949 "POLUTIL.spad" 1540102 1540131 1541117 1541122) (-948 "POLTOPOL.spad" 1538850 1538865 1540092 1540097) (-947 "POINT.spad" 1537688 1537698 1537775 1537802) (-946 "PNTHEORY.spad" 1534390 1534398 1537678 1537683) (-945 "PMTOOLS.spad" 1533165 1533179 1534380 1534385) (-944 "PMSYM.spad" 1532714 1532724 1533155 1533160) (-943 "PMQFCAT.spad" 1532305 1532319 1532704 1532709) (-942 "PMPREDFS.spad" 1531759 1531781 1532295 1532300) (-941 "PMPRED.spad" 1531238 1531252 1531749 1531754) (-940 "PMPLCAT.spad" 1530318 1530336 1531170 1531175) (-939 "PMLSAGG.spad" 1529903 1529917 1530308 1530313) (-938 "PMKERNEL.spad" 1529482 1529494 1529893 1529898) (-937 "PMINS.spad" 1529062 1529072 1529472 1529477) (-936 "PMFS.spad" 1528639 1528657 1529052 1529057) (-935 "PMDOWN.spad" 1527929 1527943 1528629 1528634) (-934 "PMASSFS.spad" 1526896 1526912 1527919 1527924) (-933 "PMASS.spad" 1525906 1525914 1526886 1526891) (-932 "PLOTTOOL.spad" 1525686 1525694 1525896 1525901) (-931 "PLOT3D.spad" 1522150 1522158 1525676 1525681) (-930 "PLOT1.spad" 1521307 1521317 1522140 1522145) (-929 "PLOT.spad" 1516230 1516238 1521297 1521302) (-928 "PLEQN.spad" 1503520 1503547 1516220 1516225) (-927 "PINTERPA.spad" 1503304 1503320 1503510 1503515) (-926 "PINTERP.spad" 1502926 1502945 1503294 1503299) (-925 "PID.spad" 1501896 1501904 1502852 1502921) (-924 "PICOERCE.spad" 1501553 1501563 1501886 1501891) (-923 "PI.spad" 1501162 1501170 1501527 1501548) (-922 "PGROEB.spad" 1499763 1499777 1501152 1501157) (-921 "PGE.spad" 1491380 1491388 1499753 1499758) (-920 "PGCD.spad" 1490270 1490287 1491370 1491375) (-919 "PFRPAC.spad" 1489419 1489429 1490260 1490265) (-918 "PFR.spad" 1486082 1486092 1489321 1489414) (-917 "PFOTOOLS.spad" 1485340 1485356 1486072 1486077) (-916 "PFOQ.spad" 1484710 1484728 1485330 1485335) (-915 "PFO.spad" 1484129 1484156 1484700 1484705) (-914 "PFECAT.spad" 1481811 1481819 1484055 1484124) (-913 "PFECAT.spad" 1479521 1479531 1481767 1481772) (-912 "PFBRU.spad" 1477409 1477421 1479511 1479516) (-911 "PFBR.spad" 1474969 1474992 1477399 1477404) (-910 "PF.spad" 1474543 1474555 1474774 1474867) (-909 "PERMGRP.spad" 1469305 1469315 1474533 1474538) (-908 "PERMCAT.spad" 1467863 1467873 1469285 1469300) (-907 "PERMAN.spad" 1466395 1466409 1467853 1467858) (-906 "PERM.spad" 1462080 1462090 1466225 1466240) (-905 "PENDTREE.spad" 1461421 1461431 1461709 1461714) (-904 "PDRING.spad" 1459972 1459982 1461401 1461416) (-903 "PDRING.spad" 1458531 1458543 1459962 1459967) (-902 "PDEPROB.spad" 1457546 1457554 1458521 1458526) (-901 "PDEPACK.spad" 1451586 1451594 1457536 1457541) (-900 "PDECOMP.spad" 1451056 1451073 1451576 1451581) (-899 "PDECAT.spad" 1449412 1449420 1451046 1451051) (-898 "PCOMP.spad" 1449265 1449278 1449402 1449407) (-897 "PBWLB.spad" 1447853 1447870 1449255 1449260) (-896 "PATTERN2.spad" 1447591 1447603 1447843 1447848) (-895 "PATTERN1.spad" 1445927 1445943 1447581 1447586) (-894 "PATTERN.spad" 1440466 1440476 1445917 1445922) (-893 "PATRES2.spad" 1440138 1440152 1440456 1440461) (-892 "PATRES.spad" 1437713 1437725 1440128 1440133) (-891 "PATMATCH.spad" 1435910 1435941 1437421 1437426) (-890 "PATMAB.spad" 1435339 1435349 1435900 1435905) (-889 "PATLRES.spad" 1434425 1434439 1435329 1435334) (-888 "PATAB.spad" 1434189 1434199 1434415 1434420) (-887 "PARTPERM.spad" 1431589 1431597 1434179 1434184) (-886 "PARSURF.spad" 1431023 1431051 1431579 1431584) (-885 "PARSU2.spad" 1430820 1430836 1431013 1431018) (-884 "script-parser.spad" 1430340 1430348 1430810 1430815) (-883 "PARSCURV.spad" 1429774 1429802 1430330 1430335) (-882 "PARSC2.spad" 1429565 1429581 1429764 1429769) (-881 "PARPCURV.spad" 1429027 1429055 1429555 1429560) (-880 "PARPC2.spad" 1428818 1428834 1429017 1429022) (-879 "PARAMAST.spad" 1427946 1427954 1428808 1428813) (-878 "PAN2EXPR.spad" 1427358 1427366 1427936 1427941) (-877 "PALETTE.spad" 1426328 1426336 1427348 1427353) (-876 "PAIR.spad" 1425315 1425328 1425916 1425921) (-875 "PADICRC.spad" 1422649 1422667 1423820 1423913) (-874 "PADICRAT.spad" 1420664 1420676 1420885 1420978) (-873 "PADICCT.spad" 1419213 1419225 1420590 1420659) (-872 "PADIC.spad" 1418908 1418920 1419139 1419208) (-871 "PADEPAC.spad" 1417597 1417616 1418898 1418903) (-870 "PADE.spad" 1416349 1416365 1417587 1417592) (-869 "OWP.spad" 1415589 1415619 1416207 1416274) (-868 "OVERSET.spad" 1415162 1415170 1415579 1415584) (-867 "OVAR.spad" 1414943 1414966 1415152 1415157) (-866 "OUTFORM.spad" 1404335 1404343 1414933 1414938) (-865 "OUTBFILE.spad" 1403753 1403761 1404325 1404330) (-864 "OUTBCON.spad" 1402759 1402767 1403743 1403748) (-863 "OUTBCON.spad" 1401763 1401773 1402749 1402754) (-862 "OUT.spad" 1400849 1400857 1401753 1401758) (-861 "OSI.spad" 1400324 1400332 1400839 1400844) (-860 "OSGROUP.spad" 1400242 1400250 1400314 1400319) (-859 "ORTHPOL.spad" 1398727 1398737 1400159 1400164) (-858 "OREUP.spad" 1398180 1398208 1398407 1398446) (-857 "ORESUP.spad" 1397481 1397505 1397860 1397899) (-856 "OREPCTO.spad" 1395338 1395350 1397401 1397406) (-855 "OREPCAT.spad" 1389485 1389495 1395294 1395333) (-854 "OREPCAT.spad" 1383522 1383534 1389333 1389338) (-853 "ORDSET.spad" 1382694 1382702 1383512 1383517) (-852 "ORDSET.spad" 1381864 1381874 1382684 1382689) (-851 "ORDRING.spad" 1381254 1381262 1381844 1381859) (-850 "ORDRING.spad" 1380652 1380662 1381244 1381249) (-849 "ORDMON.spad" 1380507 1380515 1380642 1380647) (-848 "ORDFUNS.spad" 1379639 1379655 1380497 1380502) (-847 "ORDFIN.spad" 1379459 1379467 1379629 1379634) (-846 "ORDCOMP2.spad" 1378752 1378764 1379449 1379454) (-845 "ORDCOMP.spad" 1377217 1377227 1378299 1378328) (-844 "OPTPROB.spad" 1375855 1375863 1377207 1377212) (-843 "OPTPACK.spad" 1368264 1368272 1375845 1375850) (-842 "OPTCAT.spad" 1365943 1365951 1368254 1368259) (-841 "OPSIG.spad" 1365597 1365605 1365933 1365938) (-840 "OPQUERY.spad" 1365146 1365154 1365587 1365592) (-839 "OPERCAT.spad" 1364612 1364622 1365136 1365141) (-838 "OPERCAT.spad" 1364076 1364088 1364602 1364607) (-837 "OP.spad" 1363818 1363828 1363898 1363965) (-836 "ONECOMP2.spad" 1363242 1363254 1363808 1363813) (-835 "ONECOMP.spad" 1361987 1361997 1362789 1362818) (-834 "OMSERVER.spad" 1360993 1361001 1361977 1361982) (-833 "OMSAGG.spad" 1360781 1360791 1360949 1360988) (-832 "OMPKG.spad" 1359397 1359405 1360771 1360776) (-831 "OMLO.spad" 1358822 1358834 1359283 1359322) (-830 "OMEXPR.spad" 1358656 1358666 1358812 1358817) (-829 "OMERRK.spad" 1357690 1357698 1358646 1358651) (-828 "OMERR.spad" 1357235 1357243 1357680 1357685) (-827 "OMENC.spad" 1356579 1356587 1357225 1357230) (-826 "OMDEV.spad" 1350888 1350896 1356569 1356574) (-825 "OMCONN.spad" 1350297 1350305 1350878 1350883) (-824 "OM.spad" 1349270 1349278 1350287 1350292) (-823 "OINTDOM.spad" 1349033 1349041 1349196 1349265) (-822 "OFMONOID.spad" 1347156 1347166 1348989 1348994) (-821 "ODVAR.spad" 1346417 1346427 1347146 1347151) (-820 "ODR.spad" 1346061 1346087 1346229 1346378) (-819 "ODPOL.spad" 1343443 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253909) (-221 "DEFINTRF.spad" 251113 251123 253521 253526) (-220 "DEFINTEF.spad" 249651 249667 251103 251108) (-219 "DEFAST.spad" 249019 249027 249641 249646) (-218 "DECIMAL.spad" 247125 247133 247486 247579) (-217 "DDFACT.spad" 244938 244955 247115 247120) (-216 "DBLRESP.spad" 244538 244562 244928 244933) (-215 "DBASE.spad" 243202 243212 244528 244533) (-214 "DATAARY.spad" 242664 242677 243192 243197) (-213 "D03FAFA.spad" 242492 242500 242654 242659) (-212 "D03EEFA.spad" 242312 242320 242482 242487) (-211 "D03AGNT.spad" 241398 241406 242302 242307) (-210 "D02EJFA.spad" 240860 240868 241388 241393) (-209 "D02CJFA.spad" 240338 240346 240850 240855) (-208 "D02BHFA.spad" 239828 239836 240328 240333) (-207 "D02BBFA.spad" 239318 239326 239818 239823) (-206 "D02AGNT.spad" 234132 234140 239308 239313) (-205 "D01WGTS.spad" 232451 232459 234122 234127) (-204 "D01TRNS.spad" 232428 232436 232441 232446) (-203 "D01GBFA.spad" 231950 231958 232418 232423) (-202 "D01FCFA.spad" 231472 231480 231940 231945) (-201 "D01ASFA.spad" 230940 230948 231462 231467) (-200 "D01AQFA.spad" 230386 230394 230930 230935) (-199 "D01APFA.spad" 229810 229818 230376 230381) (-198 "D01ANFA.spad" 229304 229312 229800 229805) (-197 "D01AMFA.spad" 228814 228822 229294 229299) (-196 "D01ALFA.spad" 228354 228362 228804 228809) (-195 "D01AKFA.spad" 227880 227888 228344 228349) (-194 "D01AJFA.spad" 227403 227411 227870 227875) (-193 "D01AGNT.spad" 223470 223478 227393 227398) (-192 "CYCLOTOM.spad" 222976 222984 223460 223465) (-191 "CYCLES.spad" 219832 219840 222966 222971) (-190 "CVMP.spad" 219249 219259 219822 219827) (-189 "CTRIGMNP.spad" 217749 217765 219239 219244) (-188 "CTORKIND.spad" 217352 217360 217739 217744) (-187 "CTORCAT.spad" 216601 216609 217342 217347) (-186 "CTORCAT.spad" 215848 215858 216591 216596) (-185 "CTORCALL.spad" 215437 215447 215838 215843) (-184 "CTOR.spad" 215128 215136 215427 215432) (-183 "CSTTOOLS.spad" 214373 214386 215118 215123) (-182 "CRFP.spad" 208097 208110 214363 214368) (-181 "CRCEAST.spad" 207817 207825 208087 208092) (-180 "CRAPACK.spad" 206868 206878 207807 207812) (-179 "CPMATCH.spad" 206372 206387 206793 206798) (-178 "CPIMA.spad" 206077 206096 206362 206367) (-177 "COORDSYS.spad" 201086 201096 206067 206072) (-176 "CONTOUR.spad" 200497 200505 201076 201081) (-175 "CONTFRAC.spad" 196247 196257 200399 200492) (-174 "CONDUIT.spad" 196005 196013 196237 196242) (-173 "COMRING.spad" 195679 195687 195943 196000) (-172 "COMPPROP.spad" 195197 195205 195669 195674) (-171 "COMPLPAT.spad" 194964 194979 195187 195192) (-170 "COMPLEX2.spad" 194679 194691 194954 194959) (-169 "COMPLEX.spad" 188816 188826 189060 189321) (-168 "COMPILER.spad" 188528 188536 188806 188811) (-167 "COMPFACT.spad" 188130 188144 188518 188523) (-166 "COMPCAT.spad" 186202 186212 187864 188125) (-165 "COMPCAT.spad" 184002 184014 185666 185671) (-164 "COMMUPC.spad" 183750 183768 183992 183997) (-163 "COMMONOP.spad" 183283 183291 183740 183745) (-162 "COMMAAST.spad" 183046 183054 183273 183278) (-161 "COMM.spad" 182857 182865 183036 183041) (-160 "COMBOPC.spad" 181772 181780 182847 182852) (-159 "COMBINAT.spad" 180539 180549 181762 181767) (-158 "COMBF.spad" 177921 177937 180529 180534) (-157 "COLOR.spad" 176758 176766 177911 177916) (-156 "COLONAST.spad" 176424 176432 176748 176753) (-155 "CMPLXRT.spad" 176135 176152 176414 176419) (-154 "CLLCTAST.spad" 175797 175805 176125 176130) (-153 "CLIP.spad" 171905 171913 175787 175792) (-152 "CLIF.spad" 170560 170576 171861 171900) (-151 "CLAGG.spad" 167065 167075 170550 170555) (-150 "CLAGG.spad" 163441 163453 166928 166933) (-149 "CINTSLPE.spad" 162772 162785 163431 163436) (-148 "CHVAR.spad" 160910 160932 162762 162767) (-147 "CHARZ.spad" 160825 160833 160890 160905) (-146 "CHARPOL.spad" 160335 160345 160815 160820) (-145 "CHARNZ.spad" 160088 160096 160315 160330) (-144 "CHAR.spad" 157962 157970 160078 160083) (-143 "CFCAT.spad" 157290 157298 157952 157957) (-142 "CDEN.spad" 156486 156500 157280 157285) (-141 "CCLASS.spad" 154635 154643 155897 155936) (-140 "CATEGORY.spad" 153677 153685 154625 154630) (-139 "CATCTOR.spad" 153568 153576 153667 153672) (-138 "CATAST.spad" 153186 153194 153558 153563) (-137 "CASEAST.spad" 152900 152908 153176 153181) (-136 "CARTEN2.spad" 152290 152317 152890 152895) (-135 "CARTEN.spad" 147577 147601 152280 152285) (-134 "CARD.spad" 144872 144880 147551 147572) (-133 "CAPSLAST.spad" 144646 144654 144862 144867) (-132 "CACHSET.spad" 144270 144278 144636 144641) (-131 "CABMON.spad" 143825 143833 144260 144265) (-130 "BYTEORD.spad" 143500 143508 143815 143820) (-129 "BYTEBUF.spad" 141359 141367 142669 142696) (-128 "BYTE.spad" 140786 140794 141349 141354) (-127 "BTREE.spad" 139859 139869 140393 140420) (-126 "BTOURN.spad" 138864 138874 139466 139493) (-125 "BTCAT.spad" 138256 138266 138832 138859) (-124 "BTCAT.spad" 137668 137680 138246 138251) (-123 "BTAGG.spad" 136796 136804 137636 137663) (-122 "BTAGG.spad" 135944 135954 136786 136791) (-121 "BSTREE.spad" 134685 134695 135551 135578) (-120 "BRILL.spad" 132882 132893 134675 134680) (-119 "BRAGG.spad" 131822 131832 132872 132877) (-118 "BRAGG.spad" 130726 130738 131778 131783) (-117 "BPADICRT.spad" 128707 128719 128962 129055) (-116 "BPADIC.spad" 128371 128383 128633 128702) (-115 "BOUNDZRO.spad" 128027 128044 128361 128366) (-114 "BOP1.spad" 125493 125503 128017 128022) (-113 "BOP.spad" 120675 120683 125483 125488) (-112 "BOOLEAN.spad" 120113 120121 120665 120670) (-111 "BMODULE.spad" 119825 119837 120081 120108) (-110 "BITS.spad" 119246 119254 119461 119488) (-109 "BINDING.spad" 118659 118667 119236 119241) (-108 "BINARY.spad" 116770 116778 117126 117219) (-107 "BGAGG.spad" 115975 115985 116750 116765) (-106 "BGAGG.spad" 115188 115200 115965 115970) (-105 "BFUNCT.spad" 114752 114760 115168 115183) (-104 "BEZOUT.spad" 113892 113919 114702 114707) (-103 "BBTREE.spad" 110737 110747 113499 113526) (-102 "BASTYPE.spad" 110409 110417 110727 110732) (-101 "BASTYPE.spad" 110079 110089 110399 110404) (-100 "BALFACT.spad" 109538 109551 110069 110074) (-99 "AUTOMOR.spad" 108989 108998 109518 109533) (-98 "ATTREG.spad" 105712 105719 108741 108984) (-97 "ATTRBUT.spad" 101735 101742 105692 105707) (-96 "ATTRAST.spad" 101452 101459 101725 101730) (-95 "ATRIG.spad" 100922 100929 101442 101447) (-94 "ATRIG.spad" 100390 100399 100912 100917) (-93 "ASTCAT.spad" 100294 100301 100380 100385) (-92 "ASTCAT.spad" 100196 100205 100284 100289) (-91 "ASTACK.spad" 99535 99544 99803 99830) (-90 "ASSOCEQ.spad" 98361 98372 99491 99496) (-89 "ASP9.spad" 97442 97455 98351 98356) (-88 "ASP80.spad" 96764 96777 97432 97437) (-87 "ASP8.spad" 95807 95820 96754 96759) (-86 "ASP78.spad" 95258 95271 95797 95802) (-85 "ASP77.spad" 94627 94640 95248 95253) (-84 "ASP74.spad" 93719 93732 94617 94622) (-83 "ASP73.spad" 92990 93003 93709 93714) (-82 "ASP7.spad" 92150 92163 92980 92985) (-81 "ASP6.spad" 91017 91030 92140 92145) (-80 "ASP55.spad" 89526 89539 91007 91012) (-79 "ASP50.spad" 87343 87356 89516 89521) (-78 "ASP49.spad" 86342 86355 87333 87338) (-77 "ASP42.spad" 84749 84788 86332 86337) (-76 "ASP41.spad" 83328 83367 84739 84744) (-75 "ASP4.spad" 82623 82636 83318 83323) (-74 "ASP35.spad" 81611 81624 82613 82618) (-73 "ASP34.spad" 80912 80925 81601 81606) (-72 "ASP33.spad" 80472 80485 80902 80907) (-71 "ASP31.spad" 79612 79625 80462 80467) (-70 "ASP30.spad" 78504 78517 79602 79607) (-69 "ASP29.spad" 77970 77983 78494 78499) (-68 "ASP28.spad" 69243 69256 77960 77965) (-67 "ASP27.spad" 68140 68153 69233 69238) (-66 "ASP24.spad" 67227 67240 68130 68135) (-65 "ASP20.spad" 66691 66704 67217 67222) (-64 "ASP19.spad" 61377 61390 66681 66686) (-63 "ASP12.spad" 60791 60804 61367 61372) (-62 "ASP10.spad" 60062 60075 60781 60786) (-61 "ASP1.spad" 59443 59456 60052 60057) (-60 "ARRAY2.spad" 58803 58812 59050 59077) (-59 "ARRAY12.spad" 57516 57527 58793 58798) (-58 "ARRAY1.spad" 56353 56362 56699 56726) (-57 "ARR2CAT.spad" 52127 52148 56321 56348) (-56 "ARR2CAT.spad" 47921 47944 52117 52122) (-55 "ARITY.spad" 47293 47300 47911 47916) (-54 "APPRULE.spad" 46553 46575 47283 47288) (-53 "APPLYORE.spad" 46172 46185 46543 46548) (-52 "ANY1.spad" 45243 45252 46162 46167) (-51 "ANY.spad" 44102 44109 45233 45238) (-50 "ANTISYM.spad" 42547 42563 44082 44097) (-49 "ANON.spad" 42240 42247 42537 42542) (-48 "AN.spad" 40549 40556 42056 42149) (-47 "AMR.spad" 38734 38745 40447 40544) (-46 "AMR.spad" 36756 36769 38471 38476) (-45 "ALIST.spad" 34168 34189 34518 34545) (-44 "ALGSC.spad" 33303 33329 34040 34093) (-43 "ALGPKG.spad" 29086 29097 33259 33264) (-42 "ALGMFACT.spad" 28279 28293 29076 29081) (-41 "ALGMANIP.spad" 25753 25768 28112 28117) (-40 "ALGFF.spad" 24068 24095 24285 24441) (-39 "ALGFACT.spad" 23195 23205 24058 24063) (-38 "ALGEBRA.spad" 23028 23037 23151 23190) (-37 "ALGEBRA.spad" 22893 22904 23018 23023) (-36 "ALAGG.spad" 22405 22426 22861 22888) (-35 "AHYP.spad" 21786 21793 22395 22400) (-34 "AGG.spad" 20103 20110 21776 21781) (-33 "AGG.spad" 18384 18393 20059 20064) (-32 "AF.spad" 16815 16830 18319 18324) (-31 "ADDAST.spad" 16493 16500 16805 16810) (-30 "ACPLOT.spad" 15084 15091 16483 16488) (-29 "ACFS.spad" 12893 12902 14986 15079) (-28 "ACFS.spad" 10788 10799 12883 12888) (-27 "ACF.spad" 7470 7477 10690 10783) (-26 "ACF.spad" 4238 4247 7460 7465) (-25 "ABELSG.spad" 3779 3786 4228 4233) (-24 "ABELSG.spad" 3318 3327 3769 3774) (-23 "ABELMON.spad" 2861 2868 3308 3313) (-22 "ABELMON.spad" 2402 2411 2851 2856) (-21 "ABELGRP.spad" 2067 2074 2392 2397) (-20 "ABELGRP.spad" 1730 1739 2057 2062) (-19 "A1AGG.spad" 870 879 1698 1725) (-18 "A1AGG.spad" 30 41 860 865)) 
\ No newline at end of file
+((-3 NIL 2264799 2264804 2264809 2264814) (-2 NIL 2264779 2264784 2264789 2264794) (-1 NIL 2264759 2264764 2264769 2264774) (0 NIL 2264739 2264744 2264749 2264754) (-1301 "ZMOD.spad" 2264548 2264561 2264677 2264734) (-1300 "ZLINDEP.spad" 2263614 2263625 2264538 2264543) (-1299 "ZDSOLVE.spad" 2253559 2253581 2263604 2263609) (-1298 "YSTREAM.spad" 2253054 2253065 2253549 2253554) (-1297 "XRPOLY.spad" 2252274 2252294 2252910 2252979) (-1296 "XPR.spad" 2250069 2250082 2251992 2252091) (-1295 "XPOLYC.spad" 2249388 2249404 2249995 2250064) (-1294 "XPOLY.spad" 2248943 2248954 2249244 2249313) (-1293 "XPBWPOLY.spad" 2247380 2247400 2248723 2248792) (-1292 "XFALG.spad" 2244428 2244444 2247306 2247375) (-1291 "XF.spad" 2242891 2242906 2244330 2244423) (-1290 "XF.spad" 2241334 2241351 2242775 2242780) (-1289 "XEXPPKG.spad" 2240585 2240611 2241324 2241329) (-1288 "XDPOLY.spad" 2240199 2240215 2240441 2240510) (-1287 "XALG.spad" 2239859 2239870 2240155 2240194) (-1286 "WUTSET.spad" 2235698 2235715 2239505 2239532) (-1285 "WP.spad" 2234897 2234941 2235556 2235623) (-1284 "WHILEAST.spad" 2234695 2234704 2234887 2234892) (-1283 "WHEREAST.spad" 2234366 2234375 2234685 2234690) (-1282 "WFFINTBS.spad" 2232029 2232051 2234356 2234361) (-1281 "WEIER.spad" 2230251 2230262 2232019 2232024) (-1280 "VSPACE.spad" 2229924 2229935 2230219 2230246) (-1279 "VSPACE.spad" 2229617 2229630 2229914 2229919) (-1278 "VOID.spad" 2229294 2229303 2229607 2229612) (-1277 "VIEWDEF.spad" 2224495 2224504 2229284 2229289) (-1276 "VIEW3D.spad" 2208456 2208465 2224485 2224490) (-1275 "VIEW2D.spad" 2196347 2196356 2208446 2208451) (-1274 "VIEW.spad" 2194027 2194036 2196337 2196342) (-1273 "VECTOR2.spad" 2192666 2192679 2194017 2194022) (-1272 "VECTOR.spad" 2191340 2191351 2191591 2191618) (-1271 "VECTCAT.spad" 2189244 2189255 2191308 2191335) (-1270 "VECTCAT.spad" 2186955 2186968 2189021 2189026) (-1269 "VARIABLE.spad" 2186735 2186750 2186945 2186950) (-1268 "UTYPE.spad" 2186379 2186388 2186725 2186730) (-1267 "UTSODETL.spad" 2185674 2185698 2186335 2186340) (-1266 "UTSODE.spad" 2183890 2183910 2185664 2185669) (-1265 "UTSCAT.spad" 2181369 2181385 2183788 2183885) (-1264 "UTSCAT.spad" 2178492 2178510 2180913 2180918) (-1263 "UTS2.spad" 2178087 2178122 2178482 2178487) (-1262 "UTS.spad" 2172891 2172919 2176554 2176651) (-1261 "URAGG.spad" 2167564 2167575 2172881 2172886) (-1260 "URAGG.spad" 2162201 2162214 2167520 2167525) (-1259 "UPXSSING.spad" 2159846 2159872 2161282 2161415) (-1258 "UPXSCONS.spad" 2157605 2157625 2157978 2158127) (-1257 "UPXSCCA.spad" 2156176 2156196 2157451 2157600) (-1256 "UPXSCCA.spad" 2154889 2154911 2156166 2156171) (-1255 "UPXSCAT.spad" 2153478 2153494 2154735 2154884) (-1254 "UPXS2.spad" 2153021 2153074 2153468 2153473) (-1253 "UPXS.spad" 2150175 2150203 2151153 2151302) (-1252 "UPSQFREE.spad" 2148590 2148604 2150165 2150170) (-1251 "UPSCAT.spad" 2146201 2146225 2148488 2148585) (-1250 "UPSCAT.spad" 2143518 2143544 2145807 2145812) (-1249 "UPOLYC2.spad" 2142989 2143008 2143508 2143513) (-1248 "UPOLYC.spad" 2138029 2138040 2142831 2142984) (-1247 "UPOLYC.spad" 2132961 2132974 2137765 2137770) (-1246 "UPMP.spad" 2131861 2131874 2132951 2132956) (-1245 "UPDIVP.spad" 2131426 2131440 2131851 2131856) (-1244 "UPDECOMP.spad" 2129671 2129685 2131416 2131421) (-1243 "UPCDEN.spad" 2128880 2128896 2129661 2129666) (-1242 "UP2.spad" 2128244 2128265 2128870 2128875) (-1241 "UP.spad" 2125443 2125458 2125830 2125983) (-1240 "UNISEG2.spad" 2124940 2124953 2125399 2125404) (-1239 "UNISEG.spad" 2124293 2124304 2124859 2124864) (-1238 "UNIFACT.spad" 2123396 2123408 2124283 2124288) (-1237 "ULSCONS.spad" 2115792 2115812 2116162 2116311) (-1236 "ULSCCAT.spad" 2113529 2113549 2115638 2115787) (-1235 "ULSCCAT.spad" 2111374 2111396 2113485 2113490) (-1234 "ULSCAT.spad" 2109606 2109622 2111220 2111369) (-1233 "ULS2.spad" 2109120 2109173 2109596 2109601) (-1232 "ULS.spad" 2099678 2099706 2100765 2101194) (-1231 "UINT8.spad" 2099555 2099564 2099668 2099673) (-1230 "UINT64.spad" 2099431 2099440 2099545 2099550) (-1229 "UINT32.spad" 2099307 2099316 2099421 2099426) (-1228 "UINT16.spad" 2099183 2099192 2099297 2099302) (-1227 "UFD.spad" 2098248 2098257 2099109 2099178) (-1226 "UFD.spad" 2097375 2097386 2098238 2098243) (-1225 "UDVO.spad" 2096256 2096265 2097365 2097370) (-1224 "UDPO.spad" 2093749 2093760 2096212 2096217) (-1223 "TYPEAST.spad" 2093668 2093677 2093739 2093744) (-1222 "TYPE.spad" 2093600 2093609 2093658 2093663) (-1221 "TWOFACT.spad" 2092252 2092267 2093590 2093595) (-1220 "TUPLE.spad" 2091738 2091749 2092151 2092156) (-1219 "TUBETOOL.spad" 2088605 2088614 2091728 2091733) (-1218 "TUBE.spad" 2087252 2087269 2088595 2088600) (-1217 "TSETCAT.spad" 2074379 2074396 2087220 2087247) (-1216 "TSETCAT.spad" 2061492 2061511 2074335 2074340) (-1215 "TS.spad" 2060091 2060107 2061057 2061154) (-1214 "TRMANIP.spad" 2054457 2054474 2059797 2059802) (-1213 "TRIMAT.spad" 2053420 2053445 2054447 2054452) (-1212 "TRIGMNIP.spad" 2051947 2051964 2053410 2053415) (-1211 "TRIGCAT.spad" 2051459 2051468 2051937 2051942) (-1210 "TRIGCAT.spad" 2050969 2050980 2051449 2051454) (-1209 "TREE.spad" 2049544 2049555 2050576 2050603) (-1208 "TRANFUN.spad" 2049383 2049392 2049534 2049539) (-1207 "TRANFUN.spad" 2049220 2049231 2049373 2049378) (-1206 "TOPSP.spad" 2048894 2048903 2049210 2049215) (-1205 "TOOLSIGN.spad" 2048557 2048568 2048884 2048889) (-1204 "TEXTFILE.spad" 2047118 2047127 2048547 2048552) (-1203 "TEX1.spad" 2046674 2046685 2047108 2047113) (-1202 "TEX.spad" 2043820 2043829 2046664 2046669) (-1201 "TEMUTL.spad" 2043375 2043384 2043810 2043815) (-1200 "TBCMPPK.spad" 2041468 2041491 2043365 2043370) (-1199 "TBAGG.spad" 2040518 2040541 2041448 2041463) (-1198 "TBAGG.spad" 2039576 2039601 2040508 2040513) (-1197 "TANEXP.spad" 2038984 2038995 2039566 2039571) (-1196 "TABLEAU.spad" 2038465 2038476 2038974 2038979) (-1195 "TABLE.spad" 2036876 2036899 2037146 2037173) (-1194 "TABLBUMP.spad" 2033679 2033690 2036866 2036871) (-1193 "SYSTEM.spad" 2032907 2032916 2033669 2033674) (-1192 "SYSSOLP.spad" 2030390 2030401 2032897 2032902) (-1191 "SYSPTR.spad" 2030289 2030298 2030380 2030385) (-1190 "SYSNNI.spad" 2029471 2029482 2030279 2030284) (-1189 "SYSINT.spad" 2028875 2028886 2029461 2029466) (-1188 "SYNTAX.spad" 2025081 2025090 2028865 2028870) (-1187 "SYMTAB.spad" 2023149 2023158 2025071 2025076) (-1186 "SYMS.spad" 2019178 2019187 2023139 2023144) (-1185 "SYMPOLY.spad" 2018185 2018196 2018267 2018394) (-1184 "SYMFUNC.spad" 2017686 2017697 2018175 2018180) (-1183 "SYMBOL.spad" 2015189 2015198 2017676 2017681) (-1182 "SWITCH.spad" 2011960 2011969 2015179 2015184) (-1181 "SUTS.spad" 2008865 2008893 2010427 2010524) (-1180 "SUPXS.spad" 2006006 2006034 2006997 2007146) (-1179 "SUPFRACF.spad" 2005111 2005129 2005996 2006001) (-1178 "SUP2.spad" 2004503 2004516 2005101 2005106) (-1177 "SUP.spad" 2001316 2001327 2002089 2002242) (-1176 "SUMRF.spad" 2000290 2000301 2001306 2001311) (-1175 "SUMFS.spad" 1999927 1999944 2000280 2000285) (-1174 "SULS.spad" 1990472 1990500 1991572 1992001) (-1173 "SUCHTAST.spad" 1990241 1990250 1990462 1990467) (-1172 "SUCH.spad" 1989923 1989938 1990231 1990236) (-1171 "SUBSPACE.spad" 1982038 1982053 1989913 1989918) (-1170 "SUBRESP.spad" 1981208 1981222 1981994 1981999) (-1169 "STTFNC.spad" 1977676 1977692 1981198 1981203) (-1168 "STTF.spad" 1973775 1973791 1977666 1977671) (-1167 "STTAYLOR.spad" 1966410 1966421 1973656 1973661) (-1166 "STRTBL.spad" 1964915 1964932 1965064 1965091) (-1165 "STRING.spad" 1964324 1964333 1964338 1964365) (-1164 "STRICAT.spad" 1964112 1964121 1964292 1964319) (-1163 "STREAM3.spad" 1963685 1963700 1964102 1964107) (-1162 "STREAM2.spad" 1962813 1962826 1963675 1963680) (-1161 "STREAM1.spad" 1962519 1962530 1962803 1962808) (-1160 "STREAM.spad" 1959437 1959448 1962044 1962059) (-1159 "STINPROD.spad" 1958373 1958389 1959427 1959432) (-1158 "STEPAST.spad" 1957607 1957616 1958363 1958368) (-1157 "STEP.spad" 1956808 1956817 1957597 1957602) (-1156 "STBL.spad" 1955334 1955362 1955501 1955516) (-1155 "STAGG.spad" 1954409 1954420 1955324 1955329) (-1154 "STAGG.spad" 1953482 1953495 1954399 1954404) (-1153 "STACK.spad" 1952839 1952850 1953089 1953116) (-1152 "SREGSET.spad" 1950543 1950560 1952485 1952512) (-1151 "SRDCMPK.spad" 1949104 1949124 1950533 1950538) (-1150 "SRAGG.spad" 1944247 1944256 1949072 1949099) (-1149 "SRAGG.spad" 1939410 1939421 1944237 1944242) (-1148 "SQMATRIX.spad" 1937026 1937044 1937942 1938029) (-1147 "SPLTREE.spad" 1931578 1931591 1936462 1936489) (-1146 "SPLNODE.spad" 1928166 1928179 1931568 1931573) (-1145 "SPFCAT.spad" 1926975 1926984 1928156 1928161) (-1144 "SPECOUT.spad" 1925527 1925536 1926965 1926970) (-1143 "SPADXPT.spad" 1917122 1917131 1925517 1925522) (-1142 "spad-parser.spad" 1916587 1916596 1917112 1917117) (-1141 "SPADAST.spad" 1916288 1916297 1916577 1916582) (-1140 "SPACEC.spad" 1900487 1900498 1916278 1916283) (-1139 "SPACE3.spad" 1900263 1900274 1900477 1900482) (-1138 "SORTPAK.spad" 1899812 1899825 1900219 1900224) (-1137 "SOLVETRA.spad" 1897575 1897586 1899802 1899807) (-1136 "SOLVESER.spad" 1896103 1896114 1897565 1897570) (-1135 "SOLVERAD.spad" 1892129 1892140 1896093 1896098) (-1134 "SOLVEFOR.spad" 1890591 1890609 1892119 1892124) (-1133 "SNTSCAT.spad" 1890191 1890208 1890559 1890586) (-1132 "SMTS.spad" 1888463 1888489 1889756 1889853) (-1131 "SMP.spad" 1885938 1885958 1886328 1886455) (-1130 "SMITH.spad" 1884783 1884808 1885928 1885933) (-1129 "SMATCAT.spad" 1882893 1882923 1884727 1884778) (-1128 "SMATCAT.spad" 1880935 1880967 1882771 1882776) (-1127 "SKAGG.spad" 1879898 1879909 1880903 1880930) (-1126 "SINT.spad" 1878730 1878739 1879764 1879893) (-1125 "SIMPAN.spad" 1878458 1878467 1878720 1878725) (-1124 "SIGNRF.spad" 1877583 1877594 1878448 1878453) (-1123 "SIGNEF.spad" 1876869 1876886 1877573 1877578) (-1122 "SIGAST.spad" 1876254 1876263 1876859 1876864) (-1121 "SIG.spad" 1875584 1875593 1876244 1876249) (-1120 "SHP.spad" 1873512 1873527 1875540 1875545) (-1119 "SHDP.spad" 1863223 1863250 1863732 1863863) (-1118 "SGROUP.spad" 1862831 1862840 1863213 1863218) (-1117 "SGROUP.spad" 1862437 1862448 1862821 1862826) (-1116 "SGCF.spad" 1855600 1855609 1862427 1862432) (-1115 "SFRTCAT.spad" 1854530 1854547 1855568 1855595) (-1114 "SFRGCD.spad" 1853593 1853613 1854520 1854525) (-1113 "SFQCMPK.spad" 1848230 1848250 1853583 1853588) (-1112 "SFORT.spad" 1847669 1847683 1848220 1848225) (-1111 "SEXOF.spad" 1847512 1847552 1847659 1847664) (-1110 "SEXCAT.spad" 1845113 1845153 1847502 1847507) (-1109 "SEX.spad" 1845005 1845014 1845103 1845108) (-1108 "SETMN.spad" 1843457 1843474 1844995 1845000) (-1107 "SETCAT.spad" 1842779 1842788 1843447 1843452) (-1106 "SETCAT.spad" 1842099 1842110 1842769 1842774) (-1105 "SETAGG.spad" 1838648 1838659 1842079 1842094) (-1104 "SETAGG.spad" 1835205 1835218 1838638 1838643) (-1103 "SET.spad" 1833529 1833540 1834626 1834665) (-1102 "SEQAST.spad" 1833232 1833241 1833519 1833524) (-1101 "SEGXCAT.spad" 1832388 1832401 1833222 1833227) (-1100 "SEGCAT.spad" 1831313 1831324 1832378 1832383) (-1099 "SEGBIND2.spad" 1831011 1831024 1831303 1831308) (-1098 "SEGBIND.spad" 1830769 1830780 1830958 1830963) (-1097 "SEGAST.spad" 1830483 1830492 1830759 1830764) (-1096 "SEG2.spad" 1829918 1829931 1830439 1830444) (-1095 "SEG.spad" 1829731 1829742 1829837 1829842) (-1094 "SDVAR.spad" 1829007 1829018 1829721 1829726) (-1093 "SDPOL.spad" 1826433 1826444 1826724 1826851) (-1092 "SCPKG.spad" 1824522 1824533 1826423 1826428) (-1091 "SCOPE.spad" 1823675 1823684 1824512 1824517) (-1090 "SCACHE.spad" 1822371 1822382 1823665 1823670) (-1089 "SASTCAT.spad" 1822280 1822289 1822361 1822366) (-1088 "SAOS.spad" 1822152 1822161 1822270 1822275) (-1087 "SAERFFC.spad" 1821865 1821885 1822142 1822147) (-1086 "SAEFACT.spad" 1821566 1821586 1821855 1821860) (-1085 "SAE.spad" 1819741 1819757 1820352 1820487) (-1084 "RURPK.spad" 1817400 1817416 1819731 1819736) (-1083 "RULESET.spad" 1816853 1816877 1817390 1817395) (-1082 "RULECOLD.spad" 1816705 1816718 1816843 1816848) (-1081 "RULE.spad" 1814945 1814969 1816695 1816700) (-1080 "RTVALUE.spad" 1814680 1814689 1814935 1814940) (-1079 "RSTRCAST.spad" 1814397 1814406 1814670 1814675) (-1078 "RSETGCD.spad" 1810775 1810795 1814387 1814392) (-1077 "RSETCAT.spad" 1800711 1800728 1810743 1810770) (-1076 "RSETCAT.spad" 1790667 1790686 1800701 1800706) (-1075 "RSDCMPK.spad" 1789119 1789139 1790657 1790662) (-1074 "RRCC.spad" 1787503 1787533 1789109 1789114) (-1073 "RRCC.spad" 1785885 1785917 1787493 1787498) (-1072 "RPTAST.spad" 1785587 1785596 1785875 1785880) (-1071 "RPOLCAT.spad" 1764947 1764962 1785455 1785582) (-1070 "RPOLCAT.spad" 1744021 1744038 1764531 1764536) (-1069 "ROUTINE.spad" 1739904 1739913 1742668 1742695) (-1068 "ROMAN.spad" 1739232 1739241 1739770 1739899) (-1067 "ROIRC.spad" 1738312 1738344 1739222 1739227) (-1066 "RNS.spad" 1737215 1737224 1738214 1738307) (-1065 "RNS.spad" 1736204 1736215 1737205 1737210) (-1064 "RNGBIND.spad" 1735364 1735378 1736159 1736164) (-1063 "RNG.spad" 1735099 1735108 1735354 1735359) (-1062 "RMODULE.spad" 1734864 1734875 1735089 1735094) (-1061 "RMCAT2.spad" 1734284 1734341 1734854 1734859) (-1060 "RMATRIX.spad" 1733108 1733127 1733451 1733490) (-1059 "RMATCAT.spad" 1728687 1728718 1733064 1733103) (-1058 "RMATCAT.spad" 1724156 1724189 1728535 1728540) (-1057 "RLINSET.spad" 1723550 1723561 1724146 1724151) (-1056 "RINTERP.spad" 1723438 1723458 1723540 1723545) (-1055 "RING.spad" 1722908 1722917 1723418 1723433) (-1054 "RING.spad" 1722386 1722397 1722898 1722903) (-1053 "RIDIST.spad" 1721778 1721787 1722376 1722381) (-1052 "RGCHAIN.spad" 1720361 1720377 1721263 1721290) (-1051 "RGBCSPC.spad" 1720142 1720154 1720351 1720356) (-1050 "RGBCMDL.spad" 1719672 1719684 1720132 1720137) (-1049 "RFFACTOR.spad" 1719134 1719145 1719662 1719667) (-1048 "RFFACT.spad" 1718869 1718881 1719124 1719129) (-1047 "RFDIST.spad" 1717865 1717874 1718859 1718864) (-1046 "RF.spad" 1715507 1715518 1717855 1717860) (-1045 "RETSOL.spad" 1714926 1714939 1715497 1715502) (-1044 "RETRACT.spad" 1714354 1714365 1714916 1714921) (-1043 "RETRACT.spad" 1713780 1713793 1714344 1714349) (-1042 "RETAST.spad" 1713592 1713601 1713770 1713775) (-1041 "RESULT.spad" 1711652 1711661 1712239 1712266) (-1040 "RESRING.spad" 1710999 1711046 1711590 1711647) (-1039 "RESLATC.spad" 1710323 1710334 1710989 1710994) (-1038 "REPSQ.spad" 1710054 1710065 1710313 1710318) (-1037 "REPDB.spad" 1709761 1709772 1710044 1710049) (-1036 "REP2.spad" 1699419 1699430 1709603 1709608) (-1035 "REP1.spad" 1693615 1693626 1699369 1699374) (-1034 "REP.spad" 1691169 1691178 1693605 1693610) (-1033 "REGSET.spad" 1688966 1688983 1690815 1690842) (-1032 "REF.spad" 1688301 1688312 1688921 1688926) (-1031 "REDORDER.spad" 1687507 1687524 1688291 1688296) (-1030 "RECLOS.spad" 1686290 1686310 1686994 1687087) (-1029 "REALSOLV.spad" 1685430 1685439 1686280 1686285) (-1028 "REAL0Q.spad" 1682728 1682743 1685420 1685425) (-1027 "REAL0.spad" 1679572 1679587 1682718 1682723) (-1026 "REAL.spad" 1679444 1679453 1679562 1679567) (-1025 "RDUCEAST.spad" 1679165 1679174 1679434 1679439) (-1024 "RDIV.spad" 1678820 1678845 1679155 1679160) (-1023 "RDIST.spad" 1678387 1678398 1678810 1678815) (-1022 "RDETRS.spad" 1677251 1677269 1678377 1678382) (-1021 "RDETR.spad" 1675390 1675408 1677241 1677246) (-1020 "RDEEFS.spad" 1674489 1674506 1675380 1675385) (-1019 "RDEEF.spad" 1673499 1673516 1674479 1674484) (-1018 "RCFIELD.spad" 1670685 1670694 1673401 1673494) (-1017 "RCFIELD.spad" 1667957 1667968 1670675 1670680) (-1016 "RCAGG.spad" 1665885 1665896 1667947 1667952) (-1015 "RCAGG.spad" 1663740 1663753 1665804 1665809) (-1014 "RATRET.spad" 1663100 1663111 1663730 1663735) (-1013 "RATFACT.spad" 1662792 1662804 1663090 1663095) (-1012 "RANDSRC.spad" 1662111 1662120 1662782 1662787) (-1011 "RADUTIL.spad" 1661867 1661876 1662101 1662106) (-1010 "RADIX.spad" 1658788 1658802 1660334 1660427) (-1009 "RADFF.spad" 1657201 1657238 1657320 1657476) (-1008 "RADCAT.spad" 1656796 1656805 1657191 1657196) (-1007 "RADCAT.spad" 1656389 1656400 1656786 1656791) (-1006 "QUEUE.spad" 1655737 1655748 1655996 1656023) (-1005 "QUATCT2.spad" 1655357 1655376 1655727 1655732) (-1004 "QUATCAT.spad" 1653527 1653538 1655287 1655352) (-1003 "QUATCAT.spad" 1651448 1651461 1653210 1653215) (-1002 "QUAT.spad" 1650029 1650040 1650372 1650437) (-1001 "QUAGG.spad" 1648856 1648867 1649997 1650024) (-1000 "QQUTAST.spad" 1648624 1648633 1648846 1648851) (-999 "QFORM.spad" 1648089 1648103 1648614 1648619) (-998 "QFCAT2.spad" 1647782 1647798 1648079 1648084) (-997 "QFCAT.spad" 1646485 1646495 1647684 1647777) (-996 "QFCAT.spad" 1644779 1644791 1645980 1645985) (-995 "QEQUAT.spad" 1644338 1644346 1644769 1644774) (-994 "QCMPACK.spad" 1639085 1639104 1644328 1644333) (-993 "QALGSET2.spad" 1637081 1637099 1639075 1639080) (-992 "QALGSET.spad" 1633162 1633194 1636995 1637000) (-991 "PWFFINTB.spad" 1630578 1630599 1633152 1633157) (-990 "PUSHVAR.spad" 1629917 1629936 1630568 1630573) (-989 "PTRANFN.spad" 1626045 1626055 1629907 1629912) (-988 "PTPACK.spad" 1623133 1623143 1626035 1626040) (-987 "PTFUNC2.spad" 1622956 1622970 1623123 1623128) (-986 "PTCAT.spad" 1622211 1622221 1622924 1622951) (-985 "PSQFR.spad" 1621518 1621542 1622201 1622206) (-984 "PSEUDLIN.spad" 1620404 1620414 1621508 1621513) (-983 "PSETPK.spad" 1605837 1605853 1620282 1620287) (-982 "PSETCAT.spad" 1599757 1599780 1605817 1605832) (-981 "PSETCAT.spad" 1593651 1593676 1599713 1599718) (-980 "PSCURVE.spad" 1592634 1592642 1593641 1593646) (-979 "PSCAT.spad" 1591417 1591446 1592532 1592629) (-978 "PSCAT.spad" 1590290 1590321 1591407 1591412) (-977 "PRTITION.spad" 1589251 1589259 1590280 1590285) (-976 "PRTDAST.spad" 1588970 1588978 1589241 1589246) (-975 "PRS.spad" 1578532 1578549 1588926 1588931) (-974 "PRQAGG.spad" 1577967 1577977 1578500 1578527) (-973 "PROPLOG.spad" 1577266 1577274 1577957 1577962) (-972 "PROPFRML.spad" 1575834 1575845 1577256 1577261) (-971 "PROPERTY.spad" 1575322 1575330 1575824 1575829) (-970 "PRODUCT.spad" 1573004 1573016 1573288 1573343) (-969 "PRINT.spad" 1572756 1572764 1572994 1572999) (-968 "PRIMES.spad" 1571009 1571019 1572746 1572751) (-967 "PRIMELT.spad" 1569090 1569104 1570999 1571004) (-966 "PRIMCAT.spad" 1568717 1568725 1569080 1569085) (-965 "PRIMARR2.spad" 1567484 1567496 1568707 1568712) (-964 "PRIMARR.spad" 1566489 1566499 1566667 1566694) (-963 "PREASSOC.spad" 1565871 1565883 1566479 1566484) (-962 "PR.spad" 1564263 1564275 1564962 1565089) (-961 "PPCURVE.spad" 1563400 1563408 1564253 1564258) (-960 "PORTNUM.spad" 1563175 1563183 1563390 1563395) (-959 "POLYROOT.spad" 1562024 1562046 1563131 1563136) (-958 "POLYLIFT.spad" 1561289 1561312 1562014 1562019) (-957 "POLYCATQ.spad" 1559407 1559429 1561279 1561284) (-956 "POLYCAT.spad" 1552877 1552898 1559275 1559402) (-955 "POLYCAT.spad" 1545685 1545708 1552085 1552090) (-954 "POLY2UP.spad" 1545137 1545151 1545675 1545680) (-953 "POLY2.spad" 1544734 1544746 1545127 1545132) (-952 "POLY.spad" 1542069 1542079 1542584 1542711) (-951 "POLUTIL.spad" 1541010 1541039 1542025 1542030) (-950 "POLTOPOL.spad" 1539758 1539773 1541000 1541005) (-949 "POINT.spad" 1538596 1538606 1538683 1538710) (-948 "PNTHEORY.spad" 1535298 1535306 1538586 1538591) (-947 "PMTOOLS.spad" 1534073 1534087 1535288 1535293) (-946 "PMSYM.spad" 1533622 1533632 1534063 1534068) (-945 "PMQFCAT.spad" 1533213 1533227 1533612 1533617) (-944 "PMPREDFS.spad" 1532667 1532689 1533203 1533208) (-943 "PMPRED.spad" 1532146 1532160 1532657 1532662) (-942 "PMPLCAT.spad" 1531226 1531244 1532078 1532083) (-941 "PMLSAGG.spad" 1530811 1530825 1531216 1531221) (-940 "PMKERNEL.spad" 1530390 1530402 1530801 1530806) (-939 "PMINS.spad" 1529970 1529980 1530380 1530385) (-938 "PMFS.spad" 1529547 1529565 1529960 1529965) (-937 "PMDOWN.spad" 1528837 1528851 1529537 1529542) (-936 "PMASSFS.spad" 1527804 1527820 1528827 1528832) (-935 "PMASS.spad" 1526814 1526822 1527794 1527799) (-934 "PLOTTOOL.spad" 1526594 1526602 1526804 1526809) (-933 "PLOT3D.spad" 1523058 1523066 1526584 1526589) (-932 "PLOT1.spad" 1522215 1522225 1523048 1523053) (-931 "PLOT.spad" 1517138 1517146 1522205 1522210) (-930 "PLEQN.spad" 1504428 1504455 1517128 1517133) (-929 "PINTERPA.spad" 1504212 1504228 1504418 1504423) (-928 "PINTERP.spad" 1503834 1503853 1504202 1504207) (-927 "PID.spad" 1502804 1502812 1503760 1503829) (-926 "PICOERCE.spad" 1502461 1502471 1502794 1502799) (-925 "PI.spad" 1502070 1502078 1502435 1502456) (-924 "PGROEB.spad" 1500671 1500685 1502060 1502065) (-923 "PGE.spad" 1492288 1492296 1500661 1500666) (-922 "PGCD.spad" 1491178 1491195 1492278 1492283) (-921 "PFRPAC.spad" 1490327 1490337 1491168 1491173) (-920 "PFR.spad" 1486990 1487000 1490229 1490322) (-919 "PFOTOOLS.spad" 1486248 1486264 1486980 1486985) (-918 "PFOQ.spad" 1485618 1485636 1486238 1486243) (-917 "PFO.spad" 1485037 1485064 1485608 1485613) (-916 "PFECAT.spad" 1482719 1482727 1484963 1485032) (-915 "PFECAT.spad" 1480429 1480439 1482675 1482680) (-914 "PFBRU.spad" 1478317 1478329 1480419 1480424) (-913 "PFBR.spad" 1475877 1475900 1478307 1478312) (-912 "PF.spad" 1475451 1475463 1475682 1475775) (-911 "PERMGRP.spad" 1470213 1470223 1475441 1475446) (-910 "PERMCAT.spad" 1468771 1468781 1470193 1470208) (-909 "PERMAN.spad" 1467303 1467317 1468761 1468766) (-908 "PERM.spad" 1462988 1462998 1467133 1467148) (-907 "PENDTREE.spad" 1462329 1462339 1462617 1462622) (-906 "PDRING.spad" 1460880 1460890 1462309 1462324) (-905 "PDRING.spad" 1459439 1459451 1460870 1460875) (-904 "PDEPROB.spad" 1458454 1458462 1459429 1459434) (-903 "PDEPACK.spad" 1452494 1452502 1458444 1458449) (-902 "PDECOMP.spad" 1451964 1451981 1452484 1452489) (-901 "PDECAT.spad" 1450320 1450328 1451954 1451959) (-900 "PCOMP.spad" 1450173 1450186 1450310 1450315) (-899 "PBWLB.spad" 1448761 1448778 1450163 1450168) (-898 "PATTERN2.spad" 1448499 1448511 1448751 1448756) (-897 "PATTERN1.spad" 1446835 1446851 1448489 1448494) (-896 "PATTERN.spad" 1441374 1441384 1446825 1446830) (-895 "PATRES2.spad" 1441046 1441060 1441364 1441369) (-894 "PATRES.spad" 1438621 1438633 1441036 1441041) (-893 "PATMATCH.spad" 1436818 1436849 1438329 1438334) (-892 "PATMAB.spad" 1436247 1436257 1436808 1436813) (-891 "PATLRES.spad" 1435333 1435347 1436237 1436242) (-890 "PATAB.spad" 1435097 1435107 1435323 1435328) (-889 "PARTPERM.spad" 1432497 1432505 1435087 1435092) (-888 "PARSURF.spad" 1431931 1431959 1432487 1432492) (-887 "PARSU2.spad" 1431728 1431744 1431921 1431926) (-886 "script-parser.spad" 1431248 1431256 1431718 1431723) (-885 "PARSCURV.spad" 1430682 1430710 1431238 1431243) (-884 "PARSC2.spad" 1430473 1430489 1430672 1430677) (-883 "PARPCURV.spad" 1429935 1429963 1430463 1430468) (-882 "PARPC2.spad" 1429726 1429742 1429925 1429930) (-881 "PARAMAST.spad" 1428854 1428862 1429716 1429721) (-880 "PAN2EXPR.spad" 1428266 1428274 1428844 1428849) (-879 "PALETTE.spad" 1427236 1427244 1428256 1428261) (-878 "PAIR.spad" 1426223 1426236 1426824 1426829) (-877 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(-858 "OREPCTO.spad" 1396246 1396258 1398309 1398314) (-857 "OREPCAT.spad" 1390393 1390403 1396202 1396241) (-856 "OREPCAT.spad" 1384430 1384442 1390241 1390246) (-855 "ORDSET.spad" 1383602 1383610 1384420 1384425) (-854 "ORDSET.spad" 1382772 1382782 1383592 1383597) (-853 "ORDRING.spad" 1382162 1382170 1382752 1382767) (-852 "ORDRING.spad" 1381560 1381570 1382152 1382157) (-851 "ORDMON.spad" 1381415 1381423 1381550 1381555) (-850 "ORDFUNS.spad" 1380547 1380563 1381405 1381410) (-849 "ORDFIN.spad" 1380367 1380375 1380537 1380542) (-848 "ORDCOMP2.spad" 1379660 1379672 1380357 1380362) (-847 "ORDCOMP.spad" 1378125 1378135 1379207 1379236) (-846 "OPTPROB.spad" 1376763 1376771 1378115 1378120) (-845 "OPTPACK.spad" 1369172 1369180 1376753 1376758) (-844 "OPTCAT.spad" 1366851 1366859 1369162 1369167) (-843 "OPSIG.spad" 1366505 1366513 1366841 1366846) (-842 "OPQUERY.spad" 1366054 1366062 1366495 1366500) (-841 "OPERCAT.spad" 1365520 1365530 1366044 1366049) (-840 "OPERCAT.spad" 1364984 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(-764 "NAGSP.spad" 1220187 1220195 1221100 1221105) (-763 "NAGS.spad" 1209848 1209856 1220177 1220182) (-762 "NAGF07.spad" 1208279 1208287 1209838 1209843) (-761 "NAGF04.spad" 1202681 1202689 1208269 1208274) (-760 "NAGF02.spad" 1196750 1196758 1202671 1202676) (-759 "NAGF01.spad" 1192511 1192519 1196740 1196745) (-758 "NAGE04.spad" 1186211 1186219 1192501 1192506) (-757 "NAGE02.spad" 1176871 1176879 1186201 1186206) (-756 "NAGE01.spad" 1172873 1172881 1176861 1176866) (-755 "NAGD03.spad" 1170877 1170885 1172863 1172868) (-754 "NAGD02.spad" 1163624 1163632 1170867 1170872) (-753 "NAGD01.spad" 1157917 1157925 1163614 1163619) (-752 "NAGC06.spad" 1153792 1153800 1157907 1157912) (-751 "NAGC05.spad" 1152293 1152301 1153782 1153787) (-750 "NAGC02.spad" 1151560 1151568 1152283 1152288) (-749 "NAALG.spad" 1151101 1151111 1151528 1151555) (-748 "NAALG.spad" 1150662 1150674 1151091 1151096) (-747 "MULTSQFR.spad" 1147620 1147637 1150652 1150657) (-746 "MULTFACT.spad" 1147003 1147020 1147610 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(-102 "BASTYPE.spad" 110409 110417 110727 110732) (-101 "BASTYPE.spad" 110079 110089 110399 110404) (-100 "BALFACT.spad" 109538 109551 110069 110074) (-99 "AUTOMOR.spad" 108989 108998 109518 109533) (-98 "ATTREG.spad" 105712 105719 108741 108984) (-97 "ATTRBUT.spad" 101735 101742 105692 105707) (-96 "ATTRAST.spad" 101452 101459 101725 101730) (-95 "ATRIG.spad" 100922 100929 101442 101447) (-94 "ATRIG.spad" 100390 100399 100912 100917) (-93 "ASTCAT.spad" 100294 100301 100380 100385) (-92 "ASTCAT.spad" 100196 100205 100284 100289) (-91 "ASTACK.spad" 99535 99544 99803 99830) (-90 "ASSOCEQ.spad" 98361 98372 99491 99496) (-89 "ASP9.spad" 97442 97455 98351 98356) (-88 "ASP80.spad" 96764 96777 97432 97437) (-87 "ASP8.spad" 95807 95820 96754 96759) (-86 "ASP78.spad" 95258 95271 95797 95802) (-85 "ASP77.spad" 94627 94640 95248 95253) (-84 "ASP74.spad" 93719 93732 94617 94622) (-83 "ASP73.spad" 92990 93003 93709 93714) (-82 "ASP7.spad" 92150 92163 92980 92985) (-81 "ASP6.spad" 91017 91030 92140 92145) (-80 "ASP55.spad" 89526 89539 91007 91012) (-79 "ASP50.spad" 87343 87356 89516 89521) (-78 "ASP49.spad" 86342 86355 87333 87338) (-77 "ASP42.spad" 84749 84788 86332 86337) (-76 "ASP41.spad" 83328 83367 84739 84744) (-75 "ASP4.spad" 82623 82636 83318 83323) (-74 "ASP35.spad" 81611 81624 82613 82618) (-73 "ASP34.spad" 80912 80925 81601 81606) (-72 "ASP33.spad" 80472 80485 80902 80907) (-71 "ASP31.spad" 79612 79625 80462 80467) (-70 "ASP30.spad" 78504 78517 79602 79607) (-69 "ASP29.spad" 77970 77983 78494 78499) (-68 "ASP28.spad" 69243 69256 77960 77965) (-67 "ASP27.spad" 68140 68153 69233 69238) (-66 "ASP24.spad" 67227 67240 68130 68135) (-65 "ASP20.spad" 66691 66704 67217 67222) (-64 "ASP19.spad" 61377 61390 66681 66686) (-63 "ASP12.spad" 60791 60804 61367 61372) (-62 "ASP10.spad" 60062 60075 60781 60786) (-61 "ASP1.spad" 59443 59456 60052 60057) (-60 "ARRAY2.spad" 58803 58812 59050 59077) (-59 "ARRAY12.spad" 57516 57527 58793 58798) (-58 "ARRAY1.spad" 56353 56362 56699 56726) 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\ No newline at end of file
diff --git a/src/share/algebra/category.daase b/src/share/algebra/category.daase
index b07536d8..742345ac 100644
--- a/src/share/algebra/category.daase
+++ b/src/share/algebra/category.daase
@@ -1,299 +1,299 @@
 
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. -1290) 187244) ((-1291 . -1293) 187223) ((-1291 . -1042) 187180) ((-1291 . -619) 187109) ((-1291 . -1053) T) ((-1291 . -1061) T) ((-1291 . -1116) T) ((-1291 . -729) T) ((-1291 . -21) T) ((-1291 . -649) 187068) ((-1291 . -23) T) ((-1291 . -1105) T) ((-1291 . -616) 187050) ((-1291 . -102) T) ((-1291 . -25) T) ((-1291 . -131) T) ((-1291 . -651) 187024) ((-1291 . -1285) 187008) ((-1291 . -720) 186978) ((-1291 . -643) 186948) ((-1291 . -1060) 186932) ((-1291 . -1055) 186916) ((-1291 . -111) 186895) ((-1291 . -38) 186865) ((-1291 . -1290) 186844) ((-1291 . -387) 186816) ((-1286 . -387) 186788) ((-1286 . -619) 186737) ((-1286 . -1042) 186714) ((-1286 . -643) 186684) ((-1286 . -720) 186654) ((-1286 . -651) 186628) ((-1286 . -649) 186587) ((-1286 . -131) T) ((-1286 . -25) T) ((-1286 . -102) T) ((-1286 . -616) 186569) ((-1286 . -1105) T) ((-1286 . -23) T) ((-1286 . -21) T) ((-1286 . -1060) 186553) ((-1286 . -1055) 186537) ((-1286 . -111) 186516) ((-1286 . -1293) 186495) ((-1286 . -1053) T) ((-1286 . -1061) T) ((-1286 . -1116) T) ((-1286 . -729) T) ((-1286 . -1285) 186479) ((-1286 . -38) 186449) ((-1286 . -1290) 186428) ((-1284 . -1215) 186397) ((-1284 . -616) 186359) ((-1284 . -151) 186343) ((-1284 . -34) T) ((-1284 . -1220) T) ((-1284 . -311) 186281) ((-1284 . -518) 186214) ((-1284 . -1105) T) ((-1284 . -102) T) ((-1284 . -493) 186198) ((-1284 . -617) 186159) ((-1284 . -980) 186128) ((-1283 . -1053) T) ((-1283 . -1061) T) ((-1283 . -1116) T) ((-1283 . -729) T) ((-1283 . -21) T) ((-1283 . -649) 186073) ((-1283 . -23) T) ((-1283 . -1105) T) ((-1283 . -616) 186042) ((-1283 . -102) T) ((-1283 . -25) T) ((-1283 . -131) T) ((-1283 . -651) 186002) ((-1283 . -619) 185944) ((-1283 . -494) 185928) ((-1283 . -38) 185898) ((-1283 . -111) 185863) ((-1283 . -1055) 185833) ((-1283 . -1060) 185803) ((-1283 . -643) 185773) ((-1283 . -720) 185743) ((-1282 . -1087) T) ((-1282 . -494) 185724) ((-1282 . -616) 185690) ((-1282 . -619) 185671) ((-1282 . -1105) T) ((-1282 . -102) T) ((-1282 . -93) T) ((-1281 . -1087) T) ((-1281 . -494) 185652) ((-1281 . -616) 185618) ((-1281 . -619) 185599) ((-1281 . -1105) T) ((-1281 . -102) T) ((-1281 . -93) T) ((-1276 . -616) 185581) ((-1274 . -1105) T) ((-1274 . -616) 185563) ((-1274 . -102) T) ((-1273 . -1105) T) ((-1273 . -616) 185545) ((-1273 . -102) T) ((-1270 . -1269) 185529) ((-1270 . -375) 185513) ((-1270 . -853) 185492) ((-1270 . -151) 185476) ((-1270 . -34) T) ((-1270 . -1220) T) ((-1270 . -616) 185388) ((-1270 . -311) 185326) ((-1270 . -518) 185259) ((-1270 . -1105) 185209) ((-1270 . -102) 185159) ((-1270 . -493) 185143) ((-1270 . -617) 185104) ((-1270 . -607) 185081) ((-1270 . -288) 185058) ((-1270 . -290) 185035) ((-1270 . -654) 185019) ((-1270 . -19) 185003) ((-1267 . -1105) T) ((-1267 . -616) 184969) ((-1267 . -102) T) ((-1260 . -1263) 184953) ((-1260 . -234) 184912) ((-1260 . -619) 184794) ((-1260 . -651) 184719) ((-1260 . -649) 184629) ((-1260 . -131) T) ((-1260 . -25) T) ((-1260 . -102) T) ((-1260 . -616) 184611) ((-1260 . -1105) T) ((-1260 . -23) T) ((-1260 . -21) T) ((-1260 . -729) T) ((-1260 . -1116) T) ((-1260 . -1061) T) ((-1260 . -1053) T) ((-1260 . -288) 184596) ((-1260 . -904) 184509) ((-1260 . -977) 184478) ((-1260 . -38) 184375) ((-1260 . -111) 184244) ((-1260 . -1055) 184127) ((-1260 . -1060) 184010) ((-1260 . -643) 183907) ((-1260 . -720) 183804) ((-1260 . -145) 183783) ((-1260 . -147) 183762) ((-1260 . -173) 183713) ((-1260 . -561) 183692) ((-1260 . -292) 183671) ((-1260 . -47) 183648) ((-1260 . -1249) 183625) ((-1260 . -35) 183591) ((-1260 . -95) 183557) ((-1260 . -286) 183523) ((-1260 . -497) 183489) ((-1260 . -1209) 183455) ((-1260 . -1206) 183421) ((-1260 . -1006) 183387) ((-1257 . -328) 183331) ((-1257 . -1042) 183297) ((-1257 . -416) 183263) ((-1257 . -38) 183155) ((-1257 . -619) 183029) ((-1257 . -651) 182934) ((-1257 . -649) 182824) ((-1257 . -729) T) ((-1257 . -1116) T) ((-1257 . -1061) T) ((-1257 . -1053) T) ((-1257 . -111) 182716) ((-1257 . -1055) 182621) ((-1257 . -1060) 182526) ((-1257 . -21) T) ((-1257 . -23) T) ((-1257 . -1105) T) ((-1257 . -616) 182508) ((-1257 . -102) T) ((-1257 . -25) T) ((-1257 . -131) T) ((-1257 . -643) 182400) ((-1257 . -720) 182292) ((-1257 . -145) 182253) ((-1257 . -147) 182214) ((-1257 . -173) T) ((-1257 . -561) T) ((-1257 . -292) T) ((-1257 . -47) 182158) ((-1256 . -1255) 182137) ((-1256 . -366) 182116) ((-1256 . -1225) 182095) ((-1256 . -925) 182074) ((-1256 . -561) 182025) ((-1256 . -173) 181956) ((-1256 . -619) 181769) ((-1256 . -720) 181610) ((-1256 . -643) 181451) ((-1256 . -38) 181292) ((-1256 . -456) 181271) ((-1256 . -309) 181250) ((-1256 . -651) 181147) ((-1256 . -649) 181029) ((-1256 . -729) T) ((-1256 . -1116) T) ((-1256 . -1061) T) ((-1256 . -1053) T) ((-1256 . -111) 180850) ((-1256 . -1055) 180685) ((-1256 . -1060) 180520) ((-1256 . -21) T) ((-1256 . -23) T) ((-1256 . -1105) T) ((-1256 . -616) 180502) ((-1256 . -102) T) ((-1256 . -25) T) ((-1256 . -131) T) ((-1256 . -292) 180453) ((-1256 . -244) 180432) ((-1256 . -1006) 180398) ((-1256 . -1206) 180364) ((-1256 . -1209) 180330) ((-1256 . -497) 180296) ((-1256 . -286) 180262) ((-1256 . -95) 180228) ((-1256 . -35) 180194) ((-1256 . -1249) 180164) ((-1256 . -47) 180134) ((-1256 . -147) 180113) ((-1256 . -145) 180092) ((-1256 . -977) 180054) ((-1256 . -904) 179960) ((-1256 . -288) 179945) ((-1256 . -234) 179897) ((-1256 . -1253) 179881) ((-1256 . -1042) 179865) ((-1251 . -1255) 179826) ((-1251 . -366) 179805) ((-1251 . -1225) 179784) ((-1251 . -925) 179763) ((-1251 . -561) 179714) ((-1251 . -173) 179645) ((-1251 . -619) 179388) ((-1251 . -720) 179229) ((-1251 . -643) 179070) ((-1251 . -38) 178911) ((-1251 . -456) 178890) ((-1251 . -309) 178869) ((-1251 . -651) 178766) ((-1251 . -649) 178648) ((-1251 . -729) T) ((-1251 . -1116) T) ((-1251 . -1061) T) ((-1251 . -1053) T) ((-1251 . -111) 178469) ((-1251 . -1055) 178304) ((-1251 . -1060) 178139) ((-1251 . -21) T) ((-1251 . -23) T) ((-1251 . -1105) T) ((-1251 . -616) 178121) ((-1251 . -102) T) ((-1251 . -25) T) ((-1251 . -131) T) ((-1251 . -292) 178072) ((-1251 . -244) 178051) ((-1251 . -1006) 178017) ((-1251 . -1206) 177983) ((-1251 . -1209) 177949) ((-1251 . -497) 177915) ((-1251 . -286) 177881) ((-1251 . -95) 177847) ((-1251 . -35) 177813) ((-1251 . -1249) 177783) ((-1251 . -47) 177753) ((-1251 . -147) 177732) ((-1251 . -145) 177711) ((-1251 . -977) 177673) ((-1251 . -904) 177579) ((-1251 . -288) 177564) ((-1251 . -234) 177516) ((-1251 . -1253) 177500) ((-1251 . -1042) 177435) ((-1239 . -1246) 177419) ((-1239 . -1155) 177397) ((-1239 . -617) NIL) ((-1239 . -311) 177384) ((-1239 . -518) 177331) ((-1239 . -328) 177308) ((-1239 . -1042) 177188) ((-1239 . -416) 177172) ((-1239 . -38) 177001) ((-1239 . -111) 176810) ((-1239 . -1055) 176633) ((-1239 . -1060) 176456) ((-1239 . -649) 176366) ((-1239 . -651) 176291) ((-1239 . -643) 176120) ((-1239 . -720) 175949) ((-1239 . -619) 175697) ((-1239 . -145) 175676) ((-1239 . -147) 175655) ((-1239 . -47) 175632) ((-1239 . -380) 175616) ((-1239 . -642) 175564) ((-1239 . -904) 175507) ((-1239 . -890) NIL) ((-1239 . -914) 175486) ((-1239 . -1225) 175465) ((-1239 . -954) 175434) ((-1239 . -925) 175413) ((-1239 . -561) 175324) ((-1239 . -292) 175235) ((-1239 . -173) 175126) ((-1239 . -456) 175057) ((-1239 . -309) 175036) ((-1239 . -288) 174963) ((-1239 . -234) T) ((-1239 . -131) T) ((-1239 . -25) T) ((-1239 . -102) T) ((-1239 . -616) 174945) ((-1239 . -1105) T) ((-1239 . -23) T) ((-1239 . -21) T) ((-1239 . -729) T) ((-1239 . -1116) T) ((-1239 . -1061) T) ((-1239 . -1053) T) ((-1239 . -232) 174929) ((-1237 . -1098) 174913) ((-1237 . -621) 174897) ((-1237 . -1105) 174875) ((-1237 . -616) 174842) ((-1237 . -102) 174820) ((-1237 . -1099) 174777) ((-1235 . -1234) 174756) ((-1235 . -1006) 174722) ((-1235 . -1206) 174688) ((-1235 . -1209) 174654) ((-1235 . -497) 174620) ((-1235 . -286) 174586) ((-1235 . -95) 174552) ((-1235 . -35) 174518) ((-1235 . -1249) 174495) ((-1235 . -47) 174472) ((-1235 . -619) 174220) ((-1235 . -720) 174034) ((-1235 . -643) 173848) ((-1235 . -651) 173718) ((-1235 . -649) 173573) ((-1235 . -1060) 173381) ((-1235 . -1055) 173189) ((-1235 . -111) 172978) ((-1235 . -38) 172792) ((-1235 . -977) 172761) ((-1235 . -288) 172681) ((-1235 . -1232) 172665) ((-1235 . -729) T) ((-1235 . -1116) T) ((-1235 . -1061) T) ((-1235 . -1053) T) ((-1235 . -21) T) ((-1235 . -23) T) ((-1235 . -1105) T) ((-1235 . -616) 172647) ((-1235 . -102) T) ((-1235 . -25) T) ((-1235 . -131) T) ((-1235 . -145) 172572) ((-1235 . -147) 172497) ((-1235 . -617) 172168) ((-1235 . -232) 172138) ((-1235 . -904) 171989) ((-1235 . -234) 171894) ((-1235 . -366) 171873) ((-1235 . -1225) 171852) ((-1235 . -925) 171831) ((-1235 . -561) 171782) ((-1235 . -173) 171713) ((-1235 . -456) 171692) ((-1235 . -309) 171671) ((-1235 . -292) 171622) ((-1235 . -244) 171601) ((-1235 . -341) 171571) ((-1235 . -518) 171431) ((-1235 . -311) 171370) ((-1235 . -380) 171340) ((-1235 . -642) 171248) ((-1235 . -404) 171218) ((-1235 . -1220) 171197) ((-1235 . -890) 171070) ((-1235 . -823) 171023) ((-1235 . -794) 170976) ((-1235 . -795) 170929) ((-1235 . -853) 170828) ((-1235 . -797) 170781) ((-1235 . -800) 170734) ((-1235 . -851) 170687) ((-1235 . -888) 170657) ((-1235 . -914) 170610) ((-1235 . -1024) 170562) ((-1235 . -1042) 170348) ((-1235 . -1155) 170300) ((-1235 . -995) 170270) ((-1230 . -1234) 170231) ((-1230 . -1006) 170197) ((-1230 . -1206) 170163) ((-1230 . -1209) 170129) ((-1230 . -497) 170095) ((-1230 . -286) 170061) ((-1230 . -95) 170027) ((-1230 . -35) 169993) ((-1230 . -1249) 169970) ((-1230 . -47) 169947) ((-1230 . -619) 169742) ((-1230 . -720) 169538) ((-1230 . -643) 169334) ((-1230 . -651) 169186) ((-1230 . -649) 169023) ((-1230 . -1060) 168813) ((-1230 . -1055) 168603) ((-1230 . -111) 168372) ((-1230 . -38) 168168) ((-1230 . -977) 168137) ((-1230 . -288) 167985) ((-1230 . -1232) 167969) ((-1230 . -729) T) ((-1230 . -1116) T) ((-1230 . -1061) T) ((-1230 . -1053) T) ((-1230 . -21) T) ((-1230 . -23) T) ((-1230 . -1105) T) ((-1230 . -616) 167951) ((-1230 . -102) T) ((-1230 . -25) T) ((-1230 . -131) T) ((-1230 . -145) 167858) ((-1230 . -147) 167765) ((-1230 . -617) NIL) ((-1230 . -232) 167717) ((-1230 . -904) 167550) ((-1230 . -234) 167437) ((-1230 . -366) 167416) ((-1230 . -1225) 167395) ((-1230 . -925) 167374) ((-1230 . -561) 167325) ((-1230 . -173) 167256) ((-1230 . -456) 167235) ((-1230 . -309) 167214) ((-1230 . -292) 167165) ((-1230 . -244) 167144) ((-1230 . -341) 167096) ((-1230 . -518) 166865) ((-1230 . -311) 166750) ((-1230 . -380) 166702) ((-1230 . -642) 166654) ((-1230 . -404) 166606) ((-1230 . -1220) 166585) ((-1230 . -890) NIL) ((-1230 . -823) NIL) ((-1230 . -794) NIL) ((-1230 . -795) NIL) ((-1230 . -853) NIL) ((-1230 . -797) NIL) ((-1230 . -800) NIL) ((-1230 . -851) NIL) ((-1230 . -888) 166537) ((-1230 . -914) NIL) ((-1230 . -1024) NIL) ((-1230 . -1042) 166503) ((-1230 . -1155) NIL) ((-1230 . -995) 166455) ((-1229 . -847) T) ((-1229 . -853) T) ((-1229 . -1105) T) ((-1229 . -616) 166437) ((-1229 . -102) T) ((-1229 . -371) T) ((-1228 . -847) T) ((-1228 . -853) T) ((-1228 . -1105) T) ((-1228 . -616) 166419) ((-1228 . -102) T) ((-1228 . -371) T) ((-1227 . -847) T) ((-1227 . -853) T) ((-1227 . -1105) T) ((-1227 . -616) 166401) ((-1227 . -102) T) ((-1227 . -371) T) ((-1226 . -847) T) ((-1226 . -853) T) ((-1226 . -1105) T) ((-1226 . -616) 166383) ((-1226 . -102) T) ((-1226 . -371) T) ((-1221 . -1087) T) ((-1221 . -494) 166364) ((-1221 . -616) 166330) ((-1221 . -619) 166311) ((-1221 . -1105) T) ((-1221 . -102) T) ((-1221 . -93) T) ((-1218 . -494) 166288) ((-1218 . -616) 166200) ((-1218 . -619) 166177) ((-1218 . -1105) 166155) ((-1218 . -102) 166133) ((-1213 . -743) 166109) ((-1213 . -35) 166075) ((-1213 . -95) 166041) ((-1213 . -286) 166007) ((-1213 . -497) 165973) ((-1213 . -1209) 165939) ((-1213 . -1206) 165905) ((-1213 . -1006) 165871) ((-1213 . -47) 165840) ((-1213 . -38) 165737) ((-1213 . -643) 165634) ((-1213 . -720) 165531) ((-1213 . -619) 165413) ((-1213 . -292) 165392) ((-1213 . -561) 165371) ((-1213 . -111) 165240) ((-1213 . -1055) 165123) ((-1213 . -1060) 165006) ((-1213 . -173) 164957) ((-1213 . -147) 164936) ((-1213 . -145) 164915) ((-1213 . -651) 164840) ((-1213 . -649) 164750) ((-1213 . -977) 164712) ((-1213 . -1053) T) ((-1213 . -1061) T) ((-1213 . -1116) T) ((-1213 . -729) T) ((-1213 . -21) T) ((-1213 . -23) T) ((-1213 . -1105) T) ((-1213 . -616) 164694) ((-1213 . -102) T) ((-1213 . -25) T) ((-1213 . -131) T) ((-1213 . -904) 164675) ((-1213 . -518) 164642) ((-1213 . -311) 164629) ((-1207 . -1014) 164613) ((-1207 . -34) T) ((-1207 . -1220) T) ((-1207 . -616) 164545) ((-1207 . -311) 164483) ((-1207 . -518) 164416) ((-1207 . -1105) 164394) ((-1207 . -102) 164372) ((-1207 . -493) 164356) ((-1202 . -368) 164330) ((-1202 . -102) T) ((-1202 . -616) 164312) ((-1202 . -1105) T) ((-1200 . -1105) T) ((-1200 . -616) 164294) ((-1200 . -102) T) ((-1200 . -619) 164276) ((-1193 . -1197) 164255) ((-1193 . -230) 164205) ((-1193 . -107) 164155) ((-1193 . -311) 163959) ((-1193 . -518) 163751) ((-1193 . -493) 163688) ((-1193 . -151) 163638) ((-1193 . -617) NIL) ((-1193 . -236) 163588) ((-1193 . -613) 163567) ((-1193 . -290) 163546) ((-1193 . -288) 163525) ((-1193 . -102) T) ((-1193 . -1105) T) ((-1193 . -616) 163507) ((-1193 . -1220) T) ((-1193 . -34) T) ((-1193 . -607) 163486) ((-1191 . -1220) T) ((-1189 . -1105) T) ((-1189 . -616) 163468) ((-1189 . -102) T) ((-1188 . -847) T) ((-1188 . -853) T) ((-1188 . -1105) T) ((-1188 . -616) 163450) ((-1188 . -102) T) ((-1188 . -371) T) ((-1187 . -847) T) ((-1187 . -853) T) ((-1187 . -1105) T) ((-1187 . -616) 163432) ((-1187 . -102) T) ((-1187 . -371) T) ((-1186 . -1266) T) ((-1186 . -1105) T) ((-1186 . -616) 163399) ((-1186 . -102) T) ((-1186 . -1042) 163335) ((-1186 . -619) 163271) ((-1185 . -616) 163253) ((-1184 . -616) 163235) ((-1183 . -328) 163212) ((-1183 . -1042) 163108) ((-1183 . -416) 163092) ((-1183 . -38) 162989) ((-1183 . -619) 162842) ((-1183 . -651) 162767) ((-1183 . -649) 162677) ((-1183 . -729) T) ((-1183 . -1116) T) ((-1183 . -1061) T) ((-1183 . -1053) T) ((-1183 . -111) 162546) ((-1183 . -1055) 162429) ((-1183 . -1060) 162312) ((-1183 . -21) T) ((-1183 . -23) T) ((-1183 . -1105) T) ((-1183 . -616) 162294) ((-1183 . -102) T) ((-1183 . -25) T) ((-1183 . -131) T) ((-1183 . -643) 162191) ((-1183 . -720) 162088) ((-1183 . -145) 162067) ((-1183 . -147) 162046) ((-1183 . -173) 161997) ((-1183 . -561) 161976) ((-1183 . -292) 161955) ((-1183 . -47) 161932) ((-1181 . -853) T) ((-1181 . -102) T) ((-1181 . -616) 161914) ((-1181 . -1105) T) ((-1181 . -617) 161836) ((-1181 . -824) T) ((-1181 . -619) 161817) ((-1181 . -890) 161784) ((-1180 . -616) 161766) ((-1179 . -1263) 161750) ((-1179 . -234) 161709) ((-1179 . -619) 161591) ((-1179 . -651) 161516) ((-1179 . -649) 161426) ((-1179 . -131) T) ((-1179 . -25) T) ((-1179 . -102) T) ((-1179 . -616) 161408) ((-1179 . -1105) T) ((-1179 . -23) T) ((-1179 . -21) T) ((-1179 . -729) T) ((-1179 . -1116) T) ((-1179 . -1061) T) ((-1179 . -1053) T) ((-1179 . -288) 161393) ((-1179 . -904) 161306) ((-1179 . -977) 161275) ((-1179 . -38) 161172) ((-1179 . -111) 161041) ((-1179 . -1055) 160924) ((-1179 . -1060) 160807) ((-1179 . -643) 160704) ((-1179 . -720) 160601) ((-1179 . -145) 160580) ((-1179 . -147) 160559) ((-1179 . -173) 160510) ((-1179 . -561) 160489) ((-1179 . -292) 160468) ((-1179 . -47) 160445) ((-1179 . -1249) 160422) ((-1179 . -35) 160388) ((-1179 . -95) 160354) ((-1179 . -286) 160320) ((-1179 . -497) 160286) ((-1179 . -1209) 160252) ((-1179 . -1206) 160218) ((-1179 . -1006) 160184) ((-1178 . -1255) 160145) ((-1178 . -366) 160124) ((-1178 . -1225) 160103) ((-1178 . -925) 160082) ((-1178 . -561) 160033) ((-1178 . -173) 159964) ((-1178 . -619) 159707) ((-1178 . -720) 159548) ((-1178 . -643) 159389) ((-1178 . -38) 159230) ((-1178 . -456) 159209) ((-1178 . -309) 159188) ((-1178 . -651) 159085) ((-1178 . -649) 158967) ((-1178 . -729) T) ((-1178 . -1116) T) ((-1178 . -1061) T) ((-1178 . -1053) T) ((-1178 . -111) 158788) ((-1178 . -1055) 158623) ((-1178 . -1060) 158458) ((-1178 . -21) T) ((-1178 . -23) T) ((-1178 . -1105) T) ((-1178 . -616) 158440) ((-1178 . -102) T) ((-1178 . -25) T) ((-1178 . -131) T) ((-1178 . -292) 158391) ((-1178 . -244) 158370) ((-1178 . -1006) 158336) ((-1178 . -1206) 158302) ((-1178 . -1209) 158268) ((-1178 . -497) 158234) ((-1178 . -286) 158200) ((-1178 . -95) 158166) ((-1178 . -35) 158132) ((-1178 . -1249) 158102) ((-1178 . -47) 158072) ((-1178 . -147) 158051) ((-1178 . -145) 158030) ((-1178 . -977) 157992) ((-1178 . -904) 157898) ((-1178 . -288) 157883) ((-1178 . -234) 157835) ((-1178 . -1253) 157819) ((-1178 . -1042) 157754) ((-1175 . -1246) 157738) ((-1175 . -1155) 157716) ((-1175 . -617) NIL) ((-1175 . -311) 157703) ((-1175 . -518) 157650) ((-1175 . -328) 157627) ((-1175 . -1042) 157507) ((-1175 . -416) 157491) ((-1175 . -38) 157320) ((-1175 . -111) 157129) ((-1175 . -1055) 156952) ((-1175 . -1060) 156775) ((-1175 . -649) 156685) ((-1175 . -651) 156610) ((-1175 . -643) 156439) ((-1175 . -720) 156268) ((-1175 . -619) 156037) ((-1175 . -145) 156016) ((-1175 . -147) 155995) ((-1175 . -47) 155972) ((-1175 . -380) 155956) ((-1175 . -642) 155904) ((-1175 . -904) 155847) ((-1175 . -890) NIL) ((-1175 . -914) 155826) ((-1175 . -1225) 155805) ((-1175 . -954) 155774) ((-1175 . -925) 155753) ((-1175 . -561) 155664) ((-1175 . -292) 155575) ((-1175 . -173) 155466) ((-1175 . -456) 155397) ((-1175 . -309) 155376) ((-1175 . -288) 155303) ((-1175 . -234) T) ((-1175 . -131) T) ((-1175 . -25) T) ((-1175 . -102) T) ((-1175 . -616) 155285) ((-1175 . -1105) T) ((-1175 . -23) T) ((-1175 . -21) T) ((-1175 . -729) T) ((-1175 . -1116) T) ((-1175 . -1061) T) ((-1175 . -1053) T) ((-1175 . -232) 155269) ((-1172 . -1234) 155230) ((-1172 . -1006) 155196) ((-1172 . -1206) 155162) ((-1172 . -1209) 155128) ((-1172 . -497) 155094) ((-1172 . -286) 155060) ((-1172 . -95) 155026) ((-1172 . -35) 154992) ((-1172 . -1249) 154969) ((-1172 . -47) 154946) ((-1172 . -619) 154741) ((-1172 . -720) 154537) ((-1172 . -643) 154333) ((-1172 . -651) 154185) ((-1172 . -649) 154022) ((-1172 . -1060) 153812) ((-1172 . -1055) 153602) ((-1172 . -111) 153371) ((-1172 . -38) 153167) ((-1172 . -977) 153136) ((-1172 . -288) 152984) ((-1172 . -1232) 152968) ((-1172 . -729) T) ((-1172 . -1116) T) ((-1172 . -1061) T) ((-1172 . -1053) T) ((-1172 . -21) T) ((-1172 . -23) T) ((-1172 . -1105) T) ((-1172 . -616) 152950) ((-1172 . -102) T) ((-1172 . -25) T) ((-1172 . -131) T) ((-1172 . -145) 152857) ((-1172 . -147) 152764) ((-1172 . -617) NIL) ((-1172 . -232) 152716) ((-1172 . -904) 152549) ((-1172 . -234) 152436) ((-1172 . -366) 152415) ((-1172 . -1225) 152394) ((-1172 . -925) 152373) ((-1172 . -561) 152324) ((-1172 . -173) 152255) ((-1172 . -456) 152234) ((-1172 . -309) 152213) ((-1172 . -292) 152164) ((-1172 . -244) 152143) ((-1172 . -341) 152095) ((-1172 . -518) 151864) ((-1172 . -311) 151749) ((-1172 . -380) 151701) ((-1172 . -642) 151653) ((-1172 . -404) 151605) ((-1172 . -1220) 151584) ((-1172 . -890) NIL) ((-1172 . -823) NIL) ((-1172 . -794) NIL) ((-1172 . -795) NIL) ((-1172 . -853) NIL) ((-1172 . -797) NIL) ((-1172 . -800) NIL) ((-1172 . -851) NIL) ((-1172 . -888) 151536) ((-1172 . -914) NIL) ((-1172 . -1024) NIL) ((-1172 . -1042) 151502) ((-1172 . -1155) NIL) ((-1172 . -995) 151454) ((-1171 . -1087) T) ((-1171 . -494) 151435) ((-1171 . -616) 151401) ((-1171 . -619) 151382) ((-1171 . -1105) T) ((-1171 . -102) T) ((-1171 . -93) T) ((-1170 . -1105) T) ((-1170 . -616) 151364) ((-1170 . -102) T) ((-1169 . -1105) T) ((-1169 . -616) 151346) ((-1169 . -102) T) ((-1164 . -1197) 151322) ((-1164 . -230) 151269) ((-1164 . -107) 151216) ((-1164 . -311) 151011) ((-1164 . -518) 150794) ((-1164 . -493) 150728) ((-1164 . -151) 150675) ((-1164 . -617) NIL) ((-1164 . -236) 150622) ((-1164 . -613) 150598) ((-1164 . -290) 150574) ((-1164 . -288) 150550) ((-1164 . -102) T) ((-1164 . -1105) T) ((-1164 . -616) 150532) ((-1164 . -1220) T) ((-1164 . -34) T) ((-1164 . -607) 150508) ((-1163 . -1162) T) ((-1163 . -19) 150490) ((-1163 . -654) 150472) ((-1163 . -290) 150447) ((-1163 . -288) 150422) ((-1163 . -607) 150397) ((-1163 . -617) NIL) ((-1163 . -493) 150379) ((-1163 . -518) NIL) ((-1163 . -311) NIL) ((-1163 . -1220) T) ((-1163 . -34) T) ((-1163 . -151) 150361) ((-1163 . -853) T) ((-1163 . -375) 150343) ((-1163 . -1148) T) ((-1163 . -102) T) ((-1163 . -616) 150325) ((-1163 . -1105) T) ((-1163 . -824) T) ((-1158 . -677) 150309) ((-1158 . -654) 150293) ((-1158 . -290) 150270) ((-1158 . -288) 150247) ((-1158 . -607) 150224) ((-1158 . -617) 150185) ((-1158 . -493) 150169) ((-1158 . -102) 150147) ((-1158 . -1105) 150125) ((-1158 . -518) 150058) ((-1158 . -311) 149996) ((-1158 . -616) 149928) ((-1158 . -1220) T) ((-1158 . -34) T) ((-1158 . -151) 149912) ((-1158 . -1259) 149896) ((-1158 . -1014) 149880) ((-1158 . -1153) 149864) ((-1158 . -619) 149841) ((-1156 . -1087) T) ((-1156 . -494) 149822) ((-1156 . -616) 149788) ((-1156 . -619) 149769) ((-1156 . -1105) T) ((-1156 . -102) T) ((-1156 . -93) T) ((-1154 . -1197) 149748) ((-1154 . -230) 149698) ((-1154 . -107) 149648) ((-1154 . -311) 149452) ((-1154 . -518) 149244) ((-1154 . -493) 149181) ((-1154 . -151) 149131) ((-1154 . -617) NIL) ((-1154 . -236) 149081) ((-1154 . -613) 149060) ((-1154 . -290) 149039) ((-1154 . -288) 149018) ((-1154 . -102) T) ((-1154 . -1105) T) ((-1154 . -616) 149000) ((-1154 . -1220) T) ((-1154 . -34) T) ((-1154 . -607) 148979) ((-1151 . -1125) 148963) ((-1151 . -493) 148947) ((-1151 . -102) 148925) ((-1151 . -1105) 148903) ((-1151 . -518) 148836) ((-1151 . -311) 148774) ((-1151 . -616) 148706) ((-1151 . -1220) T) ((-1151 . -34) T) ((-1151 . -107) 148690) ((-1150 . -1113) 148659) ((-1150 . -1215) 148628) ((-1150 . -616) 148590) ((-1150 . -151) 148574) ((-1150 . -34) T) ((-1150 . -1220) T) ((-1150 . -311) 148512) ((-1150 . -518) 148445) ((-1150 . -1105) T) ((-1150 . -102) T) ((-1150 . -493) 148429) ((-1150 . -617) 148390) ((-1150 . -980) 148359) ((-1150 . -1075) 148328) ((-1146 . -1127) 148273) ((-1146 . -493) 148257) ((-1146 . -518) 148190) ((-1146 . -311) 148128) ((-1146 . -1220) T) ((-1146 . -34) T) ((-1146 . -1057) 148068) ((-1146 . -1042) 147964) ((-1146 . -619) 147882) ((-1146 . -416) 147866) ((-1146 . -642) 147814) ((-1146 . -380) 147798) ((-1146 . -234) 147777) ((-1146 . -904) 147736) ((-1146 . -232) 147720) ((-1146 . -720) 147652) ((-1146 . -643) 147584) ((-1146 . -651) 147558) ((-1146 . -649) 147517) ((-1146 . -131) T) ((-1146 . -25) T) ((-1146 . -102) T) ((-1146 . -616) 147479) ((-1146 . -1105) T) ((-1146 . -23) T) ((-1146 . -21) T) ((-1146 . -1060) 147463) ((-1146 . -1055) 147447) ((-1146 . -111) 147426) ((-1146 . -1053) T) ((-1146 . -1061) T) ((-1146 . -1116) T) ((-1146 . -729) T) ((-1146 . -38) 147386) ((-1146 . -617) 147347) ((-1145 . -1014) 147318) ((-1145 . -34) T) ((-1145 . -1220) T) ((-1145 . -616) 147300) ((-1145 . -311) 147226) ((-1145 . -518) 147145) ((-1145 . -1105) T) ((-1145 . -102) T) ((-1145 . -493) 147116) ((-1144 . -1105) T) ((-1144 . -616) 147098) ((-1144 . -102) T) ((-1139 . -1141) T) ((-1139 . -1266) T) ((-1139 . -93) T) ((-1139 . -102) T) ((-1139 . -616) 147064) ((-1139 . -1105) T) ((-1139 . -619) 147045) ((-1139 . -494) 147026) ((-1139 . -1087) T) ((-1137 . -1138) 147010) ((-1137 . -102) T) ((-1137 . -616) 146992) ((-1137 . -1105) T) ((-1130 . -743) 146971) ((-1130 . -35) 146937) ((-1130 . -95) 146903) ((-1130 . -286) 146869) ((-1130 . -497) 146835) ((-1130 . -1209) 146801) ((-1130 . -1206) 146767) ((-1130 . -1006) 146733) ((-1130 . -47) 146705) ((-1130 . -38) 146602) ((-1130 . -643) 146499) ((-1130 . -720) 146396) ((-1130 . -619) 146278) ((-1130 . -292) 146257) ((-1130 . -561) 146236) ((-1130 . -111) 146105) ((-1130 . -1055) 145988) ((-1130 . -1060) 145871) ((-1130 . -173) 145822) ((-1130 . -147) 145801) ((-1130 . -145) 145780) ((-1130 . -651) 145705) ((-1130 . -649) 145615) ((-1130 . -977) 145582) ((-1130 . -1053) T) ((-1130 . -1061) T) ((-1130 . -1116) T) ((-1130 . -729) T) ((-1130 . -21) T) ((-1130 . -23) T) ((-1130 . -1105) T) ((-1130 . -616) 145564) ((-1130 . -102) T) ((-1130 . -25) T) ((-1130 . -131) T) ((-1130 . -904) 145548) ((-1130 . -518) 145518) ((-1130 . -311) 145505) ((-1129 . -954) 145472) ((-1129 . -619) 145264) ((-1129 . -1042) 145147) ((-1129 . -1225) 145126) ((-1129 . -914) 145105) ((-1129 . -890) 144964) ((-1129 . -904) 144948) ((-1129 . -518) 144900) ((-1129 . -456) 144851) ((-1129 . -642) 144799) ((-1129 . -380) 144783) ((-1129 . -47) 144755) ((-1129 . -38) 144604) ((-1129 . -643) 144453) ((-1129 . -720) 144302) ((-1129 . -292) 144233) ((-1129 . -561) 144164) ((-1129 . -111) 143993) ((-1129 . -1055) 143836) ((-1129 . -1060) 143679) ((-1129 . -173) 143590) ((-1129 . -147) 143569) ((-1129 . -145) 143548) ((-1129 . -651) 143473) ((-1129 . -649) 143383) ((-1129 . -131) T) ((-1129 . -25) T) ((-1129 . -102) T) ((-1129 . -616) 143365) ((-1129 . -1105) T) ((-1129 . -23) T) ((-1129 . -21) T) ((-1129 . -1053) T) ((-1129 . -1061) T) ((-1129 . -1116) T) ((-1129 . -729) T) ((-1129 . -416) 143349) ((-1129 . -328) 143321) ((-1129 . -311) 143308) ((-1129 . -617) 143056) ((-1124 . -549) T) ((-1124 . -1225) T) ((-1124 . -1155) T) ((-1124 . -1042) 143038) ((-1124 . -617) 142953) ((-1124 . -1024) T) ((-1124 . -890) 142935) ((-1124 . -851) T) ((-1124 . -800) T) ((-1124 . -797) T) ((-1124 . -853) T) ((-1124 . -795) T) ((-1124 . -794) T) ((-1124 . -823) T) ((-1124 . -642) 142917) ((-1124 . -925) T) ((-1124 . -561) T) ((-1124 . -292) T) ((-1124 . -173) T) ((-1124 . -619) 142889) ((-1124 . -720) 142876) ((-1124 . -643) 142863) ((-1124 . -1060) 142850) ((-1124 . -1055) 142837) ((-1124 . -111) 142822) ((-1124 . -38) 142809) ((-1124 . -456) T) ((-1124 . -309) T) ((-1124 . -234) T) ((-1124 . -143) T) ((-1124 . -1053) T) ((-1124 . -1061) T) ((-1124 . -1116) T) ((-1124 . -729) T) ((-1124 . -21) T) ((-1124 . -649) 142781) ((-1124 . -23) T) ((-1124 . -1105) T) ((-1124 . -616) 142763) ((-1124 . -102) T) ((-1124 . -25) T) ((-1124 . -131) T) ((-1124 . -651) 142750) ((-1124 . -147) T) ((-1124 . -847) T) ((-1124 . -371) T) ((-1124 . -665) T) ((-1124 . -824) T) ((-1120 . -1087) T) ((-1120 . -494) 142731) ((-1120 . -616) 142697) ((-1120 . -619) 142678) ((-1120 . -1105) T) ((-1120 . -102) T) ((-1120 . -93) T) ((-1119 . -1105) T) ((-1119 . -616) 142660) ((-1119 . -102) T) ((-1117 . -239) 142639) ((-1117 . -1278) 142609) ((-1117 . -794) 142588) ((-1117 . -851) 142567) ((-1117 . -800) 142518) ((-1117 . -797) 142469) ((-1117 . -853) 142420) ((-1117 . -795) 142371) ((-1117 . -796) 142350) ((-1117 . -290) 142327) ((-1117 . -288) 142304) ((-1117 . -493) 142288) ((-1117 . -518) 142221) ((-1117 . -311) 142159) ((-1117 . -1220) T) ((-1117 . -34) T) ((-1117 . -607) 142136) ((-1117 . -1042) 141963) ((-1117 . -619) 141693) ((-1117 . -416) 141662) ((-1117 . -642) 141568) ((-1117 . -380) 141537) ((-1117 . -371) 141516) ((-1117 . -234) 141468) ((-1117 . -904) 141400) ((-1117 . -232) 141369) ((-1117 . -111) 141259) ((-1117 . -1055) 141156) ((-1117 . -1060) 141053) ((-1117 . -173) 141032) ((-1117 . -616) 140763) ((-1117 . -720) 140705) ((-1117 . -643) 140647) ((-1117 . -651) 140495) ((-1117 . -649) 140245) ((-1117 . -131) 140115) ((-1117 . -23) 139985) ((-1117 . -21) 139895) ((-1117 . -1053) 139825) ((-1117 . -1061) 139755) ((-1117 . -1116) 139665) ((-1117 . -729) 139575) ((-1117 . -38) 139545) ((-1117 . -1105) 139335) ((-1117 . -102) 139125) ((-1117 . -25) 138976) ((-1110 . -400) T) ((-1110 . -1220) T) ((-1110 . -616) 138958) ((-1109 . -1108) 138922) ((-1109 . -102) T) ((-1109 . -616) 138904) ((-1109 . -1105) T) ((-1109 . -621) 138819) ((-1107 . -1108) 138771) ((-1107 . -102) T) ((-1107 . -616) 138753) ((-1107 . -1105) T) ((-1107 . -621) 138656) ((-1106 . -371) T) ((-1106 . -102) T) ((-1106 . -616) 138638) ((-1106 . -1105) T) ((-1101 . -430) 138622) ((-1101 . -1103) 138606) ((-1101 . -371) 138585) ((-1101 . -236) 138569) ((-1101 . -617) 138530) ((-1101 . -151) 138514) ((-1101 . -493) 138498) ((-1101 . -102) T) ((-1101 . -1105) T) ((-1101 . -518) 138431) ((-1101 . -311) 138369) ((-1101 . -616) 138351) ((-1101 . -1220) T) ((-1101 . -34) T) ((-1101 . -107) 138335) ((-1101 . -230) 138319) ((-1100 . -1087) T) ((-1100 . -494) 138300) ((-1100 . -616) 138266) ((-1100 . -619) 138247) ((-1100 . -1105) T) ((-1100 . -102) T) ((-1100 . -93) T) ((-1096 . -1220) T) ((-1096 . -1105) 138217) ((-1096 . -616) 138176) ((-1096 . -102) 138146) ((-1095 . -1087) T) ((-1095 . -494) 138127) ((-1095 . -616) 138093) ((-1095 . -619) 138074) ((-1095 . -1105) T) ((-1095 . -102) T) ((-1095 . -93) T) ((-1093 . -1098) 138058) ((-1093 . -621) 138042) ((-1093 . -1105) 138020) ((-1093 . -616) 137987) ((-1093 . -102) 137965) ((-1093 . -1099) 137923) ((-1092 . -268) 137907) ((-1092 . -619) 137891) ((-1092 . -1042) 137875) ((-1092 . -1105) T) ((-1092 . -616) 137857) ((-1092 . -102) T) ((-1092 . -853) T) ((-1091 . -255) 137794) ((-1091 . -619) 137530) ((-1091 . -1042) 137357) ((-1091 . -617) NIL) ((-1091 . -328) 137318) ((-1091 . -416) 137302) ((-1091 . -38) 137151) ((-1091 . -111) 136980) ((-1091 . -1055) 136823) ((-1091 . -1060) 136666) ((-1091 . -649) 136576) ((-1091 . -651) 136501) ((-1091 . -643) 136350) ((-1091 . -720) 136199) ((-1091 . -145) 136178) ((-1091 . -147) 136157) ((-1091 . -173) 136068) ((-1091 . -561) 135999) ((-1091 . -292) 135930) ((-1091 . -47) 135891) ((-1091 . -380) 135875) ((-1091 . -642) 135823) ((-1091 . -456) 135774) ((-1091 . -518) 135641) ((-1091 . -904) 135576) ((-1091 . -890) NIL) ((-1091 . -914) 135555) ((-1091 . -1225) 135534) ((-1091 . -954) 135479) ((-1091 . -311) 135466) ((-1091 . -234) 135445) ((-1091 . -131) T) ((-1091 . -25) T) ((-1091 . -102) T) ((-1091 . -616) 135427) ((-1091 . -1105) T) ((-1091 . -23) T) ((-1091 . -21) T) ((-1091 . -729) T) ((-1091 . -1116) T) ((-1091 . -1061) T) ((-1091 . -1053) T) ((-1091 . -232) 135411) ((-1089 . -616) 135393) ((-1086 . -853) T) ((-1086 . -102) T) ((-1086 . -616) 135375) ((-1086 . -1105) T) ((-1086 . -617) 135356) ((-1083 . -727) 135335) ((-1083 . -1042) 135231) ((-1083 . -416) 135215) ((-1083 . -642) 135163) ((-1083 . -380) 135147) ((-1083 . -373) 135126) ((-1083 . -147) 135105) ((-1083 . -619) 134923) ((-1083 . -720) 134791) ((-1083 . -643) 134659) ((-1083 . -651) 134569) ((-1083 . -649) 134464) ((-1083 . -1060) 134374) ((-1083 . -1055) 134284) ((-1083 . -111) 134180) ((-1083 . -38) 134048) ((-1083 . -414) 134027) ((-1083 . -406) 134006) ((-1083 . -145) 133957) ((-1083 . -1155) 133936) ((-1083 . -353) 133915) ((-1083 . -371) 133866) ((-1083 . -244) 133817) ((-1083 . -292) 133768) ((-1083 . -309) 133719) ((-1083 . -456) 133670) ((-1083 . -561) 133621) ((-1083 . -925) 133572) ((-1083 . -1225) 133523) ((-1083 . -366) 133474) ((-1083 . -234) 133399) ((-1083 . -904) 133332) ((-1083 . -232) 133302) ((-1083 . -617) 133286) ((-1083 . -21) T) ((-1083 . -23) T) ((-1083 . -1105) T) ((-1083 . -616) 133268) ((-1083 . -102) T) ((-1083 . -25) T) ((-1083 . -131) T) ((-1083 . -1053) T) ((-1083 . -1061) T) ((-1083 . -1116) T) ((-1083 . -729) T) ((-1083 . -173) T) ((-1081 . -1105) T) ((-1081 . -616) 133250) ((-1081 . -102) T) ((-1081 . -288) 133229) ((-1080 . -1105) T) ((-1080 . -616) 133211) ((-1080 . -102) T) ((-1079 . -1105) T) ((-1079 . -616) 133193) ((-1079 . -102) T) ((-1079 . -288) 133172) ((-1079 . -1042) 133149) ((-1079 . -619) 133126) ((-1078 . -1220) T) ((-1077 . -1087) T) ((-1077 . -494) 133107) ((-1077 . -616) 133073) ((-1077 . -619) 133054) ((-1077 . -1105) T) ((-1077 . -102) T) ((-1077 . -93) T) ((-1070 . -1087) T) ((-1070 . -494) 133035) ((-1070 . -616) 133001) ((-1070 . -619) 132982) ((-1070 . -1105) T) ((-1070 . -102) T) ((-1070 . -93) T) ((-1067 . -1197) 132957) ((-1067 . -230) 132903) ((-1067 . -107) 132849) ((-1067 . -311) 132700) ((-1067 . -518) 132544) ((-1067 . -493) 132475) ((-1067 . -151) 132421) ((-1067 . -617) NIL) ((-1067 . -236) 132367) ((-1067 . -613) 132342) ((-1067 . -290) 132317) ((-1067 . -288) 132292) ((-1067 . -102) T) ((-1067 . -1105) T) ((-1067 . -616) 132274) ((-1067 . -1220) T) ((-1067 . -34) T) ((-1067 . 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115138) ((-875 . -147) 115117) ((-875 . -145) 115096) ((-875 . -131) T) ((-875 . -25) T) ((-875 . -102) T) ((-875 . -616) 115078) ((-875 . -1105) T) ((-875 . -23) T) ((-875 . -21) T) ((-875 . -1053) T) ((-875 . -1061) T) ((-875 . -1116) T) ((-875 . -729) T) ((-874 . -995) 115055) ((-874 . -1155) NIL) ((-874 . -1042) 115032) ((-874 . -619) 114962) ((-874 . -617) NIL) ((-874 . -1024) NIL) ((-874 . -914) NIL) ((-874 . -888) 114939) ((-874 . -851) NIL) ((-874 . -800) NIL) ((-874 . -797) NIL) ((-874 . -853) NIL) ((-874 . -795) NIL) ((-874 . -794) NIL) ((-874 . -823) NIL) ((-874 . -890) NIL) ((-874 . -1220) T) ((-874 . -404) 114916) ((-874 . -642) 114893) ((-874 . -380) 114870) ((-874 . -288) 114821) ((-874 . -311) 114778) ((-874 . -518) 114686) ((-874 . -341) 114663) ((-874 . -244) T) ((-874 . -111) 114592) ((-874 . -1055) 114537) ((-874 . -1060) 114482) ((-874 . -292) T) ((-874 . -720) 114427) ((-874 . -643) 114372) ((-874 . -651) 114317) ((-874 . -649) 114247) ((-874 . -38) 114192) ((-874 . -309) T) ((-874 . -456) T) ((-874 . -173) T) ((-874 . -561) T) ((-874 . -925) T) ((-874 . -1225) T) ((-874 . -366) T) ((-874 . -234) NIL) ((-874 . -904) NIL) ((-874 . -232) 114169) ((-874 . -147) T) ((-874 . -145) NIL) ((-874 . -131) T) ((-874 . -25) T) ((-874 . -102) T) ((-874 . -616) 114151) ((-874 . -1105) T) ((-874 . -23) T) ((-874 . -21) T) ((-874 . -1053) T) ((-874 . -1061) T) ((-874 . -1116) T) ((-874 . -729) T) ((-872 . -873) 114135) ((-872 . -925) T) ((-872 . -561) T) ((-872 . -292) T) ((-872 . -173) T) ((-872 . -619) 114107) ((-872 . -720) 114094) ((-872 . -643) 114081) ((-872 . -1060) 114068) ((-872 . -1055) 114055) ((-872 . -111) 114040) ((-872 . -38) 114027) ((-872 . -456) T) ((-872 . -309) T) ((-872 . -1053) T) ((-872 . -1061) T) ((-872 . -1116) T) ((-872 . -729) T) ((-872 . -21) T) ((-872 . -649) 113999) ((-872 . -23) T) ((-872 . -1105) T) ((-872 . -616) 113981) ((-872 . -102) T) ((-872 . -25) T) ((-872 . -131) T) ((-872 . -651) 113968) ((-872 . -147) T) ((-869 . -1053) T) ((-869 . -1061) T) ((-869 . -1116) T) ((-869 . -729) T) ((-869 . -21) T) ((-869 . -649) 113913) ((-869 . -23) T) ((-869 . -1105) T) ((-869 . -616) 113875) ((-869 . -102) T) ((-869 . -25) T) ((-869 . -131) T) ((-869 . -651) 113835) ((-869 . -619) 113770) ((-869 . -494) 113747) ((-869 . -38) 113717) ((-869 . -111) 113682) ((-869 . -1055) 113652) ((-869 . -1060) 113622) ((-869 . -643) 113592) ((-869 . -720) 113562) ((-868 . -1105) T) ((-868 . -616) 113544) ((-868 . -102) T) ((-867 . -847) T) ((-867 . -853) T) ((-867 . -1105) T) ((-867 . -616) 113526) ((-867 . -102) T) ((-867 . -371) T) ((-867 . -617) 113448) ((-866 . -1105) T) ((-866 . -616) 113430) ((-866 . -102) T) ((-865 . -864) T) ((-865 . -174) T) ((-865 . -616) 113412) ((-861 . -853) T) ((-861 . -102) T) ((-861 . -616) 113394) ((-861 . -1105) T) ((-858 . -855) 113378) ((-858 . -1042) 113274) ((-858 . -619) 113171) ((-858 . -416) 113155) ((-858 . -720) 113125) ((-858 . -643) 113095) ((-858 . -651) 113069) ((-858 . -649) 113028) ((-858 . -131) T) ((-858 . -25) T) ((-858 . -102) T) ((-858 . -616) 113010) ((-858 . -1105) T) ((-858 . -23) T) ((-858 . -21) T) ((-858 . -1060) 112994) ((-858 . -1055) 112978) ((-858 . -111) 112957) ((-858 . -1053) T) ((-858 . -1061) T) ((-858 . -1116) T) ((-858 . -729) T) ((-858 . -38) 112927) ((-857 . -855) 112911) ((-857 . -1042) 112807) ((-857 . -619) 112725) ((-857 . -416) 112709) ((-857 . -720) 112679) ((-857 . -643) 112649) ((-857 . -651) 112623) ((-857 . -649) 112582) ((-857 . -131) T) ((-857 . -25) T) ((-857 . -102) T) ((-857 . -616) 112564) ((-857 . -1105) T) ((-857 . -23) T) ((-857 . -21) T) ((-857 . -1060) 112548) ((-857 . -1055) 112532) ((-857 . -111) 112511) ((-857 . -1053) T) ((-857 . -1061) T) ((-857 . -1116) T) ((-857 . -729) T) ((-857 . -38) 112481) ((-845 . -1105) T) ((-845 . -616) 112463) ((-845 . -102) T) ((-845 . -416) 112447) ((-845 . -619) 112315) ((-845 . -1042) 112211) ((-845 . -21) 112163) ((-845 . -649) 112080) ((-845 . -23) 112032) ((-845 . -25) 111984) ((-845 . -131) 111936) ((-845 . -851) 111915) ((-845 . -651) 111888) ((-845 . -1061) 111867) ((-845 . -1053) 111846) ((-845 . -800) 111825) ((-845 . -797) 111804) ((-845 . -853) 111783) ((-845 . -795) 111762) ((-845 . -794) 111741) ((-845 . -1116) 111720) ((-845 . -729) 111699) ((-844 . -1105) T) ((-844 . -616) 111681) ((-844 . -102) T) ((-841 . -839) 111663) ((-841 . -102) T) ((-841 . -616) 111645) ((-841 . -1105) T) ((-837 . -1053) T) ((-837 . -1061) T) ((-837 . -1116) T) ((-837 . -729) T) ((-837 . -21) T) ((-837 . -649) 111590) ((-837 . -23) T) ((-837 . -1105) T) ((-837 . -616) 111572) ((-837 . -102) T) ((-837 . -25) T) ((-837 . -131) T) ((-837 . -651) 111532) ((-837 . -619) 111486) ((-837 . -1042) 111455) ((-837 . -288) 111434) ((-837 . -147) 111413) ((-837 . -145) 111392) ((-837 . -38) 111362) ((-837 . -111) 111327) ((-837 . -1055) 111297) ((-837 . -1060) 111267) ((-837 . -643) 111237) ((-837 . -720) 111207) ((-835 . -1105) T) ((-835 . -616) 111189) ((-835 . -102) T) ((-835 . -416) 111173) ((-835 . -619) 111041) ((-835 . -1042) 110937) ((-835 . -21) 110889) ((-835 . -649) 110806) ((-835 . -23) 110758) ((-835 . -25) 110710) ((-835 . -131) 110662) ((-835 . -851) 110641) ((-835 . -651) 110614) ((-835 . -1061) 110593) ((-835 . -1053) 110572) ((-835 . -800) 110551) ((-835 . -797) 110530) ((-835 . -853) 110509) ((-835 . -795) 110488) ((-835 . -794) 110467) ((-835 . -1116) 110446) ((-835 . -729) 110425) ((-831 . -711) 110409) ((-831 . -619) 110364) ((-831 . -720) 110334) ((-831 . -643) 110304) ((-831 . -651) 110278) ((-831 . -649) 110237) ((-831 . -131) T) ((-831 . -25) T) ((-831 . -102) T) ((-831 . -616) 110219) ((-831 . -1105) T) ((-831 . -23) T) ((-831 . -21) T) ((-831 . -1060) 110203) ((-831 . -1055) 110187) ((-831 . -111) 110166) ((-831 . -1053) T) ((-831 . -1061) T) ((-831 . -1116) T) ((-831 . -729) T) ((-831 . -38) 110136) ((-831 . -234) 110115) ((-829 . -1105) T) ((-829 . -616) 110097) ((-829 . -102) T) ((-828 . -1105) T) ((-828 . -616) 110079) ((-828 . -102) T) ((-827 . -1105) T) ((-827 . -616) 110061) ((-827 . -102) T) ((-822 . -389) 110045) ((-822 . -619) 110029) ((-822 . -1042) 110013) ((-822 . -853) T) ((-822 . -1116) T) ((-822 . -102) T) ((-822 . -616) 109995) ((-822 . -1105) T) ((-822 . -729) T) ((-822 . -849) T) ((-822 . -860) T) ((-821 . -268) 109979) ((-821 . -619) 109963) ((-821 . -1042) 109947) ((-821 . -1105) T) ((-821 . -616) 109929) ((-821 . -102) T) ((-821 . -853) T) ((-820 . -111) 109871) ((-820 . -1055) 109822) ((-820 . -1060) 109773) ((-820 . -21) T) ((-820 . -649) 109709) ((-820 . -23) T) ((-820 . -1105) T) ((-820 . -616) 109678) ((-820 . -102) T) ((-820 . -25) T) ((-820 . -131) T) ((-820 . -651) 109629) ((-820 . -234) T) ((-820 . -619) 109543) ((-820 . -729) T) ((-820 . -1116) T) ((-820 . -1061) T) ((-820 . -1053) T) ((-820 . -494) 109527) ((-820 . -366) 109506) ((-820 . -1225) 109485) ((-820 . -925) 109464) ((-820 . -561) 109443) ((-820 . -173) 109422) ((-820 . -720) 109364) ((-820 . -643) 109306) ((-820 . -38) 109248) ((-820 . -456) 109227) ((-820 . -309) 109206) ((-820 . -292) 109185) ((-820 . -244) 109164) ((-819 . -255) 109103) ((-819 . -619) 108840) ((-819 . -1042) 108668) ((-819 . -617) NIL) ((-819 . -328) 108630) ((-819 . -416) 108614) ((-819 . -38) 108463) ((-819 . -111) 108292) ((-819 . -1055) 108135) ((-819 . -1060) 107978) ((-819 . -649) 107888) ((-819 . -651) 107813) ((-819 . -643) 107662) ((-819 . -720) 107511) ((-819 . -145) 107490) ((-819 . -147) 107469) ((-819 . -173) 107380) ((-819 . -561) 107311) ((-819 . -292) 107242) ((-819 . -47) 107204) ((-819 . -380) 107188) ((-819 . -642) 107136) ((-819 . -456) 107087) ((-819 . -518) 106955) ((-819 . -904) 106891) ((-819 . -890) NIL) ((-819 . -914) 106870) ((-819 . -1225) 106849) ((-819 . -954) 106796) ((-819 . -311) 106783) ((-819 . -234) 106762) ((-819 . -131) T) ((-819 . -25) T) ((-819 . -102) T) ((-819 . -616) 106744) ((-819 . -1105) T) ((-819 . -23) T) ((-819 . -21) T) ((-819 . -729) T) ((-819 . -1116) T) ((-819 . -1061) T) ((-819 . -1053) T) ((-819 . -232) 106728) ((-818 . -239) 106707) ((-818 . -1278) 106677) ((-818 . -794) 106656) ((-818 . -851) 106635) ((-818 . -800) 106586) ((-818 . -797) 106537) ((-818 . -853) 106488) ((-818 . -795) 106439) ((-818 . -796) 106418) ((-818 . -290) 106395) ((-818 . -288) 106372) ((-818 . -493) 106356) ((-818 . -518) 106289) ((-818 . -311) 106227) ((-818 . -1220) T) ((-818 . -34) T) ((-818 . -607) 106204) ((-818 . -1042) 106031) ((-818 . -619) 105761) ((-818 . -416) 105730) ((-818 . -642) 105636) ((-818 . -380) 105605) ((-818 . -371) 105584) ((-818 . -234) 105536) ((-818 . -904) 105468) ((-818 . -232) 105437) ((-818 . -111) 105327) ((-818 . -1055) 105224) ((-818 . -1060) 105121) ((-818 . -173) 105100) ((-818 . -616) 104831) ((-818 . -720) 104773) ((-818 . -643) 104715) ((-818 . -651) 104563) ((-818 . -649) 104313) ((-818 . -131) 104183) ((-818 . -23) 104053) ((-818 . -21) 103963) ((-818 . -1053) 103893) ((-818 . -1061) 103823) ((-818 . -1116) 103733) ((-818 . -729) 103643) ((-818 . -38) 103613) ((-818 . -1105) 103403) ((-818 . -102) 103193) ((-818 . -25) 103044) ((-811 . -1105) T) ((-811 . -616) 103026) ((-811 . -102) T) ((-801 . -799) 103010) ((-801 . -853) 102989) ((-801 . -1042) 102769) ((-801 . -619) 102615) ((-801 . -416) 102578) ((-801 . -288) 102536) ((-801 . -311) 102501) ((-801 . -518) 102413) ((-801 . -341) 102397) ((-801 . -371) 102376) ((-801 . -617) 102337) ((-801 . -147) 102316) ((-801 . -145) 102295) ((-801 . -720) 102279) ((-801 . -643) 102263) ((-801 . -651) 102237) ((-801 . -649) 102196) ((-801 . -131) T) ((-801 . -25) T) ((-801 . -102) T) ((-801 . -616) 102178) ((-801 . -1105) T) ((-801 . -23) T) ((-801 . -21) T) ((-801 . -1060) 102162) ((-801 . -1055) 102146) ((-801 . -111) 102125) ((-801 . -1053) T) ((-801 . -1061) T) ((-801 . -1116) T) ((-801 . -729) T) ((-801 . -38) 102109) ((-784 . -1246) 102093) ((-784 . -1155) 102071) ((-784 . -617) NIL) ((-784 . -311) 102058) ((-784 . -518) 102005) ((-784 . -328) 101982) ((-784 . -1042) 101841) ((-784 . -416) 101825) ((-784 . -38) 101654) ((-784 . -111) 101463) ((-784 . -1055) 101286) ((-784 . -1060) 101109) ((-784 . -649) 101019) ((-784 . -651) 100944) ((-784 . -643) 100773) ((-784 . -720) 100602) ((-784 . -619) 100350) ((-784 . -145) 100329) ((-784 . -147) 100308) ((-784 . -47) 100285) ((-784 . -380) 100269) ((-784 . -642) 100217) ((-784 . -904) 100160) ((-784 . -890) NIL) ((-784 . -914) 100139) ((-784 . -1225) 100118) ((-784 . -954) 100087) ((-784 . -925) 100066) ((-784 . -561) 99977) ((-784 . -292) 99888) ((-784 . -173) 99779) ((-784 . -456) 99710) ((-784 . -309) 99689) ((-784 . -288) 99616) ((-784 . -234) T) ((-784 . -131) T) ((-784 . -25) T) ((-784 . -102) T) ((-784 . -616) 99577) ((-784 . -1105) T) ((-784 . -23) T) ((-784 . -21) T) ((-784 . -729) T) ((-784 . -1116) T) ((-784 . -1061) T) ((-784 . -1053) T) ((-784 . -232) 99561) ((-783 . -1069) 99528) ((-783 . -617) 99162) ((-783 . -311) 99149) ((-783 . -518) 99101) ((-783 . -328) 99073) ((-783 . -1042) 98930) ((-783 . -416) 98914) ((-783 . -38) 98763) ((-783 . -619) 98529) ((-783 . -651) 98454) ((-783 . -649) 98364) ((-783 . -729) T) ((-783 . -1116) T) ((-783 . -1061) T) ((-783 . -1053) T) ((-783 . -111) 98193) ((-783 . -1055) 98036) ((-783 . -1060) 97879) ((-783 . -21) T) ((-783 . -23) T) ((-783 . -1105) T) ((-783 . -616) 97793) ((-783 . -102) T) ((-783 . -25) T) ((-783 . -131) T) ((-783 . -643) 97642) ((-783 . -720) 97491) ((-783 . -145) 97470) ((-783 . -147) 97449) ((-783 . -173) 97360) ((-783 . -561) 97291) ((-783 . -292) 97222) ((-783 . -47) 97194) ((-783 . -380) 97178) ((-783 . -642) 97126) ((-783 . -456) 97077) ((-783 . -904) 97061) ((-783 . -890) 96920) ((-783 . -914) 96899) ((-783 . -1225) 96878) ((-783 . -954) 96845) ((-776 . -1105) T) ((-776 . -616) 96827) ((-776 . -102) T) ((-774 . -796) T) ((-774 . -131) T) ((-774 . -25) T) ((-774 . -102) T) ((-774 . -616) 96809) ((-774 . -1105) T) ((-774 . -23) T) ((-774 . -795) T) ((-774 . -853) T) ((-774 . -797) T) ((-774 . -800) T) ((-774 . -729) T) ((-774 . -1116) T) ((-772 . -1105) T) ((-772 . -616) 96791) ((-772 . -102) T) ((-739 . -740) 96775) ((-739 . -1103) 96759) ((-739 . -236) 96743) ((-739 . -617) 96704) ((-739 . -151) 96688) ((-739 . -493) 96672) ((-739 . -102) T) ((-739 . -1105) T) ((-739 . -518) 96605) ((-739 . -311) 96543) ((-739 . -616) 96525) ((-739 . -1220) T) ((-739 . -34) T) ((-739 . -107) 96509) ((-739 . -698) 96493) ((-738 . -1053) T) ((-738 . -1061) T) ((-738 . -1116) T) ((-738 . -729) T) ((-738 . -21) T) ((-738 . -649) 96438) ((-738 . -23) T) ((-738 . -1105) T) ((-738 . -616) 96420) ((-738 . -102) T) ((-738 . -25) T) ((-738 . -131) T) ((-738 . -651) 96380) ((-738 . -619) 96336) ((-738 . -1042) 96307) ((-738 . -147) 96286) ((-738 . -145) 96265) ((-738 . -38) 96235) ((-738 . -111) 96200) ((-738 . -1055) 96170) ((-738 . -1060) 96140) ((-738 . -643) 96110) ((-738 . -720) 96080) ((-738 . -371) 96033) ((-734 . -954) 95986) ((-734 . -619) 95771) ((-734 . -1042) 95647) ((-734 . -1225) 95626) ((-734 . -914) 95605) ((-734 . -890) NIL) ((-734 . -904) 95582) ((-734 . -518) 95525) ((-734 . -456) 95476) ((-734 . -642) 95424) ((-734 . -380) 95408) ((-734 . -47) 95373) ((-734 . -38) 95222) ((-734 . -643) 95071) ((-734 . -720) 94920) ((-734 . -292) 94851) ((-734 . -561) 94782) ((-734 . -111) 94611) ((-734 . -1055) 94454) ((-734 . -1060) 94297) ((-734 . -173) 94208) ((-734 . -147) 94187) ((-734 . -145) 94166) ((-734 . -651) 94091) ((-734 . -649) 94001) ((-734 . -131) T) ((-734 . -25) T) ((-734 . -102) T) ((-734 . -616) 93983) ((-734 . -1105) T) ((-734 . -23) T) ((-734 . -21) T) ((-734 . -1053) T) ((-734 . -1061) T) ((-734 . -1116) T) ((-734 . -729) T) ((-734 . -416) 93967) ((-734 . -328) 93932) ((-734 . -311) 93919) ((-734 . -617) 93780) ((-721 . -477) T) ((-721 . -1116) T) ((-721 . -102) T) ((-721 . -616) 93762) ((-721 . -1105) T) ((-721 . -729) T) ((-718 . -1053) T) ((-718 . -1061) T) ((-718 . -1116) T) ((-718 . -729) T) ((-718 . -21) T) ((-718 . -649) 93734) ((-718 . -23) T) ((-718 . -1105) T) 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. -1055) 85048) ((-657 . -1060) 85032) ((-657 . -21) T) ((-657 . -23) T) ((-657 . -1105) T) ((-657 . -616) 85014) ((-657 . -102) T) ((-657 . -25) T) ((-657 . -131) T) ((-657 . -643) 84984) ((-657 . -720) 84954) ((-657 . -416) 84938) ((-657 . -1042) 84834) ((-657 . -855) 84818) ((-657 . -288) 84797) ((-655 . -720) 84781) ((-655 . -643) 84765) ((-655 . -651) 84749) ((-655 . -649) 84718) ((-655 . -131) T) ((-655 . -25) T) ((-655 . -102) T) ((-655 . -616) 84700) ((-655 . -1105) T) ((-655 . -23) T) ((-655 . -21) T) ((-655 . -1060) 84684) ((-655 . -1055) 84668) ((-655 . -111) 84647) ((-655 . -794) 84626) ((-655 . -795) 84605) ((-655 . -853) 84584) ((-655 . -797) 84563) ((-655 . -800) 84542) ((-652 . -1105) T) ((-652 . -616) 84524) ((-652 . -102) T) ((-652 . -1042) 84508) ((-652 . -619) 84492) ((-650 . -698) 84476) ((-650 . -107) 84460) ((-650 . -34) T) ((-650 . -1220) T) ((-650 . -616) 84392) ((-650 . -311) 84330) ((-650 . -518) 84263) ((-650 . -1105) 84241) ((-650 . -102) 84219) ((-650 . 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-651) 82215) ((-626 . -619) 82110) ((-626 . -234) 82089) ((-626 . -561) T) ((-626 . -292) T) ((-626 . -173) T) ((-626 . -720) 82076) ((-626 . -643) 82063) ((-626 . -1060) 82050) ((-626 . -1055) 82037) ((-626 . -111) 82022) ((-626 . -38) 82009) ((-626 . -617) 81986) ((-626 . -416) 81970) ((-626 . -1042) 81853) ((-626 . -147) 81832) ((-626 . -145) 81811) ((-626 . -309) 81790) ((-626 . -456) 81769) ((-626 . -925) 81748) ((-622 . -38) 81732) ((-622 . -619) 81701) ((-622 . -651) 81675) ((-622 . -649) 81634) ((-622 . -729) T) ((-622 . -1116) T) ((-622 . -1061) T) ((-622 . -1053) T) ((-622 . -111) 81613) ((-622 . -1055) 81597) ((-622 . -1060) 81581) ((-622 . -21) T) ((-622 . -23) T) ((-622 . -1105) T) ((-622 . -616) 81563) ((-622 . -102) T) ((-622 . -25) T) ((-622 . -131) T) ((-622 . -643) 81547) ((-622 . -720) 81531) ((-622 . -851) 81510) ((-622 . -800) 81489) ((-622 . -797) 81468) ((-622 . -853) 81447) ((-622 . -795) 81426) ((-622 . -794) 81405) ((-620 . -971) T) ((-620 . -102) T) ((-620 . 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-102) T) ((-1 . -1105) T) ((-1 . -616) 30) ((-1 . -102) T)) 
\ No newline at end of file
+((((-551)) . T)) 
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. -1292) 187334) ((-1293 . -1295) 187313) ((-1293 . -1044) 187270) ((-1293 . -621) 187199) ((-1293 . -1055) T) ((-1293 . -1063) T) ((-1293 . -1118) T) ((-1293 . -731) T) ((-1293 . -21) T) ((-1293 . -651) 187158) ((-1293 . -23) T) ((-1293 . -1107) T) ((-1293 . -618) 187140) ((-1293 . -102) T) ((-1293 . -25) T) ((-1293 . -131) T) ((-1293 . -653) 187114) ((-1293 . -1287) 187098) ((-1293 . -722) 187068) ((-1293 . -645) 187038) ((-1293 . -1062) 187022) ((-1293 . -1057) 187006) ((-1293 . -111) 186985) ((-1293 . -38) 186955) ((-1293 . -1292) 186934) ((-1293 . -388) 186906) ((-1288 . -388) 186878) ((-1288 . -621) 186827) ((-1288 . -1044) 186804) ((-1288 . -645) 186774) ((-1288 . -722) 186744) ((-1288 . -653) 186718) ((-1288 . -651) 186677) ((-1288 . -131) T) ((-1288 . -25) T) ((-1288 . -102) T) ((-1288 . -618) 186659) ((-1288 . -1107) T) ((-1288 . -23) T) ((-1288 . -21) T) ((-1288 . -1062) 186643) ((-1288 . -1057) 186627) ((-1288 . -111) 186606) ((-1288 . -1295) 186585) ((-1288 . -1055) T) 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-93) T) ((-1283 . -1089) T) ((-1283 . -495) 185742) ((-1283 . -618) 185708) ((-1283 . -621) 185689) ((-1283 . -1107) T) ((-1283 . -102) T) ((-1283 . -93) T) ((-1278 . -618) 185671) ((-1276 . -1107) T) ((-1276 . -618) 185653) ((-1276 . -102) T) ((-1275 . -1107) T) ((-1275 . -618) 185635) ((-1275 . -102) T) ((-1272 . -1271) 185619) ((-1272 . -376) 185603) ((-1272 . -855) 185582) ((-1272 . -151) 185566) ((-1272 . -34) T) ((-1272 . -1222) T) ((-1272 . -618) 185478) ((-1272 . -312) 185416) ((-1272 . -519) 185349) ((-1272 . -1107) 185299) ((-1272 . -102) 185249) ((-1272 . -494) 185233) ((-1272 . -619) 185194) ((-1272 . -609) 185171) ((-1272 . -289) 185148) ((-1272 . -291) 185125) ((-1272 . -656) 185109) ((-1272 . -19) 185093) ((-1269 . -1107) T) ((-1269 . -618) 185059) ((-1269 . -102) T) ((-1262 . -1265) 185043) ((-1262 . -234) 185002) ((-1262 . -621) 184884) ((-1262 . -653) 184809) ((-1262 . -651) 184719) ((-1262 . -131) T) ((-1262 . -25) T) ((-1262 . -102) T) ((-1262 . -618) 184701) 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166527) ((-1231 . -102) T) ((-1231 . -372) T) ((-1230 . -849) T) ((-1230 . -855) T) ((-1230 . -1107) T) ((-1230 . -618) 166509) ((-1230 . -102) T) ((-1230 . -372) T) ((-1229 . -849) T) ((-1229 . -855) T) ((-1229 . -1107) T) ((-1229 . -618) 166491) ((-1229 . -102) T) ((-1229 . -372) T) ((-1228 . -849) T) ((-1228 . -855) T) ((-1228 . -1107) T) ((-1228 . -618) 166473) ((-1228 . -102) T) ((-1228 . -372) T) ((-1223 . -1089) T) ((-1223 . -495) 166454) ((-1223 . -618) 166420) ((-1223 . -621) 166401) ((-1223 . -1107) T) ((-1223 . -102) T) ((-1223 . -93) T) ((-1220 . -495) 166378) ((-1220 . -618) 166290) ((-1220 . -621) 166267) ((-1220 . -1107) 166245) ((-1220 . -102) 166223) ((-1215 . -745) 166199) ((-1215 . -35) 166165) ((-1215 . -95) 166131) ((-1215 . -287) 166097) ((-1215 . -498) 166063) ((-1215 . -1211) 166029) ((-1215 . -1208) 165995) ((-1215 . -1008) 165961) ((-1215 . -47) 165930) ((-1215 . -38) 165827) ((-1215 . -645) 165724) ((-1215 . -722) 165621) ((-1215 . -621) 165503) ((-1215 . -293) 165482) ((-1215 . -562) 165461) ((-1215 . -111) 165330) ((-1215 . -1057) 165213) ((-1215 . -1062) 165096) ((-1215 . -173) 165047) ((-1215 . -147) 165026) ((-1215 . -145) 165005) ((-1215 . -653) 164930) ((-1215 . -651) 164840) ((-1215 . -979) 164802) ((-1215 . -1055) T) ((-1215 . -1063) T) ((-1215 . -1118) T) ((-1215 . -731) T) ((-1215 . -21) T) ((-1215 . -23) T) ((-1215 . -1107) T) ((-1215 . -618) 164784) ((-1215 . -102) T) ((-1215 . -25) T) ((-1215 . -131) T) ((-1215 . -906) 164765) ((-1215 . -519) 164732) ((-1215 . -312) 164719) ((-1209 . -1016) 164703) ((-1209 . -34) T) ((-1209 . -1222) T) ((-1209 . -618) 164635) ((-1209 . -312) 164573) ((-1209 . -519) 164506) ((-1209 . -1107) 164484) ((-1209 . -102) 164462) ((-1209 . -494) 164446) ((-1204 . -369) 164420) ((-1204 . -102) T) ((-1204 . -618) 164402) ((-1204 . -1107) T) ((-1202 . -1107) T) ((-1202 . -618) 164384) ((-1202 . -102) T) ((-1202 . -621) 164366) ((-1195 . -1199) 164345) ((-1195 . -230) 164295) ((-1195 . -107) 164245) ((-1195 . -312) 164049) ((-1195 . -519) 163841) ((-1195 . -494) 163778) ((-1195 . -151) 163728) ((-1195 . -619) NIL) ((-1195 . -236) 163678) ((-1195 . -615) 163657) ((-1195 . -291) 163636) ((-1195 . -289) 163615) ((-1195 . -102) T) ((-1195 . -1107) T) ((-1195 . -618) 163597) ((-1195 . -1222) T) ((-1195 . -34) T) ((-1195 . -609) 163576) ((-1193 . -1222) T) ((-1191 . -1107) T) ((-1191 . -618) 163558) ((-1191 . -102) T) ((-1190 . -849) T) ((-1190 . -855) T) ((-1190 . -1107) T) ((-1190 . -618) 163540) ((-1190 . -102) T) ((-1190 . -372) T) ((-1189 . -849) T) ((-1189 . -855) T) ((-1189 . -1107) T) ((-1189 . -618) 163522) ((-1189 . -102) T) ((-1189 . -372) T) ((-1188 . -1268) T) ((-1188 . -1107) T) ((-1188 . -618) 163489) ((-1188 . -102) T) ((-1188 . -1044) 163425) ((-1188 . -621) 163361) ((-1187 . -618) 163343) ((-1186 . -618) 163325) ((-1185 . -329) 163302) ((-1185 . -1044) 163198) ((-1185 . -417) 163182) ((-1185 . -38) 163079) ((-1185 . -621) 162932) ((-1185 . -653) 162857) ((-1185 . -651) 162767) ((-1185 . -731) T) ((-1185 . -1118) T) ((-1185 . -1063) T) ((-1185 . -1055) T) ((-1185 . -111) 162636) ((-1185 . -1057) 162519) ((-1185 . -1062) 162402) ((-1185 . -21) T) ((-1185 . -23) T) ((-1185 . -1107) T) ((-1185 . -618) 162384) ((-1185 . -102) T) ((-1185 . -25) T) ((-1185 . -131) T) ((-1185 . -645) 162281) ((-1185 . -722) 162178) ((-1185 . -145) 162157) ((-1185 . -147) 162136) ((-1185 . -173) 162087) ((-1185 . -562) 162066) ((-1185 . -293) 162045) ((-1185 . -47) 162022) ((-1183 . -855) T) ((-1183 . -102) T) ((-1183 . -618) 162004) ((-1183 . -1107) T) ((-1183 . -619) 161926) ((-1183 . -826) T) ((-1183 . -621) 161907) ((-1183 . -892) 161874) ((-1182 . -618) 161856) ((-1181 . -1265) 161840) ((-1181 . -234) 161799) ((-1181 . -621) 161681) ((-1181 . -653) 161606) ((-1181 . -651) 161516) ((-1181 . -131) T) ((-1181 . -25) T) ((-1181 . -102) T) ((-1181 . -618) 161498) ((-1181 . -1107) T) ((-1181 . -23) T) ((-1181 . -21) T) ((-1181 . -731) T) ((-1181 . -1118) T) ((-1181 . -1063) T) ((-1181 . -1055) T) ((-1181 . -289) 161483) ((-1181 . -906) 161396) ((-1181 . -979) 161365) ((-1181 . -38) 161262) ((-1181 . -111) 161131) ((-1181 . -1057) 161014) ((-1181 . -1062) 160897) ((-1181 . -645) 160794) ((-1181 . -722) 160691) ((-1181 . -145) 160670) ((-1181 . -147) 160649) ((-1181 . -173) 160600) ((-1181 . -562) 160579) ((-1181 . -293) 160558) ((-1181 . -47) 160535) ((-1181 . -1251) 160512) ((-1181 . -35) 160478) ((-1181 . -95) 160444) ((-1181 . -287) 160410) ((-1181 . -498) 160376) ((-1181 . -1211) 160342) ((-1181 . -1208) 160308) ((-1181 . -1008) 160274) ((-1180 . -1257) 160235) ((-1180 . -367) 160214) ((-1180 . -1227) 160193) ((-1180 . -927) 160172) ((-1180 . -562) 160123) ((-1180 . -173) 160054) ((-1180 . -621) 159797) ((-1180 . -722) 159638) ((-1180 . -645) 159479) ((-1180 . -38) 159320) ((-1180 . -457) 159299) ((-1180 . -310) 159278) ((-1180 . -653) 159175) ((-1180 . -651) 159057) ((-1180 . -731) T) ((-1180 . -1118) T) ((-1180 . -1063) T) ((-1180 . -1055) T) ((-1180 . -111) 158878) ((-1180 . -1057) 158713) ((-1180 . -1062) 158548) ((-1180 . -21) T) ((-1180 . -23) T) ((-1180 . -1107) T) ((-1180 . -618) 158530) ((-1180 . -102) T) ((-1180 . -25) T) ((-1180 . -131) T) ((-1180 . -293) 158481) ((-1180 . -244) 158460) ((-1180 . -1008) 158426) ((-1180 . -1208) 158392) ((-1180 . -1211) 158358) ((-1180 . -498) 158324) ((-1180 . -287) 158290) ((-1180 . -95) 158256) ((-1180 . -35) 158222) ((-1180 . -1251) 158192) ((-1180 . -47) 158162) ((-1180 . -147) 158141) ((-1180 . -145) 158120) ((-1180 . -979) 158082) ((-1180 . -906) 157988) ((-1180 . -289) 157973) ((-1180 . -234) 157925) ((-1180 . -1255) 157909) ((-1180 . -1044) 157844) ((-1177 . -1248) 157828) ((-1177 . -1157) 157806) ((-1177 . -619) NIL) ((-1177 . -312) 157793) ((-1177 . -519) 157740) ((-1177 . -329) 157717) ((-1177 . -1044) 157597) ((-1177 . -417) 157581) ((-1177 . -38) 157410) ((-1177 . -111) 157219) ((-1177 . -1057) 157042) ((-1177 . -1062) 156865) ((-1177 . -651) 156775) ((-1177 . -653) 156700) ((-1177 . -645) 156529) ((-1177 . -722) 156358) ((-1177 . -621) 156127) ((-1177 . -145) 156106) ((-1177 . -147) 156085) ((-1177 . -47) 156062) ((-1177 . -381) 156046) ((-1177 . -644) 155994) ((-1177 . -906) 155937) ((-1177 . -892) NIL) ((-1177 . -916) 155916) ((-1177 . -1227) 155895) ((-1177 . -956) 155864) ((-1177 . -927) 155843) ((-1177 . -562) 155754) ((-1177 . -293) 155665) ((-1177 . -173) 155556) ((-1177 . -457) 155487) ((-1177 . -310) 155466) ((-1177 . -289) 155393) ((-1177 . -234) T) ((-1177 . -131) T) ((-1177 . -25) T) ((-1177 . -102) T) ((-1177 . -618) 155375) ((-1177 . -1107) T) ((-1177 . -23) T) ((-1177 . -21) T) ((-1177 . -731) T) ((-1177 . -1118) T) ((-1177 . -1063) T) ((-1177 . -1055) T) ((-1177 . -232) 155359) ((-1174 . -1236) 155320) ((-1174 . -1008) 155286) ((-1174 . -1208) 155252) ((-1174 . -1211) 155218) ((-1174 . -498) 155184) ((-1174 . -287) 155150) ((-1174 . -95) 155116) ((-1174 . -35) 155082) ((-1174 . -1251) 155059) ((-1174 . -47) 155036) ((-1174 . -621) 154831) ((-1174 . -722) 154627) ((-1174 . -645) 154423) ((-1174 . -653) 154275) ((-1174 . -651) 154112) ((-1174 . -1062) 153902) ((-1174 . -1057) 153692) ((-1174 . -111) 153461) ((-1174 . -38) 153257) ((-1174 . -979) 153226) ((-1174 . -289) 153074) ((-1174 . -1234) 153058) ((-1174 . -731) T) ((-1174 . -1118) T) ((-1174 . -1063) T) ((-1174 . -1055) T) ((-1174 . -21) T) ((-1174 . -23) T) ((-1174 . -1107) T) ((-1174 . -618) 153040) ((-1174 . -102) T) ((-1174 . -25) T) ((-1174 . -131) T) ((-1174 . -145) 152947) ((-1174 . -147) 152854) ((-1174 . -619) NIL) ((-1174 . -232) 152806) ((-1174 . -906) 152639) ((-1174 . -234) 152526) ((-1174 . -367) 152505) ((-1174 . -1227) 152484) ((-1174 . -927) 152463) ((-1174 . -562) 152414) ((-1174 . -173) 152345) ((-1174 . -457) 152324) ((-1174 . -310) 152303) ((-1174 . -293) 152254) ((-1174 . -244) 152233) ((-1174 . -342) 152185) ((-1174 . -519) 151954) ((-1174 . -312) 151839) ((-1174 . -381) 151791) ((-1174 . -644) 151743) ((-1174 . -405) 151695) ((-1174 . -1222) 151674) ((-1174 . -892) NIL) ((-1174 . -825) NIL) ((-1174 . -796) NIL) ((-1174 . -797) NIL) ((-1174 . -855) NIL) ((-1174 . -799) NIL) ((-1174 . -802) NIL) ((-1174 . -853) NIL) ((-1174 . -890) 151626) ((-1174 . -916) NIL) ((-1174 . -1026) NIL) ((-1174 . -1044) 151592) ((-1174 . -1157) NIL) ((-1174 . -997) 151544) ((-1173 . -1089) T) ((-1173 . -495) 151525) ((-1173 . -618) 151491) ((-1173 . -621) 151472) ((-1173 . -1107) T) ((-1173 . -102) T) ((-1173 . -93) T) ((-1172 . -1107) T) ((-1172 . -618) 151454) ((-1172 . -102) T) ((-1171 . -1107) T) ((-1171 . -618) 151436) ((-1171 . -102) T) ((-1166 . -1199) 151412) ((-1166 . -230) 151359) ((-1166 . -107) 151306) ((-1166 . -312) 151101) ((-1166 . -519) 150884) ((-1166 . -494) 150818) ((-1166 . -151) 150765) ((-1166 . -619) NIL) ((-1166 . -236) 150712) ((-1166 . -615) 150688) ((-1166 . -291) 150664) ((-1166 . -289) 150640) ((-1166 . -102) T) ((-1166 . -1107) T) ((-1166 . -618) 150622) ((-1166 . -1222) T) ((-1166 . -34) T) ((-1166 . -609) 150598) ((-1165 . -1164) T) ((-1165 . -19) 150580) ((-1165 . -656) 150562) ((-1165 . -291) 150537) ((-1165 . -289) 150512) ((-1165 . -609) 150487) ((-1165 . -619) NIL) ((-1165 . -494) 150469) ((-1165 . -519) NIL) ((-1165 . -312) NIL) ((-1165 . -1222) T) ((-1165 . -34) T) ((-1165 . -151) 150451) ((-1165 . -855) T) ((-1165 . -376) 150433) ((-1165 . -1150) T) ((-1165 . -102) T) ((-1165 . -618) 150415) ((-1165 . -1107) T) ((-1165 . -826) T) ((-1160 . -679) 150399) ((-1160 . -656) 150383) ((-1160 . -291) 150360) ((-1160 . -289) 150337) ((-1160 . -609) 150314) ((-1160 . -619) 150275) ((-1160 . -494) 150259) ((-1160 . -102) 150237) ((-1160 . -1107) 150215) ((-1160 . -519) 150148) ((-1160 . -312) 150086) ((-1160 . -618) 150018) ((-1160 . -1222) T) ((-1160 . -34) T) ((-1160 . -151) 150002) ((-1160 . -1261) 149986) ((-1160 . -1016) 149970) ((-1160 . -1155) 149954) ((-1160 . -621) 149931) ((-1158 . -1089) T) ((-1158 . -495) 149912) ((-1158 . -618) 149878) ((-1158 . -621) 149859) ((-1158 . -1107) T) ((-1158 . -102) T) ((-1158 . -93) T) ((-1156 . -1199) 149838) ((-1156 . -230) 149788) ((-1156 . -107) 149738) ((-1156 . -312) 149542) ((-1156 . -519) 149334) ((-1156 . -494) 149271) ((-1156 . -151) 149221) ((-1156 . -619) NIL) ((-1156 . -236) 149171) ((-1156 . -615) 149150) ((-1156 . -291) 149129) ((-1156 . -289) 149108) ((-1156 . -102) T) ((-1156 . -1107) T) ((-1156 . -618) 149090) ((-1156 . -1222) T) ((-1156 . -34) T) ((-1156 . -609) 149069) ((-1153 . -1127) 149053) ((-1153 . -494) 149037) ((-1153 . -102) 149015) ((-1153 . -1107) 148993) ((-1153 . -519) 148926) ((-1153 . -312) 148864) ((-1153 . -618) 148796) ((-1153 . -1222) T) ((-1153 . -34) T) ((-1153 . -107) 148780) ((-1152 . -1115) 148749) ((-1152 . -1217) 148718) ((-1152 . -618) 148680) ((-1152 . -151) 148664) ((-1152 . -34) T) ((-1152 . -1222) T) ((-1152 . -312) 148602) ((-1152 . -519) 148535) ((-1152 . -1107) T) ((-1152 . -102) T) ((-1152 . -494) 148519) ((-1152 . -619) 148480) ((-1152 . -982) 148449) ((-1152 . -1077) 148418) ((-1148 . -1129) 148363) ((-1148 . -494) 148347) ((-1148 . -519) 148280) ((-1148 . -312) 148218) ((-1148 . -1222) T) ((-1148 . -34) T) ((-1148 . -1059) 148158) ((-1148 . -1044) 148054) ((-1148 . -621) 147972) ((-1148 . -417) 147956) ((-1148 . -644) 147904) ((-1148 . -381) 147888) ((-1148 . -234) 147867) ((-1148 . -906) 147826) ((-1148 . -232) 147810) ((-1148 . -722) 147742) ((-1148 . -645) 147674) ((-1148 . -653) 147648) ((-1148 . -651) 147607) ((-1148 . -131) T) ((-1148 . -25) T) ((-1148 . -102) T) ((-1148 . -618) 147569) ((-1148 . -1107) T) ((-1148 . -23) T) ((-1148 . -21) T) ((-1148 . -1062) 147553) ((-1148 . -1057) 147537) ((-1148 . -111) 147516) ((-1148 . -1055) T) ((-1148 . -1063) T) ((-1148 . -1118) T) ((-1148 . -731) T) ((-1148 . -38) 147476) ((-1148 . -619) 147437) ((-1147 . -1016) 147408) ((-1147 . -34) T) ((-1147 . -1222) T) ((-1147 . -618) 147390) ((-1147 . -312) 147316) ((-1147 . -519) 147235) ((-1147 . -1107) T) ((-1147 . -102) T) ((-1147 . -494) 147206) ((-1146 . -1107) T) ((-1146 . -618) 147188) ((-1146 . -102) T) ((-1141 . -1143) T) ((-1141 . -1268) T) ((-1141 . -93) T) ((-1141 . -102) T) ((-1141 . -618) 147154) ((-1141 . -1107) T) ((-1141 . -621) 147135) ((-1141 . -495) 147116) ((-1141 . -1089) T) ((-1139 . -1140) 147100) ((-1139 . -102) T) ((-1139 . -618) 147082) ((-1139 . -1107) T) ((-1132 . -745) 147061) ((-1132 . -35) 147027) ((-1132 . -95) 146993) ((-1132 . -287) 146959) ((-1132 . -498) 146925) ((-1132 . -1211) 146891) ((-1132 . -1208) 146857) ((-1132 . -1008) 146823) ((-1132 . -47) 146795) ((-1132 . -38) 146692) ((-1132 . -645) 146589) ((-1132 . -722) 146486) ((-1132 . -621) 146368) ((-1132 . -293) 146347) ((-1132 . -562) 146326) ((-1132 . -111) 146195) ((-1132 . -1057) 146078) ((-1132 . -1062) 145961) ((-1132 . -173) 145912) ((-1132 . -147) 145891) ((-1132 . -145) 145870) ((-1132 . -653) 145795) ((-1132 . -651) 145705) ((-1132 . -979) 145672) ((-1132 . -1055) T) ((-1132 . -1063) T) ((-1132 . -1118) T) ((-1132 . -731) T) ((-1132 . -21) T) ((-1132 . -23) T) ((-1132 . -1107) T) ((-1132 . -618) 145654) ((-1132 . -102) T) ((-1132 . -25) T) ((-1132 . -131) T) ((-1132 . -906) 145638) ((-1132 . -519) 145608) ((-1132 . -312) 145595) ((-1131 . -956) 145562) ((-1131 . -621) 145354) ((-1131 . -1044) 145237) ((-1131 . -1227) 145216) ((-1131 . -916) 145195) ((-1131 . -892) 145054) ((-1131 . -906) 145038) ((-1131 . -519) 144990) ((-1131 . -457) 144941) ((-1131 . -644) 144889) ((-1131 . -381) 144873) ((-1131 . -47) 144845) ((-1131 . -38) 144694) ((-1131 . -645) 144543) ((-1131 . -722) 144392) ((-1131 . -293) 144323) ((-1131 . -562) 144254) ((-1131 . -111) 144083) ((-1131 . -1057) 143926) ((-1131 . -1062) 143769) ((-1131 . -173) 143680) ((-1131 . -147) 143659) ((-1131 . -145) 143638) ((-1131 . -653) 143563) ((-1131 . -651) 143473) ((-1131 . -131) T) ((-1131 . -25) T) ((-1131 . -102) T) ((-1131 . -618) 143455) ((-1131 . -1107) T) ((-1131 . -23) T) ((-1131 . -21) T) ((-1131 . -1055) T) ((-1131 . -1063) T) ((-1131 . -1118) T) ((-1131 . -731) T) ((-1131 . -417) 143439) ((-1131 . -329) 143411) ((-1131 . -312) 143398) ((-1131 . -619) 143146) ((-1126 . -550) T) ((-1126 . -1227) T) ((-1126 . -1157) T) ((-1126 . -1044) 143128) ((-1126 . -619) 143043) ((-1126 . -1026) T) ((-1126 . -892) 143025) ((-1126 . -853) T) ((-1126 . -802) T) ((-1126 . -799) T) ((-1126 . -855) T) ((-1126 . -797) T) ((-1126 . -796) T) ((-1126 . -825) T) ((-1126 . -644) 143007) ((-1126 . -927) T) ((-1126 . -562) T) ((-1126 . -293) T) ((-1126 . -173) T) ((-1126 . -621) 142979) ((-1126 . -722) 142966) ((-1126 . -645) 142953) ((-1126 . -1062) 142940) ((-1126 . -1057) 142927) ((-1126 . -111) 142912) ((-1126 . -38) 142899) ((-1126 . -457) T) ((-1126 . -310) T) ((-1126 . -234) T) ((-1126 . -143) T) ((-1126 . -1055) T) ((-1126 . -1063) T) ((-1126 . -1118) T) ((-1126 . -731) T) ((-1126 . -21) T) ((-1126 . -651) 142871) ((-1126 . -23) T) ((-1126 . -1107) T) ((-1126 . -618) 142853) ((-1126 . -102) T) ((-1126 . -25) T) ((-1126 . -131) T) ((-1126 . -653) 142840) ((-1126 . -147) T) ((-1126 . -849) T) ((-1126 . -372) T) ((-1126 . -667) T) ((-1126 . -826) T) ((-1122 . -1089) T) ((-1122 . -495) 142821) ((-1122 . -618) 142787) ((-1122 . -621) 142768) ((-1122 . -1107) T) ((-1122 . -102) T) ((-1122 . -93) T) ((-1121 . -1107) T) ((-1121 . -618) 142750) ((-1121 . -102) T) ((-1119 . -239) 142729) ((-1119 . -1280) 142699) ((-1119 . -796) 142678) ((-1119 . -853) 142657) ((-1119 . -802) 142608) ((-1119 . -799) 142559) ((-1119 . -855) 142510) ((-1119 . -797) 142461) ((-1119 . -798) 142440) ((-1119 . -291) 142417) ((-1119 . -289) 142394) ((-1119 . -494) 142378) ((-1119 . -519) 142311) ((-1119 . -312) 142249) ((-1119 . -1222) T) ((-1119 . -34) T) ((-1119 . -609) 142226) ((-1119 . -1044) 142053) ((-1119 . -621) 141783) ((-1119 . -417) 141752) ((-1119 . -644) 141658) ((-1119 . -381) 141627) ((-1119 . -372) 141606) ((-1119 . -234) 141558) ((-1119 . -906) 141490) ((-1119 . -232) 141459) ((-1119 . -111) 141349) ((-1119 . -1057) 141246) ((-1119 . -1062) 141143) ((-1119 . -173) 141122) ((-1119 . -618) 140853) ((-1119 . -722) 140795) ((-1119 . -645) 140737) ((-1119 . -653) 140585) ((-1119 . -651) 140335) ((-1119 . -131) 140205) ((-1119 . -23) 140075) ((-1119 . -21) 139985) ((-1119 . -1055) 139915) ((-1119 . -1063) 139845) ((-1119 . -1118) 139755) ((-1119 . -731) 139665) ((-1119 . -38) 139635) ((-1119 . -1107) 139425) ((-1119 . -102) 139215) ((-1119 . -25) 139066) ((-1112 . -401) T) ((-1112 . -1222) T) ((-1112 . -618) 139048) ((-1111 . -1110) 139012) ((-1111 . -102) T) ((-1111 . -618) 138994) ((-1111 . -1107) T) ((-1111 . -623) 138909) ((-1109 . -1110) 138861) ((-1109 . -102) T) ((-1109 . -618) 138843) ((-1109 . -1107) T) ((-1109 . -623) 138746) ((-1108 . -372) T) ((-1108 . -102) T) ((-1108 . -618) 138728) ((-1108 . -1107) T) ((-1103 . -431) 138712) ((-1103 . -1105) 138696) ((-1103 . -372) 138675) ((-1103 . -236) 138659) ((-1103 . -619) 138620) ((-1103 . -151) 138604) ((-1103 . -494) 138588) ((-1103 . -102) T) ((-1103 . -1107) T) ((-1103 . -519) 138521) ((-1103 . -312) 138459) ((-1103 . -618) 138441) ((-1103 . -1222) T) ((-1103 . -34) T) ((-1103 . -107) 138425) ((-1103 . -230) 138409) ((-1102 . -1089) T) ((-1102 . -495) 138390) ((-1102 . -618) 138356) ((-1102 . -621) 138337) ((-1102 . -1107) T) ((-1102 . -102) T) ((-1102 . -93) T) ((-1098 . -1222) T) ((-1098 . -1107) 138307) ((-1098 . -618) 138266) ((-1098 . -102) 138236) ((-1097 . -1089) T) ((-1097 . -495) 138217) ((-1097 . -618) 138183) ((-1097 . -621) 138164) ((-1097 . -1107) T) ((-1097 . -102) T) ((-1097 . -93) T) ((-1095 . -1100) 138148) ((-1095 . -623) 138132) ((-1095 . -1107) 138110) ((-1095 . -618) 138077) ((-1095 . -102) 138055) ((-1095 . -1101) 138013) ((-1094 . -268) 137997) ((-1094 . -621) 137981) ((-1094 . -1044) 137965) ((-1094 . -1107) T) ((-1094 . -618) 137947) ((-1094 . -102) T) ((-1094 . -855) T) ((-1093 . -255) 137884) ((-1093 . -621) 137620) ((-1093 . -1044) 137447) ((-1093 . -619) NIL) ((-1093 . -329) 137408) ((-1093 . -417) 137392) ((-1093 . -38) 137241) ((-1093 . -111) 137070) ((-1093 . -1057) 136913) ((-1093 . -1062) 136756) ((-1093 . -651) 136666) ((-1093 . -653) 136591) ((-1093 . -645) 136440) ((-1093 . -722) 136289) ((-1093 . -145) 136268) ((-1093 . -147) 136247) ((-1093 . -173) 136158) ((-1093 . -562) 136089) ((-1093 . -293) 136020) ((-1093 . -47) 135981) ((-1093 . -381) 135965) ((-1093 . -644) 135913) ((-1093 . -457) 135864) ((-1093 . -519) 135731) ((-1093 . -906) 135666) ((-1093 . -892) NIL) ((-1093 . -916) 135645) ((-1093 . -1227) 135624) ((-1093 . -956) 135569) ((-1093 . -312) 135556) ((-1093 . -234) 135535) ((-1093 . -131) T) ((-1093 . -25) T) ((-1093 . -102) T) ((-1093 . -618) 135517) ((-1093 . -1107) T) ((-1093 . -23) T) ((-1093 . -21) T) ((-1093 . -731) T) ((-1093 . -1118) T) ((-1093 . -1063) T) ((-1093 . -1055) T) ((-1093 . -232) 135501) ((-1091 . -618) 135483) ((-1088 . -855) T) ((-1088 . -102) T) ((-1088 . -618) 135465) ((-1088 . -1107) T) ((-1088 . -619) 135446) ((-1085 . -729) 135425) ((-1085 . -1044) 135321) ((-1085 . -417) 135305) ((-1085 . -644) 135253) ((-1085 . -381) 135237) ((-1085 . -374) 135216) ((-1085 . -147) 135195) ((-1085 . -621) 135013) ((-1085 . -722) 134881) ((-1085 . -645) 134749) ((-1085 . -653) 134659) ((-1085 . -651) 134554) ((-1085 . -1062) 134464) ((-1085 . -1057) 134374) ((-1085 . -111) 134270) ((-1085 . -38) 134138) ((-1085 . -415) 134117) ((-1085 . -407) 134096) ((-1085 . -145) 134047) ((-1085 . -1157) 134026) ((-1085 . -354) 134005) ((-1085 . -372) 133956) ((-1085 . -244) 133907) ((-1085 . -293) 133858) ((-1085 . -310) 133809) ((-1085 . -457) 133760) ((-1085 . -562) 133711) ((-1085 . -927) 133662) ((-1085 . -1227) 133613) ((-1085 . -367) 133564) ((-1085 . -234) 133489) ((-1085 . -906) 133422) ((-1085 . -232) 133392) ((-1085 . -619) 133376) ((-1085 . -21) T) ((-1085 . -23) T) ((-1085 . -1107) T) ((-1085 . -618) 133358) ((-1085 . -102) T) ((-1085 . -25) T) ((-1085 . -131) T) ((-1085 . -1055) T) ((-1085 . -1063) T) ((-1085 . -1118) T) ((-1085 . -731) T) ((-1085 . -173) T) ((-1083 . -1107) T) ((-1083 . -618) 133340) ((-1083 . -102) T) ((-1083 . -289) 133319) ((-1082 . -1107) T) ((-1082 . -618) 133301) ((-1082 . -102) T) ((-1081 . -1107) T) ((-1081 . -618) 133283) ((-1081 . -102) T) ((-1081 . -289) 133262) ((-1081 . -1044) 133239) ((-1081 . -621) 133216) ((-1080 . -1222) T) ((-1079 . -1089) T) ((-1079 . -495) 133197) ((-1079 . -618) 133163) ((-1079 . -621) 133144) ((-1079 . -1107) T) ((-1079 . -102) T) ((-1079 . -93) T) ((-1072 . -1089) T) ((-1072 . -495) 133125) ((-1072 . -618) 133091) ((-1072 . -621) 133072) ((-1072 . -1107) T) ((-1072 . -102) T) ((-1072 . -93) T) ((-1069 . -1199) 133047) ((-1069 . -230) 132993) ((-1069 . -107) 132939) ((-1069 . -312) 132790) ((-1069 . -519) 132634) ((-1069 . -494) 132565) ((-1069 . -151) 132511) ((-1069 . -619) NIL) ((-1069 . -236) 132457) ((-1069 . -615) 132432) ((-1069 . -291) 132407) ((-1069 . -289) 132382) ((-1069 . -102) T) ((-1069 . -1107) T) ((-1069 . -618) 132364) ((-1069 . -1222) T) ((-1069 . -34) T) ((-1069 . -609) 132339) ((-1068 . -550) T) ((-1068 . -1227) T) ((-1068 . -1157) T) ((-1068 . -1044) 132321) ((-1068 . -619) 132236) ((-1068 . -1026) T) ((-1068 . -892) 132218) ((-1068 . -853) T) ((-1068 . -802) T) ((-1068 . -799) T) ((-1068 . -855) T) ((-1068 . -797) T) ((-1068 . -796) T) ((-1068 . -825) T) ((-1068 . -644) 132200) ((-1068 . -927) T) ((-1068 . -562) T) ((-1068 . -293) T) ((-1068 . -173) T) ((-1068 . -621) 132172) ((-1068 . -722) 132159) ((-1068 . -645) 132146) ((-1068 . -1062) 132133) ((-1068 . -1057) 132120) ((-1068 . -111) 132105) ((-1068 . -38) 132092) ((-1068 . -457) T) ((-1068 . -310) T) ((-1068 . -234) T) ((-1068 . -143) T) ((-1068 . -1055) T) ((-1068 . -1063) T) ((-1068 . -1118) T) ((-1068 . -731) T) ((-1068 . -21) T) ((-1068 . -651) 132064) ((-1068 . -23) T) ((-1068 . -1107) T) ((-1068 . -618) 132046) ((-1068 . -102) T) ((-1068 . -25) T) ((-1068 . -131) T) ((-1068 . -653) 132033) ((-1068 . -147) T) ((-1068 . -623) 132014) ((-1067 . -1074) 131993) ((-1067 . -102) T) ((-1067 . -618) 131975) ((-1067 . -1107) T) ((-1064 . -1222) T) ((-1064 . -1107) 131953) ((-1064 . -618) 131920) ((-1064 . -102) 131898) ((-1060 . -1059) 131838) ((-1060 . -645) 131780) ((-1060 . -722) 131722) ((-1060 . -34) T) ((-1060 . -1222) T) ((-1060 . -312) 131660) ((-1060 . -519) 131593) ((-1060 . -494) 131577) ((-1060 . -653) 131561) ((-1060 . -651) 131530) ((-1060 . -131) T) ((-1060 . -25) T) ((-1060 . -102) T) ((-1060 . -618) 131492) ((-1060 . -1107) T) ((-1060 . -23) T) ((-1060 . -21) T) ((-1060 . -1062) 131476) ((-1060 . -1057) 131460) ((-1060 . -111) 131439) ((-1060 . -1280) 131409) ((-1060 . -619) 131370) ((-1052 . -1077) 131299) ((-1052 . -982) 131228) ((-1052 . -619) 131170) ((-1052 . -494) 131135) ((-1052 . -102) T) ((-1052 . -1107) T) ((-1052 . -519) 131036) ((-1052 . -312) 130944) ((-1052 . -618) 130887) ((-1052 . -1222) T) ((-1052 . -34) T) ((-1052 . -151) 130852) ((-1052 . -1217) 130781) ((-1042 . -1089) T) ((-1042 . -495) 130762) ((-1042 . -618) 130728) ((-1042 . -621) 130709) ((-1042 . -1107) T) ((-1042 . -102) T) ((-1042 . -93) T) ((-1041 . -1199) 130684) ((-1041 . -230) 130630) ((-1041 . -107) 130576) ((-1041 . -312) 130427) ((-1041 . -519) 130271) ((-1041 . -494) 130202) ((-1041 . -151) 130148) ((-1041 . -619) NIL) ((-1041 . -236) 130094) ((-1041 . -615) 130069) ((-1041 . -291) 130044) ((-1041 . -289) 130019) ((-1041 . -102) T) ((-1041 . -1107) T) ((-1041 . -618) 130001) ((-1041 . -1222) T) ((-1041 . -34) T) ((-1041 . -609) 129976) ((-1040 . -173) T) ((-1040 . -621) 129945) ((-1040 . -731) T) ((-1040 . -1118) T) ((-1040 . -1063) T) ((-1040 . -1055) T) ((-1040 . -653) 129919) ((-1040 . -651) 129878) ((-1040 . -131) T) ((-1040 . -25) T) ((-1040 . -102) T) ((-1040 . -618) 129860) ((-1040 . -1107) T) ((-1040 . -23) T) ((-1040 . -21) T) ((-1040 . -1062) 129834) ((-1040 . -1057) 129808) ((-1040 . -111) 129775) ((-1040 . -38) 129759) ((-1040 . -645) 129743) ((-1040 . -722) 129727) ((-1033 . -1077) 129696) ((-1033 . -982) 129665) ((-1033 . -619) 129626) ((-1033 . -494) 129610) ((-1033 . -102) T) ((-1033 . -1107) T) ((-1033 . -519) 129543) ((-1033 . -312) 129481) ((-1033 . -618) 129443) ((-1033 . -1222) T) ((-1033 . -34) T) ((-1033 . -151) 129427) ((-1033 . -1217) 129396) ((-1032 . -1222) T) ((-1032 . -1107) 129374) ((-1032 . -618) 129341) ((-1032 . -102) 129319) ((-1030 . -1018) T) ((-1030 . -1008) T) ((-1030 . -796) T) ((-1030 . -797) T) ((-1030 . -855) T) ((-1030 . -799) T) ((-1030 . -802) T) ((-1030 . -853) T) ((-1030 . -1044) 129199) ((-1030 . -417) 129161) ((-1030 . -244) T) ((-1030 . -293) T) ((-1030 . -310) T) ((-1030 . -457) T) ((-1030 . -38) 129098) ((-1030 . -645) 129035) ((-1030 . -722) 128972) ((-1030 . -621) 128909) ((-1030 . -562) T) ((-1030 . -927) T) ((-1030 . -1227) T) ((-1030 . -367) T) ((-1030 . -111) 128825) ((-1030 . -1057) 128762) ((-1030 . -1062) 128699) ((-1030 . -173) T) ((-1030 . -147) T) ((-1030 . -653) 128636) ((-1030 . -651) 128573) ((-1030 . -131) T) ((-1030 . -25) T) ((-1030 . -102) T) ((-1030 . -618) 128555) ((-1030 . -1107) T) ((-1030 . -23) T) ((-1030 . -21) T) ((-1030 . -1055) T) ((-1030 . -1063) T) ((-1030 . -1118) T) ((-1030 . -731) T) ((-1025 . -1089) T) ((-1025 . -495) 128536) ((-1025 . -618) 128502) ((-1025 . -621) 128483) ((-1025 . -1107) T) ((-1025 . -102) T) ((-1025 . -93) T) ((-1010 . -997) 128465) ((-1010 . -1157) T) ((-1010 . -621) 128415) ((-1010 . -1044) 128375) ((-1010 . -619) 128305) ((-1010 . -1026) T) ((-1010 . -916) NIL) ((-1010 . -890) 128287) ((-1010 . -853) T) ((-1010 . -802) T) ((-1010 . -799) T) ((-1010 . -855) T) ((-1010 . -797) T) ((-1010 . -796) T) ((-1010 . -825) T) ((-1010 . -892) 128269) ((-1010 . -1222) T) ((-1010 . -405) 128251) ((-1010 . -644) 128233) ((-1010 . -381) 128215) ((-1010 . -289) NIL) ((-1010 . -312) NIL) ((-1010 . -519) NIL) ((-1010 . -342) 128197) ((-1010 . -244) T) ((-1010 . -111) 128131) ((-1010 . -1057) 128081) ((-1010 . -1062) 128031) ((-1010 . -293) T) ((-1010 . -722) 127981) ((-1010 . -645) 127931) ((-1010 . -653) 127881) ((-1010 . -651) 127831) ((-1010 . -38) 127781) ((-1010 . -310) T) ((-1010 . -457) T) ((-1010 . -173) T) ((-1010 . -562) T) ((-1010 . -927) T) ((-1010 . -1227) T) ((-1010 . -367) T) ((-1010 . -234) T) ((-1010 . -906) NIL) ((-1010 . -232) 127763) ((-1010 . -147) T) ((-1010 . -145) NIL) ((-1010 . -131) T) ((-1010 . -25) T) ((-1010 . -102) T) ((-1010 . -618) 127723) ((-1010 . -1107) T) ((-1010 . -23) T) ((-1010 . -21) T) ((-1010 . -1055) T) ((-1010 . -1063) T) ((-1010 . -1118) T) ((-1010 . -731) T) ((-1009 . -346) 127697) ((-1009 . -173) T) ((-1009 . -621) 127627) ((-1009 . -731) T) ((-1009 . -1118) T) ((-1009 . -1063) T) ((-1009 . -1055) T) ((-1009 . -653) 127572) ((-1009 . -651) 127502) ((-1009 . -131) T) ((-1009 . -25) T) ((-1009 . -102) T) ((-1009 . -618) 127484) ((-1009 . -1107) T) ((-1009 . -23) T) ((-1009 . -21) T) ((-1009 . -1062) 127429) ((-1009 . -1057) 127374) ((-1009 . -111) 127303) ((-1009 . -619) 127287) ((-1009 . -232) 127264) ((-1009 . -906) 127216) ((-1009 . -234) 127188) ((-1009 . -367) T) ((-1009 . -1227) T) ((-1009 . -927) T) ((-1009 . -562) T) ((-1009 . -722) 127133) ((-1009 . -645) 127078) ((-1009 . -38) 127023) ((-1009 . -457) T) ((-1009 . -310) T) ((-1009 . -293) T) ((-1009 . -244) T) ((-1009 . -372) NIL) ((-1009 . -354) NIL) ((-1009 . -1157) NIL) ((-1009 . -145) 126995) ((-1009 . -407) NIL) ((-1009 . -415) 126967) ((-1009 . -147) 126939) ((-1009 . -374) 126911) ((-1009 . -381) 126888) ((-1009 . -644) 126827) ((-1009 . -417) 126804) ((-1009 . -1044) 126692) ((-1009 . -729) 126664) ((-1006 . -1001) 126648) ((-1006 . -494) 126632) ((-1006 . -102) 126610) ((-1006 . -1107) 126588) ((-1006 . -519) 126521) ((-1006 . -312) 126459) ((-1006 . -618) 126391) ((-1006 . -1222) T) ((-1006 . -34) T) ((-1006 . -107) 126375) ((-1002 . -1004) 126359) ((-1002 . -855) 126338) ((-1002 . -1044) 126234) ((-1002 . -417) 126218) ((-1002 . -644) 126166) ((-1002 . -381) 126150) ((-1002 . -289) 126108) ((-1002 . -312) 126073) ((-1002 . -519) 125985) ((-1002 . -342) 125969) ((-1002 . -38) 125917) ((-1002 . -111) 125799) ((-1002 . -1057) 125695) ((-1002 . -1062) 125591) ((-1002 . -651) 125514) ((-1002 . -653) 125452) ((-1002 . -645) 125400) ((-1002 . -722) 125348) ((-1002 . -621) 125238) ((-1002 . -293) 125189) ((-1002 . -244) 125168) ((-1002 . -234) 125147) ((-1002 . -906) 125106) ((-1002 . -232) 125090) ((-1002 . -619) 125051) ((-1002 . -147) 125030) ((-1002 . -145) 125009) ((-1002 . -131) T) ((-1002 . -25) T) ((-1002 . -102) T) ((-1002 . -618) 124991) ((-1002 . -1107) T) ((-1002 . -23) T) ((-1002 . -21) T) ((-1002 . -1055) T) ((-1002 . -1063) T) ((-1002 . -1118) T) ((-1002 . -731) T) ((-1000 . -1089) T) ((-1000 . -495) 124972) ((-1000 . -618) 124938) ((-1000 . -621) 124919) ((-1000 . -1107) T) ((-1000 . -102) T) ((-1000 . -93) T) ((-999 . -21) T) ((-999 . -651) 124901) ((-999 . -23) T) ((-999 . -1107) T) ((-999 . -618) 124883) ((-999 . -102) T) ((-999 . -25) T) ((-999 . -131) T) ((-995 . -618) 124865) ((-992 . -1107) T) ((-992 . -618) 124847) ((-992 . -102) T) ((-977 . -802) T) ((-977 . -799) T) ((-977 . -855) T) ((-977 . -797) T) ((-977 . -23) T) ((-977 . -1107) T) ((-977 . -618) 124807) ((-977 . -102) T) ((-977 . -25) T) ((-977 . -131) T) ((-977 . -619) 124782) ((-976 . -1089) T) ((-976 . -495) 124763) ((-976 . -618) 124729) ((-976 . -621) 124710) ((-976 . -1107) T) ((-976 . -102) T) ((-976 . -93) T) ((-972 . -973) T) ((-972 . -102) T) ((-972 . -618) 124692) ((-972 . -1107) T) ((-972 . -621) 124676) ((-971 . -618) 124658) ((-970 . -1107) T) ((-970 . -618) 124640) ((-970 . -102) T) ((-970 . -372) 124593) ((-970 . -731) 124492) ((-970 . -1118) 124391) ((-970 . -23) 124202) ((-970 . -25) 124013) ((-970 . -131) 123868) ((-970 . -478) 123821) ((-970 . -21) 123776) ((-970 . -651) 123720) ((-970 . -798) 123673) ((-970 . -797) 123626) ((-970 . -855) 123525) ((-970 . -799) 123478) ((-970 . -802) 123431) ((-964 . -19) 123415) ((-964 . -656) 123399) ((-964 . -291) 123376) ((-964 . -289) 123353) ((-964 . -609) 123330) ((-964 . -619) 123291) 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. -1227) 121181) ((-952 . -916) 121160) ((-952 . -892) 121085) ((-952 . -906) 121066) ((-952 . -519) 121013) ((-952 . -457) 120964) ((-952 . -644) 120912) ((-952 . -381) 120896) ((-952 . -47) 120865) ((-952 . -38) 120714) ((-952 . -645) 120563) ((-952 . -722) 120412) ((-952 . -293) 120343) ((-952 . -562) 120274) ((-952 . -111) 120103) ((-952 . -1057) 119946) ((-952 . -1062) 119789) ((-952 . -173) 119700) ((-952 . -147) 119679) ((-952 . -145) 119658) ((-952 . -653) 119583) ((-952 . -651) 119493) ((-952 . -131) T) ((-952 . -25) T) ((-952 . -102) T) ((-952 . -618) 119475) ((-952 . -1107) T) ((-952 . -23) T) ((-952 . -21) T) ((-952 . -1055) T) ((-952 . -1063) T) ((-952 . -1118) T) ((-952 . -731) T) ((-952 . -417) 119459) ((-952 . -329) 119428) ((-952 . -312) 119415) ((-952 . -619) 119276) ((-949 . -986) 119260) ((-949 . -19) 119244) ((-949 . -656) 119228) ((-949 . -291) 119205) ((-949 . -289) 119182) ((-949 . -609) 119159) ((-949 . -619) 119120) ((-949 . -494) 119104) ((-949 . -102) 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. -38) 109338) ((-822 . -457) 109317) ((-822 . -310) 109296) ((-822 . -293) 109275) ((-822 . -244) 109254) ((-821 . -255) 109193) ((-821 . -621) 108930) ((-821 . -1044) 108758) ((-821 . -619) NIL) ((-821 . -329) 108720) ((-821 . -417) 108704) ((-821 . -38) 108553) ((-821 . -111) 108382) ((-821 . -1057) 108225) ((-821 . -1062) 108068) ((-821 . -651) 107978) ((-821 . -653) 107903) ((-821 . -645) 107752) ((-821 . -722) 107601) ((-821 . -145) 107580) ((-821 . -147) 107559) ((-821 . -173) 107470) ((-821 . -562) 107401) ((-821 . -293) 107332) ((-821 . -47) 107294) ((-821 . -381) 107278) ((-821 . -644) 107226) ((-821 . -457) 107177) ((-821 . -519) 107045) ((-821 . -906) 106981) ((-821 . -892) NIL) ((-821 . -916) 106960) ((-821 . -1227) 106939) ((-821 . -956) 106886) ((-821 . -312) 106873) ((-821 . -234) 106852) ((-821 . -131) T) ((-821 . -25) T) ((-821 . -102) T) ((-821 . -618) 106834) ((-821 . -1107) T) ((-821 . -23) T) ((-821 . -21) T) ((-821 . -731) T) ((-821 . -1118) T) ((-821 . -1063) T) 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103733) ((-820 . -38) 103703) ((-820 . -1107) 103493) ((-820 . -102) 103283) ((-820 . -25) 103134) ((-813 . -1107) T) ((-813 . -618) 103116) ((-813 . -102) T) ((-803 . -801) 103100) ((-803 . -855) 103079) ((-803 . -1044) 102859) ((-803 . -621) 102705) ((-803 . -417) 102668) ((-803 . -289) 102626) ((-803 . -312) 102591) ((-803 . -519) 102503) ((-803 . -342) 102487) ((-803 . -372) 102466) ((-803 . -619) 102427) ((-803 . -147) 102406) ((-803 . -145) 102385) ((-803 . -722) 102369) ((-803 . -645) 102353) ((-803 . -653) 102327) ((-803 . -651) 102286) ((-803 . -131) T) ((-803 . -25) T) ((-803 . -102) T) ((-803 . -618) 102268) ((-803 . -1107) T) ((-803 . -23) T) ((-803 . -21) T) ((-803 . -1062) 102252) ((-803 . -1057) 102236) ((-803 . -111) 102215) ((-803 . -1055) T) ((-803 . -1063) T) ((-803 . -1118) T) ((-803 . -731) T) ((-803 . -38) 102199) ((-786 . -1248) 102183) ((-786 . -1157) 102161) ((-786 . -619) NIL) ((-786 . -312) 102148) ((-786 . -519) 102095) ((-786 . -329) 102072) ((-786 . -1044) 101931) ((-786 . -417) 101915) ((-786 . -38) 101744) ((-786 . -111) 101553) ((-786 . -1057) 101376) ((-786 . -1062) 101199) ((-786 . -651) 101109) ((-786 . -653) 101034) ((-786 . -645) 100863) ((-786 . -722) 100692) ((-786 . -621) 100440) ((-786 . -145) 100419) ((-786 . -147) 100398) ((-786 . -47) 100375) ((-786 . -381) 100359) ((-786 . -644) 100307) ((-786 . -906) 100250) ((-786 . -892) NIL) ((-786 . -916) 100229) ((-786 . -1227) 100208) ((-786 . -956) 100177) ((-786 . -927) 100156) ((-786 . -562) 100067) ((-786 . -293) 99978) ((-786 . -173) 99869) ((-786 . -457) 99800) ((-786 . -310) 99779) ((-786 . -289) 99706) ((-786 . -234) T) ((-786 . -131) T) ((-786 . -25) T) ((-786 . -102) T) ((-786 . -618) 99667) ((-786 . -1107) T) ((-786 . -23) T) ((-786 . -21) T) ((-786 . -731) T) ((-786 . -1118) T) ((-786 . -1063) T) ((-786 . -1055) T) ((-786 . -232) 99651) ((-785 . -1071) 99618) ((-785 . -619) 99252) ((-785 . -312) 99239) ((-785 . -519) 99191) ((-785 . -329) 99163) ((-785 . -1044) 99020) 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-731) T) ((-776 . -1118) T) ((-774 . -1107) T) ((-774 . -618) 96881) ((-774 . -102) T) ((-741 . -742) 96865) ((-741 . -1105) 96849) ((-741 . -236) 96833) ((-741 . -619) 96794) ((-741 . -151) 96778) ((-741 . -494) 96762) ((-741 . -102) T) ((-741 . -1107) T) ((-741 . -519) 96695) ((-741 . -312) 96633) ((-741 . -618) 96615) ((-741 . -1222) T) ((-741 . -34) T) ((-741 . -107) 96599) ((-741 . -700) 96583) ((-740 . -1055) T) ((-740 . -1063) T) ((-740 . -1118) T) ((-740 . -731) T) ((-740 . -21) T) ((-740 . -651) 96528) ((-740 . -23) T) ((-740 . -1107) T) ((-740 . -618) 96510) ((-740 . -102) T) ((-740 . -25) T) ((-740 . -131) T) ((-740 . -653) 96470) ((-740 . -621) 96426) ((-740 . -1044) 96397) ((-740 . -147) 96376) ((-740 . -145) 96355) ((-740 . -38) 96325) ((-740 . -111) 96290) ((-740 . -1057) 96260) ((-740 . -1062) 96230) ((-740 . -645) 96200) ((-740 . -722) 96170) ((-740 . -372) 96123) ((-736 . -956) 96076) ((-736 . -621) 95861) ((-736 . -1044) 95737) ((-736 . -1227) 95716) ((-736 . -916) 95695) ((-736 . -892) NIL) ((-736 . -906) 95672) ((-736 . -519) 95615) ((-736 . -457) 95566) ((-736 . -644) 95514) ((-736 . -381) 95498) ((-736 . -47) 95463) ((-736 . -38) 95312) ((-736 . -645) 95161) ((-736 . -722) 95010) ((-736 . -293) 94941) ((-736 . -562) 94872) ((-736 . -111) 94701) ((-736 . -1057) 94544) ((-736 . -1062) 94387) ((-736 . -173) 94298) ((-736 . -147) 94277) ((-736 . -145) 94256) ((-736 . -653) 94181) ((-736 . -651) 94091) ((-736 . -131) T) ((-736 . -25) T) ((-736 . -102) T) ((-736 . -618) 94073) ((-736 . -1107) T) ((-736 . -23) T) ((-736 . -21) T) ((-736 . -1055) T) ((-736 . -1063) T) ((-736 . -1118) T) ((-736 . -731) T) ((-736 . -417) 94057) ((-736 . -329) 94022) ((-736 . -312) 94009) ((-736 . -619) 93870) ((-723 . -478) T) ((-723 . -1118) T) ((-723 . -102) T) ((-723 . -618) 93852) ((-723 . -1107) T) ((-723 . -731) T) ((-720 . -1055) T) ((-720 . -1063) T) ((-720 . -1118) T) ((-720 . -731) T) ((-720 . -21) T) ((-720 . -651) 93824) ((-720 . -23) T) ((-720 . -1107) T) ((-720 . -618) 93806) ((-720 . -102) T) ((-720 . -25) T) ((-720 . -131) T) ((-720 . -653) 93793) ((-720 . -621) 93775) ((-719 . -1055) T) ((-719 . -1063) T) ((-719 . -1118) T) ((-719 . -731) T) ((-719 . -21) T) ((-719 . -651) 93720) ((-719 . -23) T) ((-719 . -1107) T) ((-719 . -618) 93702) ((-719 . -102) T) ((-719 . -25) T) ((-719 . -131) T) ((-719 . -653) 93662) ((-719 . -621) 93616) ((-719 . -1044) 93585) ((-719 . -289) 93564) ((-719 . -147) 93543) ((-719 . -145) 93522) ((-719 . -38) 93492) ((-719 . -111) 93457) ((-719 . -1057) 93427) ((-719 . -1062) 93397) ((-719 . -645) 93367) ((-719 . -722) 93337) ((-718 . -855) T) ((-718 . -102) T) ((-718 . -618) 93272) ((-718 . -1107) T) ((-718 . -495) 93222) ((-718 . -621) 93172) ((-717 . -1248) 93156) ((-717 . -1157) 93134) ((-717 . -619) NIL) ((-717 . -312) 93121) ((-717 . -519) 93068) ((-717 . -329) 93045) ((-717 . -1044) 92925) ((-717 . -417) 92909) ((-717 . -38) 92738) ((-717 . -111) 92547) ((-717 . -1057) 92370) ((-717 . -1062) 92193) 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90413) ((-716 . -731) T) ((-716 . -1118) T) ((-716 . -1063) T) ((-716 . -1055) T) ((-716 . -111) 90369) ((-716 . -1057) 90334) ((-716 . -1062) 90299) ((-716 . -21) T) ((-716 . -23) T) ((-716 . -1107) T) ((-716 . -618) 90281) ((-716 . -102) T) ((-716 . -25) T) ((-716 . -131) T) ((-716 . -293) T) ((-716 . -244) T) ((-715 . -1107) T) ((-715 . -618) 90263) ((-715 . -102) T) ((-706 . -392) T) ((-706 . -1044) 90245) ((-706 . -855) T) ((-706 . -38) 90232) ((-706 . -621) 90204) ((-706 . -731) T) ((-706 . -1118) T) ((-706 . -1063) T) ((-706 . -1055) T) ((-706 . -111) 90189) ((-706 . -1057) 90176) ((-706 . -1062) 90163) ((-706 . -21) T) ((-706 . -651) 90135) ((-706 . -23) T) ((-706 . -1107) T) ((-706 . -618) 90117) ((-706 . -102) T) ((-706 . -25) T) ((-706 . -131) T) ((-706 . -653) 90104) ((-706 . -645) 90091) ((-706 . -722) 90078) ((-706 . -173) T) ((-706 . -293) T) ((-706 . -562) T) ((-706 . -550) T) ((-706 . -1227) T) ((-706 . -1157) T) ((-706 . -619) 89993) ((-706 . -1026) T) ((-706 . -892) 89975) ((-706 . -853) T) ((-706 . -802) T) ((-706 . -799) T) ((-706 . -797) T) ((-706 . -796) T) ((-706 . -825) T) ((-706 . -644) 89957) ((-706 . -927) T) ((-706 . -457) T) ((-706 . -310) T) ((-706 . -234) T) ((-706 . -143) T) ((-706 . -147) T) ((-704 . -409) T) ((-704 . -147) T) ((-704 . -621) 89892) ((-704 . -653) 89857) ((-704 . -651) 89807) ((-704 . -131) T) ((-704 . -25) T) ((-704 . -102) T) ((-704 . -618) 89789) ((-704 . -1107) T) ((-704 . -23) T) ((-704 . -21) T) ((-704 . -731) T) ((-704 . -1118) T) ((-704 . -1063) T) ((-704 . -1055) T) ((-704 . -619) 89734) ((-704 . -367) T) ((-704 . -1227) T) ((-704 . -927) T) ((-704 . -562) T) ((-704 . -173) T) ((-704 . -722) 89699) ((-704 . -645) 89664) ((-704 . -38) 89629) ((-704 . -457) T) ((-704 . -310) T) ((-704 . -111) 89585) ((-704 . -1057) 89550) ((-704 . -1062) 89515) ((-704 . -293) T) ((-704 . -244) T) ((-704 . -853) T) ((-704 . -802) T) ((-704 . -799) T) ((-704 . -855) T) ((-704 . -797) T) ((-704 . -796) T) ((-704 . -892) 89497) ((-704 . -1008) T) ((-704 . -1026) T) ((-704 . -1044) 89442) ((-704 . -1066) T) ((-704 . -392) T) ((-699 . -392) T) ((-699 . -1044) 89387) ((-699 . -855) T) ((-699 . -38) 89337) ((-699 . -621) 89272) ((-699 . -731) T) ((-699 . -1118) T) ((-699 . -1063) T) ((-699 . -1055) T) ((-699 . -111) 89206) ((-699 . -1057) 89156) ((-699 . -1062) 89106) ((-699 . -21) T) ((-699 . -651) 89041) ((-699 . -23) T) ((-699 . -1107) T) ((-699 . -618) 89023) ((-699 . -102) T) ((-699 . -25) T) ((-699 . -131) T) ((-699 . -653) 88973) ((-699 . -645) 88923) ((-699 . -722) 88873) ((-699 . -173) T) ((-699 . -293) T) ((-699 . -562) T) ((-699 . -166) 88855) ((-699 . -35) NIL) ((-699 . -95) NIL) ((-699 . -287) NIL) ((-699 . -498) NIL) ((-699 . -1211) NIL) ((-699 . -1208) NIL) ((-699 . -1008) NIL) ((-699 . -916) NIL) ((-699 . -619) 88763) ((-699 . -890) 88745) ((-699 . -372) NIL) ((-699 . -354) NIL) ((-699 . -1157) NIL) ((-699 . -407) NIL) ((-699 . -415) 88712) ((-699 . -374) 88679) ((-699 . -729) 88646) ((-699 . -417) 88628) ((-699 . -892) 88610) ((-699 . -1222) T) ((-699 . -405) 88592) ((-699 . -644) 88574) ((-699 . -381) 88556) ((-699 . -289) NIL) ((-699 . -312) NIL) ((-699 . -519) NIL) ((-699 . -342) 88538) ((-699 . -244) T) ((-699 . -1227) T) ((-699 . -367) T) ((-699 . -927) T) ((-699 . -457) T) ((-699 . -310) T) ((-699 . -234) NIL) ((-699 . -906) NIL) ((-699 . -232) 88520) ((-699 . -147) T) ((-699 . -145) NIL) ((-696 . -1268) T) ((-696 . -1044) 88504) ((-696 . -621) 88488) ((-696 . -618) 88470) ((-694 . -691) 88428) ((-694 . -494) 88412) ((-694 . -102) 88390) ((-694 . -1107) 88368) ((-694 . -519) 88301) ((-694 . -312) 88239) ((-694 . -618) 88171) ((-694 . -1222) T) ((-694 . -34) T) ((-694 . -57) 88129) ((-694 . -619) 88090) ((-686 . -1089) T) ((-686 . -495) 88071) ((-686 . -618) 88021) ((-686 . -621) 88002) ((-686 . -1107) T) ((-686 . -102) T) ((-686 . -93) T) ((-682 . -855) T) ((-682 . -102) T) ((-682 . -618) 87984) ((-682 . -1107) T) ((-682 . -1044) 87968) ((-682 . -621) 87952) ((-681 . -1089) T) ((-681 . -495) 87933) ((-681 . -618) 87899) ((-681 . -621) 87880) ((-681 . -1107) T) ((-681 . -102) T) ((-681 . -93) T) ((-680 . -494) 87864) ((-680 . -102) 87842) ((-680 . -1107) 87820) ((-680 . -519) 87753) ((-680 . -312) 87691) ((-680 . -618) 87623) ((-680 . -1222) T) ((-680 . -34) T) ((-677 . -855) T) ((-677 . -102) T) ((-677 . -618) 87605) ((-677 . -1107) T) ((-677 . -1044) 87589) ((-677 . -621) 87573) ((-676 . -1089) T) ((-676 . -495) 87554) ((-676 . -618) 87520) ((-676 . -621) 87501) ((-676 . -1107) T) ((-676 . -102) T) ((-676 . -93) T) ((-675 . -1129) 87446) ((-675 . -494) 87430) ((-675 . -519) 87363) ((-675 . -312) 87301) ((-675 . -1222) T) ((-675 . -34) T) ((-675 . -1059) 87241) ((-675 . -1044) 87137) ((-675 . -621) 87055) ((-675 . -417) 87039) ((-675 . -644) 86987) ((-675 . -381) 86971) ((-675 . -234) 86950) ((-675 . -906) 86909) ((-675 . -232) 86893) ((-675 . -722) 86877) ((-675 . -645) 86861) ((-675 . -653) 86835) ((-675 . -651) 86794) ((-675 . -131) T) ((-675 . -25) T) ((-675 . -102) T) ((-675 . -618) 86756) ((-675 . -1107) T) ((-675 . -23) T) ((-675 . -21) T) ((-675 . -1062) 86740) ((-675 . -1057) 86724) ((-675 . -111) 86703) ((-675 . -1055) T) ((-675 . -1063) T) ((-675 . -1118) T) ((-675 . -731) T) ((-675 . -38) 86663) ((-675 . -423) 86647) ((-675 . -749) 86631) ((-675 . -725) T) ((-675 . -766) T) ((-675 . -371) 86615) ((-669 . -378) 86594) ((-669 . -722) 86578) ((-669 . -645) 86562) ((-669 . -653) 86546) ((-669 . -651) 86515) ((-669 . -131) T) ((-669 . -25) T) ((-669 . -102) T) ((-669 . -618) 86497) ((-669 . -1107) T) ((-669 . -23) T) ((-669 . -21) T) ((-669 . -1062) 86481) ((-669 . -1057) 86465) ((-669 . -111) 86444) ((-669 . -640) 86428) ((-669 . -388) 86400) ((-669 . -621) 86377) ((-669 . -1044) 86354) ((-661 . -663) 86338) ((-661 . -38) 86308) ((-661 . -621) 86226) ((-661 . -653) 86200) ((-661 . -651) 86159) ((-661 . -731) T) ((-661 . -1118) T) ((-661 . -1063) T) ((-661 . -1055) T) ((-661 . -111) 86138) ((-661 . -1057) 86122) ((-661 . -1062) 86106) ((-661 . -21) T) ((-661 . -23) T) ((-661 . -1107) T) ((-661 . -618) 86088) ((-661 . -102) T) ((-661 . -25) T) ((-661 . -131) T) ((-661 . -645) 86058) ((-661 . -722) 86028) ((-661 . -417) 86012) ((-661 . -1044) 85908) ((-661 . -857) 85892) ((-661 . -289) 85853) ((-660 . -663) 85837) ((-660 . -38) 85807) ((-660 . -621) 85725) ((-660 . -653) 85699) ((-660 . -651) 85658) ((-660 . -731) T) ((-660 . -1118) T) ((-660 . -1063) T) ((-660 . -1055) T) ((-660 . -111) 85637) ((-660 . -1057) 85621) ((-660 . -1062) 85605) ((-660 . -21) T) ((-660 . -23) T) ((-660 . -1107) T) ((-660 . -618) 85587) ((-660 . -102) T) ((-660 . -25) T) ((-660 . -131) T) ((-660 . -645) 85557) ((-660 . -722) 85527) ((-660 . -417) 85511) ((-660 . -1044) 85407) ((-660 . -857) 85391) ((-660 . -289) 85370) ((-659 . -663) 85354) ((-659 . -38) 85324) ((-659 . -621) 85242) ((-659 . -653) 85216) ((-659 . -651) 85175) ((-659 . -731) T) ((-659 . -1118) T) ((-659 . -1063) T) ((-659 . -1055) T) ((-659 . -111) 85154) ((-659 . -1057) 85138) ((-659 . -1062) 85122) ((-659 . -21) T) ((-659 . -23) T) ((-659 . -1107) T) ((-659 . -618) 85104) ((-659 . -102) T) ((-659 . -25) T) ((-659 . -131) T) ((-659 . -645) 85074) ((-659 . -722) 85044) ((-659 . -417) 85028) ((-659 . -1044) 84924) ((-659 . -857) 84908) ((-659 . -289) 84887) ((-657 . -722) 84871) ((-657 . -645) 84855) ((-657 . -653) 84839) ((-657 . -651) 84808) ((-657 . -131) T) ((-657 . -25) T) ((-657 . -102) T) ((-657 . -618) 84790) ((-657 . -1107) T) ((-657 . -23) T) ((-657 . -21) T) ((-657 . -1062) 84774) ((-657 . -1057) 84758) ((-657 . -111) 84737) ((-657 . -796) 84716) ((-657 . -797) 84695) ((-657 . -855) 84674) ((-657 . -799) 84653) ((-657 . -802) 84632) ((-654 . -1107) T) ((-654 . -618) 84614) ((-654 . -102) T) ((-654 . -1044) 84598) ((-654 . -621) 84582) ((-652 . -700) 84566) ((-652 . -107) 84550) ((-652 . -34) T) ((-652 . -1222) T) ((-652 . -618) 84482) ((-652 . -312) 84420) ((-652 . -519) 84353) ((-652 . -1107) 84331) ((-652 . -102) 84309) ((-652 . -494) 84293) ((-652 . -151) 84277) ((-652 . -619) 84238) ((-652 . -236) 84222) ((-650 . -1089) T) ((-650 . -495) 84203) ((-650 . -618) 84156) ((-650 . -621) 84137) ((-650 . -1107) T) ((-650 . -102) T) ((-650 . -93) T) ((-646 . -671) 84121) ((-646 . -1261) 84105) ((-646 . -1016) 84089) ((-646 . -1155) 84073) ((-646 . -855) 84052) ((-646 . -376) 84036) ((-646 . -656) 84020) ((-646 . -291) 83997) ((-646 . -289) 83974) ((-646 . -609) 83951) ((-646 . -619) 83912) ((-646 . -494) 83896) ((-646 . -102) 83846) ((-646 . -1107) 83796) ((-646 . -519) 83729) ((-646 . -312) 83667) ((-646 . -618) 83579) ((-646 . -1222) T) ((-646 . -34) T) ((-646 . -151) 83563) ((-646 . -285) 83547) ((-646 . -826) 83526) ((-638 . -749) 83510) ((-638 . -725) T) ((-638 . -766) T) ((-638 . -111) 83489) ((-638 . -1057) 83473) ((-638 . -1062) 83457) ((-638 . -21) T) ((-638 . -651) 83426) ((-638 . -23) T) ((-638 . -1107) T) ((-638 . -618) 83395) ((-638 . -102) T) ((-638 . -25) T) ((-638 . -131) T) ((-638 . -653) 83379) 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. -102) 5358) ((-91 . -1107) 5336) ((-91 . -519) 5269) ((-91 . -312) 5207) ((-91 . -618) 5139) ((-91 . -1222) T) ((-91 . -34) T) ((-91 . -107) 5123) ((-89 . -402) T) ((-89 . -618) 5105) ((-89 . -1222) T) ((-89 . -401) T) ((-88 . -389) T) ((-88 . -618) 5087) ((-88 . -1222) T) ((-88 . -401) T) ((-87 . -445) T) ((-87 . -618) 5069) ((-87 . -1222) T) ((-87 . -401) T) ((-86 . -446) T) ((-86 . -618) 5051) ((-86 . -1222) T) ((-86 . -401) T) ((-85 . -389) T) ((-85 . -618) 5033) ((-85 . -1222) T) ((-85 . -401) T) ((-84 . -389) T) ((-84 . -618) 5015) ((-84 . -1222) T) ((-84 . -401) T) ((-83 . -446) T) ((-83 . -618) 4997) ((-83 . -1222) T) ((-83 . -401) T) ((-82 . -446) T) ((-82 . -618) 4979) ((-82 . -1222) T) ((-82 . -401) T) ((-81 . -446) T) ((-81 . -618) 4961) ((-81 . -1222) T) ((-81 . -401) T) ((-81 . -621) 4902) ((-80 . -446) T) ((-80 . -618) 4884) ((-80 . -1222) T) ((-80 . -401) T) ((-79 . -446) T) ((-79 . -618) 4866) ((-79 . -1222) T) ((-79 . -401) T) ((-78 . -402) T) ((-78 . -618) 4848) ((-78 . -1222) T) ((-78 . -401) T) ((-77 . -446) T) ((-77 . -618) 4830) ((-77 . -1222) T) ((-77 . -401) T) ((-76 . -446) T) ((-76 . -618) 4812) ((-76 . -1222) T) ((-76 . -401) T) ((-75 . -402) T) ((-75 . -618) 4794) ((-75 . -1222) T) ((-75 . -401) T) ((-74 . -446) T) ((-74 . -618) 4776) ((-74 . -1222) T) ((-74 . -401) T) ((-73 . -387) T) ((-73 . -618) 4758) ((-73 . -1222) T) ((-73 . -401) T) ((-72 . -401) T) ((-72 . -1222) T) ((-72 . -618) 4740) ((-71 . -446) T) ((-71 . -618) 4722) ((-71 . -1222) T) ((-71 . -401) T) ((-70 . -387) T) ((-70 . -618) 4704) ((-70 . -1222) T) ((-70 . -401) T) ((-69 . -401) T) ((-69 . -1222) T) ((-69 . -618) 4686) ((-68 . -387) T) ((-68 . -618) 4668) ((-68 . -1222) T) ((-68 . -401) T) ((-67 . -387) T) ((-67 . -618) 4650) ((-67 . -1222) T) ((-67 . -401) T) ((-66 . -402) T) ((-66 . -618) 4632) ((-66 . -1222) T) ((-66 . -401) T) ((-65 . -389) T) ((-65 . -618) 4614) ((-65 . -1222) T) ((-65 . -401) T) ((-65 . -621) 4543) ((-64 . -446) T) ((-64 . -618) 4525) ((-64 . -1222) T) ((-64 . -401) T) ((-63 . -401) T) ((-63 . -1222) T) ((-63 . -618) 4507) ((-62 . -446) T) ((-62 . -618) 4489) ((-62 . -1222) T) ((-62 . -401) T) ((-61 . -402) T) ((-61 . -618) 4471) ((-61 . -1222) T) ((-61 . -401) T) ((-60 . -57) 4433) ((-60 . -34) T) ((-60 . -1222) T) ((-60 . -618) 4365) ((-60 . -312) 4303) ((-60 . -519) 4236) ((-60 . -1107) 4214) ((-60 . -102) 4192) ((-60 . -494) 4176) ((-58 . -19) 4160) ((-58 . -656) 4144) ((-58 . -291) 4121) ((-58 . -289) 4098) ((-58 . -609) 4075) ((-58 . -619) 4036) ((-58 . -494) 4020) ((-58 . -102) 3970) ((-58 . -1107) 3920) ((-58 . -519) 3853) ((-58 . -312) 3791) ((-58 . -618) 3703) ((-58 . -1222) T) ((-58 . -34) T) ((-58 . -151) 3687) ((-58 . -855) 3666) ((-58 . -376) 3650) ((-55 . -1107) T) ((-55 . -618) 3632) ((-55 . -102) T) ((-55 . -1044) 3614) ((-55 . -621) 3596) ((-51 . -1107) T) ((-51 . -618) 3578) ((-51 . -102) T) ((-50 . -626) 3562) ((-50 . -621) 3531) ((-50 . -653) 3505) ((-50 . -651) 3464) ((-50 . -731) T) ((-50 . -1118) T) ((-50 . -1063) T) ((-50 . -1055) T) ((-50 . -21) T) ((-50 . -23) T) ((-50 . -1107) T) ((-50 . -618) 3446) ((-50 . -102) T) ((-50 . -25) T) ((-50 . -131) T) ((-50 . -1044) 3430) ((-49 . -1107) T) ((-49 . -618) 3412) ((-49 . -102) T) ((-48 . -301) T) ((-48 . -102) T) ((-48 . -618) 3394) ((-48 . -1107) T) ((-48 . -621) 3327) ((-48 . -1044) 3270) ((-48 . -519) 3236) ((-48 . -312) 3223) ((-48 . -27) T) ((-48 . -1008) T) ((-48 . -244) T) ((-48 . -111) 3179) ((-48 . -1057) 3144) ((-48 . -1062) 3109) ((-48 . -293) T) ((-48 . -722) 3074) ((-48 . -645) 3039) ((-48 . -653) 3004) ((-48 . -651) 2954) ((-48 . -131) T) ((-48 . -25) T) ((-48 . -23) T) ((-48 . -21) T) ((-48 . -1055) T) ((-48 . -1063) T) ((-48 . -1118) T) ((-48 . -731) T) ((-48 . -38) 2919) ((-48 . -310) T) ((-48 . -457) T) ((-48 . -173) T) ((-48 . -562) T) ((-48 . -927) T) ((-48 . -1227) T) ((-48 . -367) T) ((-48 . -644) 2879) ((-48 . -1026) T) ((-48 . -619) 2824) ((-48 . -147) T) ((-48 . -234) T) ((-45 . -36) 2803) ((-45 . -609) 2728) ((-45 . -312) 2532) ((-45 . -519) 2324) ((-45 . -494) 2261) ((-45 . -289) 2186) ((-45 . -291) 2111) ((-45 . -615) 2090) ((-45 . -236) 2040) ((-45 . -107) 1990) ((-45 . -230) 1940) ((-45 . -1199) 1919) ((-45 . -285) 1869) ((-45 . -151) 1819) ((-45 . -34) T) ((-45 . -1222) T) ((-45 . -618) 1801) ((-45 . -1107) T) ((-45 . -102) T) ((-45 . -619) NIL) ((-45 . -656) 1751) ((-45 . -376) 1701) ((-45 . -855) NIL) ((-45 . -1155) 1651) ((-45 . -1016) 1601) ((-45 . -1261) 1551) ((-45 . -671) 1501) ((-44 . -423) 1485) ((-44 . -749) 1469) ((-44 . -725) T) ((-44 . -766) T) ((-44 . -111) 1448) ((-44 . -1057) 1432) ((-44 . -1062) 1416) ((-44 . -21) T) ((-44 . -651) 1359) ((-44 . -23) T) ((-44 . -1107) T) ((-44 . -618) 1341) ((-44 . -102) T) ((-44 . -25) T) ((-44 . -131) T) ((-44 . -653) 1299) ((-44 . -645) 1283) ((-44 . -722) 1267) ((-44 . -371) 1251) ((-40 . -346) 1225) ((-40 . -173) T) ((-40 . -621) 1155) ((-40 . -731) T) ((-40 . -1118) T) ((-40 . -1063) T) ((-40 . -1055) T) ((-40 . -653) 1100) ((-40 . -651) 1030) ((-40 . -131) T) ((-40 . -25) T) ((-40 . -102) T) ((-40 . -618) 1012) ((-40 . -1107) T) ((-40 . -23) T) ((-40 . -21) T) ((-40 . -1062) 957) ((-40 . -1057) 902) ((-40 . -111) 831) ((-40 . -619) 815) ((-40 . -232) 792) ((-40 . -906) 744) ((-40 . -234) 716) ((-40 . -367) T) ((-40 . -1227) T) ((-40 . -927) T) ((-40 . -562) T) ((-40 . -722) 661) ((-40 . -645) 606) ((-40 . -38) 551) ((-40 . -457) T) ((-40 . -310) T) ((-40 . -293) T) ((-40 . -244) T) ((-40 . -372) NIL) ((-40 . -354) NIL) ((-40 . -1157) NIL) ((-40 . -145) 523) ((-40 . -407) NIL) ((-40 . -415) 495) ((-40 . -147) 467) ((-40 . -374) 439) ((-40 . -381) 416) ((-40 . -644) 355) ((-40 . -417) 332) ((-40 . -1044) 220) ((-40 . -729) 192) ((-31 . -1089) T) ((-31 . -495) 173) ((-31 . -618) 139) ((-31 . -621) 120) ((-31 . -1107) T) ((-31 . -102) T) ((-31 . -93) T) ((-30 . -961) T) ((-30 . -618) 102) ((0 . |EnumerationCategory|) T) ((0 . -618) 84) ((0 . -1107) T) ((0 . -102) T) ((-2 . |RecordCategory|) T) ((-2 . -618) 66) ((-2 . -1107) T) ((-2 . -102) T) ((-3 . |UnionCategory|) T) ((-3 . -618) 48) ((-3 . -1107) T) ((-3 . -102) T) ((-1 . -1107) T) ((-1 . -618) 30) ((-1 . -102) T)) 
\ No newline at end of file
diff --git a/src/share/algebra/compress.daase b/src/share/algebra/compress.daase
index 9b36e7fb..2e9add15 100644
--- a/src/share/algebra/compress.daase
+++ b/src/share/algebra/compress.daase
@@ -1,6 +1,6 @@
 
-(30 . 3477420784)            
-(4430 |Enumeration| |Mapping| |Record| |Union| |ofCategory| |isDomain|
+(30 . 3477425182)            
+(4437 |Enumeration| |Mapping| |Record| |Union| |ofCategory| |isDomain|
  ATTRIBUTE |package| |domain| |category| CATEGORY |nobranch| AND |Join|
  |ofType| SIGNATURE "failed" "algebra" |OneDimensionalArrayAggregate&|
  |OneDimensionalArrayAggregate| |AbelianGroup&| |AbelianGroup| |AbelianMonoid&|
@@ -77,7 +77,7 @@
  |ExtAlgBasis| |ElementaryFunction| |ElementaryFunctionStructurePackage|
  |ElementaryFunctionsUnivariateLaurentSeries|
  |ElementaryFunctionsUnivariatePuiseuxSeries| |ElaboratedExpression|
- |ExtensibleLinearAggregate&| |ExtensibleLinearAggregate|
+ |Elaboration| |ExtensibleLinearAggregate&| |ExtensibleLinearAggregate|
  |ElementaryFunctionCategory&| |ElementaryFunctionCategory|
  |EllipticFunctionsUnivariateTaylorSeries| |Eltable| |EltableAggregate&|
  |EltableAggregate| |EuclideanModularRing| |EntireRing| |Environment|
@@ -187,7 +187,7 @@
  |IntegrationResultRFToFunction| |IrrRepSymNatPackage|
  |InternalRationalUnivariateRepresentationPackage| |IsAst| |IndexedString|
  |InnerPolySum| |InnerSparseUnivariatePowerSeries| |InnerTaylorSeries|
- |InfiniteTupleFunctions2| |InfiniteTupleFunctions3|
+ |InternalTypeForm| |InfiniteTupleFunctions2| |InfiniteTupleFunctions3|
  |InnerTrigonometricManipulations| |InfiniteTuple| |IndexedVector|
  |IndexedAggregate&| |IndexedAggregate| |JavaBytecode| |JoinAst|
  |AssociatedJordanAlgebra| |KeyedAccessFile| |KeyedDictionary&|
@@ -452,9 +452,9 @@
  |clipParametric| |clipWithRanges| |numberOfHues| |yellow| |iifact| |iibinom|
  |iiperm| |iipow| |iidsum| |iidprod| |ipow| |factorial| |multinomial|
  |permutation| |stirling1| |stirling2| |summation| |factorials| |mkcomm|
- |polarCoordinates| |complex| |imaginary| |macroExpand| |solid| |solid?|
- |denominators| |numerators| |convergents| |approximants| |reducedForm|
- |partialQuotients| |partialDenominators| |partialNumerators|
+ |polarCoordinates| |complex| |imaginary| |elaborate| |macroExpand| |solid|
+ |solid?| |denominators| |numerators| |convergents| |approximants|
+ |reducedForm| |partialQuotients| |partialDenominators| |partialNumerators|
  |reducedContinuedFraction| |push| |bindings| |cartesian| |polar| |cylindrical|
  |spherical| |parabolic| |parabolicCylindrical| |paraboloidal|
  |ellipticCylindrical| |prolateSpheroidal| |oblateSpheroidal| |bipolar|
@@ -494,9 +494,10 @@
  |iiasinh| |iiacosh| |iiatanh| |iiacoth| |iiasech| |iiacsch| |specialTrigs|
  |localReal?| |rischNormalize| |realElementary| |validExponential|
  |rootNormalize| |tanQ| |callForm?| |getIdentifier| |variable?| |getConstant|
- |type| |select!| |delete!| |sn| |cn| |dn| |sncndn| |qsetelt!| |categoryFrame|
- |interactiveEnv| |currentEnv| |setProperties!| |getProperties| |setProperty!|
- |getProperty| |scopes| |eigenvalues| |eigenvector| |generalizedEigenvector|
+ |type| |environment| |typeForm| |irForm| |elaboration| |select!| |delete!|
+ |sn| |cn| |dn| |sncndn| |qsetelt!| |categoryFrame| |interactiveEnv|
+ |currentEnv| |setProperties!| |getProperties| |setProperty!| |getProperty|
+ |scopes| |eigenvalues| |eigenvector| |generalizedEigenvector|
  |generalizedEigenvectors| |eigenvectors| |factorAndSplit| |rightOne| |leftOne|
  |rightZero| |leftZero| |swap| |error| |minPoly| |freeOf?| |operators| |tower|
  |kernels| |mainKernel| |distribute| |subst| |functionIsFracPolynomial?|
diff --git a/src/share/algebra/interp.daase b/src/share/algebra/interp.daase
index 86b9f042..4d8cafb6 100644
--- a/src/share/algebra/interp.daase
+++ b/src/share/algebra/interp.daase
@@ -1,5361 +1,5369 @@
 
-(3214087 . 3477420805)       
-((-1902 (((-112) (-1 (-112) |#2| |#2|) $) 87) (((-112) $) NIL)) (-1900 (($ (-1 (-112) |#2| |#2|) $) 18) (($ $) NIL)) (-4221 ((|#2| $ (-550) |#2|) NIL) ((|#2| $ (-1237 (-550)) |#2|) 44)) (-2444 (($ $) 81)) (-4276 ((|#2| (-1 |#2| |#2| |#2|) $ |#2| |#2|) 52) ((|#2| (-1 |#2| |#2| |#2|) $ |#2|) 50) ((|#2| (-1 |#2| |#2| |#2|) $) 49)) (-3845 (((-550) (-1 (-112) |#2|) $) 27) (((-550) |#2| $) NIL) (((-550) |#2| $ (-550)) 97)) (-2126 (((-644 |#2|) $) 13)) (-3943 (($ (-1 (-112) |#2| |#2|) $ $) 64) (($ $ $) NIL)) (-2130 (($ (-1 |#2| |#2|) $) 37)) (-4392 (($ (-1 |#2| |#2|) $) NIL) (($ (-1 |#2| |#2| |#2|) $ $) 60)) (-2451 (($ |#2| $ (-550)) NIL) (($ $ $ (-550)) 67)) (-1442 (((-3 |#2| "failed") (-1 (-112) |#2|) $) 29)) (-2128 (((-112) (-1 (-112) |#2|) $) 23)) (-4233 ((|#2| $ (-550) |#2|) NIL) ((|#2| $ (-550)) NIL) (($ $ (-1237 (-550))) 66)) (-2452 (($ $ (-550)) 76) (($ $ (-1237 (-550))) 75)) (-2127 (((-774) (-1 (-112) |#2|) $) 34) (((-774) |#2| $) NIL)) (-1901 (($ $ $ (-550)) 69)) (-3826 (($ $) 68)) (-3955 (($ (-644 |#2|)) 73)) (-4235 (($ $ |#2|) NIL) (($ |#2| $) NIL) (($ $ $) 88) (($ (-644 $)) 86)) (-4380 (((-866) $) 93)) (-2129 (((-112) (-1 (-112) |#2|) $) 22)) (-3457 (((-112) $ $) 96)) (-3090 (((-112) $ $) 100))) 
-(((-18 |#1| |#2|) (-10 -8 (-15 -3457 ((-112) |#1| |#1|)) (-15 -4380 ((-866) |#1|)) (-15 -3090 ((-112) |#1| |#1|)) (-15 -1900 (|#1| |#1|)) (-15 -1900 (|#1| (-1 (-112) |#2| |#2|) |#1|)) (-15 -2444 (|#1| |#1|)) (-15 -1901 (|#1| |#1| |#1| (-550))) (-15 -1902 ((-112) |#1|)) (-15 -3943 (|#1| |#1| |#1|)) (-15 -3845 ((-550) |#2| |#1| (-550))) (-15 -3845 ((-550) |#2| |#1|)) (-15 -3845 ((-550) (-1 (-112) |#2|) |#1|)) (-15 -1902 ((-112) (-1 (-112) |#2| |#2|) |#1|)) (-15 -3943 (|#1| (-1 (-112) |#2| |#2|) |#1| |#1|)) (-15 -4221 (|#2| |#1| (-1237 (-550)) |#2|)) (-15 -2451 (|#1| |#1| |#1| (-550))) (-15 -2451 (|#1| |#2| |#1| (-550))) (-15 -2452 (|#1| |#1| (-1237 (-550)))) (-15 -2452 (|#1| |#1| (-550))) (-15 -4233 (|#1| |#1| (-1237 (-550)))) (-15 -4392 (|#1| (-1 |#2| |#2| |#2|) |#1| |#1|)) (-15 -4235 (|#1| (-644 |#1|))) (-15 -4235 (|#1| |#1| |#1|)) (-15 -4235 (|#1| |#2| |#1|)) (-15 -4235 (|#1| |#1| |#2|)) (-15 -3955 (|#1| (-644 |#2|))) (-15 -1442 ((-3 |#2| "failed") (-1 (-112) |#2|) |#1|)) (-15 -4276 (|#2| (-1 |#2| |#2| |#2|) |#1|)) (-15 -4276 (|#2| (-1 |#2| |#2| |#2|) |#1| |#2|)) (-15 -4276 (|#2| (-1 |#2| |#2| |#2|) |#1| |#2| |#2|)) (-15 -4233 (|#2| |#1| (-550))) (-15 -4233 (|#2| |#1| (-550) |#2|)) (-15 -4221 (|#2| |#1| (-550) |#2|)) (-15 -2127 ((-774) |#2| |#1|)) (-15 -2126 ((-644 |#2|) |#1|)) (-15 -2127 ((-774) (-1 (-112) |#2|) |#1|)) (-15 -2128 ((-112) (-1 (-112) |#2|) |#1|)) (-15 -2129 ((-112) (-1 (-112) |#2|) |#1|)) (-15 -2130 (|#1| (-1 |#2| |#2|) |#1|)) (-15 -4392 (|#1| (-1 |#2| |#2|) |#1|)) (-15 -3826 (|#1| |#1|))) (-19 |#2|) (-1220)) (T -18)) 
+(3215100 . 3477425203)       
+((-1909 (((-112) (-1 (-112) |#2| |#2|) $) 87) (((-112) $) NIL)) (-1907 (($ (-1 (-112) |#2| |#2|) $) 18) (($ $) NIL)) (-4228 ((|#2| $ (-551) |#2|) NIL) ((|#2| $ (-1239 (-551)) |#2|) 44)) (-2451 (($ $) 81)) (-4283 ((|#2| (-1 |#2| |#2| |#2|) $ |#2| |#2|) 52) ((|#2| (-1 |#2| |#2| |#2|) $ |#2|) 50) ((|#2| (-1 |#2| |#2| |#2|) $) 49)) (-3852 (((-551) (-1 (-112) |#2|) $) 27) (((-551) |#2| $) NIL) (((-551) |#2| $ (-551)) 97)) (-2133 (((-646 |#2|) $) 13)) (-3950 (($ (-1 (-112) |#2| |#2|) $ $) 64) (($ $ $) NIL)) (-2137 (($ (-1 |#2| |#2|) $) 37)) (-4399 (($ (-1 |#2| |#2|) $) NIL) (($ (-1 |#2| |#2| |#2|) $ $) 60)) (-2458 (($ |#2| $ (-551)) NIL) (($ $ $ (-551)) 67)) (-1444 (((-3 |#2| "failed") (-1 (-112) |#2|) $) 29)) (-2135 (((-112) (-1 (-112) |#2|) $) 23)) (-4240 ((|#2| $ (-551) |#2|) NIL) ((|#2| $ (-551)) NIL) (($ $ (-1239 (-551))) 66)) (-2459 (($ $ (-551)) 76) (($ $ (-1239 (-551))) 75)) (-2134 (((-776) (-1 (-112) |#2|) $) 34) (((-776) |#2| $) NIL)) (-1908 (($ $ $ (-551)) 69)) (-3833 (($ $) 68)) (-3962 (($ (-646 |#2|)) 73)) (-4242 (($ $ |#2|) NIL) (($ |#2| $) NIL) (($ $ $) 88) (($ (-646 $)) 86)) (-4387 (((-868) $) 93)) (-2136 (((-112) (-1 (-112) |#2|) $) 22)) (-3464 (((-112) $ $) 96)) (-3097 (((-112) $ $) 100))) 
+(((-18 |#1| |#2|) (-10 -8 (-15 -3464 ((-112) |#1| |#1|)) (-15 -4387 ((-868) |#1|)) (-15 -3097 ((-112) |#1| |#1|)) (-15 -1907 (|#1| |#1|)) (-15 -1907 (|#1| (-1 (-112) |#2| |#2|) |#1|)) (-15 -2451 (|#1| |#1|)) (-15 -1908 (|#1| |#1| |#1| (-551))) (-15 -1909 ((-112) |#1|)) (-15 -3950 (|#1| |#1| |#1|)) (-15 -3852 ((-551) |#2| |#1| (-551))) (-15 -3852 ((-551) |#2| |#1|)) (-15 -3852 ((-551) (-1 (-112) |#2|) |#1|)) (-15 -1909 ((-112) (-1 (-112) |#2| |#2|) |#1|)) (-15 -3950 (|#1| (-1 (-112) |#2| |#2|) |#1| |#1|)) (-15 -4228 (|#2| |#1| (-1239 (-551)) |#2|)) (-15 -2458 (|#1| |#1| |#1| (-551))) (-15 -2458 (|#1| |#2| |#1| (-551))) (-15 -2459 (|#1| |#1| (-1239 (-551)))) (-15 -2459 (|#1| |#1| (-551))) (-15 -4240 (|#1| |#1| (-1239 (-551)))) (-15 -4399 (|#1| (-1 |#2| |#2| |#2|) |#1| |#1|)) (-15 -4242 (|#1| (-646 |#1|))) (-15 -4242 (|#1| |#1| |#1|)) (-15 -4242 (|#1| |#2| |#1|)) (-15 -4242 (|#1| |#1| |#2|)) (-15 -3962 (|#1| (-646 |#2|))) (-15 -1444 ((-3 |#2| "failed") (-1 (-112) |#2|) |#1|)) (-15 -4283 (|#2| (-1 |#2| |#2| |#2|) |#1|)) (-15 -4283 (|#2| (-1 |#2| |#2| |#2|) |#1| |#2|)) (-15 -4283 (|#2| (-1 |#2| |#2| |#2|) |#1| |#2| |#2|)) (-15 -4240 (|#2| |#1| (-551))) (-15 -4240 (|#2| |#1| (-551) |#2|)) (-15 -4228 (|#2| |#1| (-551) |#2|)) (-15 -2134 ((-776) |#2| |#1|)) (-15 -2133 ((-646 |#2|) |#1|)) (-15 -2134 ((-776) (-1 (-112) |#2|) |#1|)) (-15 -2135 ((-112) (-1 (-112) |#2|) |#1|)) (-15 -2136 ((-112) (-1 (-112) |#2|) |#1|)) (-15 -2137 (|#1| (-1 |#2| |#2|) |#1|)) (-15 -4399 (|#1| (-1 |#2| |#2|) |#1|)) (-15 -3833 (|#1| |#1|))) (-19 |#2|) (-1222)) (T -18)) 
 NIL 
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-(((-19 |#1|) (-140) (-1220)) (T -19)) 
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+(((-19 |#1|) (-140) (-1222)) (T -19)) 
 NIL 
-(-13 (-375 |t#1|) (-10 -7 (-6 -4428))) 
-(((-34) . T) ((-102) -3962 (|has| |#1| (-1105)) (|has| |#1| (-853))) ((-616 (-866)) -3962 (|has| |#1| (-1105)) (|has| |#1| (-853)) (|has| |#1| (-616 (-866)))) ((-151 |#1|) . T) ((-617 (-539)) |has| |#1| (-617 (-539))) ((-288 #1=(-550) |#1|) . T) ((-290 #1# |#1|) . T) ((-311 |#1|) -12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105))) ((-375 |#1|) . T) ((-493 |#1|) . T) ((-607 #1# |#1|) . T) ((-518 |#1| |#1|) -12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105))) ((-654 |#1|) . T) ((-853) |has| |#1| (-853)) ((-1105) -3962 (|has| |#1| (-1105)) (|has| |#1| (-853))) ((-1220) . T)) 
-((-1408 (((-3 $ "failed") $ $) 12)) (-4271 (($ $) NIL) (($ $ $) 9)) (* (($ (-923) $) NIL) (($ (-774) $) 16) (($ (-550) $) 26))) 
-(((-20 |#1|) (-10 -8 (-15 -4271 (|#1| |#1| |#1|)) (-15 -4271 (|#1| |#1|)) (-15 * (|#1| (-550) |#1|)) (-15 -1408 ((-3 |#1| "failed") |#1| |#1|)) (-15 * (|#1| (-774) |#1|)) (-15 * (|#1| (-923) |#1|))) (-21)) (T -20)) 
+(-13 (-376 |t#1|) (-10 -7 (-6 -4435))) 
+(((-34) . T) ((-102) -3969 (|has| |#1| (-1107)) (|has| |#1| (-855))) ((-618 (-868)) -3969 (|has| |#1| (-1107)) (|has| |#1| (-855)) (|has| |#1| (-618 (-868)))) ((-151 |#1|) . T) ((-619 (-540)) |has| |#1| (-619 (-540))) ((-289 #1=(-551) |#1|) . T) ((-291 #1# |#1|) . T) ((-312 |#1|) -12 (|has| |#1| (-312 |#1|)) (|has| |#1| (-1107))) ((-376 |#1|) . T) ((-494 |#1|) . T) ((-609 #1# |#1|) . T) ((-519 |#1| |#1|) -12 (|has| |#1| (-312 |#1|)) (|has| |#1| (-1107))) ((-656 |#1|) . T) ((-855) |has| |#1| (-855)) ((-1107) -3969 (|has| |#1| (-1107)) (|has| |#1| (-855))) ((-1222) . T)) 
+((-1410 (((-3 $ "failed") $ $) 12)) (-4278 (($ $) NIL) (($ $ $) 9)) (* (($ (-925) $) NIL) (($ (-776) $) 16) (($ (-551) $) 26))) 
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 NIL 
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-NIL 
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-NIL 
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-NIL 
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+((-1359 (*1 *2 *3 *3) (-12 (-5 *3 (-1165)) (-5 *2 (-382)) (-5 *1 (-97)))) (-1359 (*1 *2 *3) (-12 (-5 *3 (-1165)) (-5 *2 (-382)) (-5 *1 (-97)))) (-1358 (*1 *2 *3 *3) (-12 (-5 *3 (-1165)) (-5 *2 (-382)) (-5 *1 (-97)))) (-1358 (*1 *2 *3) (-12 (-5 *3 (-1165)) (-5 *2 (-382)) (-5 *1 (-97)))) (-1357 (*1 *2 *3 *3) (-12 (-5 *3 (-1165)) (-5 *2 (-382)) (-5 *1 (-97)))) (-1356 (*1 *2) (-12 (-5 *2 (-1278)) (-5 *1 (-97)))) (-1355 (*1 *2 *2) (-12 (-5 *2 (-382)) (-5 *1 (-97)))) (-1354 (*1 *2 *3 *2) (-12 (-5 *2 (-382)) (-5 *3 (-1165)) (-5 *1 (-97)))) (-1354 (*1 *2 *3 *3 *2) (-12 (-5 *2 (-382)) (-5 *3 (-1165)) (-5 *1 (-97))))) 
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 NIL 
 (((-98) (-140)) (T -98)) 
 NIL 
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-((-1359 (((-409 |#2|) |#2| (-644 |#2|)) 10) (((-409 |#2|) |#2| |#2|) 11))) 
-(((-100 |#1| |#2|) (-10 -7 (-15 -1359 ((-409 |#2|) |#2| |#2|)) (-15 -1359 ((-409 |#2|) |#2| (-644 |#2|)))) (-13 (-456) (-147)) (-1246 |#1|)) (T -100)) 
-((-1359 (*1 *2 *3 *4) (-12 (-5 *4 (-644 *3)) (-4 *3 (-1246 *5)) (-4 *5 (-13 (-456) (-147))) (-5 *2 (-409 *3)) (-5 *1 (-100 *5 *3)))) (-1359 (*1 *2 *3 *3) (-12 (-4 *4 (-13 (-456) (-147))) (-5 *2 (-409 *3)) (-5 *1 (-100 *4 *3)) (-4 *3 (-1246 *4))))) 
-(-10 -7 (-15 -1359 ((-409 |#2|) |#2| |#2|)) (-15 -1359 ((-409 |#2|) |#2| (-644 |#2|)))) 
-((-2970 (((-112) $ $) 10))) 
-(((-101 |#1|) (-10 -8 (-15 -2970 ((-112) |#1| |#1|))) (-102)) (T -101)) 
-NIL 
-(-10 -8 (-15 -2970 ((-112) |#1| |#1|))) 
-((-2970 (((-112) $ $) 7)) (-3457 (((-112) $ $) 6))) 
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+(((-99 |#1|) (-13 (-478) (-289 |#1| |#1|) (-10 -8 (-15 -1360 ($ (-1 |#1| |#1|))) (-15 -1360 ($ (-1 |#1| |#1|) (-1 |#1| |#1|))) (-15 -1360 ($ (-1 |#1| |#1| (-551)))))) (-1055)) (T -99)) 
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+(-13 (-478) (-289 |#1| |#1|) (-10 -8 (-15 -1360 ($ (-1 |#1| |#1|))) (-15 -1360 ($ (-1 |#1| |#1|) (-1 |#1| |#1|))) (-15 -1360 ($ (-1 |#1| |#1| (-551)))))) 
+((-1361 (((-410 |#2|) |#2| (-646 |#2|)) 10) (((-410 |#2|) |#2| |#2|) 11))) 
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+((-1361 (*1 *2 *3 *4) (-12 (-5 *4 (-646 *3)) (-4 *3 (-1248 *5)) (-4 *5 (-13 (-457) (-147))) (-5 *2 (-410 *3)) (-5 *1 (-100 *5 *3)))) (-1361 (*1 *2 *3 *3) (-12 (-4 *4 (-13 (-457) (-147))) (-5 *2 (-410 *3)) (-5 *1 (-100 *4 *3)) (-4 *3 (-1248 *4))))) 
+(-10 -7 (-15 -1361 ((-410 |#2|) |#2| |#2|)) (-15 -1361 ((-410 |#2|) |#2| (-646 |#2|)))) 
+((-2977 (((-112) $ $) 10))) 
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+NIL 
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+((-2977 (((-112) $ $) 7)) (-3464 (((-112) $ $) 6))) 
 (((-102) (-140)) (T -102)) 
-((-2970 (*1 *2 *1 *1) (-12 (-4 *1 (-102)) (-5 *2 (-112)))) (-3457 (*1 *2 *1 *1) (-12 (-4 *1 (-102)) (-5 *2 (-112))))) 
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-((-1364 ((|#3| |#2| |#2|) 36)) (-1366 ((|#1| |#2| |#2|) 53 (|has| |#1| (-6 (-4429 #1="*"))))) (-1365 ((|#3| |#2| |#2|) 38)) (-1367 ((|#1| |#2|) 58 (|has| |#1| (-6 (-4429 #1#)))))) 
-(((-104 |#1| |#2| |#3| |#4| |#5|) (-10 -7 (-15 -1364 (|#3| |#2| |#2|)) (-15 -1365 (|#3| |#2| |#2|)) (IF (|has| |#1| (-6 (-4429 "*"))) (PROGN (-15 -1366 (|#1| |#2| |#2|)) (-15 -1367 (|#1| |#2|))) |%noBranch|)) (-1053) (-1246 |#1|) (-689 |#1| |#4| |#5|) (-375 |#1|) (-375 |#1|)) (T -104)) 
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-(-10 -7 (-15 -1364 (|#3| |#2| |#2|)) (-15 -1365 (|#3| |#2| |#2|)) (IF (|has| |#1| (-6 (-4429 "*"))) (PROGN (-15 -1366 (|#1| |#2| |#2|)) (-15 -1367 (|#1| |#2|))) |%noBranch|)) 
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-(((-105) (-13 (-1105) (-10 -7 (-15 -1369 ((-644 (-1181)))) (-15 -1368 ((-2 (|:| |zeros| (-1158 (-226))) (|:| |ones| (-1158 (-226))) (|:| |singularities| (-1158 (-226)))) (-1181))) (-6 -4427)))) (T -105)) 
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-(-13 (-1105) (-10 -7 (-15 -1369 ((-644 (-1181)))) (-15 -1368 ((-2 (|:| |zeros| (-1158 (-226))) (|:| |ones| (-1158 (-226))) (|:| |singularities| (-1158 (-226)))) (-1181))) (-6 -4427))) 
-((-1372 (($ (-644 |#2|)) 11))) 
-(((-106 |#1| |#2|) (-10 -8 (-15 -1372 (|#1| (-644 |#2|)))) (-107 |#2|) (-1220)) (T -106)) 
-NIL 
-(-10 -8 (-15 -1372 (|#1| (-644 |#2|)))) 
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-(((-107 |#1|) (-140) (-1220)) (T -107)) 
-((-1372 (*1 *1 *2) (-12 (-5 *2 (-644 *3)) (-4 *3 (-1220)) (-4 *1 (-107 *3)))) (-1371 (*1 *2 *1) (-12 (-4 *1 (-107 *2)) (-4 *2 (-1220)))) (-4041 (*1 *1 *2 *1) (-12 (-4 *1 (-107 *2)) (-4 *2 (-1220)))) (-1370 (*1 *2 *1) (-12 (-4 *1 (-107 *2)) (-4 *2 (-1220))))) 
-(-13 (-493 |t#1|) (-10 -8 (-6 -4428) (-15 -1372 ($ (-644 |t#1|))) (-15 -1371 (|t#1| $)) (-15 -4041 ($ |t#1| $)) (-15 -1370 (|t#1| $)))) 
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-NIL 
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-(((-193) (-10 -7 (-15 -1529 ((-3 (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated")) (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226))) (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226)) (|:| |relerr| (-226))))) (-15 -1530 ((-3 (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated")) (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226))) (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226)) (|:| |relerr| (-226))))) (-15 -1740 ((-226) (-1093 (-845 (-226))))) (-15 -1741 ((-226) (-1093 (-845 (-226))))) (-15 -1531 ((-381) (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226))) (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226)) (|:| |relerr| (-226))))) (-15 -1716 ((-644 (-226)) (-316 (-226)) (-1181) (-1093 (-845 (-226))))) (-15 -1718 ((-1158 (-226)) (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226))) (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226)) (|:| |relerr| (-226))))) (-15 -1736 ((-1163) (-226))) (-15 -1738 ((-644 (-1163)) (-644 (-226)))) (-15 -1739 ((-644 (-1163)) (-1158 (-226)))) (-15 -2973 ((-1163) (-644 (-1163)))) (-15 -1532 ((-1039) (-1181) (-1181) (-1039))))) (T -193)) 
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-(((-194) (-790)) (T -194)) 
-NIL 
-(-790) 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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+NIL 
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+(((-230 |#1|) (-140) (-1107)) (T -230)) 
 NIL 
 (-13 (-236 |t#1|)) 
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-NIL 
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-(((-232 |#1|) (-140) (-1053)) (T -232)) 
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-(((-21) . T) ((-23) . T) ((-25) . T) ((-102) . T) ((-131) . T) ((-619 (-550)) . T) ((-616 (-866)) . T) ((-234) |has| |#1| (-234)) ((-649 (-550)) . T) ((-649 $) . T) ((-651 $) . T) ((-729) . T) ((-904 (-1181)) |has| |#1| (-904 (-1181))) ((-1053) . T) ((-1061) . T) ((-1116) . T) ((-1105) . T)) 
-((-4244 (($ $) NIL) (($ $ (-774)) 13)) (-3074 (($ $) 8) (($ $ (-774)) 15))) 
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-NIL 
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+NIL 
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+NIL 
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-NIL 
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+NIL 
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 (((-244) (-140)) (T -244)) 
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-(-13 (-292) (-38 (-411 (-550))) (-10 -8 (-15 ** ($ $ (-550))) (-15 -2808 ($ $)))) 
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-NIL 
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-NIL 
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-NIL 
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-(((-102) . T) ((-619 |#1|) . T) ((-616 (-866)) . T) ((-853) . T) ((-1042 |#1|) . T) ((-1105) . T)) 
-((-3487 (((-644 (-1181)) (-2 (|:| |lfn| (-644 (-316 (-226)))) (|:| -3871 (-644 (-226))))) 54)) (-4368 (((-644 (-1181)) (-316 (-226)) (-774)) 96)) (-1629 (((-3 (-316 (-226)) "failed") (-316 (-226))) 64)) (-1630 (((-316 (-226)) (-316 (-226))) 82)) (-1628 (((-2 (|:| |fn| (-316 (-226))) (|:| -3871 (-644 (-226))) (|:| |lb| (-644 (-845 (-226)))) (|:| |cf| (-644 (-316 (-226)))) (|:| |ub| (-644 (-845 (-226))))) (-2 (|:| |fn| (-316 (-226))) (|:| -3871 (-644 (-226))) (|:| |lb| (-644 (-845 (-226)))) (|:| |cf| (-644 (-316 (-226)))) (|:| |ub| (-644 (-845 (-226)))))) 39)) (-1631 (((-112) (-644 (-316 (-226)))) 106)) (-1635 (((-112) (-316 (-226))) 37)) (-1637 (((-644 (-1163)) (-3 (|:| |noa| (-2 (|:| |fn| (-316 (-226))) (|:| -3871 (-644 (-226))) (|:| |lb| (-644 (-845 (-226)))) (|:| |cf| (-644 (-316 (-226)))) (|:| |ub| (-644 (-845 (-226)))))) (|:| |lsa| (-2 (|:| |lfn| (-644 (-316 (-226)))) (|:| -3871 (-644 (-226))))))) 134)) (-1634 (((-644 (-316 (-226))) (-644 (-316 (-226)))) 110)) (-1633 (((-644 (-316 (-226))) (-644 (-316 (-226)))) 108)) (-1632 (((-692 (-226)) (-644 (-316 (-226))) (-774)) 122)) (-3330 (((-112) (-316 (-226))) 32) (((-112) (-644 (-316 (-226)))) 107)) (-1627 (((-644 (-226)) (-644 (-845 (-226))) (-226)) 15)) (-1727 (((-381) (-2 (|:| |lfn| (-644 (-316 (-226)))) (|:| -3871 (-644 (-226))))) 128)) (-1636 (((-1039) (-1181) (-1039)) 47))) 
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-(((-270) (-842)) (T -270)) 
-NIL 
-(-842) 
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-(((-271) (-842)) (T -271)) 
-NIL 
-(-842) 
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-(((-272) (-842)) (T -272)) 
-NIL 
-(-842) 
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-(((-273) (-842)) (T -273)) 
-NIL 
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-NIL 
-(-842) 
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-(((-275) (-842)) (T -275)) 
-NIL 
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-(((-276) (-842)) (T -276)) 
-NIL 
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 NIL 
 (-10 -8 (-15 ** (|#1| |#1| |#1|))) 
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-NIL 
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-(((-292) (-140)) (T -292)) 
-NIL 
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-(((-21) . T) ((-23) . T) ((-25) . T) ((-102) . T) ((-111 $ $) . T) ((-131) . T) ((-619 (-550)) . T) ((-616 (-866)) . T) ((-649 (-550)) . T) ((-649 $) . T) ((-651 $) . T) ((-729) . T) ((-1055 $) . T) ((-1060 $) . T) ((-1053) . T) ((-1061) . T) ((-1116) . T) ((-1105) . T)) 
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-NIL 
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-(((-102) . T) ((-616 (-866)) . T) ((-1105) . T)) 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
-(-1215 |#1| |#2| |#3| |#4|) 
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-(((-481) (-13 (-366) (-147) (-1042 (-550)) (-1042 (-411 (-550))) (-1024) (-616 (-226)) (-616 (-381)) (-617 (-411 (-1175 (-550)))) (-617 (-539)) (-10 -8 (-15 -4061 ($))))) (T -481)) 
-((-4061 (*1 *1) (-5 *1 (-481)))) 
-(-13 (-366) (-147) (-1042 (-550)) (-1042 (-411 (-550))) (-1024) (-616 (-226)) (-616 (-381)) (-617 (-411 (-1175 (-550)))) (-617 (-539)) (-10 -8 (-15 -4061 ($)))) 
-((-2970 (((-112) $ $) NIL)) (-3953 (((-1139) $) 11)) (-3954 (((-1139) $) 9)) (-3665 (((-1163) $) NIL)) (-3666 (((-1124) $) NIL)) (-4380 (((-866) $) 17) (($ (-1186)) NIL) (((-1186) $) NIL)) (-3664 (((-112) $ $) NIL)) (-3457 (((-112) $ $) NIL))) 
-(((-482) (-13 (-1087) (-10 -8 (-15 -3954 ((-1139) $)) (-15 -3953 ((-1139) $))))) (T -482)) 
-((-3954 (*1 *2 *1) (-12 (-5 *2 (-1139)) (-5 *1 (-482)))) (-3953 (*1 *2 *1) (-12 (-5 *2 (-1139)) (-5 *1 (-482))))) 
-(-13 (-1087) (-10 -8 (-15 -3954 ((-1139) $)) (-15 -3953 ((-1139) $)))) 
-((-2970 (((-112) $ $) NIL (-3962 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)) (|has| |#2| (-1105))))) (-4031 (($) NIL) (($ (-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) NIL)) (-2374 (((-1276) $ |#1| |#1|) NIL (|has| $ (-6 -4428)))) (-1310 (((-112) $ (-774)) NIL)) (-4221 ((|#2| $ |#1| |#2|) 16)) (-1680 (($ (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427)))) (-4144 (($ (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427)))) (-2383 (((-3 |#2| #1="failed") |#1| $) 20)) (-4158 (($) NIL T CONST)) (-1441 (($ $) NIL (-12 (|has| $ (-6 -4427)) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105))))) (-3831 (($ (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) $) NIL (|has| $ (-6 -4427))) (($ (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427))) (((-3 |#2| #1#) |#1| $) 18)) (-3832 (($ (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) $) NIL (-12 (|has| $ (-6 -4427)) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (($ (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427)))) (-4276 (((-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $ (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) NIL (-12 (|has| $ (-6 -4427)) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (((-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $ (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) NIL (|has| $ (-6 -4427))) (((-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427)))) (-1686 ((|#2| $ |#1| |#2|) NIL (|has| $ (-6 -4428)))) (-3519 ((|#2| $ |#1|) NIL)) (-2126 (((-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427))) (((-644 |#2|) $) NIL (|has| $ (-6 -4427)))) (-4153 (((-112) $ (-774)) NIL)) (-2376 ((|#1| $) NIL (|has| |#1| (-853)))) (-3010 (((-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427))) (((-644 |#2|) $) NIL (|has| $ (-6 -4427)))) (-3668 (((-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) $) NIL (-12 (|has| $ (-6 -4427)) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (((-112) |#2| $) NIL (-12 (|has| $ (-6 -4427)) (|has| |#2| (-1105))))) (-2377 ((|#1| $) NIL (|has| |#1| (-853)))) (-2130 (($ (-1 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4428))) (($ (-1 |#2| |#2|) $) NIL (|has| $ (-6 -4428)))) (-4392 (($ (-1 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL) (($ (-1 |#2| |#2|) $) NIL) (($ (-1 |#2| |#2| |#2|) $ $) NIL)) (-4150 (((-112) $ (-774)) NIL)) (-3665 (((-1163) $) NIL (-3962 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)) (|has| |#2| (-1105))))) (-2818 (((-644 |#1|) $) 13)) (-2384 (((-112) |#1| $) NIL)) (-1370 (((-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) $) NIL)) (-4041 (($ (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) $) NIL)) (-2379 (((-644 |#1|) $) NIL)) (-2380 (((-112) |#1| $) NIL)) (-3666 (((-1124) $) NIL (-3962 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)) (|has| |#2| (-1105))))) (-4234 ((|#2| $) NIL (|has| |#1| (-853)))) (-1442 (((-3 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) "failed") (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL)) (-2375 (($ $ |#2|) NIL (|has| $ (-6 -4428)))) (-1371 (((-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) $) NIL)) (-2128 (((-112) (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427))) (((-112) (-1 (-112) |#2|) $) NIL (|has| $ (-6 -4427)))) (-4201 (($ $ (-644 (-295 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))))) NIL (-12 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-311 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (($ $ (-295 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) NIL (-12 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-311 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (($ $ (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) NIL (-12 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-311 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (($ $ (-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) (-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) NIL (-12 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-311 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (($ $ (-644 |#2|) (-644 |#2|)) NIL (-12 (|has| |#2| (-311 |#2|)) (|has| |#2| (-1105)))) (($ $ |#2| |#2|) NIL (-12 (|has| |#2| (-311 |#2|)) (|has| |#2| (-1105)))) (($ $ (-295 |#2|)) NIL (-12 (|has| |#2| (-311 |#2|)) (|has| |#2| (-1105)))) (($ $ (-644 (-295 |#2|))) NIL (-12 (|has| |#2| (-311 |#2|)) (|has| |#2| (-1105))))) (-1311 (((-112) $ $) NIL)) (-2378 (((-112) |#2| $) NIL (-12 (|has| $ (-6 -4427)) (|has| |#2| (-1105))))) (-2381 (((-644 |#2|) $) NIL)) (-3829 (((-112) $) NIL)) (-3998 (($) 19)) (-4233 ((|#2| $ |#1|) NIL) ((|#2| $ |#1| |#2|) NIL)) (-1569 (($) NIL) (($ (-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) NIL)) (-2127 (((-774) (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427))) (((-774) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) $) NIL (-12 (|has| $ (-6 -4427)) (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)))) (((-774) |#2| $) NIL (-12 (|has| $ (-6 -4427)) (|has| |#2| (-1105)))) (((-774) (-1 (-112) |#2|) $) NIL (|has| $ (-6 -4427)))) (-3826 (($ $) NIL)) (-4404 (((-539) $) NIL (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-617 (-539))))) (-3955 (($ (-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) NIL)) (-4380 (((-866) $) NIL (-3962 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-616 (-866))) (|has| |#2| (-616 (-866)))))) (-3664 (((-112) $ $) NIL (-3962 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)) (|has| |#2| (-1105))))) (-1372 (($ (-644 (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)))) NIL)) (-2129 (((-112) (-1 (-112) (-2 (|:| -4294 |#1|) (|:| -2256 |#2|))) $) NIL (|has| $ (-6 -4427))) (((-112) (-1 (-112) |#2|) $) NIL (|has| $ (-6 -4427)))) (-3457 (((-112) $ $) 11 (-3962 (|has| (-2 (|:| -4294 |#1|) (|:| -2256 |#2|)) (-1105)) (|has| |#2| (-1105))))) (-4391 (((-774) $) 15 (|has| $ (-6 -4427))))) 
-(((-483 |#1| |#2| |#3|) (-13 (-1197 |#1| |#2|) (-10 -7 (-6 -4427))) (-1105) (-1105) (-1163)) (T -483)) 
-NIL 
-(-13 (-1197 |#1| |#2|) (-10 -7 (-6 -4427))) 
-((-2115 (((-550) (-550) (-550)) 19)) (-2116 (((-112) (-550) (-550) (-550) (-550)) 28)) (-3882 (((-1270 (-644 (-550))) (-774) (-774)) 44))) 
-(((-484) (-10 -7 (-15 -2115 ((-550) (-550) (-550))) (-15 -2116 ((-112) (-550) (-550) (-550) (-550))) (-15 -3882 ((-1270 (-644 (-550))) (-774) (-774))))) (T -484)) 
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+NIL 
+(-1217 |#1| |#2| |#3| |#4|) 
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+((-4068 (*1 *1) (-5 *1 (-482)))) 
+(-13 (-367) (-147) (-1044 (-551)) (-1044 (-412 (-551))) (-1026) (-618 (-226)) (-618 (-382)) (-619 (-412 (-1177 (-551)))) (-619 (-540)) (-10 -8 (-15 -4068 ($)))) 
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+(((-483) (-13 (-1089) (-10 -8 (-15 -3961 ((-1141) $)) (-15 -3960 ((-1141) $))))) (T -483)) 
+((-3961 (*1 *2 *1) (-12 (-5 *2 (-1141)) (-5 *1 (-483)))) (-3960 (*1 *2 *1) (-12 (-5 *2 (-1141)) (-5 *1 (-483))))) 
+(-13 (-1089) (-10 -8 (-15 -3961 ((-1141) $)) (-15 -3960 ((-1141) $)))) 
+((-2977 (((-112) $ $) NIL (-3969 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)) (|has| |#2| (-1107))))) (-4038 (($) NIL) (($ (-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) NIL)) (-2381 (((-1278) $ |#1| |#1|) NIL (|has| $ (-6 -4435)))) (-1312 (((-112) $ (-776)) NIL)) (-4228 ((|#2| $ |#1| |#2|) 16)) (-1687 (($ (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434)))) (-4151 (($ (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434)))) (-2390 (((-3 |#2| #1="failed") |#1| $) 20)) (-4165 (($) NIL T CONST)) (-1443 (($ $) NIL (-12 (|has| $ (-6 -4434)) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107))))) (-3838 (($ (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) $) NIL (|has| $ (-6 -4434))) (($ (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434))) (((-3 |#2| #1#) |#1| $) 18)) (-3839 (($ (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) $) NIL (-12 (|has| $ (-6 -4434)) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (($ (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434)))) (-4283 (((-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $ (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) NIL (-12 (|has| $ (-6 -4434)) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (((-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $ (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) NIL (|has| $ (-6 -4434))) (((-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434)))) (-1693 ((|#2| $ |#1| |#2|) NIL (|has| $ (-6 -4435)))) (-3526 ((|#2| $ |#1|) NIL)) (-2133 (((-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434))) (((-646 |#2|) $) NIL (|has| $ (-6 -4434)))) (-4160 (((-112) $ (-776)) NIL)) (-2383 ((|#1| $) NIL (|has| |#1| (-855)))) (-3017 (((-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434))) (((-646 |#2|) $) NIL (|has| $ (-6 -4434)))) (-3675 (((-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) $) NIL (-12 (|has| $ (-6 -4434)) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (((-112) |#2| $) NIL (-12 (|has| $ (-6 -4434)) (|has| |#2| (-1107))))) (-2384 ((|#1| $) NIL (|has| |#1| (-855)))) (-2137 (($ (-1 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4435))) (($ (-1 |#2| |#2|) $) NIL (|has| $ (-6 -4435)))) (-4399 (($ (-1 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL) (($ (-1 |#2| |#2|) $) NIL) (($ (-1 |#2| |#2| |#2|) $ $) NIL)) (-4157 (((-112) $ (-776)) NIL)) (-3672 (((-1165) $) NIL (-3969 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)) (|has| |#2| (-1107))))) (-2825 (((-646 |#1|) $) 13)) (-2391 (((-112) |#1| $) NIL)) (-1372 (((-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) $) NIL)) (-4048 (($ (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) $) NIL)) (-2386 (((-646 |#1|) $) NIL)) (-2387 (((-112) |#1| $) NIL)) (-3673 (((-1126) $) NIL (-3969 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)) (|has| |#2| (-1107))))) (-4241 ((|#2| $) NIL (|has| |#1| (-855)))) (-1444 (((-3 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) "failed") (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL)) (-2382 (($ $ |#2|) NIL (|has| $ (-6 -4435)))) (-1373 (((-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) $) NIL)) (-2135 (((-112) (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434))) (((-112) (-1 (-112) |#2|) $) NIL (|has| $ (-6 -4434)))) (-4208 (($ $ (-646 (-296 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))))) NIL (-12 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-312 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (($ $ (-296 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) NIL (-12 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-312 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (($ $ (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) NIL (-12 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-312 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (($ $ (-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) (-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) NIL (-12 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-312 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (($ $ (-646 |#2|) (-646 |#2|)) NIL (-12 (|has| |#2| (-312 |#2|)) (|has| |#2| (-1107)))) (($ $ |#2| |#2|) NIL (-12 (|has| |#2| (-312 |#2|)) (|has| |#2| (-1107)))) (($ $ (-296 |#2|)) NIL (-12 (|has| |#2| (-312 |#2|)) (|has| |#2| (-1107)))) (($ $ (-646 (-296 |#2|))) NIL (-12 (|has| |#2| (-312 |#2|)) (|has| |#2| (-1107))))) (-1313 (((-112) $ $) NIL)) (-2385 (((-112) |#2| $) NIL (-12 (|has| $ (-6 -4434)) (|has| |#2| (-1107))))) (-2388 (((-646 |#2|) $) NIL)) (-3836 (((-112) $) NIL)) (-4005 (($) 19)) (-4240 ((|#2| $ |#1|) NIL) ((|#2| $ |#1| |#2|) NIL)) (-1572 (($) NIL) (($ (-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) NIL)) (-2134 (((-776) (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434))) (((-776) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) $) NIL (-12 (|has| $ (-6 -4434)) (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)))) (((-776) |#2| $) NIL (-12 (|has| $ (-6 -4434)) (|has| |#2| (-1107)))) (((-776) (-1 (-112) |#2|) $) NIL (|has| $ (-6 -4434)))) (-3833 (($ $) NIL)) (-4411 (((-540) $) NIL (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-619 (-540))))) (-3962 (($ (-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) NIL)) (-4387 (((-868) $) NIL (-3969 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-618 (-868))) (|has| |#2| (-618 (-868)))))) (-3671 (((-112) $ $) NIL (-3969 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)) (|has| |#2| (-1107))))) (-1374 (($ (-646 (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)))) NIL)) (-2136 (((-112) (-1 (-112) (-2 (|:| -4301 |#1|) (|:| -2263 |#2|))) $) NIL (|has| $ (-6 -4434))) (((-112) (-1 (-112) |#2|) $) NIL (|has| $ (-6 -4434)))) (-3464 (((-112) $ $) 11 (-3969 (|has| (-2 (|:| -4301 |#1|) (|:| -2263 |#2|)) (-1107)) (|has| |#2| (-1107))))) (-4398 (((-776) $) 15 (|has| $ (-6 -4434))))) 
+(((-484 |#1| |#2| |#3|) (-13 (-1199 |#1| |#2|) (-10 -7 (-6 -4434))) (-1107) (-1107) (-1165)) (T -484)) 
+NIL 
+(-13 (-1199 |#1| |#2|) (-10 -7 (-6 -4434))) 
+((-2122 (((-551) (-551) (-551)) 19)) (-2123 (((-112) (-551) (-551) (-551) (-551)) 28)) (-3889 (((-1272 (-646 (-551))) (-776) (-776)) 44))) 
+(((-485) (-10 -7 (-15 -2122 ((-551) (-551) (-551))) (-15 -2123 ((-112) (-551) (-551) (-551) (-551))) (-15 -3889 ((-1272 (-646 (-551))) (-776) (-776))))) (T -485)) 
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 NIL 
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-NIL 
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-NIL 
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 NIL 
 (-19 |#1|) 
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-NIL 
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-NIL 
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+NIL 
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 NIL 
 (-57 |#1| |#4| |#5|) 
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 (((-174) . T)) 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-NIL 
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-(((-765 |#1|) (-10 -8 (-15 -4380 (|#1| (-550))) (-15 -4380 ((-866) |#1|))) (-766)) (T -765)) 
-NIL 
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+NIL 
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 (((-797) (-140)) (T -797)) 
 NIL 
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-(((-800) (-140)) (T -800)) 
-NIL 
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 NIL 
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 NIL 
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-NIL 
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 NIL 
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 NIL 
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-NIL 
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-NIL 
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-(((-860) (-140)) (T -860)) 
-NIL 
-(-13 (-853) (-1116)) 
-(((-102) . T) ((-616 (-866)) . T) ((-853) . T) ((-1116) . T) ((-1105) . T)) 
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-(((-861) (-13 (-853) (-10 -8 (-15 -4380 ($ (-550))) (-15 -3828 ((-550) $))))) (T -861)) 
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-(-10 -7 (-15 -3885 ((-1276) (-1163))) (-15 -3885 ((-1276) (-866))) (-15 -3885 ((-1276) (-1163) (-866))) (-15 -2958 ((-1276) (-644 (-51))))) 
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-NIL 
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+NIL 
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+(((-855) (-140)) (T -855)) 
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+(((-102) . T) ((-618 (-868)) . T) ((-1107) . T)) 
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+NIL 
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-NIL 
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-NIL 
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+NIL 
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 NIL 
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-NIL 
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-NIL 
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-((-2970 (((-112) $ $) 7)) (-3656 (($ $ $) 19) (($ $ |#1|) 18) (($ |#1| $) 17)) (-3658 (($ $ $) 21)) (-3657 (((-112) $ $) 20)) (-1310 (((-112) $ (-774)) 36)) (-3661 (($) 26) (($ (-644 |#1|)) 25)) (-4144 (($ (-1 (-112) |#1|) $) 57 (|has| $ (-6 -4427)))) (-4158 (($) 37 T CONST)) (-1441 (($ $) 60 (-12 (|has| |#1| (-1105)) (|has| $ (-6 -4427))))) (-3832 (($ |#1| $) 59 (-12 (|has| |#1| (-1105)) (|has| $ (-6 -4427)))) (($ (-1 (-112) |#1|) $) 56 (|has| $ (-6 -4427)))) (-4276 ((|#1| (-1 |#1| |#1| |#1|) $ |#1| |#1|) 58 (-12 (|has| |#1| (-1105)) (|has| $ (-6 -4427)))) ((|#1| (-1 |#1| |#1| |#1|) $ |#1|) 55 (|has| $ (-6 -4427))) ((|#1| (-1 |#1| |#1| |#1|) $) 54 (|has| $ (-6 -4427)))) (-2126 (((-644 |#1|) $) 44 (|has| $ (-6 -4427)))) (-3663 (((-112) $ $) 29)) (-4153 (((-112) $ (-774)) 35)) (-3010 (((-644 |#1|) $) 45 (|has| $ (-6 -4427)))) (-3668 (((-112) |#1| $) 47 (-12 (|has| |#1| (-1105)) (|has| $ (-6 -4427))))) (-2130 (($ (-1 |#1| |#1|) $) 40 (|has| $ (-6 -4428)))) (-4392 (($ (-1 |#1| |#1|) $) 39)) (-4150 (((-112) $ (-774)) 34)) (-3665 (((-1163) $) 10)) (-3660 (($ $ $) 24)) (-3666 (((-1124) $) 11)) (-1442 (((-3 |#1| "failed") (-1 (-112) |#1|) $) 53)) (-2128 (((-112) (-1 (-112) |#1|) $) 42 (|has| $ (-6 -4427)))) (-4201 (($ $ (-644 |#1|) (-644 |#1|)) 51 (-12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105)))) (($ $ |#1| |#1|) 50 (-12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105)))) (($ $ (-295 |#1|)) 49 (-12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105)))) (($ $ (-644 (-295 |#1|))) 48 (-12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105))))) (-1311 (((-112) $ $) 30)) (-3829 (((-112) $) 33)) (-3998 (($) 32)) (-3659 (($ $ $) 23) (($ $ |#1|) 22)) (-2127 (((-774) |#1| $) 46 (-12 (|has| |#1| (-1105)) (|has| $ (-6 -4427)))) (((-774) (-1 (-112) |#1|) $) 43 (|has| $ (-6 -4427)))) (-3826 (($ $) 31)) (-4404 (((-539) $) 61 (|has| |#1| (-617 (-539))))) (-3955 (($ (-644 |#1|)) 52)) (-4380 (((-866) $) 12)) (-3662 (($) 28) (($ (-644 |#1|)) 27)) (-3664 (((-112) $ $) 9)) (-2129 (((-112) (-1 (-112) |#1|) $) 41 (|has| $ (-6 -4427)))) (-3457 (((-112) $ $) 6)) (-4391 (((-774) $) 38 (|has| $ (-6 -4427))))) 
-(((-1103 |#1|) (-140) (-1105)) (T -1103)) 
-((-3663 (*1 *2 *1 *1) (-12 (-4 *1 (-1103 *3)) (-4 *3 (-1105)) (-5 *2 (-112)))) (-3662 (*1 *1) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3662 (*1 *1 *2) (-12 (-5 *2 (-644 *3)) (-4 *3 (-1105)) (-4 *1 (-1103 *3)))) (-3661 (*1 *1) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3661 (*1 *1 *2) (-12 (-5 *2 (-644 *3)) (-4 *3 (-1105)) (-4 *1 (-1103 *3)))) (-3660 (*1 *1 *1 *1) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3659 (*1 *1 *1 *1) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3659 (*1 *1 *1 *2) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3658 (*1 *1 *1 *1) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3657 (*1 *2 *1 *1) (-12 (-4 *1 (-1103 *3)) (-4 *3 (-1105)) (-5 *2 (-112)))) (-3656 (*1 *1 *1 *1) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3656 (*1 *1 *1 *2) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105)))) (-3656 (*1 *1 *2 *1) (-12 (-4 *1 (-1103 *2)) (-4 *2 (-1105))))) 
-(-13 (-1105) (-151 |t#1|) (-10 -8 (-6 -4417) (-15 -3663 ((-112) $ $)) (-15 -3662 ($)) (-15 -3662 ($ (-644 |t#1|))) (-15 -3661 ($)) (-15 -3661 ($ (-644 |t#1|))) (-15 -3660 ($ $ $)) (-15 -3659 ($ $ $)) (-15 -3659 ($ $ |t#1|)) (-15 -3658 ($ $ $)) (-15 -3657 ((-112) $ $)) (-15 -3656 ($ $ $)) (-15 -3656 ($ $ |t#1|)) (-15 -3656 ($ |t#1| $)))) 
-(((-34) . T) ((-102) . T) ((-616 (-866)) . T) ((-151 |#1|) . T) ((-617 (-539)) |has| |#1| (-617 (-539))) ((-311 |#1|) -12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105))) ((-493 |#1|) . T) ((-518 |#1| |#1|) -12 (|has| |#1| (-311 |#1|)) (|has| |#1| (-1105))) ((-1105) . T) ((-1220) . T)) 
-((-3665 (((-1163) $) 10)) (-3666 (((-1124) $) 8)) (-3664 (((-112) $ $) 12))) 
-(((-1104 |#1|) (-10 -8 (-15 -3664 ((-112) |#1| |#1|)) (-15 -3665 ((-1163) |#1|)) (-15 -3666 ((-1124) |#1|))) (-1105)) (T -1104)) 
-NIL 
-(-10 -8 (-15 -3664 ((-112) |#1| |#1|)) (-15 -3665 ((-1163) |#1|)) (-15 -3666 ((-1124) |#1|))) 
-((-2970 (((-112) $ $) 7)) (-3665 (((-1163) $) 10)) (-3666 (((-1124) $) 11)) (-4380 (((-866) $) 12)) (-3664 (((-112) $ $) 9)) (-3457 (((-112) $ $) 6))) 
-(((-1105) (-140)) (T -1105)) 
-((-3666 (*1 *2 *1) (-12 (-4 *1 (-1105)) (-5 *2 (-1124)))) (-3665 (*1 *2 *1) (-12 (-4 *1 (-1105)) (-5 *2 (-1163)))) (-3664 (*1 *2 *1 *1) (-12 (-4 *1 (-1105)) (-5 *2 (-112))))) 
-(-13 (-102) (-616 (-866)) (-10 -8 (-15 -3666 ((-1124) $)) (-15 -3665 ((-1163) $)) (-15 -3664 ((-112) $ $)))) 
-(((-102) . T) ((-616 (-866)) . T)) 
-((-2970 (((-112) $ $) NIL)) (-3542 (((-774)) 36)) (-3670 (($ (-644 (-923))) 72)) (-3672 (((-3 $ #1="failed") $ (-923) (-923)) 83)) (-3397 (($) 40)) (-3668 (((-112) (-923) $) 44)) (-2190 (((-923) $) 66)) (-3665 (((-1163) $) NIL)) (-2565 (($ (-923)) 39)) (-3673 (((-3 $ #1#) $ (-923)) 79)) (-3666 (((-1124) $) NIL)) (-3669 (((-1270 $)) 49)) (-3671 (((-644 (-923)) $) 27)) (-3667 (((-774) $ (-923) (-923)) 80)) (-4380 (((-866) $) 32)) (-3664 (((-112) $ $) NIL)) (-3457 (((-112) $ $) 24))) 
-(((-1106 |#1| |#2|) (-13 (-371) (-10 -8 (-15 -3673 ((-3 $ #1="failed") $ (-923))) (-15 -3672 ((-3 $ #1#) $ (-923) (-923))) (-15 -3671 ((-644 (-923)) $)) (-15 -3670 ($ (-644 (-923)))) (-15 -3669 ((-1270 $))) (-15 -3668 ((-112) (-923) $)) (-15 -3667 ((-774) $ (-923) (-923))))) (-923) (-923)) (T -1106)) 
-((-3673 (*1 *1 *1 *2) (|partial| -12 (-5 *2 (-923)) (-5 *1 (-1106 *3 *4)) (-14 *3 *2) (-14 *4 *2))) (-3672 (*1 *1 *1 *2 *2) (|partial| -12 (-5 *2 (-923)) (-5 *1 (-1106 *3 *4)) (-14 *3 *2) (-14 *4 *2))) (-3671 (*1 *2 *1) (-12 (-5 *2 (-644 (-923))) (-5 *1 (-1106 *3 *4)) (-14 *3 (-923)) (-14 *4 (-923)))) (-3670 (*1 *1 *2) (-12 (-5 *2 (-644 (-923))) (-5 *1 (-1106 *3 *4)) (-14 *3 (-923)) (-14 *4 (-923)))) (-3669 (*1 *2) (-12 (-5 *2 (-1270 (-1106 *3 *4))) (-5 *1 (-1106 *3 *4)) (-14 *3 (-923)) (-14 *4 (-923)))) (-3668 (*1 *2 *3 *1) (-12 (-5 *3 (-923)) (-5 *2 (-112)) (-5 *1 (-1106 *4 *5)) (-14 *4 *3) (-14 *5 *3))) (-3667 (*1 *2 *1 *3 *3) (-12 (-5 *3 (-923)) (-5 *2 (-774)) (-5 *1 (-1106 *4 *5)) (-14 *4 *3) (-14 *5 *3)))) 
-(-13 (-371) (-10 -8 (-15 -3673 ((-3 $ #1="failed") $ (-923))) (-15 -3672 ((-3 $ #1#) $ (-923) (-923))) (-15 -3671 ((-644 (-923)) $)) (-15 -3670 ($ (-644 (-923)))) (-15 -3669 ((-1270 $))) (-15 -3668 ((-112) (-923) $)) (-15 -3667 ((-774) $ (-923) (-923))))) 
-((-2970 (((-112) $ $) NIL)) (-3683 (((-112) $) NIL)) (-3679 (((-1181) $) NIL)) (-3684 (((-112) $) NIL)) (-3968 (((-1163) $) NIL)) (-3686 (((-112) $) NIL)) (-3688 (((-112) $) NIL)) (-3685 (((-112) $) NIL)) (-3665 (((-1163) $) NIL)) (-3682 (((-112) $) NIL)) (-3678 (((-550) $) NIL)) (-3666 (((-1124) $) NIL)) (-3681 (((-112) $) NIL)) (-3677 (((-226) $) NIL)) (-3676 (((-866) $) NIL)) (-3689 (((-112) $ $) NIL)) (-4233 (($ $ (-550)) NIL) (($ $ (-644 (-550))) NIL)) (-3680 (((-644 $) $) NIL)) (-4404 (($ (-1163)) NIL) (($ (-1181)) NIL) (($ (-550)) NIL) (($ (-226)) NIL) (($ (-866)) NIL) (($ (-644 $)) NIL)) (-4380 (((-866) $) NIL)) (-3674 (($ $) NIL)) (-3675 (($ $) NIL)) (-3664 (((-112) $ $) NIL)) (-3687 (((-112) $) NIL)) (-3457 (((-112) $ $) NIL)) (-4391 (((-550) $) NIL))) 
-(((-1107) (-1108 (-1163) (-1181) (-550) (-226) (-866))) (T -1107)) 
-NIL 
-(-1108 (-1163) (-1181) (-550) (-226) (-866)) 
-((-2970 (((-112) $ $) 7)) (-3683 (((-112) $) 33)) (-3679 ((|#2| $) 28)) (-3684 (((-112) $) 34)) (-3968 ((|#1| $) 29)) (-3686 (((-112) $) 36)) (-3688 (((-112) $) 38)) (-3685 (((-112) $) 35)) (-3665 (((-1163) $) 10)) (-3682 (((-112) $) 32)) (-3678 ((|#3| $) 27)) (-3666 (((-1124) $) 11)) (-3681 (((-112) $) 31)) (-3677 ((|#4| $) 26)) (-3676 ((|#5| $) 25)) (-3689 (((-112) $ $) 39)) (-4233 (($ $ (-550)) 21) (($ $ (-644 (-550))) 20)) (-3680 (((-644 $) $) 30)) (-4404 (($ |#1|) 45) (($ |#2|) 44) (($ |#3|) 43) (($ |#4|) 42) (($ |#5|) 41) (($ (-644 $)) 40)) (-4380 (((-866) $) 12)) (-3674 (($ $) 23)) (-3675 (($ $) 24)) (-3664 (((-112) $ $) 9)) (-3687 (((-112) $) 37)) (-3457 (((-112) $ $) 6)) (-4391 (((-550) $) 22))) 
-(((-1108 |#1| |#2| |#3| |#4| |#5|) (-140) (-1105) (-1105) (-1105) (-1105) (-1105)) (T -1108)) 
-((-3689 (*1 *2 *1 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3688 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3687 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3686 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3685 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3684 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3683 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3682 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3681 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-112)))) (-3680 (*1 *2 *1) (-12 (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-644 *1)) (-4 *1 (-1108 *3 *4 *5 *6 *7)))) (-3968 (*1 *2 *1) (-12 (-4 *1 (-1108 *2 *3 *4 *5 *6)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *2 (-1105)))) (-3679 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *2 *4 *5 *6)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *2 (-1105)))) (-3678 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *2 *5 *6)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *2 (-1105)))) (-3677 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *2 *6)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *2 (-1105)))) (-3676 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *2)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *2 (-1105)))) (-3675 (*1 *1 *1) (-12 (-4 *1 (-1108 *2 *3 *4 *5 *6)) (-4 *2 (-1105)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)))) (-3674 (*1 *1 *1) (-12 (-4 *1 (-1108 *2 *3 *4 *5 *6)) (-4 *2 (-1105)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)))) (-4391 (*1 *2 *1) (-12 (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)) (-5 *2 (-550)))) (-4233 (*1 *1 *1 *2) (-12 (-5 *2 (-550)) (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105)))) (-4233 (*1 *1 *1 *2) (-12 (-5 *2 (-644 (-550))) (-4 *1 (-1108 *3 *4 *5 *6 *7)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-4 *5 (-1105)) (-4 *6 (-1105)) (-4 *7 (-1105))))) 
-(-13 (-1105) (-621 |t#1|) (-621 |t#2|) (-621 |t#3|) (-621 |t#4|) (-621 |t#4|) (-621 |t#5|) (-621 (-644 $)) (-10 -8 (-15 -3689 ((-112) $ $)) (-15 -3688 ((-112) $)) (-15 -3687 ((-112) $)) (-15 -3686 ((-112) $)) (-15 -3685 ((-112) $)) (-15 -3684 ((-112) $)) (-15 -3683 ((-112) $)) (-15 -3682 ((-112) $)) (-15 -3681 ((-112) $)) (-15 -3680 ((-644 $) $)) (-15 -3968 (|t#1| $)) (-15 -3679 (|t#2| $)) (-15 -3678 (|t#3| $)) (-15 -3677 (|t#4| $)) (-15 -3676 (|t#5| $)) (-15 -3675 ($ $)) (-15 -3674 ($ $)) (-15 -4391 ((-550) $)) (-15 -4233 ($ $ (-550))) (-15 -4233 ($ $ (-644 (-550)))))) 
-(((-102) . T) ((-616 (-866)) . T) ((-621 (-644 $)) . T) ((-621 |#1|) . T) ((-621 |#2|) . T) ((-621 |#3|) . T) ((-621 |#4|) . T) ((-621 |#5|) . T) ((-1105) . T)) 
-((-2970 (((-112) $ $) NIL)) (-3683 (((-112) $) 45)) (-3679 ((|#2| $) 48)) (-3684 (((-112) $) 20)) (-3968 ((|#1| $) 21)) (-3686 (((-112) $) 42)) (-3688 (((-112) $) 14)) (-3685 (((-112) $) 44)) (-3665 (((-1163) $) NIL)) (-3682 (((-112) $) 46)) (-3678 ((|#3| $) 50)) (-3666 (((-1124) $) NIL)) (-3681 (((-112) $) 47)) (-3677 ((|#4| $) 49)) (-3676 ((|#5| $) 51)) (-3689 (((-112) $ $) 41)) (-4233 (($ $ (-550)) 62) (($ $ (-644 (-550))) 64)) (-3680 (((-644 $) $) 27)) (-4404 (($ |#1|) 53) (($ |#2|) 54) (($ |#3|) 55) (($ |#4|) 56) (($ |#5|) 57) (($ (-644 $)) 52)) (-4380 (((-866) $) 28)) (-3674 (($ $) 26)) (-3675 (($ $) 58)) (-3664 (((-112) $ $) NIL)) (-3687 (((-112) $) 23)) (-3457 (((-112) $ $) 40)) (-4391 (((-550) $) 60))) 
-(((-1109 |#1| |#2| |#3| |#4| |#5|) (-1108 |#1| |#2| |#3| |#4| |#5|) (-1105) (-1105) (-1105) (-1105) (-1105)) (T -1109)) 
-NIL 
-(-1108 |#1| |#2| |#3| |#4| |#5|) 
-((-3806 (((-1276) $) 22)) (-3690 (($ (-1181) (-438) |#2|) 11)) (-4380 (((-866) $) 16))) 
-(((-1110 |#1| |#2|) (-13 (-400) (-10 -8 (-15 -3690 ($ (-1181) (-438) |#2|)))) (-1105) (-425 |#1|)) (T -1110)) 
-((-3690 (*1 *1 *2 *3 *4) (-12 (-5 *2 (-1181)) (-5 *3 (-438)) (-4 *5 (-1105)) (-5 *1 (-1110 *5 *4)) (-4 *4 (-425 *5))))) 
-(-13 (-400) (-10 -8 (-15 -3690 ($ (-1181) (-438) |#2|)))) 
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-NIL 
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+(((-21) . T) ((-23) . T) ((-25) . T) ((-34) . T) ((-38 |#2|) |has| |#2| (-6 (-4436 #1="*"))) ((-102) . T) ((-111 |#2| |#2|) . T) ((-131) . T) ((-621 #2=(-412 (-551))) |has| |#2| (-1044 (-412 (-551)))) ((-621 (-551)) . T) ((-621 |#2|) . T) ((-618 (-868)) . T) ((-232 |#2|) . T) ((-234) |has| |#2| (-234)) ((-312 |#2|) -12 (|has| |#2| (-312 |#2|)) (|has| |#2| (-1107))) ((-381 |#2|) . T) ((-417 |#2|) . T) ((-494 |#2|) . T) ((-519 |#2| |#2|) -12 (|has| |#2| (-312 |#2|)) (|has| |#2| (-1107))) ((-651 (-551)) . T) ((-651 |#2|) . T) ((-651 $) . T) ((-653 |#2|) . T) ((-653 $) . T) ((-645 |#2|) -3969 (|has| |#2| (-173)) (|has| |#2| (-6 (-4436 #1#)))) ((-644 (-551)) |has| |#2| (-644 (-551))) ((-644 |#2|) . T) ((-722 |#2|) -3969 (|has| |#2| (-173)) (|has| |#2| (-6 (-4436 #1#)))) ((-731) . T) ((-906 (-1183)) |has| |#2| (-906 (-1183))) ((-1059 |#1| |#1| |#2| |#3| |#4|) . T) ((-1044 #2#) |has| |#2| (-1044 (-412 (-551)))) ((-1044 (-551)) |has| |#2| (-1044 (-551))) ((-1044 |#2|) . T) ((-1057 |#2|) . T) ((-1062 |#2|) . T) ((-1055) . T) ((-1063) . T) ((-1118) . T) ((-1107) . T) ((-1222) . T)) 
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-NIL 
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T) ((-794) -12 (|has| |#1| (-366)) (|has| |#2| (-823))) ((-795) -12 (|has| |#1| (-366)) (|has| |#2| (-823))) ((-797) -12 (|has| |#1| (-366)) (|has| |#2| (-823))) ((-800) -12 (|has| |#1| (-366)) (|has| |#2| (-823))) ((-823) -12 (|has| |#1| (-366)) (|has| |#2| (-823))) ((-851) -12 (|has| |#1| (-366)) (|has| |#2| (-823))) ((-853) -3962 (-12 (|has| |#1| (-366)) (|has| |#2| (-853))) (-12 (|has| |#1| (-366)) (|has| |#2| (-823)))) ((-904 (-1181)) -3962 (-12 (|has| |#1| (-904 (-1181))) (|has| |#1| (-15 * (|#1| (-550) |#1|)))) (-12 (|has| |#1| (-366)) (|has| |#2| (-904 (-1181))))) ((-890 (-381)) -12 (|has| |#1| (-366)) (|has| |#2| (-890 (-381)))) ((-890 (-550)) -12 (|has| |#1| (-366)) (|has| |#2| (-890 (-550)))) ((-888 |#2|) |has| |#1| (-366)) ((-914) -12 (|has| |#1| (-366)) (|has| |#2| (-914))) ((-977 |#1| #1# (-1086)) . T) ((-925) |has| |#1| (-366)) ((-995 |#2|) |has| |#1| (-366)) ((-1006) |has| |#1| (-38 (-411 (-550)))) ((-1024) -12 (|has| |#1| (-366)) (|has| |#2| (-1024))) ((-1042 (-411 (-550))) -12 (|has| |#1| (-366)) (|has| |#2| (-1042 (-550)))) ((-1042 (-550)) -12 (|has| |#1| (-366)) (|has| |#2| (-1042 (-550)))) ((-1042 #3#) -12 (|has| |#1| (-366)) (|has| |#2| (-1042 (-1181)))) ((-1042 |#2|) . T) ((-1055 #2#) -3962 (|has| |#1| (-366)) (|has| |#1| (-38 (-411 (-550))))) ((-1055 |#1|) . T) ((-1055 |#2|) |has| |#1| (-366)) ((-1055 $) -3962 (|has| |#1| (-561)) (|has| |#1| (-366)) (|has| |#1| (-173))) ((-1060 #2#) -3962 (|has| |#1| (-366)) (|has| |#1| (-38 (-411 (-550))))) ((-1060 |#1|) . T) ((-1060 |#2|) |has| |#1| (-366)) ((-1060 $) -3962 (|has| |#1| (-561)) (|has| |#1| (-366)) (|has| |#1| (-173))) ((-1053) . T) ((-1061) . T) ((-1116) . T) ((-1105) . T) ((-1155) -12 (|has| |#1| (-366)) (|has| |#2| (-1155))) ((-1206) |has| |#1| (-38 (-411 (-550)))) ((-1209) |has| |#1| (-38 (-411 (-550)))) ((-1220) |has| |#1| (-366)) ((-1225) |has| |#1| (-366)) ((-1232 |#1|) . T) ((-1249 |#1| #1#) . T)) 
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-NIL 
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-((-4407 (*1 *2 *3 *4) (|partial| -12 (-5 *3 (-1270 *4)) (-4 *4 (-642 (-550))) (-5 *2 (-1270 (-411 (-550)))) (-5 *1 (-1298 *4)))) (-4406 (*1 *2 *3) (|partial| -12 (-5 *3 (-1270 *4)) (-4 *4 (-642 (-550))) (-5 *2 (-1270 (-550))) (-5 *1 (-1298 *4)))) (-4405 (*1 *2 *3) (-12 (-5 *3 (-1270 *4)) (-4 *4 (-642 (-550))) (-5 *2 (-112)) (-5 *1 (-1298 *4))))) 
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-(((-1299 |#1|) (-13 (-173) (-371) (-617 (-550)) (-1155)) (-923)) (T -1299)) 
-NIL 
-(-13 (-173) (-371) (-617 (-550)) (-1155)) 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
-NIL 
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TYPEAST (NIL) -8 NIL NIL NIL) (-1220 2933928 2933933 2933963 "TYPE" 2933968 T TYPE (NIL) -9 NIL NIL NIL) (-1219 2932899 2933101 2933341 "TWOFACT" 2933722 NIL TWOFACT (NIL T) -7 NIL NIL NIL) (-1218 2931922 2932308 2932543 "TUPLE" 2932699 NIL TUPLE (NIL T) -8 NIL NIL NIL) (-1217 2929613 2930132 2930671 "TUBETOOL" 2931405 T TUBETOOL (NIL) -7 NIL NIL NIL) (-1216 2928462 2928667 2928908 "TUBE" 2929406 NIL TUBE (NIL T) -8 NIL NIL NIL) (-1215 2917102 2921221 2921318 "TSETCAT" 2926587 NIL TSETCAT (NIL T T T T) -9 NIL 2928118 NIL) (-1214 2911834 2913434 2915325 "TSETCAT-" 2915330 NIL TSETCAT- (NIL T T T T T) -8 NIL NIL NIL) (-1213 2906563 2910806 2911089 "TS" 2911586 NIL TS (NIL T) -8 NIL NIL NIL) (-1212 2901202 2902049 2902978 "TRMANIP" 2905699 NIL TRMANIP (NIL T T) -7 NIL NIL NIL) (-1211 2900643 2900706 2900869 "TRIMAT" 2901134 NIL TRIMAT (NIL T T T T) -7 NIL NIL NIL) (-1210 2898509 2898746 2899103 "TRIGMNIP" 2900392 NIL TRIGMNIP (NIL T T) -7 NIL NIL NIL) (-1209 2898029 2898142 2898172 "TRIGCAT" 2898385 T TRIGCAT (NIL) -9 NIL NIL NIL) (-1208 2897698 2897777 2897918 "TRIGCAT-" 2897923 NIL TRIGCAT- (NIL T) -8 NIL NIL NIL) (-1207 2894544 2896556 2896837 "TREE" 2897452 NIL TREE (NIL T) -8 NIL NIL NIL) (-1206 2893818 2894346 2894376 "TRANFUN" 2894411 T TRANFUN (NIL) -9 NIL 2894477 NIL) (-1205 2893097 2893288 2893568 "TRANFUN-" 2893573 NIL TRANFUN- (NIL T) -8 NIL NIL NIL) (-1204 2892901 2892933 2892994 "TOPSP" 2893058 T TOPSP (NIL) -7 NIL NIL NIL) (-1203 2892249 2892364 2892518 "TOOLSIGN" 2892782 NIL TOOLSIGN (NIL T) -7 NIL NIL NIL) (-1202 2890883 2891426 2891665 "TEXTFILE" 2892032 T TEXTFILE (NIL) -8 NIL NIL NIL) (-1201 2890664 2890695 2890767 "TEX1" 2890846 NIL TEX1 (NIL T) -7 NIL NIL NIL) (-1200 2888576 2889117 2889546 "TEX" 2890257 T TEX (NIL) -8 NIL NIL NIL) (-1199 2888224 2888287 2888377 "TEMUTL" 2888508 T TEMUTL (NIL) -7 NIL NIL NIL) (-1198 2886378 2886658 2886983 "TBCMPPK" 2887947 NIL TBCMPPK (NIL T T) -7 NIL NIL NIL) (-1197 2878157 2884538 2884594 "TBAGG" 2884994 NIL TBAGG (NIL T T) -9 NIL 2885205 NIL) (-1196 2873227 2874715 2876469 "TBAGG-" 2876474 NIL TBAGG- (NIL T T T) -8 NIL NIL NIL) (-1195 2872611 2872718 2872863 "TANEXP" 2873116 NIL TANEXP (NIL T) -7 NIL NIL NIL) (-1194 2872023 2872122 2872260 "TABLEAU" 2872508 NIL TABLEAU (NIL T) -8 NIL NIL NIL) (-1193 2865415 2871880 2871973 "TABLE" 2871978 NIL TABLE (NIL T T) -8 NIL NIL NIL) (-1192 2860023 2861243 2862491 "TABLBUMP" 2864201 NIL TABLBUMP (NIL T) -7 NIL NIL NIL) (-1191 2859245 2859392 2859573 "SYSTEM" 2859864 T SYSTEM (NIL) -8 NIL NIL NIL) (-1190 2855704 2856403 2857186 "SYSSOLP" 2858496 NIL SYSSOLP (NIL T) -7 NIL NIL NIL) (-1189 2855502 2855659 2855690 "SYSPTR" 2855695 T SYSPTR (NIL) -8 NIL NIL NIL) (-1188 2854546 2855051 2855170 "SYSNNI" 2855356 NIL SYSNNI (NIL NIL) -8 NIL NIL 2855441) (-1187 2853853 2854312 2854391 "SYSINT" 2854451 NIL SYSINT (NIL NIL) -8 NIL NIL 2854496) (-1186 2850197 2851131 2851841 "SYNTAX" 2853165 T SYNTAX (NIL) -8 NIL NIL NIL) (-1185 2847355 2847957 2848589 "SYMTAB" 2849587 T SYMTAB (NIL) -8 NIL NIL NIL) (-1184 2842628 2843524 2844501 "SYMS" 2846400 T SYMS (NIL) -8 NIL NIL NIL) (-1183 2839873 2842089 2842319 "SYMPOLY" 2842436 NIL SYMPOLY (NIL T) -8 NIL NIL NIL) (-1182 2839390 2839465 2839588 "SYMFUNC" 2839785 NIL SYMFUNC (NIL T) -7 NIL NIL NIL) (-1181 2835410 2836702 2837515 "SYMBOL" 2838599 T SYMBOL (NIL) -8 NIL NIL NIL) (-1180 2828949 2830638 2832358 "SWITCH" 2833712 T SWITCH (NIL) -8 NIL NIL NIL) (-1179 2822183 2827770 2828073 "SUTS" 2828704 NIL SUTS (NIL T NIL NIL) -8 NIL NIL NIL) (-1178 2814253 2821430 2821703 "SUPXS" 2821968 NIL SUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-1177 2813412 2813539 2813756 "SUPFRACF" 2814121 NIL SUPFRACF (NIL T T T T) -7 NIL NIL NIL) (-1176 2813033 2813092 2813205 "SUP2" 2813347 NIL SUP2 (NIL T T) -7 NIL NIL NIL) (-1175 2804832 2812651 2812777 "SUP" 2812942 NIL SUP (NIL T) -8 NIL NIL NIL) (-1174 2803280 2803554 2803910 "SUMRF" 2804531 NIL SUMRF (NIL T) -7 NIL NIL NIL) (-1173 2802615 2802681 2802873 "SUMFS" 2803201 NIL SUMFS (NIL T T) -7 NIL NIL NIL) (-1172 2786598 2801792 2802043 "SULS" 2802422 NIL SULS (NIL T NIL NIL) -8 NIL NIL NIL) (-1171 2786200 2786420 2786490 "SUCHTAST" 2786550 T SUCHTAST (NIL) -8 NIL NIL NIL) (-1170 2785495 2785725 2785865 "SUCH" 2786108 NIL SUCH (NIL T T) -8 NIL NIL NIL) (-1169 2779361 2780401 2781360 "SUBSPACE" 2784583 NIL SUBSPACE (NIL NIL T) -8 NIL NIL NIL) (-1168 2778791 2778881 2779045 "SUBRESP" 2779249 NIL SUBRESP (NIL T T) -7 NIL NIL NIL) (-1167 2772964 2774084 2775231 "STTFNC" 2777691 NIL STTFNC (NIL T) -7 NIL NIL NIL) (-1166 2766330 2767629 2768940 "STTF" 2771700 NIL STTF (NIL T) -7 NIL NIL NIL) (-1165 2757641 2759512 2761306 "STTAYLOR" 2764571 NIL STTAYLOR (NIL T) -7 NIL NIL NIL) (-1164 2750773 2757505 2757588 "STRTBL" 2757593 NIL STRTBL (NIL T) -8 NIL NIL NIL) (-1163 2746137 2750728 2750759 "STRING" 2750764 T STRING (NIL) -8 NIL NIL NIL) (-1162 2740998 2745510 2745540 "STRICAT" 2745599 T STRICAT (NIL) -9 NIL 2745661 NIL) (-1161 2740508 2740585 2740729 "STREAM3" 2740915 NIL STREAM3 (NIL T T T) -7 NIL NIL NIL) (-1160 2739490 2739673 2739908 "STREAM2" 2740321 NIL STREAM2 (NIL T T) -7 NIL NIL NIL) (-1159 2739178 2739230 2739323 "STREAM1" 2739432 NIL STREAM1 (NIL T) -7 NIL NIL NIL) (-1158 2731933 2736797 2737408 "STREAM" 2738602 NIL STREAM (NIL T) -8 NIL NIL NIL) (-1157 2730949 2731130 2731361 "STINPROD" 2731749 NIL STINPROD (NIL T) -7 NIL NIL NIL) (-1156 2730136 2730438 2730586 "STEPAST" 2730823 T STEPAST (NIL) -8 NIL NIL NIL) (-1155 2729688 2729898 2729928 "STEP" 2730008 T STEP (NIL) -9 NIL 2730086 NIL) (-1154 2723122 2729587 2729664 "STBL" 2729669 NIL STBL (NIL T T NIL) -8 NIL NIL NIL) (-1153 2718250 2722343 2722386 "STAGG" 2722539 NIL STAGG (NIL T) -9 NIL 2722628 NIL) (-1152 2715958 2716558 2717428 "STAGG-" 2717433 NIL STAGG- (NIL T T) -8 NIL NIL NIL) (-1151 2714105 2715728 2715820 "STACK" 2715901 NIL STACK (NIL T) -8 NIL NIL NIL) (-1150 2706827 2712246 2712702 "SREGSET" 2713735 NIL SREGSET (NIL T T T T) -8 NIL NIL NIL) (-1149 2699252 2700621 2702134 "SRDCMPK" 2705433 NIL SRDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1148 2692169 2696692 2696722 "SRAGG" 2698025 T SRAGG (NIL) -9 NIL 2698633 NIL) (-1147 2691186 2691441 2691820 "SRAGG-" 2691825 NIL SRAGG- (NIL T) -8 NIL NIL NIL) (-1146 2685650 2690133 2690554 "SQMATRIX" 2690812 NIL SQMATRIX (NIL NIL T) -8 NIL NIL NIL) (-1145 2679336 2682368 2683095 "SPLTREE" 2684995 NIL SPLTREE (NIL T T) -8 NIL NIL NIL) (-1144 2675299 2675992 2676638 "SPLNODE" 2678762 NIL SPLNODE (NIL T T) -8 NIL NIL NIL) (-1143 2674346 2674579 2674609 "SPFCAT" 2675053 T SPFCAT (NIL) -9 NIL NIL NIL) (-1142 2673083 2673293 2673557 "SPECOUT" 2674104 T SPECOUT (NIL) -7 NIL NIL NIL) (-1141 2664193 2666065 2666095 "SPADXPT" 2670771 T SPADXPT (NIL) -9 NIL 2672935 NIL) (-1140 2663954 2663994 2664063 "SPADPRSR" 2664146 T SPADPRSR (NIL) -7 NIL NIL NIL) (-1139 2662003 2663909 2663940 "SPADAST" 2663945 T SPADAST (NIL) -8 NIL NIL NIL) (-1138 2653948 2655721 2655764 "SPACEC" 2660137 NIL SPACEC (NIL T) -9 NIL 2661953 NIL) (-1137 2652078 2653880 2653929 "SPACE3" 2653934 NIL SPACE3 (NIL T) -8 NIL NIL NIL) (-1136 2650830 2651001 2651292 "SORTPAK" 2651883 NIL SORTPAK (NIL T T) -7 NIL NIL NIL) (-1135 2648922 2649225 2649637 "SOLVETRA" 2650494 NIL SOLVETRA (NIL T) -7 NIL NIL NIL) (-1134 2647972 2648194 2648455 "SOLVESER" 2648695 NIL SOLVESER (NIL T) -7 NIL NIL NIL) (-1133 2643276 2644164 2645159 "SOLVERAD" 2647024 NIL SOLVERAD (NIL T) -7 NIL NIL NIL) (-1132 2639091 2639700 2640429 "SOLVEFOR" 2642643 NIL SOLVEFOR (NIL T T) -7 NIL NIL NIL) (-1131 2633388 2638440 2638537 "SNTSCAT" 2638542 NIL SNTSCAT (NIL T T T T) -9 NIL 2638612 NIL) (-1130 2627494 2631711 2632102 "SMTS" 2633078 NIL SMTS (NIL T T T) -8 NIL NIL NIL) (-1129 2622205 2627382 2627459 "SMP" 2627464 NIL SMP (NIL T T) -8 NIL NIL NIL) (-1128 2620364 2620665 2621063 "SMITH" 2621902 NIL SMITH (NIL T T T T) -7 NIL NIL NIL) (-1127 2613075 2617267 2617370 "SMATCAT" 2618724 NIL SMATCAT (NIL NIL T T T) -9 NIL 2619274 NIL) (-1126 2610036 2610852 2612023 "SMATCAT-" 2612028 NIL SMATCAT- (NIL T NIL T T T) -8 NIL NIL NIL) (-1125 2607702 2609272 2609315 "SKAGG" 2609576 NIL SKAGG (NIL T) -9 NIL 2609711 NIL) (-1124 2604015 2607118 2607313 "SINT" 2607500 T SINT (NIL) -8 NIL NIL 2607673) (-1123 2603787 2603825 2603891 "SIMPAN" 2603971 T SIMPAN (NIL) -7 NIL NIL NIL) (-1122 2602646 2602860 2603128 "SIGNRF" 2603553 NIL SIGNRF (NIL T) -7 NIL NIL NIL) (-1121 2601500 2601644 2601921 "SIGNEF" 2602482 NIL SIGNEF (NIL T T) -7 NIL NIL NIL) (-1120 2600806 2601083 2601207 "SIGAST" 2601398 T SIGAST (NIL) -8 NIL NIL NIL) (-1119 2600085 2600341 2600481 "SIG" 2600688 T SIG (NIL) -8 NIL NIL NIL) (-1118 2597775 2598229 2598735 "SHP" 2599626 NIL SHP (NIL T NIL) -7 NIL NIL NIL) (-1117 2591634 2597676 2597752 "SHDP" 2597757 NIL SHDP (NIL NIL NIL T) -8 NIL NIL NIL) (-1116 2591207 2591399 2591429 "SGROUP" 2591522 T SGROUP (NIL) -9 NIL 2591584 NIL) (-1115 2591065 2591091 2591164 "SGROUP-" 2591169 NIL SGROUP- (NIL T) -8 NIL NIL NIL) (-1114 2587900 2588598 2589321 "SGCF" 2590364 T SGCF (NIL) -7 NIL NIL NIL) (-1113 2582295 2587347 2587444 "SFRTCAT" 2587449 NIL SFRTCAT (NIL T T T T) -9 NIL 2587488 NIL) (-1112 2575716 2576734 2577870 "SFRGCD" 2581278 NIL SFRGCD (NIL T T T T T) -7 NIL NIL NIL) (-1111 2568842 2569915 2571101 "SFQCMPK" 2574649 NIL SFQCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1110 2568462 2568551 2568662 "SFORT" 2568783 NIL SFORT (NIL T T) -8 NIL NIL NIL) (-1109 2567580 2568302 2568423 "SEXOF" 2568428 NIL SEXOF (NIL T T T T T) -8 NIL NIL NIL) (-1108 2563093 2563808 2563903 "SEXCAT" 2566840 NIL SEXCAT (NIL T T T T T) -9 NIL 2567418 NIL) (-1107 2562200 2562974 2563042 "SEX" 2563047 T SEX (NIL) -8 NIL NIL NIL) (-1106 2560430 2560917 2561220 "SETMN" 2561943 NIL SETMN (NIL NIL NIL) -8 NIL NIL NIL) (-1105 2559926 2560078 2560108 "SETCAT" 2560284 T SETCAT (NIL) -9 NIL 2560394 NIL) (-1104 2559618 2559696 2559826 "SETCAT-" 2559831 NIL SETCAT- (NIL T) -8 NIL NIL NIL) (-1103 2555979 2558079 2558122 "SETAGG" 2558992 NIL SETAGG (NIL T) -9 NIL 2559332 NIL) (-1102 2555437 2555553 2555790 "SETAGG-" 2555795 NIL SETAGG- (NIL T T) -8 NIL NIL NIL) (-1101 2552590 2555371 2555419 "SET" 2555424 NIL SET (NIL T) -8 NIL NIL NIL) (-1100 2552033 2552286 2552387 "SEQAST" 2552511 T SEQAST (NIL) -8 NIL NIL NIL) (-1099 2551232 2551526 2551587 "SEGXCAT" 2551873 NIL SEGXCAT (NIL T T) -9 NIL 2551993 NIL) (-1098 2550211 2550425 2550468 "SEGCAT" 2550990 NIL SEGCAT (NIL T) -9 NIL 2551211 NIL) (-1097 2549832 2549891 2550004 "SEGBIND2" 2550146 NIL SEGBIND2 (NIL T T) -7 NIL NIL NIL) (-1096 2548764 2549195 2549403 "SEGBIND" 2549659 NIL SEGBIND (NIL T) -8 NIL NIL NIL) (-1095 2548337 2548565 2548642 "SEGAST" 2548709 T SEGAST (NIL) -8 NIL NIL NIL) (-1094 2547556 2547682 2547886 "SEG2" 2548181 NIL SEG2 (NIL T T) -7 NIL NIL NIL) (-1093 2546562 2547222 2547404 "SEG" 2547409 NIL SEG (NIL T) -8 NIL NIL NIL) (-1092 2545972 2546497 2546544 "SDVAR" 2546549 NIL SDVAR (NIL T) -8 NIL NIL NIL) (-1091 2538540 2545742 2545872 "SDPOL" 2545877 NIL SDPOL (NIL T) -8 NIL NIL NIL) (-1090 2537133 2537399 2537718 "SCPKG" 2538255 NIL SCPKG (NIL T) -7 NIL NIL NIL) (-1089 2536297 2536469 2536661 "SCOPE" 2536963 T SCOPE (NIL) -8 NIL NIL NIL) (-1088 2535517 2535651 2535830 "SCACHE" 2536152 NIL SCACHE (NIL T) -7 NIL NIL NIL) (-1087 2535163 2535349 2535379 "SASTCAT" 2535384 T SASTCAT (NIL) -9 NIL 2535397 NIL) (-1086 2534650 2534998 2535074 "SAOS" 2535109 T SAOS (NIL) -8 NIL NIL NIL) (-1085 2534215 2534250 2534423 "SAERFFC" 2534609 NIL SAERFFC (NIL T T T) -7 NIL NIL NIL) (-1084 2533808 2533843 2534002 "SAEFACT" 2534174 NIL SAEFACT (NIL T T T) -7 NIL NIL NIL) (-1083 2527756 2533705 2533785 "SAE" 2533790 NIL SAE (NIL T T NIL) -8 NIL NIL NIL) (-1082 2526077 2526391 2526792 "RURPK" 2527422 NIL RURPK (NIL T NIL) -7 NIL NIL NIL) (-1081 2524714 2525020 2525325 "RULESET" 2525911 NIL RULESET (NIL T T T) -8 NIL NIL NIL) (-1080 2524326 2524508 2524591 "RULECOLD" 2524666 NIL RULECOLD (NIL NIL) -8 NIL NIL NIL) (-1079 2521549 2522079 2522537 "RULE" 2524007 NIL RULE (NIL T T T) -8 NIL NIL NIL) (-1078 2521339 2521367 2521438 "RTVALUE" 2521500 T RTVALUE (NIL) -8 NIL NIL NIL) (-1077 2520810 2521056 2521150 "RSTRCAST" 2521267 T RSTRCAST (NIL) -8 NIL NIL NIL) (-1076 2515658 2516453 2517373 "RSETGCD" 2520009 NIL RSETGCD (NIL T T T T T) -7 NIL NIL NIL) (-1075 2504915 2509967 2510064 "RSETCAT" 2514183 NIL RSETCAT (NIL T T T T) -9 NIL 2515280 NIL) (-1074 2502842 2503381 2504205 "RSETCAT-" 2504210 NIL RSETCAT- (NIL T T T T T) -8 NIL NIL NIL) (-1073 2495228 2496604 2498124 "RSDCMPK" 2501441 NIL RSDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1072 2493207 2493674 2493748 "RRCC" 2494834 NIL RRCC (NIL T T) -9 NIL 2495178 NIL) (-1071 2492558 2492732 2493011 "RRCC-" 2493016 NIL RRCC- (NIL T T T) -8 NIL NIL NIL) (-1070 2492001 2492254 2492355 "RPTAST" 2492479 T RPTAST (NIL) -8 NIL NIL NIL) (-1069 2465883 2475209 2475276 "RPOLCAT" 2485940 NIL RPOLCAT (NIL T T T) -9 NIL 2489099 NIL) (-1068 2457417 2459745 2462855 "RPOLCAT-" 2462860 NIL RPOLCAT- (NIL T T T T) -8 NIL NIL NIL) (-1067 2448350 2455628 2456110 "ROUTINE" 2456957 T ROUTINE (NIL) -8 NIL NIL NIL) (-1066 2445150 2447976 2448116 "ROMAN" 2448232 T ROMAN (NIL) -8 NIL NIL NIL) (-1065 2443396 2444010 2444270 "ROIRC" 2444955 NIL ROIRC (NIL T T) -8 NIL NIL NIL) (-1064 2439632 2441912 2441942 "RNS" 2442246 T RNS (NIL) -9 NIL 2442520 NIL) (-1063 2438141 2438524 2439058 "RNS-" 2439133 NIL RNS- (NIL T) -8 NIL NIL NIL) (-1062 2437144 2437506 2437708 "RNGBIND" 2437992 NIL RNGBIND (NIL T T) -8 NIL NIL NIL) (-1061 2436547 2436955 2436985 "RNG" 2436990 T RNG (NIL) -9 NIL 2437011 NIL) (-1060 2435946 2436334 2436377 "RMODULE" 2436382 NIL RMODULE (NIL T) -9 NIL 2436409 NIL) (-1059 2434782 2434876 2435212 "RMCAT2" 2435847 NIL RMCAT2 (NIL NIL NIL T T T T T T T T) -7 NIL NIL NIL) (-1058 2431632 2434128 2434425 "RMATRIX" 2434544 NIL RMATRIX (NIL NIL NIL T) -8 NIL NIL NIL) (-1057 2424459 2426719 2426834 "RMATCAT" 2430193 NIL RMATCAT (NIL NIL NIL T T T) -9 NIL 2431175 NIL) (-1056 2423834 2423981 2424288 "RMATCAT-" 2424293 NIL RMATCAT- (NIL T NIL NIL T T T) -8 NIL NIL NIL) (-1055 2423235 2423456 2423499 "RLINSET" 2423693 NIL RLINSET (NIL T) -9 NIL 2423784 NIL) (-1054 2422802 2422877 2423005 "RINTERP" 2423154 NIL RINTERP (NIL NIL T) -7 NIL NIL NIL) (-1053 2421860 2422414 2422444 "RING" 2422500 T RING (NIL) -9 NIL 2422592 NIL) (-1052 2421652 2421696 2421793 "RING-" 2421798 NIL RING- (NIL T) -8 NIL NIL NIL) (-1051 2420493 2420730 2420988 "RIDIST" 2421416 T RIDIST (NIL) -7 NIL NIL NIL) (-1050 2411809 2419961 2420167 "RGCHAIN" 2420341 NIL RGCHAIN (NIL T NIL) -8 NIL NIL NIL) (-1049 2411159 2411565 2411606 "RGBCSPC" 2411664 NIL RGBCSPC (NIL T) -9 NIL 2411716 NIL) (-1048 2410317 2410698 2410739 "RGBCMDL" 2410971 NIL RGBCMDL (NIL T) -9 NIL 2411085 NIL) (-1047 2409963 2410026 2410129 "RFFACTOR" 2410248 NIL RFFACTOR (NIL T) -7 NIL NIL NIL) (-1046 2409688 2409723 2409820 "RFFACT" 2409922 NIL RFFACT (NIL T) -7 NIL NIL NIL) (-1045 2407805 2408169 2408551 "RFDIST" 2409328 T RFDIST (NIL) -7 NIL NIL NIL) (-1044 2404799 2405413 2406083 "RF" 2407169 NIL RF (NIL T) -7 NIL NIL NIL) (-1043 2404252 2404344 2404507 "RETSOL" 2404701 NIL RETSOL (NIL T T) -7 NIL NIL NIL) (-1042 2403888 2403968 2404011 "RETRACT" 2404144 NIL RETRACT (NIL T) -9 NIL 2404231 NIL) (-1041 2403737 2403762 2403849 "RETRACT-" 2403854 NIL RETRACT- (NIL T T) -8 NIL NIL NIL) (-1040 2403339 2403559 2403629 "RETAST" 2403689 T RETAST (NIL) -8 NIL NIL NIL) (-1039 2396079 2402992 2403119 "RESULT" 2403234 T RESULT (NIL) -8 NIL NIL NIL) (-1038 2394670 2395348 2395547 "RESRING" 2395982 NIL RESRING (NIL T T T T NIL) -8 NIL NIL NIL) (-1037 2394306 2394355 2394453 "RESLATC" 2394607 NIL RESLATC (NIL T) -7 NIL NIL NIL) (-1036 2394011 2394046 2394153 "REPSQ" 2394265 NIL REPSQ (NIL T) -7 NIL NIL NIL) (-1035 2393708 2393743 2393854 "REPDB" 2393970 NIL REPDB (NIL T) -7 NIL NIL NIL) (-1034 2387608 2388997 2390220 "REP2" 2392520 NIL REP2 (NIL T) -7 NIL NIL NIL) (-1033 2383985 2384666 2385474 "REP1" 2386835 NIL REP1 (NIL T) -7 NIL NIL NIL) (-1032 2381407 2381987 2382589 "REP" 2383405 T REP (NIL) -7 NIL NIL NIL) (-1031 2374130 2379548 2380004 "REGSET" 2381037 NIL REGSET (NIL T T T T) -8 NIL NIL NIL) (-1030 2372895 2373278 2373528 "REF" 2373915 NIL REF (NIL T) -8 NIL NIL NIL) (-1029 2372272 2372375 2372542 "REDORDER" 2372779 NIL REDORDER (NIL T T) -7 NIL NIL NIL) (-1028 2368271 2371485 2371712 "RECLOS" 2372100 NIL RECLOS (NIL T) -8 NIL NIL NIL) (-1027 2367323 2367504 2367719 "REALSOLV" 2368078 T REALSOLV (NIL) -7 NIL NIL NIL) (-1026 2363806 2364608 2365492 "REAL0Q" 2366488 NIL REAL0Q (NIL T) -7 NIL NIL NIL) (-1025 2359407 2360395 2361456 "REAL0" 2362787 NIL REAL0 (NIL T) -7 NIL NIL NIL) (-1024 2359253 2359294 2359324 "REAL" 2359329 T REAL (NIL) -9 NIL 2359364 NIL) (-1023 2358724 2358970 2359064 "RDUCEAST" 2359181 T RDUCEAST (NIL) -8 NIL NIL NIL) (-1022 2358129 2358201 2358408 "RDIV" 2358646 NIL RDIV (NIL T T T T T) -7 NIL NIL NIL) (-1021 2357197 2357371 2357584 "RDIST" 2357951 NIL RDIST (NIL T) -7 NIL NIL NIL) (-1020 2355794 2356081 2356453 "RDETRS" 2356905 NIL RDETRS (NIL T T) -7 NIL NIL NIL) (-1019 2353606 2354060 2354598 "RDETR" 2355336 NIL RDETR (NIL T T) -7 NIL NIL NIL) (-1018 2352231 2352509 2352906 "RDEEFS" 2353322 NIL RDEEFS (NIL T T) -7 NIL NIL NIL) (-1017 2350740 2351046 2351471 "RDEEF" 2351919 NIL RDEEF (NIL T T) -7 NIL NIL NIL) (-1016 2344810 2347721 2347751 "RCFIELD" 2349046 T RCFIELD (NIL) -9 NIL 2349777 NIL) (-1015 2342874 2343378 2344074 "RCFIELD-" 2344149 NIL RCFIELD- (NIL T) -8 NIL NIL NIL) (-1014 2339143 2340975 2341018 "RCAGG" 2342102 NIL RCAGG (NIL T) -9 NIL 2342567 NIL) (-1013 2338771 2338865 2339028 "RCAGG-" 2339033 NIL RCAGG- (NIL T T) -8 NIL NIL NIL) (-1012 2338106 2338218 2338383 "RATRET" 2338655 NIL RATRET (NIL T) -7 NIL NIL NIL) (-1011 2337659 2337726 2337847 "RATFACT" 2338034 NIL RATFACT (NIL T) -7 NIL NIL NIL) (-1010 2336967 2337087 2337239 "RANDSRC" 2337529 T RANDSRC (NIL) -7 NIL NIL NIL) (-1009 2336701 2336745 2336818 "RADUTIL" 2336916 T RADUTIL (NIL) -7 NIL NIL NIL) (-1008 2329838 2335534 2335844 "RADIX" 2336425 NIL RADIX (NIL NIL) -8 NIL NIL NIL) (-1007 2321468 2329680 2329810 "RADFF" 2329815 NIL RADFF (NIL T T T NIL NIL) -8 NIL NIL NIL) (-1006 2321115 2321190 2321220 "RADCAT" 2321380 T RADCAT (NIL) -9 NIL NIL NIL) (-1005 2320897 2320945 2321045 "RADCAT-" 2321050 NIL RADCAT- (NIL T) -8 NIL NIL NIL) (-1004 2318997 2320669 2320760 "QUEUE" 2320841 NIL QUEUE (NIL T) -8 NIL NIL NIL) (-1003 2318628 2318671 2318802 "QUATCT2" 2318948 NIL QUATCT2 (NIL T T T T) -7 NIL NIL NIL) (-1002 2312084 2315422 2315464 "QUATCAT" 2316255 NIL QUATCAT (NIL T) -9 NIL 2317021 NIL) (-1001 2308244 2309274 2310657 "QUATCAT-" 2310753 NIL QUATCAT- (NIL T T) -8 NIL NIL NIL) (-1000 2304788 2308177 2308225 "QUAT" 2308230 NIL QUAT (NIL T) -8 NIL NIL NIL) (-999 2302261 2303872 2303913 "QUAGG" 2304288 NIL QUAGG (NIL T) -9 NIL 2304463 NIL) (-998 2301866 2302086 2302154 "QQUTAST" 2302213 T QQUTAST (NIL) -8 NIL NIL NIL) (-997 2300764 2301264 2301436 "QFORM" 2301738 NIL QFORM (NIL NIL T) -8 NIL NIL NIL) (-996 2300402 2300445 2300572 "QFCAT2" 2300715 NIL QFCAT2 (NIL T T T T) -7 NIL NIL NIL) (-995 2291423 2296646 2296686 "QFCAT" 2297344 NIL QFCAT (NIL T) -9 NIL 2298345 NIL) (-994 2287031 2288220 2289799 "QFCAT-" 2289893 NIL QFCAT- (NIL T T) -8 NIL NIL NIL) (-993 2286491 2286601 2286731 "QEQUAT" 2286921 T QEQUAT (NIL) -8 NIL NIL NIL) (-992 2279637 2280710 2281894 "QCMPACK" 2285424 NIL QCMPACK (NIL T T T T T) -7 NIL NIL NIL) (-991 2278882 2279056 2279288 "QALGSET2" 2279457 NIL QALGSET2 (NIL NIL NIL) -7 NIL NIL NIL) (-990 2276437 2276883 2277309 "QALGSET" 2278539 NIL QALGSET (NIL T T T T) -8 NIL NIL NIL) (-989 2275127 2275351 2275668 "PWFFINTB" 2276210 NIL PWFFINTB (NIL T T T T) -7 NIL NIL NIL) (-988 2273326 2273494 2273848 "PUSHVAR" 2274941 NIL PUSHVAR (NIL T T T T) -7 NIL NIL NIL) (-987 2269244 2270298 2270339 "PTRANFN" 2272223 NIL PTRANFN (NIL T) -9 NIL NIL NIL) (-986 2267646 2267937 2268259 "PTPACK" 2268955 NIL PTPACK (NIL T) -7 NIL NIL NIL) (-985 2267278 2267335 2267444 "PTFUNC2" 2267583 NIL PTFUNC2 (NIL T T) -7 NIL NIL NIL) (-984 2261755 2266150 2266191 "PTCAT" 2266487 NIL PTCAT (NIL T) -9 NIL 2266640 NIL) (-983 2261413 2261448 2261572 "PSQFR" 2261714 NIL PSQFR (NIL T T T T) -7 NIL NIL NIL) (-982 2260008 2260306 2260640 "PSEUDLIN" 2261111 NIL PSEUDLIN (NIL T) -7 NIL NIL NIL) (-981 2246771 2249142 2251466 "PSETPK" 2257768 NIL PSETPK (NIL T T T T) -7 NIL NIL NIL) (-980 2239789 2242529 2242625 "PSETCAT" 2245646 NIL PSETCAT (NIL T T T T) -9 NIL 2246460 NIL) (-979 2237625 2238259 2239080 "PSETCAT-" 2239085 NIL PSETCAT- (NIL T T T T T) -8 NIL NIL NIL) (-978 2236974 2237139 2237167 "PSCURVE" 2237435 T PSCURVE (NIL) -9 NIL 2237602 NIL) (-977 2232972 2234488 2234553 "PSCAT" 2235397 NIL PSCAT (NIL T T T) -9 NIL 2235637 NIL) (-976 2232035 2232251 2232651 "PSCAT-" 2232656 NIL PSCAT- (NIL T T T T) -8 NIL NIL NIL) (-975 2230740 2231400 2231605 "PRTITION" 2231850 T PRTITION (NIL) -8 NIL NIL NIL) (-974 2230215 2230461 2230553 "PRTDAST" 2230668 T PRTDAST (NIL) -8 NIL NIL NIL) (-973 2219305 2221519 2223707 "PRS" 2228077 NIL PRS (NIL T T) -7 NIL NIL NIL) (-972 2217116 2218655 2218695 "PRQAGG" 2218878 NIL PRQAGG (NIL T) -9 NIL 2218980 NIL) (-971 2216320 2216625 2216653 "PROPLOG" 2216900 T PROPLOG (NIL) -9 NIL 2217066 NIL) (-970 2214501 2215067 2215364 "PROPFRML" 2216056 NIL PROPFRML (NIL T) -8 NIL NIL NIL) (-969 2213970 2214077 2214205 "PROPERTY" 2214393 T PROPERTY (NIL) -8 NIL NIL NIL) (-968 2208028 2212136 2212956 "PRODUCT" 2213196 NIL PRODUCT (NIL T T) -8 NIL NIL NIL) (-967 2207824 2207856 2207915 "PRINT" 2207989 T PRINT (NIL) -7 NIL NIL NIL) (-966 2207164 2207281 2207433 "PRIMES" 2207704 NIL PRIMES (NIL T) -7 NIL NIL NIL) (-965 2205229 2205630 2206096 "PRIMELT" 2206743 NIL PRIMELT (NIL T) -7 NIL NIL NIL) (-964 2204958 2205007 2205035 "PRIMCAT" 2205159 T PRIMCAT (NIL) -9 NIL NIL NIL) (-963 2203965 2204143 2204371 "PRIMARR2" 2204776 NIL PRIMARR2 (NIL T T) -7 NIL NIL NIL) (-962 2200080 2203903 2203948 "PRIMARR" 2203953 NIL PRIMARR (NIL T) -8 NIL NIL NIL) (-961 2199723 2199779 2199890 "PREASSOC" 2200018 NIL PREASSOC (NIL T T) -7 NIL NIL NIL) (-960 2197008 2199181 2199415 "PR" 2199534 NIL PR (NIL T T) -8 NIL NIL NIL) (-959 2196483 2196616 2196644 "PPCURVE" 2196849 T PPCURVE (NIL) -9 NIL 2196985 NIL) (-958 2196078 2196278 2196361 "PORTNUM" 2196420 T PORTNUM (NIL) -8 NIL NIL NIL) (-957 2193437 2193836 2194428 "POLYROOT" 2195659 NIL POLYROOT (NIL T T T T T) -7 NIL NIL NIL) (-956 2192820 2192878 2193112 "POLYLIFT" 2193373 NIL POLYLIFT (NIL T T T T T) -7 NIL NIL NIL) (-955 2189095 2189544 2190173 "POLYCATQ" 2192365 NIL POLYCATQ (NIL T T T T T) -7 NIL NIL NIL) (-954 2175821 2180935 2181000 "POLYCAT" 2184514 NIL POLYCAT (NIL T T T) -9 NIL 2186392 NIL) (-953 2169327 2171170 2173535 "POLYCAT-" 2173540 NIL POLYCAT- (NIL T T T T) -8 NIL NIL NIL) (-952 2168914 2168982 2169102 "POLY2UP" 2169253 NIL POLY2UP (NIL NIL T) -7 NIL NIL NIL) (-951 2168546 2168603 2168712 "POLY2" 2168851 NIL POLY2 (NIL T T) -7 NIL NIL NIL) (-950 2162759 2168150 2168310 "POLY" 2168419 NIL POLY (NIL T) -8 NIL NIL NIL) (-949 2161444 2161683 2161959 "POLUTIL" 2162533 NIL POLUTIL (NIL T T) -7 NIL NIL NIL) (-948 2159799 2160076 2160407 "POLTOPOL" 2161166 NIL POLTOPOL (NIL NIL T) -7 NIL NIL NIL) (-947 2155264 2159735 2159781 "POINT" 2159786 NIL POINT (NIL T) -8 NIL NIL NIL) (-946 2153451 2153808 2154183 "PNTHEORY" 2154909 T PNTHEORY (NIL) -7 NIL NIL NIL) (-945 2151909 2152206 2152605 "PMTOOLS" 2153149 NIL PMTOOLS (NIL T T T) -7 NIL NIL NIL) (-944 2151502 2151580 2151697 "PMSYM" 2151825 NIL PMSYM (NIL T) -7 NIL NIL NIL) (-943 2151012 2151081 2151255 "PMQFCAT" 2151427 NIL PMQFCAT (NIL T T T) -7 NIL NIL NIL) (-942 2150405 2150491 2150653 "PMPREDFS" 2150913 NIL PMPREDFS (NIL T T T) -7 NIL NIL NIL) (-941 2149760 2149870 2150026 "PMPRED" 2150282 NIL PMPRED (NIL T) -7 NIL NIL NIL) (-940 2148424 2148632 2149010 "PMPLCAT" 2149522 NIL PMPLCAT (NIL T T T T T) -7 NIL NIL NIL) (-939 2147956 2148035 2148187 "PMLSAGG" 2148339 NIL PMLSAGG (NIL T T T) -7 NIL NIL NIL) (-938 2147429 2147505 2147687 "PMKERNEL" 2147874 NIL PMKERNEL (NIL T T) -7 NIL NIL NIL) (-937 2147046 2147121 2147234 "PMINS" 2147348 NIL PMINS (NIL T) -7 NIL NIL NIL) (-936 2146488 2146557 2146766 "PMFS" 2146971 NIL PMFS (NIL T T T) -7 NIL NIL NIL) (-935 2145716 2145834 2146039 "PMDOWN" 2146365 NIL PMDOWN (NIL T T T) -7 NIL NIL NIL) (-934 2144989 2145099 2145262 "PMASSFS" 2145603 NIL PMASSFS (NIL T T) -7 NIL NIL NIL) (-933 2144156 2144314 2144495 "PMASS" 2144828 T PMASS (NIL) -7 NIL NIL NIL) (-932 2143811 2143879 2143973 "PLOTTOOL" 2144082 T PLOTTOOL (NIL) -7 NIL NIL NIL) (-931 2139615 2140659 2141580 "PLOT3D" 2142910 T PLOT3D (NIL) -8 NIL NIL NIL) (-930 2138527 2138704 2138939 "PLOT1" 2139419 NIL PLOT1 (NIL T) -7 NIL NIL NIL) (-929 2133134 2134338 2135486 "PLOT" 2137399 T PLOT (NIL) -8 NIL NIL NIL) (-928 2108523 2113200 2118051 "PLEQN" 2128400 NIL PLEQN (NIL T T T T) -7 NIL NIL NIL) (-927 2108216 2108263 2108366 "PINTERPA" 2108470 NIL PINTERPA (NIL T T) -7 NIL NIL NIL) (-926 2107534 2107656 2107836 "PINTERP" 2108081 NIL PINTERP (NIL NIL T) -7 NIL NIL NIL) (-925 2105831 2106806 2106834 "PID" 2107016 T PID (NIL) -9 NIL 2107150 NIL) (-924 2105582 2105619 2105694 "PICOERCE" 2105788 NIL PICOERCE (NIL T) -7 NIL NIL NIL) (-923 2104803 2105351 2105438 "PI" 2105478 T PI (NIL) -8 NIL NIL 2105545) (-922 2104123 2104262 2104438 "PGROEB" 2104659 NIL PGROEB (NIL T) -7 NIL NIL NIL) (-921 2099710 2100524 2101429 "PGE" 2103238 T PGE (NIL) -7 NIL NIL NIL) (-920 2097833 2098080 2098446 "PGCD" 2099427 NIL PGCD (NIL T T T T) -7 NIL NIL NIL) (-919 2097171 2097274 2097435 "PFRPAC" 2097717 NIL PFRPAC (NIL T) -7 NIL NIL NIL) (-918 2093813 2095719 2096072 "PFR" 2096850 NIL PFR (NIL T) -8 NIL NIL NIL) (-917 2092202 2092446 2092771 "PFOTOOLS" 2093560 NIL PFOTOOLS (NIL T T) -7 NIL NIL NIL) (-916 2090735 2090974 2091325 "PFOQ" 2091959 NIL PFOQ (NIL T T T) -7 NIL NIL NIL) (-915 2089236 2089448 2089804 "PFO" 2090519 NIL PFO (NIL T T T T T) -7 NIL NIL NIL) (-914 2086570 2087841 2087869 "PFECAT" 2088454 T PFECAT (NIL) -9 NIL 2088838 NIL) (-913 2086015 2086169 2086383 "PFECAT-" 2086388 NIL PFECAT- (NIL T) -8 NIL NIL NIL) (-912 2084618 2084870 2085171 "PFBRU" 2085764 NIL PFBRU (NIL T T) -7 NIL NIL NIL) (-911 2082484 2082836 2083268 "PFBR" 2084269 NIL PFBR (NIL T T T T) -7 NIL NIL NIL) (-910 2079039 2082373 2082442 "PF" 2082447 NIL PF (NIL NIL) -8 NIL NIL NIL) (-909 2074273 2075246 2076116 "PERMGRP" 2078202 NIL PERMGRP (NIL T) -8 NIL NIL NIL) (-908 2072379 2073336 2073377 "PERMCAT" 2073823 NIL PERMCAT (NIL T) -9 NIL 2074128 NIL) (-907 2072032 2072073 2072197 "PERMAN" 2072332 NIL PERMAN (NIL NIL T) -7 NIL NIL NIL) (-906 2067914 2069408 2070084 "PERM" 2071389 NIL PERM (NIL T) -8 NIL NIL NIL) (-905 2065404 2067579 2067701 "PENDTREE" 2067825 NIL PENDTREE (NIL T) -8 NIL NIL NIL) (-904 2063428 2064196 2064237 "PDRING" 2064894 NIL PDRING (NIL T) -9 NIL 2065180 NIL) (-903 2062531 2062749 2063111 "PDRING-" 2063116 NIL PDRING- (NIL T T) -8 NIL NIL NIL) (-902 2059746 2060524 2061192 "PDEPROB" 2061883 T PDEPROB (NIL) -8 NIL NIL NIL) (-901 2057291 2057795 2058350 "PDEPACK" 2059211 T PDEPACK (NIL) -7 NIL NIL NIL) (-900 2056203 2056393 2056644 "PDECOMP" 2057090 NIL PDECOMP (NIL T T) -7 NIL NIL NIL) (-899 2053782 2054625 2054653 "PDECAT" 2055440 T PDECAT (NIL) -9 NIL 2056153 NIL) (-898 2053533 2053566 2053656 "PCOMP" 2053743 NIL PCOMP (NIL T T) -7 NIL NIL NIL) (-897 2051711 2052334 2052631 "PBWLB" 2053262 NIL PBWLB (NIL T) -8 NIL NIL NIL) (-896 2051343 2051400 2051509 "PATTERN2" 2051648 NIL PATTERN2 (NIL T T) -7 NIL NIL NIL) (-895 2049100 2049488 2049945 "PATTERN1" 2050932 NIL PATTERN1 (NIL T T) -7 NIL NIL NIL) (-894 2041575 2043173 2044511 "PATTERN" 2047783 NIL PATTERN (NIL T) -8 NIL NIL NIL) (-893 2041139 2041206 2041338 "PATRES2" 2041502 NIL PATRES2 (NIL T T T) -7 NIL NIL NIL) (-892 2038507 2039088 2039569 "PATRES" 2040704 NIL PATRES (NIL T T) -8 NIL NIL NIL) (-891 2036390 2036795 2037202 "PATMATCH" 2038174 NIL PATMATCH (NIL T T T) -7 NIL NIL NIL) (-890 2035900 2036109 2036150 "PATMAB" 2036257 NIL PATMAB (NIL T) -9 NIL 2036340 NIL) (-889 2034418 2034754 2035012 "PATLRES" 2035705 NIL PATLRES (NIL T T T) -8 NIL NIL NIL) (-888 2033964 2034087 2034128 "PATAB" 2034133 NIL PATAB (NIL T) -9 NIL 2034305 NIL) (-887 2031445 2031977 2032550 "PARTPERM" 2033411 T PARTPERM (NIL) -7 NIL NIL NIL) (-886 2031066 2031129 2031231 "PARSURF" 2031376 NIL PARSURF (NIL T) -8 NIL NIL NIL) (-885 2030698 2030755 2030864 "PARSU2" 2031003 NIL PARSU2 (NIL T T) -7 NIL NIL NIL) (-884 2030462 2030502 2030569 "PARSER" 2030651 T PARSER (NIL) -7 NIL NIL NIL) (-883 2030083 2030146 2030248 "PARSCURV" 2030393 NIL PARSCURV (NIL T) -8 NIL NIL NIL) (-882 2029715 2029772 2029881 "PARSC2" 2030020 NIL PARSC2 (NIL T T) -7 NIL NIL NIL) (-881 2029354 2029412 2029509 "PARPCURV" 2029651 NIL PARPCURV (NIL T) -8 NIL NIL NIL) (-880 2028986 2029043 2029152 "PARPC2" 2029291 NIL PARPC2 (NIL T T) -7 NIL NIL NIL) (-879 2028047 2028359 2028541 "PARAMAST" 2028824 T PARAMAST (NIL) -8 NIL NIL NIL) (-878 2027567 2027653 2027772 "PAN2EXPR" 2027948 T PAN2EXPR (NIL) -7 NIL NIL NIL) (-877 2026344 2026688 2026916 "PALETTE" 2027359 T PALETTE (NIL) -8 NIL NIL NIL) (-876 2024737 2025349 2025709 "PAIR" 2026030 NIL PAIR (NIL T T) -8 NIL NIL NIL) (-875 2018628 2023996 2024190 "PADICRC" 2024592 NIL PADICRC (NIL NIL T) -8 NIL NIL NIL) (-874 2011878 2017974 2018158 "PADICRAT" 2018476 NIL PADICRAT (NIL NIL) -8 NIL NIL NIL) (-873 2008990 2010552 2010592 "PADICCT" 2011173 NIL PADICCT (NIL NIL) -9 NIL 2011455 NIL) (-872 2007307 2008927 2008972 "PADIC" 2008977 NIL PADIC (NIL NIL) -8 NIL NIL NIL) (-871 2006264 2006464 2006732 "PADEPAC" 2007094 NIL PADEPAC (NIL T NIL NIL) -7 NIL NIL NIL) (-870 2005476 2005609 2005815 "PADE" 2006126 NIL PADE (NIL T T T) -7 NIL NIL NIL) (-869 2003863 2004684 2004964 "OWP" 2005280 NIL OWP (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-868 2003356 2003569 2003666 "OVERSET" 2003786 T OVERSET (NIL) -8 NIL NIL NIL) (-867 2002402 2002961 2003133 "OVAR" 2003224 NIL OVAR (NIL NIL) -8 NIL NIL NIL) (-866 1991274 1993511 1995711 "OUTFORM" 2000222 T OUTFORM (NIL) -8 NIL NIL NIL) (-865 1990610 1990871 1990998 "OUTBFILE" 1991167 T OUTBFILE (NIL) -8 NIL NIL NIL) (-864 1989917 1990082 1990110 "OUTBCON" 1990428 T OUTBCON (NIL) -9 NIL 1990594 NIL) (-863 1989518 1989630 1989787 "OUTBCON-" 1989792 NIL OUTBCON- (NIL T) -8 NIL NIL NIL) (-862 1988782 1988903 1989064 "OUT" 1989377 T OUT (NIL) -7 NIL NIL NIL) (-861 1988162 1988511 1988600 "OSI" 1988713 T OSI (NIL) -8 NIL NIL NIL) (-860 1987692 1988030 1988058 "OSGROUP" 1988063 T OSGROUP (NIL) -9 NIL 1988085 NIL) (-859 1986437 1986664 1986949 "ORTHPOL" 1987439 NIL ORTHPOL (NIL T) -7 NIL NIL NIL) (-858 1984002 1986272 1986393 "OREUP" 1986398 NIL OREUP (NIL NIL T NIL NIL) -8 NIL NIL NIL) (-857 1981419 1983693 1983820 "ORESUP" 1983944 NIL ORESUP (NIL T NIL NIL) -8 NIL NIL NIL) (-856 1978947 1979447 1980008 "OREPCTO" 1980908 NIL OREPCTO (NIL T T) -7 NIL NIL NIL) (-855 1972640 1974834 1974875 "OREPCAT" 1977223 NIL OREPCAT (NIL T) -9 NIL 1978327 NIL) (-854 1969808 1970583 1971634 "OREPCAT-" 1971639 NIL OREPCAT- (NIL T T) -8 NIL NIL NIL) (-853 1968959 1969257 1969285 "ORDSET" 1969594 T ORDSET (NIL) -9 NIL 1969758 NIL) (-852 1968390 1968538 1968762 "ORDSET-" 1968767 NIL ORDSET- (NIL T) -8 NIL NIL NIL) (-851 1966955 1967746 1967774 "ORDRING" 1967976 T ORDRING (NIL) -9 NIL 1968101 NIL) (-850 1966600 1966694 1966838 "ORDRING-" 1966843 NIL ORDRING- (NIL T) -8 NIL NIL NIL) (-849 1965980 1966443 1966471 "ORDMON" 1966476 T ORDMON (NIL) -9 NIL 1966497 NIL) (-848 1965142 1965289 1965484 "ORDFUNS" 1965829 NIL ORDFUNS (NIL NIL T) -7 NIL NIL NIL) (-847 1964480 1964899 1964927 "ORDFIN" 1964992 T ORDFIN (NIL) -9 NIL 1965066 NIL) (-846 1963746 1963873 1964059 "ORDCOMP2" 1964340 NIL ORDCOMP2 (NIL T T) -7 NIL NIL NIL) (-845 1960312 1962332 1962741 "ORDCOMP" 1963370 NIL ORDCOMP (NIL T) -8 NIL NIL NIL) (-844 1956893 1957803 1958617 "OPTPROB" 1959518 T OPTPROB (NIL) -8 NIL NIL NIL) (-843 1953695 1954334 1955038 "OPTPACK" 1956209 T OPTPACK (NIL) -7 NIL NIL NIL) (-842 1951382 1952148 1952176 "OPTCAT" 1952995 T OPTCAT (NIL) -9 NIL 1953645 NIL) (-841 1950766 1951059 1951164 "OPSIG" 1951297 T OPSIG (NIL) -8 NIL NIL NIL) (-840 1950534 1950573 1950639 "OPQUERY" 1950720 T OPQUERY (NIL) -7 NIL NIL NIL) (-839 1949908 1950134 1950175 "OPERCAT" 1950387 NIL OPERCAT (NIL T) -9 NIL 1950484 NIL) (-838 1949663 1949719 1949836 "OPERCAT-" 1949841 NIL OPERCAT- (NIL T T) -8 NIL NIL NIL) (-837 1946796 1947974 1948478 "OP" 1949192 NIL OP (NIL T) -8 NIL NIL NIL) (-836 1946101 1946216 1946390 "ONECOMP2" 1946668 NIL ONECOMP2 (NIL T T) -7 NIL NIL NIL) (-835 1942921 1944898 1945267 "ONECOMP" 1945765 NIL ONECOMP (NIL T) -8 NIL NIL NIL) (-834 1942340 1942446 1942576 "OMSERVER" 1942811 T OMSERVER (NIL) -7 NIL NIL NIL) (-833 1939202 1941780 1941820 "OMSAGG" 1941881 NIL OMSAGG (NIL T) -9 NIL 1941945 NIL) (-832 1937825 1938088 1938370 "OMPKG" 1938940 T OMPKG (NIL) -7 NIL NIL NIL) (-831 1936372 1937374 1937543 "OMLO" 1937706 NIL OMLO (NIL T T) -8 NIL NIL NIL) (-830 1935332 1935479 1935699 "OMEXPR" 1936198 NIL OMEXPR (NIL T) -7 NIL NIL NIL) (-829 1934483 1934753 1934913 "OMERRK" 1935192 T OMERRK (NIL) -8 NIL NIL NIL) (-828 1933774 1934029 1934165 "OMERR" 1934367 T OMERR (NIL) -8 NIL NIL NIL) (-827 1933225 1933451 1933559 "OMENC" 1933686 T OMENC (NIL) -8 NIL NIL NIL) (-826 1927120 1928305 1929476 "OMDEV" 1932074 T OMDEV (NIL) -8 NIL NIL NIL) (-825 1926189 1926360 1926554 "OMCONN" 1926946 T OMCONN (NIL) -8 NIL NIL NIL) (-824 1925619 1925722 1925750 "OM" 1926049 T OM (NIL) -9 NIL NIL NIL) (-823 1924140 1925116 1925144 "OINTDOM" 1925149 T OINTDOM (NIL) -9 NIL 1925170 NIL) (-822 1921485 1922828 1923165 "OFMONOID" 1923835 NIL OFMONOID (NIL T) -8 NIL NIL NIL) (-821 1920896 1921422 1921467 "ODVAR" 1921472 NIL ODVAR (NIL T) -8 NIL NIL NIL) (-820 1918321 1920641 1920796 "ODR" 1920801 NIL ODR (NIL T T NIL) -8 NIL NIL NIL) (-819 1910943 1918097 1918223 "ODPOL" 1918228 NIL ODPOL (NIL T) -8 NIL NIL NIL) (-818 1904772 1910815 1910920 "ODP" 1910925 NIL ODP (NIL NIL T NIL) -8 NIL NIL NIL) (-817 1903538 1903753 1904028 "ODETOOLS" 1904546 NIL ODETOOLS (NIL T T) -7 NIL NIL NIL) (-816 1900505 1901163 1901879 "ODESYS" 1902871 NIL ODESYS (NIL T T) -7 NIL NIL NIL) (-815 1895387 1896295 1897320 "ODERTRIC" 1899580 NIL ODERTRIC (NIL T T) -7 NIL NIL NIL) (-814 1894813 1894895 1895089 "ODERED" 1895299 NIL ODERED (NIL T T T T T) -7 NIL NIL NIL) (-813 1891709 1892255 1892930 "ODERAT" 1894238 NIL ODERAT (NIL T T) -7 NIL NIL NIL) (-812 1888666 1889133 1889730 "ODEPRRIC" 1891238 NIL ODEPRRIC (NIL T T T T) -7 NIL NIL NIL) (-811 1886609 1887205 1887691 "ODEPROB" 1888200 T ODEPROB (NIL) -8 NIL NIL NIL) (-810 1883129 1883614 1884261 "ODEPRIM" 1886088 NIL ODEPRIM (NIL T T T T) -7 NIL NIL NIL) (-809 1882378 1882480 1882740 "ODEPAL" 1883021 NIL ODEPAL (NIL T T T T) -7 NIL NIL NIL) (-808 1878540 1879331 1880195 "ODEPACK" 1881534 T ODEPACK (NIL) -7 NIL NIL NIL) (-807 1877601 1877708 1877930 "ODEINT" 1878429 NIL ODEINT (NIL T T) -7 NIL NIL NIL) (-806 1871702 1873127 1874574 "ODEIFTBL" 1876174 T ODEIFTBL (NIL) -8 NIL NIL NIL) (-805 1867114 1867896 1868844 "ODEEF" 1870865 NIL ODEEF (NIL T T) -7 NIL NIL NIL) (-804 1866463 1866552 1866775 "ODECONST" 1867019 NIL ODECONST (NIL T T T) -7 NIL NIL NIL) (-803 1864588 1865249 1865277 "ODECAT" 1865882 T ODECAT (NIL) -9 NIL 1866413 NIL) (-802 1864226 1864269 1864396 "OCTCT2" 1864539 NIL OCTCT2 (NIL T T T T) -7 NIL NIL NIL) (-801 1861093 1863931 1864053 "OCT" 1864136 NIL OCT (NIL T) -8 NIL NIL NIL) (-800 1860445 1860913 1860941 "OCAMON" 1860946 T OCAMON (NIL) -9 NIL 1860967 NIL) (-799 1855101 1857529 1857569 "OC" 1858666 NIL OC (NIL T) -9 NIL 1859524 NIL) (-798 1852349 1853090 1854073 "OC-" 1854167 NIL OC- (NIL T T) -8 NIL NIL NIL) (-797 1851880 1852221 1852249 "OASGP" 1852254 T OASGP (NIL) -9 NIL 1852274 NIL) (-796 1851141 1851630 1851658 "OAMONS" 1851698 T OAMONS (NIL) -9 NIL 1851741 NIL) (-795 1850555 1850988 1851016 "OAMON" 1851021 T OAMON (NIL) -9 NIL 1851041 NIL) (-794 1849813 1850331 1850359 "OAGROUP" 1850364 T OAGROUP (NIL) -9 NIL 1850384 NIL) (-793 1849503 1849553 1849641 "NUMTUBE" 1849757 NIL NUMTUBE (NIL T) -7 NIL NIL NIL) (-792 1843076 1844594 1846130 "NUMQUAD" 1847987 T NUMQUAD (NIL) -7 NIL NIL NIL) (-791 1838832 1839820 1840845 "NUMODE" 1842071 T NUMODE (NIL) -7 NIL NIL NIL) (-790 1836187 1837067 1837095 "NUMINT" 1838018 T NUMINT (NIL) -9 NIL 1838782 NIL) (-789 1835135 1835332 1835550 "NUMFMT" 1835989 T NUMFMT (NIL) -7 NIL NIL NIL) (-788 1821494 1824439 1826971 "NUMERIC" 1832642 NIL NUMERIC (NIL T) -7 NIL NIL NIL) (-787 1815891 1820943 1821038 "NTSCAT" 1821043 NIL NTSCAT (NIL T T T T) -9 NIL 1821082 NIL) (-786 1815085 1815250 1815443 "NTPOLFN" 1815730 NIL NTPOLFN (NIL T) -7 NIL NIL NIL) (-785 1814717 1814774 1814883 "NSUP2" 1815022 NIL NSUP2 (NIL T T) -7 NIL NIL NIL) (-784 1802839 1811542 1812354 "NSUP" 1813938 NIL NSUP (NIL T) -8 NIL NIL NIL) (-783 1793115 1802613 1802746 "NSMP" 1802751 NIL NSMP (NIL T T) -8 NIL NIL NIL) (-782 1791547 1791848 1792205 "NREP" 1792803 NIL NREP (NIL T) -7 NIL NIL NIL) (-781 1790138 1790390 1790748 "NPCOEF" 1791290 NIL NPCOEF (NIL T T T T T) -7 NIL NIL NIL) (-780 1789204 1789319 1789535 "NORMRETR" 1790019 NIL NORMRETR (NIL T T T T NIL) -7 NIL NIL NIL) (-779 1787245 1787535 1787944 "NORMPK" 1788912 NIL NORMPK (NIL T T T T T) -7 NIL NIL NIL) (-778 1786930 1786958 1787082 "NORMMA" 1787211 NIL NORMMA (NIL T T T T) -7 NIL NIL NIL) (-777 1786719 1786748 1786817 "NONE1" 1786894 NIL NONE1 (NIL T) -7 NIL NIL NIL) (-776 1786519 1786676 1786705 "NONE" 1786710 T NONE (NIL) -8 NIL NIL NIL) (-775 1786016 1786078 1786257 "NODE1" 1786451 NIL NODE1 (NIL T T) -7 NIL NIL NIL) (-774 1784301 1785152 1785407 "NNI" 1785754 T NNI (NIL) -8 NIL NIL 1785989) (-773 1782721 1783034 1783398 "NLINSOL" 1783969 NIL NLINSOL (NIL T) -7 NIL NIL NIL) (-772 1778962 1779957 1780856 "NIPROB" 1781842 T NIPROB (NIL) -8 NIL NIL NIL) (-771 1777719 1777953 1778255 "NFINTBAS" 1778724 NIL NFINTBAS (NIL T T) -7 NIL NIL NIL) (-770 1776893 1777369 1777410 "NETCLT" 1777582 NIL NETCLT (NIL T) -9 NIL 1777664 NIL) (-769 1775601 1775832 1776113 "NCODIV" 1776661 NIL NCODIV (NIL T T) -7 NIL NIL NIL) (-768 1775363 1775400 1775475 "NCNTFRAC" 1775558 NIL NCNTFRAC (NIL T) -7 NIL NIL NIL) (-767 1773543 1773907 1774327 "NCEP" 1774988 NIL NCEP (NIL T) -7 NIL NIL NIL) (-766 1772401 1773167 1773195 "NASRING" 1773305 T NASRING (NIL) -9 NIL 1773385 NIL) (-765 1772196 1772240 1772334 "NASRING-" 1772339 NIL NASRING- (NIL T) -8 NIL NIL NIL) (-764 1771303 1771828 1771856 "NARNG" 1771973 T NARNG (NIL) -9 NIL 1772064 NIL) (-763 1770995 1771062 1771196 "NARNG-" 1771201 NIL NARNG- (NIL T) -8 NIL NIL NIL) (-762 1769874 1770081 1770316 "NAGSP" 1770780 T NAGSP (NIL) -7 NIL NIL NIL) (-761 1761146 1762830 1764503 "NAGS" 1768221 T NAGS (NIL) -7 NIL NIL NIL) (-760 1759694 1760002 1760333 "NAGF07" 1760835 T NAGF07 (NIL) -7 NIL NIL NIL) (-759 1754232 1755523 1756830 "NAGF04" 1758407 T NAGF04 (NIL) -7 NIL NIL NIL) (-758 1747200 1748814 1750447 "NAGF02" 1752619 T NAGF02 (NIL) -7 NIL NIL NIL) (-757 1742424 1743524 1744641 "NAGF01" 1746103 T NAGF01 (NIL) -7 NIL NIL NIL) (-756 1736052 1737618 1739203 "NAGE04" 1740859 T NAGE04 (NIL) -7 NIL NIL NIL) (-755 1727221 1729342 1731472 "NAGE02" 1733942 T NAGE02 (NIL) -7 NIL NIL NIL) (-754 1723174 1724121 1725085 "NAGE01" 1726277 T NAGE01 (NIL) -7 NIL NIL NIL) (-753 1720969 1721503 1722061 "NAGD03" 1722636 T NAGD03 (NIL) -7 NIL NIL NIL) (-752 1712719 1714647 1716601 "NAGD02" 1719035 T NAGD02 (NIL) -7 NIL NIL NIL) (-751 1706530 1707955 1709395 "NAGD01" 1711299 T NAGD01 (NIL) -7 NIL NIL NIL) (-750 1702739 1703561 1704398 "NAGC06" 1705713 T NAGC06 (NIL) -7 NIL NIL NIL) (-749 1701204 1701536 1701892 "NAGC05" 1702403 T NAGC05 (NIL) -7 NIL NIL NIL) (-748 1700580 1700699 1700843 "NAGC02" 1701080 T NAGC02 (NIL) -7 NIL NIL NIL) (-747 1699539 1700122 1700162 "NAALG" 1700241 NIL NAALG (NIL T) -9 NIL 1700302 NIL) (-746 1699374 1699403 1699493 "NAALG-" 1699498 NIL NAALG- (NIL T T) -8 NIL NIL NIL) (-745 1693324 1694432 1695619 "MULTSQFR" 1698270 NIL MULTSQFR (NIL T T T T) -7 NIL NIL NIL) (-744 1692643 1692718 1692902 "MULTFACT" 1693236 NIL MULTFACT (NIL T T T T) -7 NIL NIL NIL) (-743 1685367 1689280 1689333 "MTSCAT" 1690403 NIL MTSCAT (NIL T T) -9 NIL 1690918 NIL) (-742 1685079 1685133 1685225 "MTHING" 1685307 NIL MTHING (NIL T) -7 NIL NIL NIL) (-741 1684871 1684904 1684964 "MSYSCMD" 1685039 T MSYSCMD (NIL) -7 NIL NIL NIL) (-740 1681940 1684432 1684473 "MSETAGG" 1684478 NIL MSETAGG (NIL T) -9 NIL 1684512 NIL) (-739 1678022 1680695 1681015 "MSET" 1681653 NIL MSET (NIL T) -8 NIL NIL NIL) (-738 1673865 1675401 1676146 "MRING" 1677322 NIL MRING (NIL T T) -8 NIL NIL NIL) (-737 1673431 1673498 1673629 "MRF2" 1673792 NIL MRF2 (NIL T T T) -7 NIL NIL NIL) (-736 1673049 1673084 1673228 "MRATFAC" 1673390 NIL MRATFAC (NIL T T T T) -7 NIL NIL NIL) (-735 1670661 1670956 1671387 "MPRFF" 1672754 NIL MPRFF (NIL T T T T) -7 NIL NIL NIL) (-734 1664984 1670515 1670612 "MPOLY" 1670617 NIL MPOLY (NIL NIL T) -8 NIL NIL NIL) (-733 1664474 1664509 1664717 "MPCPF" 1664943 NIL MPCPF (NIL T T T T) -7 NIL NIL NIL) (-732 1663988 1664031 1664215 "MPC3" 1664425 NIL MPC3 (NIL T T T T T T T) -7 NIL NIL NIL) (-731 1663183 1663264 1663485 "MPC2" 1663903 NIL MPC2 (NIL T T T T T T T) -7 NIL NIL NIL) (-730 1661484 1661821 1662211 "MONOTOOL" 1662843 NIL MONOTOOL (NIL T T) -7 NIL NIL NIL) (-729 1660709 1661026 1661054 "MONOID" 1661273 T MONOID (NIL) -9 NIL 1661420 NIL) (-728 1660255 1660374 1660555 "MONOID-" 1660560 NIL MONOID- (NIL T) -8 NIL NIL NIL) (-727 1650739 1656681 1656740 "MONOGEN" 1657414 NIL MONOGEN (NIL T T) -9 NIL 1657870 NIL) (-726 1647978 1648706 1649699 "MONOGEN-" 1649818 NIL MONOGEN- (NIL T T T) -8 NIL NIL NIL) (-725 1646811 1647257 1647285 "MONADWU" 1647677 T MONADWU (NIL) -9 NIL 1647915 NIL) (-724 1646183 1646342 1646590 "MONADWU-" 1646595 NIL MONADWU- (NIL T) -8 NIL NIL NIL) (-723 1645542 1645786 1645814 "MONAD" 1646021 T MONAD (NIL) -9 NIL 1646133 NIL) (-722 1645227 1645305 1645437 "MONAD-" 1645442 NIL MONAD- (NIL T) -8 NIL NIL NIL) (-721 1643516 1644140 1644419 "MOEBIUS" 1644980 NIL MOEBIUS (NIL T) -8 NIL NIL NIL) (-720 1642794 1643198 1643238 "MODULE" 1643243 NIL MODULE (NIL T) -9 NIL 1643282 NIL) (-719 1642362 1642458 1642648 "MODULE-" 1642653 NIL MODULE- (NIL T T) -8 NIL NIL NIL) (-718 1640086 1640770 1641097 "MODRING" 1642186 NIL MODRING (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-717 1637032 1638191 1638712 "MODOP" 1639615 NIL MODOP (NIL T T) -8 NIL NIL NIL) (-716 1635620 1636099 1636376 "MODMONOM" 1636895 NIL MODMONOM (NIL T T NIL) -8 NIL NIL NIL) (-715 1625702 1633911 1634325 "MODMON" 1635257 NIL MODMON (NIL T T) -8 NIL NIL NIL) (-714 1622884 1624570 1624846 "MODFIELD" 1625577 NIL MODFIELD (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-713 1621861 1622165 1622355 "MMLFORM" 1622714 T MMLFORM (NIL) -8 NIL NIL NIL) (-712 1621387 1621430 1621609 "MMAP" 1621812 NIL MMAP (NIL T T T T T T) -7 NIL NIL NIL) (-711 1619466 1620233 1620274 "MLO" 1620697 NIL MLO (NIL T) -9 NIL 1620939 NIL) (-710 1616832 1617348 1617950 "MLIFT" 1618947 NIL MLIFT (NIL T T T T) -7 NIL NIL NIL) (-709 1616223 1616307 1616461 "MKUCFUNC" 1616743 NIL MKUCFUNC (NIL T T T) -7 NIL NIL NIL) (-708 1615822 1615892 1616015 "MKRECORD" 1616146 NIL MKRECORD (NIL T T) -7 NIL NIL NIL) (-707 1614869 1615031 1615259 "MKFUNC" 1615633 NIL MKFUNC (NIL T) -7 NIL NIL NIL) (-706 1614257 1614361 1614517 "MKFLCFN" 1614752 NIL MKFLCFN (NIL T) -7 NIL NIL NIL) (-705 1613534 1613636 1613821 "MKBCFUNC" 1614150 NIL MKBCFUNC (NIL T T T T) -7 NIL NIL NIL) (-704 1610243 1613088 1613224 "MINT" 1613418 T MINT (NIL) -8 NIL NIL NIL) (-703 1609055 1609298 1609575 "MHROWRED" 1609998 NIL MHROWRED (NIL T) -7 NIL NIL NIL) (-702 1604444 1607590 1607995 "MFLOAT" 1608670 T MFLOAT (NIL) -8 NIL NIL NIL) (-701 1603801 1603877 1604048 "MFINFACT" 1604356 NIL MFINFACT (NIL T T T T) -7 NIL NIL NIL) (-700 1600136 1600979 1601858 "MESH" 1602942 T MESH (NIL) -7 NIL NIL NIL) (-699 1598526 1598838 1599191 "MDDFACT" 1599823 NIL MDDFACT (NIL T) -7 NIL NIL NIL) (-698 1595321 1597685 1597726 "MDAGG" 1597981 NIL MDAGG (NIL T) -9 NIL 1598124 NIL) (-697 1585079 1594614 1594821 "MCMPLX" 1595134 T MCMPLX (NIL) -8 NIL NIL NIL) (-696 1584220 1584366 1584566 "MCDEN" 1584928 NIL MCDEN (NIL T T) -7 NIL NIL NIL) (-695 1582110 1582380 1582760 "MCALCFN" 1583950 NIL MCALCFN (NIL T T T T) -7 NIL NIL NIL) (-694 1581035 1581275 1581508 "MAYBE" 1581916 NIL MAYBE (NIL T) -8 NIL NIL NIL) (-693 1578647 1579170 1579732 "MATSTOR" 1580506 NIL MATSTOR (NIL T) -7 NIL NIL NIL) (-692 1574603 1578019 1578267 "MATRIX" 1578432 NIL MATRIX (NIL T) -8 NIL NIL NIL) (-691 1570367 1571076 1571812 "MATLIN" 1573960 NIL MATLIN (NIL T T T T) -7 NIL NIL NIL) (-690 1568961 1569114 1569447 "MATCAT2" 1570202 NIL MATCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-689 1559061 1562250 1562327 "MATCAT" 1567210 NIL MATCAT (NIL T T T) -9 NIL 1568627 NIL) (-688 1555417 1556438 1557794 "MATCAT-" 1557799 NIL MATCAT- (NIL T T T T) -8 NIL NIL NIL) (-687 1553529 1553853 1554237 "MAPPKG3" 1555092 NIL MAPPKG3 (NIL T T T) -7 NIL NIL NIL) (-686 1552510 1552683 1552905 "MAPPKG2" 1553353 NIL MAPPKG2 (NIL T T) -7 NIL NIL NIL) (-685 1551009 1551293 1551620 "MAPPKG1" 1552216 NIL MAPPKG1 (NIL T) -7 NIL NIL NIL) (-684 1550088 1550415 1550592 "MAPPAST" 1550852 T MAPPAST (NIL) -8 NIL NIL NIL) (-683 1549699 1549757 1549880 "MAPHACK3" 1550024 NIL MAPHACK3 (NIL T T T) -7 NIL NIL NIL) (-682 1549291 1549352 1549466 "MAPHACK2" 1549631 NIL MAPHACK2 (NIL T T) -7 NIL NIL NIL) (-681 1548728 1548832 1548974 "MAPHACK1" 1549182 NIL MAPHACK1 (NIL T) -7 NIL NIL NIL) (-680 1546807 1547428 1547732 "MAGMA" 1548456 NIL MAGMA (NIL T) -8 NIL NIL NIL) (-679 1546286 1546531 1546622 "MACROAST" 1546736 T MACROAST (NIL) -8 NIL NIL NIL) (-678 1542704 1544525 1544986 "M3D" 1545858 NIL M3D (NIL T) -8 NIL NIL NIL) (-677 1536812 1541073 1541114 "LZSTAGG" 1541896 NIL LZSTAGG (NIL T) -9 NIL 1542191 NIL) (-676 1532769 1533943 1535400 "LZSTAGG-" 1535405 NIL LZSTAGG- (NIL T T) -8 NIL NIL NIL) (-675 1529856 1530660 1531147 "LWORD" 1532314 NIL LWORD (NIL T) -8 NIL NIL NIL) (-674 1529432 1529660 1529735 "LSTAST" 1529801 T LSTAST (NIL) -8 NIL NIL NIL) (-673 1522629 1529203 1529337 "LSQM" 1529342 NIL LSQM (NIL NIL T) -8 NIL NIL NIL) (-672 1521853 1521992 1522220 "LSPP" 1522484 NIL LSPP (NIL T T T T) -7 NIL NIL NIL) (-671 1518695 1519352 1520065 "LSMP1" 1521172 NIL LSMP1 (NIL T) -7 NIL NIL NIL) (-670 1516530 1516824 1517273 "LSMP" 1518391 NIL LSMP (NIL T T T T) -7 NIL NIL NIL) (-669 1510409 1515697 1515738 "LSAGG" 1515800 NIL LSAGG (NIL T) -9 NIL 1515878 NIL) (-668 1507104 1508028 1509241 "LSAGG-" 1509246 NIL LSAGG- (NIL T T) -8 NIL NIL NIL) (-667 1504703 1506248 1506497 "LPOLY" 1506899 NIL LPOLY (NIL T T) -8 NIL NIL NIL) (-666 1504285 1504370 1504493 "LPEFRAC" 1504612 NIL LPEFRAC (NIL T) -7 NIL NIL NIL) (-665 1503937 1504049 1504077 "LOGIC" 1504188 T LOGIC (NIL) -9 NIL 1504269 NIL) (-664 1503799 1503822 1503893 "LOGIC-" 1503898 NIL LOGIC- (NIL T) -8 NIL NIL NIL) (-663 1502992 1503132 1503325 "LODOOPS" 1503655 NIL LODOOPS (NIL T T) -7 NIL NIL NIL) (-662 1501530 1501765 1502118 "LODOF" 1502739 NIL LODOF (NIL T T) -7 NIL NIL NIL) (-661 1497762 1500179 1500220 "LODOCAT" 1500658 NIL LODOCAT (NIL T) -9 NIL 1500869 NIL) (-660 1497495 1497553 1497680 "LODOCAT-" 1497685 NIL LODOCAT- (NIL T T) -8 NIL NIL NIL) (-659 1494829 1497336 1497454 "LODO2" 1497459 NIL LODO2 (NIL T T) -8 NIL NIL NIL) (-658 1492278 1494766 1494811 "LODO1" 1494816 NIL LODO1 (NIL T) -8 NIL NIL NIL) (-657 1489715 1492194 1492260 "LODO" 1492265 NIL LODO (NIL T NIL) -8 NIL NIL NIL) (-656 1488596 1488761 1489066 "LODEEF" 1489538 NIL LODEEF (NIL T T T) -7 NIL NIL NIL) (-655 1486917 1487690 1487943 "LO" 1488428 NIL LO (NIL T T T) -8 NIL NIL NIL) (-654 1482156 1485047 1485088 "LNAGG" 1486035 NIL LNAGG (NIL T) -9 NIL 1486479 NIL) (-653 1481303 1481517 1481859 "LNAGG-" 1481864 NIL LNAGG- (NIL T T) -8 NIL NIL NIL) (-652 1477439 1478228 1478867 "LMOPS" 1480718 NIL LMOPS (NIL T T NIL) -8 NIL NIL NIL) (-651 1476842 1477230 1477271 "LMODULE" 1477276 NIL LMODULE (NIL T) -9 NIL 1477302 NIL) (-650 1474040 1476487 1476610 "LMDICT" 1476752 NIL LMDICT (NIL T) -8 NIL NIL NIL) (-649 1473446 1473667 1473708 "LLINSET" 1473899 NIL LLINSET (NIL T) -9 NIL 1473990 NIL) (-648 1473145 1473354 1473414 "LITERAL" 1473419 NIL LITERAL (NIL T) -8 NIL NIL NIL) (-647 1472670 1472744 1472883 "LIST3" 1473065 NIL LIST3 (NIL T T T) -7 NIL NIL NIL) (-646 1470804 1471116 1471515 "LIST2MAP" 1472317 NIL LIST2MAP (NIL T T) -7 NIL NIL NIL) (-645 1469811 1469989 1470217 "LIST2" 1470622 NIL LIST2 (NIL T T) -7 NIL NIL NIL) (-644 1462976 1468745 1469049 "LIST" 1469540 NIL LIST (NIL T) -8 NIL NIL NIL) (-643 1462572 1462809 1462850 "LINSET" 1462855 NIL LINSET (NIL T) -9 NIL 1462889 NIL) (-642 1461233 1461903 1461944 "LINEXP" 1462199 NIL LINEXP (NIL T) -9 NIL 1462348 NIL) (-641 1459880 1460140 1460437 "LINDEP" 1460985 NIL LINDEP (NIL T T) -7 NIL NIL NIL) (-640 1456718 1457418 1458176 "LIMITRF" 1459154 NIL LIMITRF (NIL T) -7 NIL NIL NIL) (-639 1455044 1455333 1455735 "LIMITPS" 1456420 NIL LIMITPS (NIL T T) -7 NIL NIL NIL) (-638 1453992 1454461 1454501 "LIECAT" 1454641 NIL LIECAT (NIL T) -9 NIL 1454792 NIL) (-637 1453833 1453860 1453948 "LIECAT-" 1453953 NIL LIECAT- (NIL T T) -8 NIL NIL NIL) (-636 1448293 1453344 1453572 "LIE" 1453654 NIL LIE (NIL T T) -8 NIL NIL NIL) (-635 1440791 1447742 1447907 "LIB" 1448148 T LIB (NIL) -8 NIL NIL NIL) (-634 1436426 1437309 1438244 "LGROBP" 1439908 NIL LGROBP (NIL NIL T) -7 NIL NIL NIL) (-633 1435266 1435958 1435986 "LFCAT" 1436193 T LFCAT (NIL) -9 NIL 1436332 NIL) (-632 1433264 1433538 1433888 "LF" 1434987 NIL LF (NIL T T) -7 NIL NIL NIL) (-631 1430166 1430796 1431484 "LEXTRIPK" 1432628 NIL LEXTRIPK (NIL T NIL) -7 NIL NIL NIL) (-630 1426910 1427736 1428239 "LEXP" 1429746 NIL LEXP (NIL T T NIL) -8 NIL NIL NIL) (-629 1426386 1426631 1426723 "LETAST" 1426838 T LETAST (NIL) -8 NIL NIL NIL) (-628 1424784 1425097 1425498 "LEADCDET" 1426068 NIL LEADCDET (NIL T T T T) -7 NIL NIL NIL) (-627 1423974 1424048 1424277 "LAZM3PK" 1424705 NIL LAZM3PK (NIL T T T T T T) -7 NIL NIL NIL) (-626 1418905 1422051 1422589 "LAUPOL" 1423486 NIL LAUPOL (NIL T T) -8 NIL NIL NIL) (-625 1418484 1418528 1418689 "LAPLACE" 1418855 NIL LAPLACE (NIL T T) -7 NIL NIL NIL) (-624 1417478 1418062 1418103 "LALG" 1418165 NIL LALG (NIL T) -9 NIL 1418224 NIL) (-623 1417192 1417251 1417387 "LALG-" 1417392 NIL LALG- (NIL T T) -8 NIL NIL NIL) (-622 1415131 1416293 1416544 "LA" 1417025 NIL LA (NIL T T T) -8 NIL NIL NIL) (-621 1414966 1414990 1415031 "KVTFROM" 1415093 NIL KVTFROM (NIL T) -9 NIL NIL NIL) (-620 1413889 1414333 1414518 "KTVLOGIC" 1414801 T KTVLOGIC (NIL) -8 NIL NIL NIL) (-619 1413724 1413748 1413789 "KRCFROM" 1413851 NIL KRCFROM (NIL T) -9 NIL NIL NIL) (-618 1412628 1412815 1413114 "KOVACIC" 1413524 NIL KOVACIC (NIL T T) -7 NIL NIL NIL) (-617 1412463 1412487 1412528 "KONVERT" 1412590 NIL KONVERT (NIL T) -9 NIL NIL NIL) (-616 1412298 1412322 1412363 "KOERCE" 1412425 NIL KOERCE (NIL T) -9 NIL NIL NIL) (-615 1411794 1411875 1412007 "KERNEL2" 1412212 NIL KERNEL2 (NIL T T) -7 NIL NIL NIL) (-614 1409624 1410387 1410764 "KERNEL" 1411450 NIL KERNEL (NIL T) -8 NIL NIL NIL) (-613 1403394 1408163 1408217 "KDAGG" 1408594 NIL KDAGG (NIL T T) -9 NIL 1408800 NIL) (-612 1402923 1403047 1403252 "KDAGG-" 1403257 NIL KDAGG- (NIL T T T) -8 NIL NIL NIL) (-611 1396073 1402584 1402739 "KAFILE" 1402801 NIL KAFILE (NIL T) -8 NIL NIL NIL) (-610 1390533 1395584 1395812 "JORDAN" 1395894 NIL JORDAN (NIL T T) -8 NIL NIL NIL) (-609 1389912 1390182 1390303 "JOINAST" 1390432 T JOINAST (NIL) -8 NIL NIL NIL) (-608 1389758 1389817 1389872 "JAVACODE" 1389877 T JAVACODE (NIL) -8 NIL NIL NIL) (-607 1386010 1387963 1388017 "IXAGG" 1388946 NIL IXAGG (NIL T T) -9 NIL 1389405 NIL) (-606 1384929 1385235 1385654 "IXAGG-" 1385659 NIL IXAGG- (NIL T T T) -8 NIL NIL NIL) (-605 1380459 1384851 1384910 "IVECTOR" 1384915 NIL IVECTOR (NIL T NIL) -8 NIL NIL NIL) (-604 1379225 1379462 1379728 "ITUPLE" 1380226 NIL ITUPLE (NIL T) -8 NIL NIL NIL) (-603 1377727 1377904 1378199 "ITRIGMNP" 1379047 NIL ITRIGMNP (NIL T T T) -7 NIL NIL NIL) (-602 1376472 1376676 1376959 "ITFUN3" 1377503 NIL ITFUN3 (NIL T T T) -7 NIL NIL NIL) (-601 1376104 1376161 1376270 "ITFUN2" 1376409 NIL ITFUN2 (NIL T T) -7 NIL NIL NIL) (-600 1374065 1375124 1375402 "ITAYLOR" 1375859 NIL ITAYLOR (NIL T) -8 NIL NIL NIL) (-599 1363010 1368202 1369365 "ISUPS" 1372935 NIL ISUPS (NIL T) -8 NIL NIL NIL) (-598 1362114 1362254 1362490 "ISUMP" 1362857 NIL ISUMP (NIL T T T T) -7 NIL NIL NIL) (-597 1357489 1362059 1362100 "ISTRING" 1362105 NIL ISTRING (NIL NIL) -8 NIL NIL NIL) (-596 1356965 1357210 1357302 "ISAST" 1357417 T ISAST (NIL) -8 NIL NIL NIL) (-595 1356174 1356256 1356472 "IRURPK" 1356879 NIL IRURPK (NIL T T T T T) -7 NIL NIL NIL) (-594 1355110 1355311 1355551 "IRSN" 1355954 T IRSN (NIL) -7 NIL NIL NIL) (-593 1353181 1353536 1353965 "IRRF2F" 1354748 NIL IRRF2F (NIL T) -7 NIL NIL NIL) (-592 1352928 1352966 1353042 "IRREDFFX" 1353137 NIL IRREDFFX (NIL T) -7 NIL NIL NIL) (-591 1351543 1351802 1352101 "IROOT" 1352661 NIL IROOT (NIL T) -7 NIL NIL NIL) (-590 1351462 1351488 1351523 "IRFORM" 1351528 T IRFORM (NIL) -8 NIL NIL NIL) (-589 1350562 1350675 1350889 "IR2F" 1351345 NIL IR2F (NIL T T) -7 NIL NIL NIL) (-588 1348175 1348670 1349236 "IR2" 1350040 NIL IR2 (NIL T T) -7 NIL NIL NIL) (-587 1344779 1345859 1346551 "IR" 1347515 NIL IR (NIL T) -8 NIL NIL NIL) (-586 1344570 1344604 1344664 "IPRNTPK" 1344739 T IPRNTPK (NIL) -7 NIL NIL NIL) (-585 1341153 1344459 1344528 "IPF" 1344533 NIL IPF (NIL NIL) -8 NIL NIL NIL) (-584 1339482 1341078 1341135 "IPADIC" 1341140 NIL IPADIC (NIL NIL NIL) -8 NIL NIL NIL) (-583 1338794 1339042 1339172 "IP4ADDR" 1339372 T IP4ADDR (NIL) -8 NIL NIL NIL) (-582 1338267 1338498 1338608 "IOMODE" 1338704 T IOMODE (NIL) -8 NIL NIL NIL) (-581 1337340 1337864 1337991 "IOBFILE" 1338160 T IOBFILE (NIL) -8 NIL NIL NIL) (-580 1336828 1337244 1337272 "IOBCON" 1337277 T IOBCON (NIL) -9 NIL 1337298 NIL) (-579 1336339 1336397 1336580 "INVLAPLA" 1336764 NIL INVLAPLA (NIL T T) -7 NIL NIL NIL) (-578 1326035 1328377 1330751 "INTTR" 1334015 NIL INTTR (NIL T T) -7 NIL NIL NIL) (-577 1322370 1323112 1323977 "INTTOOLS" 1325220 NIL INTTOOLS (NIL T T) -7 NIL NIL NIL) (-576 1321956 1322047 1322164 "INTSLPE" 1322273 T INTSLPE (NIL) -7 NIL NIL NIL) (-575 1319909 1321879 1321938 "INTRVL" 1321943 NIL INTRVL (NIL T) -8 NIL NIL NIL) (-574 1317511 1318023 1318598 "INTRF" 1319394 NIL INTRF (NIL T) -7 NIL NIL NIL) (-573 1316922 1317019 1317161 "INTRET" 1317409 NIL INTRET (NIL T) -7 NIL NIL NIL) (-572 1314919 1315308 1315778 "INTRAT" 1316530 NIL INTRAT (NIL T T) -7 NIL NIL NIL) (-571 1312182 1312765 1313384 "INTPM" 1314404 NIL INTPM (NIL T T) -7 NIL NIL NIL) (-570 1308950 1309542 1310273 "INTPAF" 1311575 NIL INTPAF (NIL T T T) -7 NIL NIL NIL) (-569 1304129 1305091 1306142 "INTPACK" 1307919 T INTPACK (NIL) -7 NIL NIL NIL) (-568 1303381 1303533 1303741 "INTHERTR" 1303971 NIL INTHERTR (NIL T T) -7 NIL NIL NIL) (-567 1302820 1302900 1303088 "INTHERAL" 1303295 NIL INTHERAL (NIL T T T T) -7 NIL NIL NIL) (-566 1300666 1301109 1301566 "INTHEORY" 1302383 T INTHEORY (NIL) -7 NIL NIL NIL) (-565 1292130 1293733 1295487 "INTG0" 1299036 NIL INTG0 (NIL T T T) -7 NIL NIL NIL) (-564 1278403 1281768 1285153 "INTFTBL" 1288765 T INTFTBL (NIL) -8 NIL NIL NIL) (-563 1277652 1277790 1277963 "INTFACT" 1278262 NIL INTFACT (NIL T) -7 NIL NIL NIL) (-562 1275085 1275529 1276084 "INTEF" 1277208 NIL INTEF (NIL T T) -7 NIL NIL NIL) (-561 1273452 1274191 1274219 "INTDOM" 1274520 T INTDOM (NIL) -9 NIL 1274727 NIL) (-560 1272821 1272995 1273237 "INTDOM-" 1273242 NIL INTDOM- (NIL T) -8 NIL NIL NIL) (-559 1269209 1271137 1271191 "INTCAT" 1271990 NIL INTCAT (NIL T) -9 NIL 1272311 NIL) (-558 1268681 1268784 1268912 "INTBIT" 1269101 T INTBIT (NIL) -7 NIL NIL NIL) (-557 1267380 1267534 1267841 "INTALG" 1268526 NIL INTALG (NIL T T T T T) -7 NIL NIL NIL) (-556 1266863 1266953 1267110 "INTAF" 1267284 NIL INTAF (NIL T T) -7 NIL NIL NIL) (-555 1260208 1266673 1266813 "INTABL" 1266818 NIL INTABL (NIL T T T) -8 NIL NIL NIL) (-554 1259549 1260015 1260080 "INT8" 1260114 T INT8 (NIL) -8 NIL NIL 1260159) (-553 1258889 1259355 1259420 "INT64" 1259454 T INT64 (NIL) -8 NIL NIL 1259499) (-552 1258229 1258695 1258760 "INT32" 1258794 T INT32 (NIL) -8 NIL NIL 1258839) (-551 1257569 1258035 1258100 "INT16" 1258134 T INT16 (NIL) -8 NIL NIL 1258179) (-550 1254519 1257366 1257475 "INT" 1257480 T INT (NIL) -8 NIL NIL NIL) (-549 1249431 1252142 1252170 "INS" 1253104 T INS (NIL) -9 NIL 1253769 NIL) (-548 1246671 1247442 1248416 "INS-" 1248489 NIL INS- (NIL T) -8 NIL NIL NIL) (-547 1245519 1245724 1246000 "INPSIGN" 1246446 NIL INPSIGN (NIL T T) -7 NIL NIL NIL) (-546 1244637 1244754 1244951 "INPRODPF" 1245399 NIL INPRODPF (NIL T T) -7 NIL NIL NIL) (-545 1243531 1243648 1243885 "INPRODFF" 1244517 NIL INPRODFF (NIL T T T T) -7 NIL NIL NIL) (-544 1242531 1242683 1242943 "INNMFACT" 1243367 NIL INNMFACT (NIL T T T T) -7 NIL NIL NIL) (-543 1241728 1241825 1242013 "INMODGCD" 1242430 NIL INMODGCD (NIL T T NIL NIL) -7 NIL NIL NIL) (-542 1240236 1240481 1240805 "INFSP" 1241473 NIL INFSP (NIL T T T) -7 NIL NIL NIL) (-541 1239420 1239537 1239720 "INFPROD0" 1240116 NIL INFPROD0 (NIL T T) -7 NIL NIL NIL) (-540 1239030 1239090 1239188 "INFORM1" 1239355 NIL INFORM1 (NIL T) -7 NIL NIL NIL) (-539 1235885 1237095 1237610 "INFORM" 1238523 T INFORM (NIL) -8 NIL NIL NIL) (-538 1235408 1235497 1235611 "INFINITY" 1235791 T INFINITY (NIL) -7 NIL NIL NIL) (-537 1234584 1235128 1235229 "INETCLTS" 1235327 T INETCLTS (NIL) -8 NIL NIL NIL) (-536 1233200 1233450 1233771 "INEP" 1234332 NIL INEP (NIL T T T) -7 NIL NIL NIL) (-535 1232449 1233097 1233162 "INDE" 1233167 NIL INDE (NIL T) -8 NIL NIL NIL) (-534 1232013 1232081 1232198 "INCRMAPS" 1232376 NIL INCRMAPS (NIL T) -7 NIL NIL NIL) (-533 1230831 1231282 1231488 "INBFILE" 1231827 T INBFILE (NIL) -8 NIL NIL NIL) (-532 1226131 1227067 1228011 "INBFF" 1229919 NIL INBFF (NIL T) -7 NIL NIL NIL) (-531 1225039 1225308 1225336 "INBCON" 1225849 T INBCON (NIL) -9 NIL 1226115 NIL) (-530 1224291 1224514 1224790 "INBCON-" 1224795 NIL INBCON- (NIL T) -8 NIL NIL NIL) (-529 1223770 1224015 1224106 "INAST" 1224220 T INAST (NIL) -8 NIL NIL NIL) (-528 1223197 1223449 1223555 "IMPTAST" 1223684 T IMPTAST (NIL) -8 NIL NIL NIL) (-527 1219642 1223041 1223145 "IMATRIX" 1223150 NIL IMATRIX (NIL T NIL NIL) -8 NIL NIL NIL) (-526 1218354 1218477 1218792 "IMATQF" 1219498 NIL IMATQF (NIL T T T T T T T T) -7 NIL NIL NIL) (-525 1216574 1216801 1217138 "IMATLIN" 1218110 NIL IMATLIN (NIL T T T T) -7 NIL NIL NIL) (-524 1211154 1216498 1216556 "ILIST" 1216561 NIL ILIST (NIL T NIL) -8 NIL NIL NIL) (-523 1209059 1211014 1211127 "IIARRAY2" 1211132 NIL IIARRAY2 (NIL T NIL NIL T T) -8 NIL NIL NIL) (-522 1204459 1208970 1209034 "IFF" 1209039 NIL IFF (NIL NIL NIL) -8 NIL NIL NIL) (-521 1203806 1204076 1204192 "IFAST" 1204363 T IFAST (NIL) -8 NIL NIL NIL) (-520 1198801 1203098 1203286 "IFARRAY" 1203663 NIL IFARRAY (NIL T NIL) -8 NIL NIL NIL) (-519 1197981 1198705 1198778 "IFAMON" 1198783 NIL IFAMON (NIL T T NIL) -8 NIL NIL NIL) (-518 1197565 1197630 1197684 "IEVALAB" 1197891 NIL IEVALAB (NIL T T) -9 NIL NIL NIL) (-517 1197240 1197308 1197468 "IEVALAB-" 1197473 NIL IEVALAB- (NIL T T T) -8 NIL NIL NIL) (-516 1196490 1197129 1197204 "IDPOAMS" 1197209 NIL IDPOAMS (NIL T T) -8 NIL NIL NIL) (-515 1195797 1196379 1196454 "IDPOAM" 1196459 NIL IDPOAM (NIL T T) -8 NIL NIL NIL) (-514 1195428 1195711 1195774 "IDPO" 1195779 NIL IDPO (NIL T T) -8 NIL NIL NIL) (-513 1194487 1194763 1194816 "IDPC" 1195229 NIL IDPC (NIL T T) -9 NIL 1195378 NIL) (-512 1193956 1194379 1194452 "IDPAM" 1194457 NIL IDPAM (NIL T T) -8 NIL NIL NIL) (-511 1193332 1193848 1193921 "IDPAG" 1193926 NIL IDPAG (NIL T T) -8 NIL NIL NIL) (-510 1192977 1193168 1193243 "IDENT" 1193277 T IDENT (NIL) -8 NIL NIL NIL) (-509 1189232 1190080 1190975 "IDECOMP" 1192134 NIL IDECOMP (NIL NIL NIL) -7 NIL NIL NIL) (-508 1182070 1183155 1184202 "IDEAL" 1188268 NIL IDEAL (NIL T T T T) -8 NIL NIL NIL) (-507 1181234 1181346 1181545 "ICDEN" 1181954 NIL ICDEN (NIL T T T T) -7 NIL NIL NIL) (-506 1180305 1180714 1180861 "ICARD" 1181107 T ICARD (NIL) -8 NIL NIL NIL) (-505 1178365 1178678 1179083 "IBPTOOLS" 1179982 NIL IBPTOOLS (NIL T T T T) -7 NIL NIL NIL) (-504 1173972 1177985 1178098 "IBITS" 1178284 NIL IBITS (NIL NIL) -8 NIL NIL NIL) (-503 1170695 1171271 1171966 "IBATOOL" 1173389 NIL IBATOOL (NIL T T T) -7 NIL NIL NIL) (-502 1168474 1168936 1169469 "IBACHIN" 1170230 NIL IBACHIN (NIL T T T) -7 NIL NIL NIL) (-501 1166303 1168320 1168423 "IARRAY2" 1168428 NIL IARRAY2 (NIL T NIL NIL) -8 NIL NIL NIL) (-500 1162409 1166229 1166286 "IARRAY1" 1166291 NIL IARRAY1 (NIL T NIL) -8 NIL NIL NIL) (-499 1156527 1160821 1161302 "IAN" 1161948 T IAN (NIL) -8 NIL NIL NIL) (-498 1156038 1156095 1156268 "IALGFACT" 1156464 NIL IALGFACT (NIL T T T T) -7 NIL NIL NIL) (-497 1155566 1155679 1155707 "HYPCAT" 1155914 T HYPCAT (NIL) -9 NIL NIL NIL) (-496 1155104 1155221 1155407 "HYPCAT-" 1155412 NIL HYPCAT- (NIL T) -8 NIL NIL NIL) (-495 1154699 1154899 1154982 "HOSTNAME" 1155041 T HOSTNAME (NIL) -8 NIL NIL NIL) (-494 1154544 1154581 1154622 "HOMOTOP" 1154627 NIL HOMOTOP (NIL T) -9 NIL 1154660 NIL) (-493 1151176 1152554 1152595 "HOAGG" 1153576 NIL HOAGG (NIL T) -9 NIL 1154255 NIL) (-492 1149770 1150169 1150695 "HOAGG-" 1150700 NIL HOAGG- (NIL T T) -8 NIL NIL NIL) (-491 1143795 1149365 1149514 "HEXADEC" 1149641 T HEXADEC (NIL) -8 NIL NIL NIL) (-490 1142543 1142765 1143028 "HEUGCD" 1143572 NIL HEUGCD (NIL T) -7 NIL NIL NIL) (-489 1141619 1142380 1142510 "HELLFDIV" 1142515 NIL HELLFDIV (NIL T T T T) -8 NIL NIL NIL) (-488 1139798 1141396 1141484 "HEAP" 1141563 NIL HEAP (NIL T) -8 NIL NIL NIL) (-487 1139061 1139350 1139484 "HEADAST" 1139684 T HEADAST (NIL) -8 NIL NIL NIL) (-486 1132934 1138976 1139038 "HDP" 1139043 NIL HDP (NIL NIL T) -8 NIL NIL NIL) (-485 1126953 1132569 1132721 "HDMP" 1132835 NIL HDMP (NIL NIL T) -8 NIL NIL NIL) (-484 1126277 1126417 1126581 "HB" 1126809 T HB (NIL) -7 NIL NIL NIL) (-483 1119665 1126123 1126227 "HASHTBL" 1126232 NIL HASHTBL (NIL T T NIL) -8 NIL NIL NIL) (-482 1119141 1119386 1119478 "HASAST" 1119593 T HASAST (NIL) -8 NIL NIL NIL) (-481 1116923 1118763 1118945 "HACKPI" 1118979 T HACKPI (NIL) -8 NIL NIL NIL) (-480 1112618 1116776 1116889 "GTSET" 1116894 NIL GTSET (NIL T T T T) -8 NIL NIL NIL) (-479 1106035 1112496 1112594 "GSTBL" 1112599 NIL GSTBL (NIL T T T NIL) -8 NIL NIL NIL) (-478 1098315 1105066 1105331 "GSERIES" 1105826 NIL GSERIES (NIL T NIL NIL) -8 NIL NIL NIL) (-477 1097456 1097873 1097901 "GROUP" 1098104 T GROUP (NIL) -9 NIL 1098238 NIL) (-476 1096822 1096981 1097232 "GROUP-" 1097237 NIL GROUP- (NIL T) -8 NIL NIL NIL) (-475 1095189 1095510 1095897 "GROEBSOL" 1096499 NIL GROEBSOL (NIL NIL T T) -7 NIL NIL NIL) (-474 1094103 1094391 1094442 "GRMOD" 1094971 NIL GRMOD (NIL T T) -9 NIL 1095139 NIL) (-473 1093871 1093907 1094035 "GRMOD-" 1094040 NIL GRMOD- (NIL T T T) -8 NIL NIL NIL) (-472 1089161 1090225 1091225 "GRIMAGE" 1092891 T GRIMAGE (NIL) -8 NIL NIL NIL) (-471 1087627 1087888 1088212 "GRDEF" 1088857 T GRDEF (NIL) -7 NIL NIL NIL) (-470 1087071 1087187 1087328 "GRAY" 1087506 T GRAY (NIL) -7 NIL NIL NIL) (-469 1086258 1086664 1086715 "GRALG" 1086868 NIL GRALG (NIL T T) -9 NIL 1086961 NIL) (-468 1085919 1085992 1086155 "GRALG-" 1086160 NIL GRALG- (NIL T T T) -8 NIL NIL NIL) (-467 1082696 1085504 1085682 "GPOLSET" 1085826 NIL GPOLSET (NIL T T T T) -8 NIL NIL NIL) (-466 1082050 1082107 1082365 "GOSPER" 1082633 NIL GOSPER (NIL T T T T T) -7 NIL NIL NIL) (-465 1077782 1078488 1079014 "GMODPOL" 1081749 NIL GMODPOL (NIL NIL T T T NIL T) -8 NIL NIL NIL) (-464 1076787 1076971 1077209 "GHENSEL" 1077594 NIL GHENSEL (NIL T T) -7 NIL NIL NIL) (-463 1070943 1071786 1072806 "GENUPS" 1075871 NIL GENUPS (NIL T T) -7 NIL NIL NIL) (-462 1070640 1070691 1070780 "GENUFACT" 1070886 NIL GENUFACT (NIL T) -7 NIL NIL NIL) (-461 1070052 1070129 1070294 "GENPGCD" 1070558 NIL GENPGCD (NIL T T T T) -7 NIL NIL NIL) (-460 1069526 1069561 1069774 "GENMFACT" 1070011 NIL GENMFACT (NIL T T T T T) -7 NIL NIL NIL) (-459 1068092 1068349 1068656 "GENEEZ" 1069269 NIL GENEEZ (NIL T T) -7 NIL NIL NIL) (-458 1062269 1067703 1067865 "GDMP" 1068015 NIL GDMP (NIL NIL T T) -8 NIL NIL NIL) (-457 1051633 1056040 1057146 "GCNAALG" 1061252 NIL GCNAALG (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-456 1049960 1050822 1050850 "GCDDOM" 1051105 T GCDDOM (NIL) -9 NIL 1051262 NIL) (-455 1049430 1049557 1049772 "GCDDOM-" 1049777 NIL GCDDOM- (NIL T) -8 NIL NIL NIL) (-454 1038046 1040376 1042768 "GBINTERN" 1047121 NIL GBINTERN (NIL T T T T) -7 NIL NIL NIL) (-453 1035883 1036175 1036596 "GBF" 1037721 NIL GBF (NIL T T T T) -7 NIL NIL NIL) (-452 1034664 1034829 1035096 "GBEUCLID" 1035699 NIL GBEUCLID (NIL T T T T) -7 NIL NIL NIL) (-451 1033336 1033521 1033825 "GB" 1034443 NIL GB (NIL T T T T) -7 NIL NIL NIL) (-450 1032685 1032810 1032959 "GAUSSFAC" 1033207 T GAUSSFAC (NIL) -7 NIL NIL NIL) (-449 1031052 1031354 1031668 "GALUTIL" 1032404 NIL GALUTIL (NIL T) -7 NIL NIL NIL) (-448 1029360 1029634 1029958 "GALPOLYU" 1030779 NIL GALPOLYU (NIL T T) -7 NIL NIL NIL) (-447 1026725 1027015 1027422 "GALFACTU" 1029057 NIL GALFACTU (NIL T T T) -7 NIL NIL NIL) (-446 1018530 1020030 1021638 "GALFACT" 1025157 NIL GALFACT (NIL T) -7 NIL NIL NIL) (-445 1015918 1016576 1016604 "FVFUN" 1017760 T FVFUN (NIL) -9 NIL 1018480 NIL) (-444 1015184 1015366 1015394 "FVC" 1015685 T FVC (NIL) -9 NIL 1015868 NIL) (-443 1014827 1015009 1015077 "FUNDESC" 1015136 T FUNDESC (NIL) -8 NIL NIL NIL) (-442 1014442 1014624 1014705 "FUNCTION" 1014779 NIL FUNCTION (NIL NIL) -8 NIL NIL NIL) (-441 1013233 1013743 1013946 "FTEM" 1014259 T FTEM (NIL) -8 NIL NIL NIL) (-440 1010989 1011564 1012027 "FT" 1012790 T FT (NIL) -8 NIL NIL NIL) (-439 1009280 1009569 1009966 "FSUPFACT" 1010680 NIL FSUPFACT (NIL T T T) -7 NIL NIL NIL) (-438 1007677 1007966 1008298 "FST" 1008968 T FST (NIL) -8 NIL NIL NIL) (-437 1006876 1006982 1007170 "FSRED" 1007559 NIL FSRED (NIL T T) -7 NIL NIL NIL) (-436 1005575 1005831 1006178 "FSPRMELT" 1006591 NIL FSPRMELT (NIL T T) -7 NIL NIL NIL) (-435 1002881 1003319 1003805 "FSPECF" 1005138 NIL FSPECF (NIL T T) -7 NIL NIL NIL) (-434 1002409 1002463 1002633 "FSINT" 1002822 NIL FSINT (NIL T T) -7 NIL NIL NIL) (-433 1000701 1001402 1001705 "FSERIES" 1002188 NIL FSERIES (NIL T T) -8 NIL NIL NIL) (-432 999743 999859 1000083 "FSCINT" 1000581 NIL FSCINT (NIL T T) -7 NIL NIL NIL) (-431 998785 998928 999155 "FSAGG2" 999596 NIL FSAGG2 (NIL T T T T) -7 NIL NIL NIL) (-430 994993 997729 997770 "FSAGG" 998140 NIL FSAGG (NIL T) -9 NIL 998399 NIL) (-429 992755 993356 994152 "FSAGG-" 994247 NIL FSAGG- (NIL T T) -8 NIL NIL NIL) (-428 990437 990717 991264 "FS2UPS" 992473 NIL FS2UPS (NIL T T T T T NIL) -7 NIL NIL NIL) (-427 989315 989486 989788 "FS2EXPXP" 990262 NIL FS2EXPXP (NIL T T NIL NIL) -7 NIL NIL NIL) (-426 988949 988992 989121 "FS2" 989266 NIL FS2 (NIL T T T T) -7 NIL NIL NIL) (-425 970616 978918 978959 "FS" 982843 NIL FS (NIL T) -9 NIL 985132 NIL) (-424 959340 962306 966336 "FS-" 966636 NIL FS- (NIL T T) -8 NIL NIL NIL) (-423 958766 958881 959033 "FRUTIL" 959220 NIL FRUTIL (NIL T) -7 NIL NIL NIL) (-422 953767 956409 956449 "FRNAALG" 957845 NIL FRNAALG (NIL T) -9 NIL 958452 NIL) (-421 949491 950550 951808 "FRNAALG-" 952558 NIL FRNAALG- (NIL T T) -8 NIL NIL NIL) (-420 949129 949172 949299 "FRNAAF2" 949442 NIL FRNAAF2 (NIL T T T T) -7 NIL NIL NIL) (-419 947509 947983 948278 "FRMOD" 948941 NIL FRMOD (NIL T T T T NIL) -8 NIL NIL NIL) (-418 946704 946791 947080 "FRIDEAL2" 947416 NIL FRIDEAL2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-417 944455 945087 945404 "FRIDEAL" 946495 NIL FRIDEAL (NIL T T T T) -8 NIL NIL NIL) (-416 943595 944002 944043 "FRETRCT" 944048 NIL FRETRCT (NIL T) -9 NIL 944224 NIL) (-415 942728 942952 943296 "FRETRCT-" 943301 NIL FRETRCT- (NIL T T) -8 NIL NIL NIL) (-414 939816 941026 941085 "FRAMALG" 941967 NIL FRAMALG (NIL T T) -9 NIL 942259 NIL) (-413 937950 938405 939035 "FRAMALG-" 939258 NIL FRAMALG- (NIL T T T) -8 NIL NIL NIL) (-412 937586 937643 937750 "FRAC2" 937887 NIL FRAC2 (NIL T T) -7 NIL NIL NIL) (-411 931528 937061 937337 "FRAC" 937342 NIL FRAC (NIL T) -8 NIL NIL NIL) (-410 931164 931221 931328 "FR2" 931465 NIL FR2 (NIL T T) -7 NIL NIL NIL) (-409 922692 926740 928071 "FR" 929865 NIL FR (NIL T) -8 NIL NIL NIL) (-408 917209 920098 920126 "FPS" 921245 T FPS (NIL) -9 NIL 921802 NIL) (-407 916658 916767 916931 "FPS-" 917077 NIL FPS- (NIL T) -8 NIL NIL NIL) (-406 913962 915629 915657 "FPC" 915882 T FPC (NIL) -9 NIL 916024 NIL) (-405 913755 913795 913892 "FPC-" 913897 NIL FPC- (NIL T) -8 NIL NIL NIL) (-404 912545 913243 913284 "FPATMAB" 913289 NIL FPATMAB (NIL T) -9 NIL 913441 NIL) (-403 910218 910721 911147 "FPARFRAC" 912182 NIL FPARFRAC (NIL T T) -8 NIL NIL NIL) (-402 905651 906149 906831 "FORTRAN" 909650 NIL FORTRAN (NIL NIL NIL NIL NIL) -8 NIL NIL NIL) (-401 903327 903889 903917 "FORTFN" 904977 T FORTFN (NIL) -9 NIL 905601 NIL) (-400 903091 903141 903169 "FORTCAT" 903228 T FORTCAT (NIL) -9 NIL 903290 NIL) (-399 900807 901307 901846 "FORT" 902572 T FORT (NIL) -7 NIL NIL NIL) (-398 900595 900625 900694 "FORMULA1" 900771 NIL FORMULA1 (NIL T) -7 NIL NIL NIL) (-397 898701 899211 899601 "FORMULA" 900225 T FORMULA (NIL) -8 NIL NIL NIL) (-396 898224 898276 898449 "FORDER" 898643 NIL FORDER (NIL T T T T) -7 NIL NIL NIL) (-395 897320 897484 897677 "FOP" 898051 T FOP (NIL) -7 NIL NIL NIL) (-394 895901 896600 896774 "FNLA" 897202 NIL FNLA (NIL NIL NIL T) -8 NIL NIL NIL) (-393 894630 895045 895073 "FNCAT" 895533 T FNCAT (NIL) -9 NIL 895793 NIL) (-392 894169 894589 894617 "FNAME" 894622 T FNAME (NIL) -8 NIL NIL NIL) (-391 892732 893695 893723 "FMTC" 893728 T FMTC (NIL) -9 NIL 893764 NIL) (-390 891485 892668 892714 "FMONOID" 892719 NIL FMONOID (NIL T) -8 NIL NIL NIL) (-389 888313 889481 889522 "FMONCAT" 890739 NIL FMONCAT (NIL T) -9 NIL 891344 NIL) (-388 885737 886383 886411 "FMFUN" 887555 T FMFUN (NIL) -9 NIL 888263 NIL) (-387 882816 883676 883730 "FMCAT" 884925 NIL FMCAT (NIL T T) -9 NIL 885420 NIL) (-386 882085 882266 882294 "FMC" 882584 T FMC (NIL) -9 NIL 882766 NIL) (-385 880951 881851 881951 "FM1" 882030 NIL FM1 (NIL T T) -8 NIL NIL NIL) (-384 880143 880693 880842 "FM" 880847 NIL FM (NIL T T) -8 NIL NIL NIL) (-383 877917 878333 878827 "FLOATRP" 879694 NIL FLOATRP (NIL T) -7 NIL NIL NIL) (-382 875355 875855 876433 "FLOATCP" 877384 NIL FLOATCP (NIL T) -7 NIL NIL NIL) (-381 868933 873084 873705 "FLOAT" 874754 T FLOAT (NIL) -8 NIL NIL NIL) (-380 867673 868511 868552 "FLINEXP" 868557 NIL FLINEXP (NIL T) -9 NIL 868650 NIL) (-379 866827 867062 867390 "FLINEXP-" 867395 NIL FLINEXP- (NIL T T) -8 NIL NIL NIL) (-378 865903 866047 866271 "FLASORT" 866679 NIL FLASORT (NIL T T) -7 NIL NIL NIL) (-377 863019 863887 863939 "FLALG" 865166 NIL FLALG (NIL T T) -9 NIL 865633 NIL) (-376 862061 862204 862431 "FLAGG2" 862872 NIL FLAGG2 (NIL T T T T) -7 NIL NIL NIL) (-375 855797 859547 859588 "FLAGG" 860850 NIL FLAGG (NIL T) -9 NIL 861502 NIL) (-374 854523 854862 855352 "FLAGG-" 855357 NIL FLAGG- (NIL T T) -8 NIL NIL NIL) (-373 851374 852382 852441 "FINRALG" 853569 NIL FINRALG (NIL T T) -9 NIL 854077 NIL) (-372 850534 850763 851102 "FINRALG-" 851107 NIL FINRALG- (NIL T T T) -8 NIL NIL NIL) (-371 849914 850153 850181 "FINITE" 850377 T FINITE (NIL) -9 NIL 850484 NIL) (-370 842271 844458 844498 "FINAALG" 848165 NIL FINAALG (NIL T) -9 NIL 849618 NIL) (-369 837603 838653 839797 "FINAALG-" 841176 NIL FINAALG- (NIL T T) -8 NIL NIL NIL) (-368 836261 836599 836653 "FILECAT" 837337 NIL FILECAT (NIL T T) -9 NIL 837553 NIL) (-367 835629 836016 836119 "FILE" 836191 NIL FILE (NIL T) -8 NIL NIL NIL) (-366 833347 834873 834901 "FIELD" 834941 T FIELD (NIL) -9 NIL 835021 NIL) (-365 831967 832352 832863 "FIELD-" 832868 NIL FIELD- (NIL T) -8 NIL NIL NIL) (-364 829817 830602 830949 "FGROUP" 831653 NIL FGROUP (NIL T) -8 NIL NIL NIL) (-363 828907 829071 829291 "FGLMICPK" 829649 NIL FGLMICPK (NIL T NIL) -7 NIL NIL NIL) (-362 824741 828832 828889 "FFX" 828894 NIL FFX (NIL T NIL) -8 NIL NIL NIL) (-361 824342 824403 824538 "FFSLPE" 824674 NIL FFSLPE (NIL T T T) -7 NIL NIL NIL) (-360 823846 823882 824091 "FFPOLY2" 824300 NIL FFPOLY2 (NIL T T) -7 NIL NIL NIL) (-359 819836 820618 821414 "FFPOLY" 823082 NIL FFPOLY (NIL T) -7 NIL NIL NIL) (-358 815682 819755 819818 "FFP" 819823 NIL FFP (NIL T NIL) -8 NIL NIL NIL) (-357 810810 815025 815215 "FFNBX" 815536 NIL FFNBX (NIL T NIL) -8 NIL NIL NIL) (-356 805740 809945 810203 "FFNBP" 810664 NIL FFNBP (NIL T NIL) -8 NIL NIL NIL) (-355 800375 805024 805235 "FFNB" 805573 NIL FFNB (NIL NIL NIL) -8 NIL NIL NIL) (-354 799207 799405 799720 "FFINTBAS" 800172 NIL FFINTBAS (NIL T T T) -7 NIL NIL NIL) (-353 795278 797496 797524 "FFIELDC" 798144 T FFIELDC (NIL) -9 NIL 798520 NIL) (-352 793940 794311 794808 "FFIELDC-" 794813 NIL FFIELDC- (NIL T) -8 NIL NIL NIL) (-351 793509 793555 793679 "FFHOM" 793882 NIL FFHOM (NIL T T T) -7 NIL NIL NIL) (-350 791204 791691 792208 "FFF" 793024 NIL FFF (NIL T) -7 NIL NIL NIL) (-349 786824 790946 791047 "FFCGX" 791147 NIL FFCGX (NIL T NIL) -8 NIL NIL NIL) (-348 782448 786556 786663 "FFCGP" 786767 NIL FFCGP (NIL T NIL) -8 NIL NIL NIL) (-347 777633 782175 782283 "FFCG" 782384 NIL FFCG (NIL NIL NIL) -8 NIL NIL NIL) (-346 777044 777087 777322 "FFCAT2" 777584 NIL FFCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-345 758449 767521 767607 "FFCAT" 772772 NIL FFCAT (NIL T T T) -9 NIL 774223 NIL) (-344 753646 754694 756008 "FFCAT-" 757238 NIL FFCAT- (NIL T T T T) -8 NIL NIL NIL) (-343 749046 753557 753621 "FF" 753626 NIL FF (NIL NIL NIL) -8 NIL NIL NIL) (-342 738371 742018 743238 "FEXPR" 747898 NIL FEXPR (NIL NIL NIL T) -8 NIL NIL NIL) (-341 737371 737806 737847 "FEVALAB" 737931 NIL FEVALAB (NIL T) -9 NIL 738192 NIL) (-340 736530 736740 737078 "FEVALAB-" 737083 NIL FEVALAB- (NIL T T) -8 NIL NIL NIL) (-339 733550 734291 734406 "FDIVCAT" 735974 NIL FDIVCAT (NIL T T T T) -9 NIL 736411 NIL) (-338 733312 733339 733509 "FDIVCAT-" 733514 NIL FDIVCAT- (NIL T T T T T) -8 NIL NIL NIL) (-337 732532 732619 732896 "FDIV2" 733219 NIL FDIV2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-336 731098 731915 732118 "FDIV" 732431 NIL FDIV (NIL T T T T) -8 NIL NIL NIL) (-335 730072 730393 730595 "FCTRDATA" 730916 T FCTRDATA (NIL) -8 NIL NIL NIL) (-334 728758 729017 729306 "FCPAK1" 729803 T FCPAK1 (NIL) -7 NIL NIL NIL) (-333 727857 728258 728399 "FCOMP" 728649 NIL FCOMP (NIL T) -8 NIL NIL NIL) (-332 711562 715007 718545 "FC" 724339 T FC (NIL) -8 NIL NIL NIL) (-331 703927 707953 707993 "FAXF" 709795 NIL FAXF (NIL T) -9 NIL 710487 NIL) (-330 701203 701861 702686 "FAXF-" 703151 NIL FAXF- (NIL T T) -8 NIL NIL NIL) (-329 696255 700579 700755 "FARRAY" 701060 NIL FARRAY (NIL T) -8 NIL NIL NIL) (-328 691156 693216 693269 "FAMR" 694292 NIL FAMR (NIL T T) -9 NIL 694752 NIL) (-327 690046 690348 690783 "FAMR-" 690788 NIL FAMR- (NIL T T T) -8 NIL NIL NIL) (-326 689215 689968 690021 "FAMONOID" 690026 NIL FAMONOID (NIL T) -8 NIL NIL NIL) (-325 687001 687711 687764 "FAMONC" 688705 NIL FAMONC (NIL T T) -9 NIL 689091 NIL) (-324 685665 686755 686892 "FAGROUP" 686897 NIL FAGROUP (NIL T) -8 NIL NIL NIL) (-323 683460 683779 684182 "FACUTIL" 685346 NIL FACUTIL (NIL T T T T) -7 NIL NIL NIL) (-322 682559 682744 682966 "FACTFUNC" 683270 NIL FACTFUNC (NIL T) -7 NIL NIL NIL) (-321 674983 681862 682061 "EXPUPXS" 682415 NIL EXPUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-320 672466 673006 673592 "EXPRTUBE" 674417 T EXPRTUBE (NIL) -7 NIL NIL NIL) (-319 668737 669329 670059 "EXPRODE" 671805 NIL EXPRODE (NIL T T) -7 NIL NIL NIL) (-318 663291 663878 664684 "EXPR2UPS" 668035 NIL EXPR2UPS (NIL T T) -7 NIL NIL NIL) (-317 662923 662980 663089 "EXPR2" 663228 NIL EXPR2 (NIL T T) -7 NIL NIL NIL) (-316 648469 661572 662001 "EXPR" 662527 NIL EXPR (NIL T) -8 NIL NIL NIL) (-315 639885 647622 647912 "EXPEXPAN" 648306 NIL EXPEXPAN (NIL T T NIL NIL) -8 NIL NIL NIL) (-314 639365 639609 639700 "EXITAST" 639814 T EXITAST (NIL) -8 NIL NIL NIL) (-313 639165 639322 639351 "EXIT" 639356 T EXIT (NIL) -8 NIL NIL NIL) (-312 638792 638854 638967 "EVALCYC" 639097 NIL EVALCYC (NIL T) -7 NIL NIL NIL) (-311 638333 638451 638492 "EVALAB" 638662 NIL EVALAB (NIL T) -9 NIL 638766 NIL) (-310 637814 637936 638157 "EVALAB-" 638162 NIL EVALAB- (NIL T T) -8 NIL NIL NIL) (-309 635182 636484 636512 "EUCDOM" 637067 T EUCDOM (NIL) -9 NIL 637417 NIL) (-308 633587 634029 634619 "EUCDOM-" 634624 NIL EUCDOM- (NIL T) -8 NIL NIL NIL) (-307 633219 633276 633385 "ESTOOLS2" 633524 NIL ESTOOLS2 (NIL T T) -7 NIL NIL NIL) (-306 632970 633012 633092 "ESTOOLS1" 633171 NIL ESTOOLS1 (NIL T) -7 NIL NIL NIL) (-305 620508 623268 626018 "ESTOOLS" 630240 T ESTOOLS (NIL) -7 NIL NIL NIL) (-304 620253 620285 620367 "ESCONT1" 620470 NIL ESCONT1 (NIL NIL NIL) -7 NIL NIL NIL) (-303 616627 617388 618168 "ESCONT" 619493 T ESCONT (NIL) -7 NIL NIL NIL) (-302 616302 616352 616452 "ES2" 616571 NIL ES2 (NIL T T) -7 NIL NIL NIL) (-301 615932 615990 616099 "ES1" 616238 NIL ES1 (NIL T T) -7 NIL NIL NIL) (-300 609969 611577 611605 "ES" 614373 T ES (NIL) -9 NIL 615783 NIL) (-299 604916 606203 608020 "ES-" 608184 NIL ES- (NIL T) -8 NIL NIL NIL) (-298 604132 604261 604437 "ERROR" 604760 T ERROR (NIL) -7 NIL NIL NIL) (-297 597526 603991 604082 "EQTBL" 604087 NIL EQTBL (NIL T T) -8 NIL NIL NIL) (-296 597158 597215 597324 "EQ2" 597463 NIL EQ2 (NIL T T) -7 NIL NIL NIL) (-295 589661 592472 593921 "EQ" 595742 NIL -3961 (NIL T) -8 NIL NIL NIL) (-294 584951 585999 587092 "EP" 588600 NIL EP (NIL T) -7 NIL NIL NIL) (-293 583551 583842 584148 "ENV" 584665 T ENV (NIL) -8 NIL NIL NIL) (-292 582645 583199 583227 "ENTIRER" 583232 T ENTIRER (NIL) -9 NIL 583278 NIL) (-291 579168 580654 581024 "EMR" 582444 NIL EMR (NIL T T T NIL NIL NIL) -8 NIL NIL NIL) (-290 578312 578497 578551 "ELTAGG" 578931 NIL ELTAGG (NIL T T) -9 NIL 579142 NIL) (-289 578031 578093 578234 "ELTAGG-" 578239 NIL ELTAGG- (NIL T T T) -8 NIL NIL NIL) (-288 577820 577849 577903 "ELTAB" 577987 NIL ELTAB (NIL T T) -9 NIL NIL NIL) (-287 576946 577092 577291 "ELFUTS" 577671 NIL ELFUTS (NIL T T) -7 NIL NIL NIL) (-286 576688 576744 576772 "ELEMFUN" 576877 T ELEMFUN (NIL) -9 NIL NIL NIL) (-285 576558 576579 576647 "ELEMFUN-" 576652 NIL ELEMFUN- (NIL T) -8 NIL NIL NIL) (-284 571402 574658 574699 "ELAGG" 575639 NIL ELAGG (NIL T) -9 NIL 576102 NIL) (-283 569687 570121 570784 "ELAGG-" 570789 NIL ELAGG- (NIL T T) -8 NIL NIL NIL) (-282 568348 568627 568921 "ELABEXPR" 569413 T ELABEXPR (NIL) -8 NIL NIL NIL) (-281 561339 563015 563842 "EFUPXS" 567624 NIL EFUPXS (NIL T T T T) -8 NIL NIL NIL) (-280 554916 556590 557400 "EFULS" 560615 NIL EFULS (NIL T T T) -8 NIL NIL NIL) (-279 552401 552759 553231 "EFSTRUC" 554548 NIL EFSTRUC (NIL T T) -7 NIL NIL NIL) (-278 542192 543758 545306 "EF" 550916 NIL EF (NIL T T) -7 NIL NIL NIL) (-277 541266 541677 541826 "EAB" 542063 T EAB (NIL) -8 NIL NIL NIL) (-276 540448 541225 541253 "E04UCFA" 541258 T E04UCFA (NIL) -8 NIL NIL NIL) (-275 539630 540407 540435 "E04NAFA" 540440 T E04NAFA (NIL) -8 NIL NIL NIL) (-274 538812 539589 539617 "E04MBFA" 539622 T E04MBFA (NIL) -8 NIL NIL NIL) (-273 537994 538771 538799 "E04JAFA" 538804 T E04JAFA (NIL) -8 NIL NIL NIL) (-272 537178 537953 537981 "E04GCFA" 537986 T E04GCFA (NIL) -8 NIL NIL NIL) (-271 536362 537137 537165 "E04FDFA" 537170 T E04FDFA (NIL) -8 NIL NIL NIL) (-270 535544 536321 536349 "E04DGFA" 536354 T E04DGFA (NIL) -8 NIL NIL NIL) (-269 529717 531069 532433 "E04AGNT" 534200 T E04AGNT (NIL) -7 NIL NIL NIL) (-268 528397 528903 528943 "DVARCAT" 529418 NIL DVARCAT (NIL T) -9 NIL 529617 NIL) (-267 527601 527813 528127 "DVARCAT-" 528132 NIL DVARCAT- (NIL T T) -8 NIL NIL NIL) (-266 520779 527400 527529 "DSMP" 527534 NIL DSMP (NIL T T T) -8 NIL NIL NIL) (-265 520444 520503 520601 "DROPT1" 520714 NIL DROPT1 (NIL T) -7 NIL NIL NIL) (-264 515559 516685 517822 "DROPT0" 519327 T DROPT0 (NIL) -7 NIL NIL NIL) (-263 510340 511504 512572 "DROPT" 514511 T DROPT (NIL) -8 NIL NIL NIL) (-262 508685 509010 509396 "DRAWPT" 509974 T DRAWPT (NIL) -7 NIL NIL NIL) (-261 508318 508371 508489 "DRAWHACK" 508626 NIL DRAWHACK (NIL T) -7 NIL NIL NIL) (-260 507049 507318 507609 "DRAWCX" 508047 T DRAWCX (NIL) -7 NIL NIL NIL) (-259 506564 506633 506784 "DRAWCURV" 506975 NIL DRAWCURV (NIL T T) -7 NIL NIL NIL) (-258 497032 498994 501109 "DRAWCFUN" 504469 T DRAWCFUN (NIL) -7 NIL NIL NIL) (-257 491619 492542 493621 "DRAW" 496006 NIL DRAW (NIL T) -7 NIL NIL NIL) (-256 488385 490314 490355 "DQAGG" 490984 NIL DQAGG (NIL T) -9 NIL 491257 NIL) (-255 476545 482978 483061 "DPOLCAT" 484913 NIL DPOLCAT (NIL T T T T) -9 NIL 485458 NIL) (-254 471432 472764 474705 "DPOLCAT-" 474710 NIL DPOLCAT- (NIL T T T T T) -8 NIL NIL NIL) (-253 464561 471293 471391 "DPMO" 471396 NIL DPMO (NIL NIL T T) -8 NIL NIL NIL) (-252 457593 464341 464508 "DPMM" 464513 NIL DPMM (NIL NIL T T T) -8 NIL NIL NIL) (-251 457071 457285 457383 "DOMTMPLT" 457515 T DOMTMPLT (NIL) -8 NIL NIL NIL) (-250 456504 456873 456953 "DOMCTOR" 457011 T DOMCTOR (NIL) -8 NIL NIL NIL) (-249 455716 455984 456135 "DOMAIN" 456373 T DOMAIN (NIL) -8 NIL NIL NIL) (-248 449735 455351 455503 "DMP" 455617 NIL DMP (NIL NIL T) -8 NIL NIL NIL) (-247 449335 449391 449535 "DLP" 449673 NIL DLP (NIL T) -7 NIL NIL NIL) (-246 443159 448662 448852 "DLIST" 449177 NIL DLIST (NIL T) -8 NIL NIL NIL) (-245 439957 442012 442053 "DLAGG" 442603 NIL DLAGG (NIL T) -9 NIL 442833 NIL) (-244 438633 439297 439325 "DIVRING" 439417 T DIVRING (NIL) -9 NIL 439500 NIL) (-243 437870 438060 438360 "DIVRING-" 438365 NIL DIVRING- (NIL T) -8 NIL NIL NIL) (-242 435972 436329 436735 "DISPLAY" 437484 T DISPLAY (NIL) -7 NIL NIL NIL) (-241 434820 435023 435288 "DIRPROD2" 435765 NIL DIRPROD2 (NIL NIL T T) -7 NIL NIL NIL) (-240 428715 434734 434797 "DIRPROD" 434802 NIL DIRPROD (NIL NIL T) -8 NIL NIL NIL) (-239 417497 423496 423549 "DIRPCAT" 423959 NIL DIRPCAT (NIL NIL T) -9 NIL 424799 NIL) (-238 414823 415465 416346 "DIRPCAT-" 416683 NIL DIRPCAT- (NIL T NIL T) -8 NIL NIL NIL) (-237 414110 414270 414456 "DIOSP" 414657 T DIOSP (NIL) -7 NIL NIL NIL) (-236 410765 413022 413063 "DIOPS" 413497 NIL DIOPS (NIL T) -9 NIL 413726 NIL) (-235 410314 410428 410619 "DIOPS-" 410624 NIL DIOPS- (NIL T T) -8 NIL NIL NIL) (-234 409137 409765 409793 "DIFRING" 409980 T DIFRING (NIL) -9 NIL 410090 NIL) (-233 408783 408860 409012 "DIFRING-" 409017 NIL DIFRING- (NIL T) -8 NIL NIL NIL) (-232 406519 407791 407832 "DIFEXT" 408195 NIL DIFEXT (NIL T) -9 NIL 408489 NIL) (-231 404804 405232 405898 "DIFEXT-" 405903 NIL DIFEXT- (NIL T T) -8 NIL NIL NIL) (-230 402079 404336 404377 "DIAGG" 404382 NIL DIAGG (NIL T) -9 NIL 404402 NIL) (-229 401463 401620 401872 "DIAGG-" 401877 NIL DIAGG- (NIL T T) -8 NIL NIL NIL) (-228 396879 400422 400699 "DHMATRIX" 401232 NIL DHMATRIX (NIL T) -8 NIL NIL NIL) (-227 392491 393400 394410 "DFSFUN" 395889 T DFSFUN (NIL) -7 NIL NIL NIL) (-226 387574 391422 391734 "DFLOAT" 392199 T DFLOAT (NIL) -8 NIL NIL NIL) (-225 385837 386118 386507 "DFINTTLS" 387282 NIL DFINTTLS (NIL T T) -7 NIL NIL NIL) (-224 382866 383858 384258 "DERHAM" 385503 NIL DERHAM (NIL T NIL) -8 NIL NIL NIL) (-223 380667 382641 382730 "DEQUEUE" 382810 NIL DEQUEUE (NIL T) -8 NIL NIL NIL) (-222 379921 380054 380237 "DEGRED" 380529 NIL DEGRED (NIL T T) -7 NIL NIL NIL) (-221 376531 377231 378032 "DEFINTRF" 379194 NIL DEFINTRF (NIL T) -7 NIL NIL NIL) (-220 374198 374639 375203 "DEFINTEF" 376078 NIL DEFINTEF (NIL T T) -7 NIL NIL NIL) (-219 373548 373818 373933 "DEFAST" 374103 T DEFAST (NIL) -8 NIL NIL NIL) (-218 367573 373143 373292 "DECIMAL" 373419 T DECIMAL (NIL) -8 NIL NIL NIL) (-217 365085 365543 366049 "DDFACT" 367117 NIL DDFACT (NIL T T) -7 NIL NIL NIL) (-216 364681 364724 364875 "DBLRESP" 365036 NIL DBLRESP (NIL T T T T) -7 NIL NIL NIL) (-215 362553 362914 363274 "DBASE" 364448 NIL DBASE (NIL T) -8 NIL NIL NIL) (-214 361795 362033 362179 "DATAARY" 362452 NIL DATAARY (NIL NIL T) -8 NIL NIL NIL) (-213 360901 361754 361782 "D03FAFA" 361787 T D03FAFA (NIL) -8 NIL NIL NIL) (-212 360008 360860 360888 "D03EEFA" 360893 T D03EEFA (NIL) -8 NIL NIL NIL) (-211 357958 358424 358913 "D03AGNT" 359539 T D03AGNT (NIL) -7 NIL NIL NIL) (-210 357247 357917 357945 "D02EJFA" 357950 T D02EJFA (NIL) -8 NIL NIL NIL) (-209 356536 357206 357234 "D02CJFA" 357239 T D02CJFA (NIL) -8 NIL NIL NIL) (-208 355825 356495 356523 "D02BHFA" 356528 T D02BHFA (NIL) -8 NIL NIL NIL) (-207 355114 355784 355812 "D02BBFA" 355817 T D02BBFA (NIL) -8 NIL NIL NIL) (-206 348311 349900 351506 "D02AGNT" 353528 T D02AGNT (NIL) -7 NIL NIL NIL) (-205 346079 346602 347148 "D01WGTS" 347785 T D01WGTS (NIL) -7 NIL NIL NIL) (-204 345146 346038 346066 "D01TRNS" 346071 T D01TRNS (NIL) -8 NIL NIL NIL) (-203 344214 345105 345133 "D01GBFA" 345138 T D01GBFA (NIL) -8 NIL NIL NIL) (-202 343282 344173 344201 "D01FCFA" 344206 T D01FCFA (NIL) -8 NIL NIL NIL) (-201 342350 343241 343269 "D01ASFA" 343274 T D01ASFA (NIL) -8 NIL NIL NIL) (-200 341418 342309 342337 "D01AQFA" 342342 T D01AQFA (NIL) -8 NIL NIL NIL) (-199 340486 341377 341405 "D01APFA" 341410 T D01APFA (NIL) -8 NIL NIL NIL) (-198 339554 340445 340473 "D01ANFA" 340478 T D01ANFA (NIL) -8 NIL NIL NIL) (-197 338622 339513 339541 "D01AMFA" 339546 T D01AMFA (NIL) -8 NIL NIL NIL) (-196 337690 338581 338609 "D01ALFA" 338614 T D01ALFA (NIL) -8 NIL NIL NIL) (-195 336758 337649 337677 "D01AKFA" 337682 T D01AKFA (NIL) -8 NIL NIL NIL) (-194 335826 336717 336745 "D01AJFA" 336750 T D01AJFA (NIL) -8 NIL NIL NIL) (-193 329121 330674 332235 "D01AGNT" 334285 T D01AGNT (NIL) -7 NIL NIL NIL) (-192 328458 328586 328738 "CYCLOTOM" 328989 T CYCLOTOM (NIL) -7 NIL NIL NIL) (-191 325193 325906 326633 "CYCLES" 327751 T CYCLES (NIL) -7 NIL NIL NIL) (-190 324505 324639 324810 "CVMP" 325054 NIL CVMP (NIL T) -7 NIL NIL NIL) (-189 322346 322604 322973 "CTRIGMNP" 324233 NIL CTRIGMNP (NIL T T) -7 NIL NIL NIL) (-188 321855 322077 322178 "CTORKIND" 322265 T CTORKIND (NIL) -8 NIL NIL NIL) (-187 321146 321462 321490 "CTORCAT" 321672 T CTORCAT (NIL) -9 NIL 321785 NIL) (-186 320744 320855 321014 "CTORCAT-" 321019 NIL CTORCAT- (NIL T) -8 NIL NIL NIL) (-185 320206 320418 320526 "CTORCALL" 320668 NIL CTORCALL (NIL T) -8 NIL NIL NIL) (-184 319642 320000 320073 "CTOR" 320153 T CTOR (NIL) -8 NIL NIL NIL) (-183 319016 319115 319268 "CSTTOOLS" 319539 NIL CSTTOOLS (NIL T T) -7 NIL NIL NIL) (-182 314815 315472 316230 "CRFP" 318328 NIL CRFP (NIL T T) -7 NIL NIL NIL) (-181 314290 314536 314628 "CRCEAST" 314743 T CRCEAST (NIL) -8 NIL NIL NIL) (-180 313337 313522 313750 "CRAPACK" 314094 NIL CRAPACK (NIL T) -7 NIL NIL NIL) (-179 312721 312822 313026 "CPMATCH" 313213 NIL CPMATCH (NIL T T T) -7 NIL NIL NIL) (-178 312446 312474 312580 "CPIMA" 312687 NIL CPIMA (NIL T T T) -7 NIL NIL NIL) (-177 308794 309466 310185 "COORDSYS" 311781 NIL COORDSYS (NIL T) -7 NIL NIL NIL) (-176 308206 308327 308469 "CONTOUR" 308672 T CONTOUR (NIL) -8 NIL NIL NIL) (-175 304099 306209 306701 "CONTFRAC" 307746 NIL CONTFRAC (NIL T) -8 NIL NIL NIL) (-174 303979 304000 304028 "CONDUIT" 304065 T CONDUIT (NIL) -9 NIL NIL NIL) (-173 303067 303621 303649 "COMRING" 303654 T COMRING (NIL) -9 NIL 303706 NIL) (-172 302121 302425 302609 "COMPPROP" 302903 T COMPPROP (NIL) -8 NIL NIL NIL) (-171 301782 301817 301945 "COMPLPAT" 302080 NIL COMPLPAT (NIL T T T) -7 NIL NIL NIL) (-170 301418 301475 301582 "COMPLEX2" 301719 NIL COMPLEX2 (NIL T T) -7 NIL NIL NIL) (-169 291727 301227 301336 "COMPLEX" 301341 NIL COMPLEX (NIL T) -8 NIL NIL NIL) (-168 291468 291508 291589 "COMPILER" 291666 T COMPILER (NIL) -8 NIL NIL NIL) (-167 291186 291221 291319 "COMPFACT" 291427 NIL COMPFACT (NIL T T) -7 NIL NIL NIL) (-166 275275 285260 285300 "COMPCAT" 286304 NIL COMPCAT (NIL T) -9 NIL 287652 NIL) (-165 264808 267728 271348 "COMPCAT-" 271704 NIL COMPCAT- (NIL T T) -8 NIL NIL NIL) (-164 264537 264565 264668 "COMMUPC" 264774 NIL COMMUPC (NIL T T T) -7 NIL NIL NIL) (-163 264331 264365 264424 "COMMONOP" 264498 T COMMONOP (NIL) -7 NIL NIL NIL) (-162 263907 264135 264210 "COMMAAST" 264276 T COMMAAST (NIL) -8 NIL NIL NIL) (-161 263463 263658 263745 "COMM" 263840 T COMM (NIL) -8 NIL NIL NIL) (-160 262712 262906 262934 "COMBOPC" 263272 T COMBOPC (NIL) -9 NIL 263447 NIL) (-159 261608 261818 262060 "COMBINAT" 262502 NIL COMBINAT (NIL T) -7 NIL NIL NIL) (-158 258065 258639 259266 "COMBF" 261030 NIL COMBF (NIL T T) -7 NIL NIL NIL) (-157 256823 257181 257416 "COLOR" 257850 T COLOR (NIL) -8 NIL NIL NIL) (-156 256299 256544 256636 "COLONAST" 256751 T COLONAST (NIL) -8 NIL NIL NIL) (-155 255939 255986 256111 "CMPLXRT" 256246 NIL CMPLXRT (NIL T T) -7 NIL NIL NIL) (-154 255387 255639 255738 "CLLCTAST" 255860 T CLLCTAST (NIL) -8 NIL NIL NIL) (-153 250886 251917 252997 "CLIP" 254327 T CLIP (NIL) -7 NIL NIL NIL) (-152 249232 249992 250231 "CLIF" 250713 NIL CLIF (NIL NIL T NIL) -8 NIL NIL NIL) (-151 245407 247378 247419 "CLAGG" 248348 NIL CLAGG (NIL T) -9 NIL 248884 NIL) (-150 243829 244286 244869 "CLAGG-" 244874 NIL CLAGG- (NIL T T) -8 NIL NIL NIL) (-149 243373 243458 243598 "CINTSLPE" 243738 NIL CINTSLPE (NIL T T) -7 NIL NIL NIL) (-148 240874 241345 241893 "CHVAR" 242901 NIL CHVAR (NIL T T T) -7 NIL NIL NIL) (-147 240048 240602 240630 "CHARZ" 240635 T CHARZ (NIL) -9 NIL 240650 NIL) (-146 239802 239842 239920 "CHARPOL" 240002 NIL CHARPOL (NIL T) -7 NIL NIL NIL) (-145 238860 239447 239475 "CHARNZ" 239522 T CHARNZ (NIL) -9 NIL 239578 NIL) (-144 236766 237514 237867 "CHAR" 238527 T CHAR (NIL) -8 NIL NIL NIL) (-143 236492 236553 236581 "CFCAT" 236692 T CFCAT (NIL) -9 NIL NIL NIL) (-142 235737 235848 236030 "CDEN" 236376 NIL CDEN (NIL T T T) -7 NIL NIL NIL) (-141 231702 234890 235170 "CCLASS" 235477 T CCLASS (NIL) -8 NIL NIL NIL) (-140 230953 231110 231287 "CATEGORY" 231545 T -10 (NIL) -8 NIL NIL NIL) (-139 230526 230872 230920 "CATCTOR" 230925 T CATCTOR (NIL) -8 NIL NIL NIL) (-138 229977 230229 230327 "CATAST" 230448 T CATAST (NIL) -8 NIL NIL NIL) (-137 229453 229698 229790 "CASEAST" 229905 T CASEAST (NIL) -8 NIL NIL NIL) (-136 228561 228709 228930 "CARTEN2" 229300 NIL CARTEN2 (NIL NIL NIL T T) -7 NIL NIL NIL) (-135 223570 224590 225343 "CARTEN" 227864 NIL CARTEN (NIL NIL NIL T) -8 NIL NIL NIL) (-134 221886 222720 222977 "CARD" 223333 T CARD (NIL) -8 NIL NIL NIL) (-133 221462 221690 221765 "CAPSLAST" 221831 T CAPSLAST (NIL) -8 NIL NIL NIL) (-132 220966 221174 221202 "CACHSET" 221334 T CACHSET (NIL) -9 NIL 221412 NIL) (-131 220436 220758 220786 "CABMON" 220836 T CABMON (NIL) -9 NIL 220892 NIL) (-130 219909 220140 220250 "BYTEORD" 220346 T BYTEORD (NIL) -8 NIL NIL NIL) (-129 215259 219414 219586 "BYTEBUF" 219757 T BYTEBUF (NIL) -8 NIL NIL NIL) (-128 214241 214793 214935 "BYTE" 215098 T BYTE (NIL) -8 NIL NIL 215220) (-127 211752 213933 214040 "BTREE" 214167 NIL BTREE (NIL T) -8 NIL NIL NIL) (-126 209203 211400 211522 "BTOURN" 211662 NIL BTOURN (NIL T) -8 NIL NIL NIL) (-125 206575 208673 208714 "BTCAT" 208782 NIL BTCAT (NIL T) -9 NIL 208859 NIL) (-124 206242 206322 206471 "BTCAT-" 206476 NIL BTCAT- (NIL T T) -8 NIL NIL NIL) (-123 201507 205385 205413 "BTAGG" 205635 T BTAGG (NIL) -9 NIL 205796 NIL) (-122 200997 201122 201328 "BTAGG-" 201333 NIL BTAGG- (NIL T) -8 NIL NIL NIL) (-121 197994 200275 200490 "BSTREE" 200814 NIL BSTREE (NIL T) -8 NIL NIL NIL) (-120 197132 197258 197442 "BRILL" 197850 NIL BRILL (NIL T) -7 NIL NIL NIL) (-119 193785 195858 195899 "BRAGG" 196548 NIL BRAGG (NIL T) -9 NIL 196806 NIL) (-118 192317 192722 193276 "BRAGG-" 193281 NIL BRAGG- (NIL T T) -8 NIL NIL NIL) (-117 185567 191663 191847 "BPADICRT" 192165 NIL BPADICRT (NIL NIL) -8 NIL NIL NIL) (-116 183884 185504 185549 "BPADIC" 185554 NIL BPADIC (NIL NIL) -8 NIL NIL NIL) (-115 183582 183612 183726 "BOUNDZRO" 183848 NIL BOUNDZRO (NIL T T) -7 NIL NIL NIL) (-114 181363 181767 182242 "BOP1" 183140 NIL BOP1 (NIL T) -7 NIL NIL NIL) (-113 176591 177789 178701 "BOP" 180471 T BOP (NIL) -8 NIL NIL NIL) (-112 175416 176165 176314 "BOOLEAN" 176462 T BOOLEAN (NIL) -8 NIL NIL NIL) (-111 174695 175099 175153 "BMODULE" 175158 NIL BMODULE (NIL T T) -9 NIL 175223 NIL) (-110 170496 174493 174566 "BITS" 174642 T BITS (NIL) -8 NIL NIL NIL) (-109 169917 170036 170176 "BINDING" 170376 T BINDING (NIL) -8 NIL NIL NIL) (-108 163945 169514 169662 "BINARY" 169789 T BINARY 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\ No newline at end of file
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T) ((-651 |#1|) . T) ((-651 |#2|) |has| |#1| (-367)) ((-651 $) . T) ((-653 #2#) -3969 (|has| |#1| (-367)) (|has| |#1| (-38 (-412 (-551))))) ((-653 |#1|) . T) ((-653 |#2|) |has| |#1| (-367)) ((-653 $) . T) ((-645 #2#) -3969 (|has| |#1| (-367)) (|has| |#1| (-38 (-412 (-551))))) ((-645 |#1|) |has| |#1| (-173)) ((-645 |#2|) |has| |#1| (-367)) ((-645 $) -3969 (|has| |#1| (-562)) (|has| |#1| (-367))) ((-644 (-551)) -12 (|has| |#1| (-367)) (|has| |#2| (-644 (-551)))) ((-644 |#2|) |has| |#1| (-367)) ((-722 #2#) -3969 (|has| |#1| (-367)) (|has| |#1| (-38 (-412 (-551))))) ((-722 |#1|) |has| |#1| (-173)) ((-722 |#2|) |has| |#1| (-367)) ((-722 $) -3969 (|has| |#1| (-562)) (|has| |#1| (-367))) ((-731) . 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+NIL 
+(-13 (-173) (-372) (-619 (-551)) (-1157)) 
+NIL 
+NIL 
+NIL 
+NIL 
+NIL 
+NIL 
+NIL 
+NIL 
+NIL 
+NIL 
+NIL 
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XPBWPOLY (NIL T T) -8 NIL NIL NIL) (-1292 3183123 3184412 3184467 "XFALG" 3186639 NIL XFALG (NIL T T) -9 NIL 3187428 NIL) (-1291 3178820 3181113 3181155 "XF" 3181776 NIL XF (NIL T) -9 NIL 3182176 NIL) (-1290 3178441 3178529 3178698 "XF-" 3178703 NIL XF- (NIL T T) -8 NIL NIL NIL) (-1289 3177574 3177678 3177883 "XEXPPKG" 3178333 NIL XEXPPKG (NIL T T T) -7 NIL NIL NIL) (-1288 3175683 3177424 3177520 "XDPOLY" 3177525 NIL XDPOLY (NIL T T) -8 NIL NIL NIL) (-1287 3174490 3175090 3175133 "XALG" 3175138 NIL XALG (NIL T) -9 NIL 3175249 NIL) (-1286 3167959 3172467 3172961 "WUTSET" 3174082 NIL WUTSET (NIL T T T T) -8 NIL NIL NIL) (-1285 3166215 3167011 3167334 "WP" 3167770 NIL WP (NIL T T T T NIL NIL NIL) -8 NIL NIL NIL) (-1284 3165817 3166037 3166107 "WHILEAST" 3166167 T WHILEAST (NIL) -8 NIL NIL NIL) (-1283 3165289 3165534 3165628 "WHEREAST" 3165745 T WHEREAST (NIL) -8 NIL NIL NIL) (-1282 3164175 3164373 3164668 "WFFINTBS" 3165086 NIL WFFINTBS (NIL T T T T) -7 NIL NIL NIL) (-1281 3162079 3162506 3162968 "WEIER" 3163747 NIL WEIER (NIL T) -7 NIL NIL NIL) (-1280 3161125 3161575 3161617 "VSPACE" 3161753 NIL VSPACE (NIL T) -9 NIL 3161827 NIL) (-1279 3160963 3160990 3161081 "VSPACE-" 3161086 NIL VSPACE- (NIL T T) -8 NIL NIL NIL) (-1278 3160772 3160814 3160882 "VOID" 3160917 T VOID (NIL) -8 NIL NIL NIL) (-1277 3157196 3157835 3158572 "VIEWDEF" 3160057 T VIEWDEF (NIL) -7 NIL NIL NIL) (-1276 3146500 3148744 3150917 "VIEW3D" 3155045 T VIEW3D (NIL) -8 NIL NIL NIL) (-1275 3138751 3140411 3141990 "VIEW2D" 3144943 T VIEW2D (NIL) -8 NIL NIL NIL) (-1274 3136887 3137246 3137652 "VIEW" 3138367 T VIEW (NIL) -7 NIL NIL NIL) (-1273 3135464 3135723 3136041 "VECTOR2" 3136617 NIL VECTOR2 (NIL T T) -7 NIL NIL NIL) (-1272 3130817 3135234 3135326 "VECTOR" 3135407 NIL VECTOR (NIL T) -8 NIL NIL NIL) (-1271 3124291 3128598 3128641 "VECTCAT" 3129636 NIL VECTCAT (NIL T) -9 NIL 3130223 NIL) (-1270 3123305 3123559 3123949 "VECTCAT-" 3123954 NIL VECTCAT- (NIL T T) -8 NIL NIL NIL) (-1269 3122759 3122956 3123076 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NIL NIL NIL) (-1257 3068354 3075145 3075207 "UPXSCCA" 3075781 NIL UPXSCCA (NIL T T) -9 NIL 3076014 NIL) (-1256 3067992 3068077 3068251 "UPXSCCA-" 3068256 NIL UPXSCCA- (NIL T T T) -8 NIL NIL NIL) (-1255 3057591 3064155 3064198 "UPXSCAT" 3064846 NIL UPXSCAT (NIL T) -9 NIL 3065455 NIL) (-1254 3057021 3057100 3057279 "UPXS2" 3057506 NIL UPXS2 (NIL T T NIL NIL NIL NIL) -7 NIL NIL NIL) (-1253 3049091 3056268 3056541 "UPXS" 3056806 NIL UPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-1252 3047748 3048000 3048350 "UPSQFREE" 3048835 NIL UPSQFREE (NIL T T) -7 NIL NIL NIL) (-1251 3041169 3044226 3044281 "UPSCAT" 3045442 NIL UPSCAT (NIL T T) -9 NIL 3046216 NIL) (-1250 3040373 3040580 3040907 "UPSCAT-" 3040912 NIL UPSCAT- (NIL T T T) -8 NIL NIL NIL) (-1249 3040000 3040043 3040176 "UPOLYC2" 3040324 NIL UPOLYC2 (NIL T T T T) -7 NIL NIL NIL) (-1248 3025688 3033423 3033466 "UPOLYC" 3035567 NIL UPOLYC (NIL T) -9 NIL 3036788 NIL) (-1247 3017052 3019466 3022601 "UPOLYC-" 3022606 NIL UPOLYC- (NIL T T) -8 NIL NIL 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TYPEAST (NIL) -8 NIL NIL NIL) (-1222 2934941 2934946 2934976 "TYPE" 2934981 T TYPE (NIL) -9 NIL NIL NIL) (-1221 2933912 2934114 2934354 "TWOFACT" 2934735 NIL TWOFACT (NIL T) -7 NIL NIL NIL) (-1220 2932935 2933321 2933556 "TUPLE" 2933712 NIL TUPLE (NIL T) -8 NIL NIL NIL) (-1219 2930626 2931145 2931684 "TUBETOOL" 2932418 T TUBETOOL (NIL) -7 NIL NIL NIL) (-1218 2929475 2929680 2929921 "TUBE" 2930419 NIL TUBE (NIL T) -8 NIL NIL NIL) (-1217 2918115 2922234 2922331 "TSETCAT" 2927600 NIL TSETCAT (NIL T T T T) -9 NIL 2929131 NIL) (-1216 2912847 2914447 2916338 "TSETCAT-" 2916343 NIL TSETCAT- (NIL T T T T T) -8 NIL NIL NIL) (-1215 2907576 2911819 2912102 "TS" 2912599 NIL TS (NIL T) -8 NIL NIL NIL) (-1214 2902215 2903062 2903991 "TRMANIP" 2906712 NIL TRMANIP (NIL T T) -7 NIL NIL NIL) (-1213 2901656 2901719 2901882 "TRIMAT" 2902147 NIL TRIMAT (NIL T T T T) -7 NIL NIL NIL) (-1212 2899522 2899759 2900116 "TRIGMNIP" 2901405 NIL TRIGMNIP (NIL T T) -7 NIL NIL NIL) (-1211 2899042 2899155 2899185 "TRIGCAT" 2899398 T TRIGCAT (NIL) -9 NIL NIL NIL) (-1210 2898711 2898790 2898931 "TRIGCAT-" 2898936 NIL TRIGCAT- (NIL T) -8 NIL NIL NIL) (-1209 2895557 2897569 2897850 "TREE" 2898465 NIL TREE (NIL T) -8 NIL NIL NIL) (-1208 2894831 2895359 2895389 "TRANFUN" 2895424 T TRANFUN (NIL) -9 NIL 2895490 NIL) (-1207 2894110 2894301 2894581 "TRANFUN-" 2894586 NIL TRANFUN- (NIL T) -8 NIL NIL NIL) (-1206 2893914 2893946 2894007 "TOPSP" 2894071 T TOPSP (NIL) -7 NIL NIL NIL) (-1205 2893262 2893377 2893531 "TOOLSIGN" 2893795 NIL TOOLSIGN (NIL T) -7 NIL NIL NIL) (-1204 2891896 2892439 2892678 "TEXTFILE" 2893045 T TEXTFILE (NIL) -8 NIL NIL NIL) (-1203 2891677 2891708 2891780 "TEX1" 2891859 NIL TEX1 (NIL T) -7 NIL NIL NIL) (-1202 2889589 2890130 2890559 "TEX" 2891270 T TEX (NIL) -8 NIL NIL NIL) (-1201 2889237 2889300 2889390 "TEMUTL" 2889521 T TEMUTL (NIL) -7 NIL NIL NIL) (-1200 2887391 2887671 2887996 "TBCMPPK" 2888960 NIL TBCMPPK (NIL T T) -7 NIL NIL NIL) (-1199 2879170 2885551 2885607 "TBAGG" 2886007 NIL TBAGG (NIL T T) -9 NIL 2886218 NIL) (-1198 2874240 2875728 2877482 "TBAGG-" 2877487 NIL TBAGG- (NIL T T T) -8 NIL NIL NIL) (-1197 2873624 2873731 2873876 "TANEXP" 2874129 NIL TANEXP (NIL T) -7 NIL NIL NIL) (-1196 2873036 2873135 2873273 "TABLEAU" 2873521 NIL TABLEAU (NIL T) -8 NIL NIL NIL) (-1195 2866428 2872893 2872986 "TABLE" 2872991 NIL TABLE (NIL T T) -8 NIL NIL NIL) (-1194 2861036 2862256 2863504 "TABLBUMP" 2865214 NIL TABLBUMP (NIL T) -7 NIL NIL NIL) (-1193 2860258 2860405 2860586 "SYSTEM" 2860877 T SYSTEM (NIL) -8 NIL NIL NIL) (-1192 2856717 2857416 2858199 "SYSSOLP" 2859509 NIL SYSSOLP (NIL T) -7 NIL NIL NIL) (-1191 2856515 2856672 2856703 "SYSPTR" 2856708 T SYSPTR (NIL) -8 NIL NIL NIL) (-1190 2855559 2856064 2856183 "SYSNNI" 2856369 NIL SYSNNI (NIL NIL) -8 NIL NIL 2856454) (-1189 2854866 2855325 2855404 "SYSINT" 2855464 NIL SYSINT (NIL NIL) -8 NIL NIL 2855509) (-1188 2851210 2852144 2852854 "SYNTAX" 2854178 T SYNTAX (NIL) -8 NIL NIL NIL) (-1187 2848368 2848970 2849602 "SYMTAB" 2850600 T SYMTAB (NIL) -8 NIL NIL NIL) (-1186 2843641 2844537 2845514 "SYMS" 2847413 T SYMS (NIL) -8 NIL NIL NIL) (-1185 2840886 2843102 2843332 "SYMPOLY" 2843449 NIL SYMPOLY (NIL T) -8 NIL NIL NIL) (-1184 2840403 2840478 2840601 "SYMFUNC" 2840798 NIL SYMFUNC (NIL T) -7 NIL NIL NIL) (-1183 2836423 2837715 2838528 "SYMBOL" 2839612 T SYMBOL (NIL) -8 NIL NIL NIL) (-1182 2829962 2831651 2833371 "SWITCH" 2834725 T SWITCH (NIL) -8 NIL NIL NIL) (-1181 2823196 2828783 2829086 "SUTS" 2829717 NIL SUTS (NIL T NIL NIL) -8 NIL NIL NIL) (-1180 2815266 2822443 2822716 "SUPXS" 2822981 NIL SUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-1179 2814425 2814552 2814769 "SUPFRACF" 2815134 NIL SUPFRACF (NIL T T T T) -7 NIL NIL NIL) (-1178 2814046 2814105 2814218 "SUP2" 2814360 NIL SUP2 (NIL T T) -7 NIL NIL NIL) (-1177 2805845 2813664 2813790 "SUP" 2813955 NIL SUP (NIL T) -8 NIL NIL NIL) (-1176 2804293 2804567 2804923 "SUMRF" 2805544 NIL SUMRF (NIL T) -7 NIL NIL NIL) (-1175 2803628 2803694 2803886 "SUMFS" 2804214 NIL SUMFS (NIL T T) -7 NIL NIL NIL) (-1174 2787611 2802805 2803056 "SULS" 2803435 NIL SULS (NIL T NIL NIL) -8 NIL NIL NIL) (-1173 2787213 2787433 2787503 "SUCHTAST" 2787563 T SUCHTAST (NIL) -8 NIL NIL NIL) (-1172 2786508 2786738 2786878 "SUCH" 2787121 NIL SUCH (NIL T T) -8 NIL NIL NIL) (-1171 2780374 2781414 2782373 "SUBSPACE" 2785596 NIL SUBSPACE (NIL NIL T) -8 NIL NIL NIL) (-1170 2779804 2779894 2780058 "SUBRESP" 2780262 NIL SUBRESP (NIL T T) -7 NIL NIL NIL) (-1169 2773977 2775097 2776244 "STTFNC" 2778704 NIL STTFNC (NIL T) -7 NIL NIL NIL) (-1168 2767343 2768642 2769953 "STTF" 2772713 NIL STTF (NIL T) -7 NIL NIL NIL) (-1167 2758654 2760525 2762319 "STTAYLOR" 2765584 NIL STTAYLOR (NIL T) -7 NIL NIL NIL) (-1166 2751786 2758518 2758601 "STRTBL" 2758606 NIL STRTBL (NIL T) -8 NIL NIL NIL) (-1165 2747150 2751741 2751772 "STRING" 2751777 T STRING (NIL) -8 NIL NIL NIL) (-1164 2742011 2746523 2746553 "STRICAT" 2746612 T STRICAT (NIL) -9 NIL 2746674 NIL) (-1163 2741521 2741598 2741742 "STREAM3" 2741928 NIL STREAM3 (NIL T T T) -7 NIL NIL NIL) (-1162 2740503 2740686 2740921 "STREAM2" 2741334 NIL STREAM2 (NIL T T) -7 NIL NIL NIL) (-1161 2740191 2740243 2740336 "STREAM1" 2740445 NIL STREAM1 (NIL T) -7 NIL NIL NIL) (-1160 2732946 2737810 2738421 "STREAM" 2739615 NIL STREAM (NIL T) -8 NIL NIL NIL) (-1159 2731962 2732143 2732374 "STINPROD" 2732762 NIL STINPROD (NIL T) -7 NIL NIL NIL) (-1158 2731149 2731451 2731599 "STEPAST" 2731836 T STEPAST (NIL) -8 NIL NIL NIL) (-1157 2730701 2730911 2730941 "STEP" 2731021 T STEP (NIL) -9 NIL 2731099 NIL) (-1156 2724135 2730600 2730677 "STBL" 2730682 NIL STBL (NIL T T NIL) -8 NIL NIL NIL) (-1155 2719263 2723356 2723399 "STAGG" 2723552 NIL STAGG (NIL T) -9 NIL 2723641 NIL) (-1154 2716971 2717571 2718441 "STAGG-" 2718446 NIL STAGG- (NIL T T) -8 NIL NIL NIL) (-1153 2715118 2716741 2716833 "STACK" 2716914 NIL STACK (NIL T) -8 NIL NIL NIL) (-1152 2707840 2713259 2713715 "SREGSET" 2714748 NIL SREGSET (NIL T T T T) -8 NIL NIL NIL) (-1151 2700265 2701634 2703147 "SRDCMPK" 2706446 NIL SRDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1150 2693182 2697705 2697735 "SRAGG" 2699038 T SRAGG (NIL) -9 NIL 2699646 NIL) (-1149 2692199 2692454 2692833 "SRAGG-" 2692838 NIL SRAGG- (NIL T) -8 NIL NIL NIL) (-1148 2686663 2691146 2691567 "SQMATRIX" 2691825 NIL SQMATRIX (NIL NIL T) -8 NIL NIL NIL) (-1147 2680349 2683381 2684108 "SPLTREE" 2686008 NIL SPLTREE (NIL T T) -8 NIL NIL NIL) (-1146 2676312 2677005 2677651 "SPLNODE" 2679775 NIL SPLNODE (NIL T T) -8 NIL NIL NIL) (-1145 2675359 2675592 2675622 "SPFCAT" 2676066 T SPFCAT (NIL) -9 NIL NIL NIL) (-1144 2674096 2674306 2674570 "SPECOUT" 2675117 T SPECOUT (NIL) -7 NIL NIL NIL) (-1143 2665206 2667078 2667108 "SPADXPT" 2671784 T SPADXPT (NIL) -9 NIL 2673948 NIL) (-1142 2664967 2665007 2665076 "SPADPRSR" 2665159 T SPADPRSR (NIL) -7 NIL NIL NIL) (-1141 2663016 2664922 2664953 "SPADAST" 2664958 T SPADAST (NIL) -8 NIL NIL NIL) (-1140 2654961 2656734 2656777 "SPACEC" 2661150 NIL SPACEC (NIL T) -9 NIL 2662966 NIL) (-1139 2653091 2654893 2654942 "SPACE3" 2654947 NIL SPACE3 (NIL T) -8 NIL NIL NIL) (-1138 2651843 2652014 2652305 "SORTPAK" 2652896 NIL SORTPAK (NIL T T) -7 NIL NIL NIL) (-1137 2649935 2650238 2650650 "SOLVETRA" 2651507 NIL SOLVETRA (NIL T) -7 NIL NIL NIL) (-1136 2648985 2649207 2649468 "SOLVESER" 2649708 NIL SOLVESER (NIL T) -7 NIL NIL NIL) (-1135 2644289 2645177 2646172 "SOLVERAD" 2648037 NIL SOLVERAD (NIL T) -7 NIL NIL NIL) (-1134 2640104 2640713 2641442 "SOLVEFOR" 2643656 NIL SOLVEFOR (NIL T T) -7 NIL NIL NIL) (-1133 2634401 2639453 2639550 "SNTSCAT" 2639555 NIL SNTSCAT (NIL T T T T) -9 NIL 2639625 NIL) (-1132 2628507 2632724 2633115 "SMTS" 2634091 NIL SMTS (NIL T T T) -8 NIL NIL NIL) (-1131 2623218 2628395 2628472 "SMP" 2628477 NIL SMP (NIL T T) -8 NIL NIL NIL) (-1130 2621377 2621678 2622076 "SMITH" 2622915 NIL SMITH (NIL T T T T) -7 NIL NIL NIL) (-1129 2614088 2618280 2618383 "SMATCAT" 2619737 NIL SMATCAT (NIL NIL T T T) -9 NIL 2620287 NIL) (-1128 2611049 2611865 2613036 "SMATCAT-" 2613041 NIL SMATCAT- (NIL T NIL T T T) -8 NIL NIL NIL) (-1127 2608715 2610285 2610328 "SKAGG" 2610589 NIL SKAGG (NIL T) -9 NIL 2610724 NIL) (-1126 2605028 2608131 2608326 "SINT" 2608513 T SINT (NIL) -8 NIL NIL 2608686) (-1125 2604800 2604838 2604904 "SIMPAN" 2604984 T SIMPAN (NIL) -7 NIL NIL NIL) (-1124 2603659 2603873 2604141 "SIGNRF" 2604566 NIL SIGNRF (NIL T) -7 NIL NIL NIL) (-1123 2602513 2602657 2602934 "SIGNEF" 2603495 NIL SIGNEF (NIL T T) -7 NIL NIL NIL) (-1122 2601819 2602096 2602220 "SIGAST" 2602411 T SIGAST (NIL) -8 NIL NIL NIL) (-1121 2601098 2601354 2601494 "SIG" 2601701 T SIG (NIL) -8 NIL NIL NIL) (-1120 2598788 2599242 2599748 "SHP" 2600639 NIL SHP (NIL T NIL) -7 NIL NIL NIL) (-1119 2592647 2598689 2598765 "SHDP" 2598770 NIL SHDP (NIL NIL NIL T) -8 NIL NIL NIL) (-1118 2592220 2592412 2592442 "SGROUP" 2592535 T SGROUP (NIL) -9 NIL 2592597 NIL) (-1117 2592078 2592104 2592177 "SGROUP-" 2592182 NIL SGROUP- (NIL T) -8 NIL NIL NIL) (-1116 2588913 2589611 2590334 "SGCF" 2591377 T SGCF (NIL) -7 NIL NIL NIL) (-1115 2583308 2588360 2588457 "SFRTCAT" 2588462 NIL SFRTCAT (NIL T T T T) -9 NIL 2588501 NIL) (-1114 2576729 2577747 2578883 "SFRGCD" 2582291 NIL SFRGCD (NIL T T T T T) -7 NIL NIL NIL) (-1113 2569855 2570928 2572114 "SFQCMPK" 2575662 NIL SFQCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1112 2569475 2569564 2569675 "SFORT" 2569796 NIL SFORT (NIL T T) -8 NIL NIL NIL) (-1111 2568593 2569315 2569436 "SEXOF" 2569441 NIL SEXOF (NIL T T T T T) -8 NIL NIL NIL) (-1110 2564106 2564821 2564916 "SEXCAT" 2567853 NIL SEXCAT (NIL T T T T T) -9 NIL 2568431 NIL) (-1109 2563213 2563987 2564055 "SEX" 2564060 T SEX (NIL) -8 NIL NIL NIL) (-1108 2561443 2561930 2562233 "SETMN" 2562956 NIL SETMN (NIL NIL NIL) -8 NIL NIL NIL) (-1107 2560939 2561091 2561121 "SETCAT" 2561297 T SETCAT (NIL) -9 NIL 2561407 NIL) (-1106 2560631 2560709 2560839 "SETCAT-" 2560844 NIL SETCAT- (NIL T) -8 NIL NIL NIL) (-1105 2556992 2559092 2559135 "SETAGG" 2560005 NIL SETAGG (NIL T) -9 NIL 2560345 NIL) (-1104 2556450 2556566 2556803 "SETAGG-" 2556808 NIL SETAGG- (NIL T T) -8 NIL NIL NIL) (-1103 2553603 2556384 2556432 "SET" 2556437 NIL SET (NIL T) -8 NIL NIL NIL) (-1102 2553046 2553299 2553400 "SEQAST" 2553524 T SEQAST (NIL) -8 NIL NIL NIL) (-1101 2552245 2552539 2552600 "SEGXCAT" 2552886 NIL SEGXCAT (NIL T T) -9 NIL 2553006 NIL) (-1100 2551224 2551438 2551481 "SEGCAT" 2552003 NIL SEGCAT (NIL T) -9 NIL 2552224 NIL) (-1099 2550845 2550904 2551017 "SEGBIND2" 2551159 NIL SEGBIND2 (NIL T T) -7 NIL NIL NIL) (-1098 2549777 2550208 2550416 "SEGBIND" 2550672 NIL SEGBIND (NIL T) -8 NIL NIL NIL) (-1097 2549350 2549578 2549655 "SEGAST" 2549722 T SEGAST (NIL) -8 NIL NIL NIL) (-1096 2548569 2548695 2548899 "SEG2" 2549194 NIL SEG2 (NIL T T) -7 NIL NIL NIL) (-1095 2547575 2548235 2548417 "SEG" 2548422 NIL SEG (NIL T) -8 NIL NIL NIL) (-1094 2546985 2547510 2547557 "SDVAR" 2547562 NIL SDVAR (NIL T) -8 NIL NIL NIL) (-1093 2539553 2546755 2546885 "SDPOL" 2546890 NIL SDPOL (NIL T) -8 NIL NIL NIL) (-1092 2538146 2538412 2538731 "SCPKG" 2539268 NIL SCPKG (NIL T) -7 NIL NIL NIL) (-1091 2537310 2537482 2537674 "SCOPE" 2537976 T SCOPE (NIL) -8 NIL NIL NIL) (-1090 2536530 2536664 2536843 "SCACHE" 2537165 NIL SCACHE (NIL T) -7 NIL NIL NIL) (-1089 2536176 2536362 2536392 "SASTCAT" 2536397 T SASTCAT (NIL) -9 NIL 2536410 NIL) (-1088 2535663 2536011 2536087 "SAOS" 2536122 T SAOS (NIL) -8 NIL NIL NIL) (-1087 2535228 2535263 2535436 "SAERFFC" 2535622 NIL SAERFFC (NIL T T T) -7 NIL NIL NIL) (-1086 2534821 2534856 2535015 "SAEFACT" 2535187 NIL SAEFACT (NIL T T T) -7 NIL NIL NIL) (-1085 2528769 2534718 2534798 "SAE" 2534803 NIL SAE (NIL T T NIL) -8 NIL NIL NIL) (-1084 2527090 2527404 2527805 "RURPK" 2528435 NIL RURPK (NIL T NIL) -7 NIL NIL NIL) (-1083 2525727 2526033 2526338 "RULESET" 2526924 NIL RULESET (NIL T T T) -8 NIL NIL NIL) (-1082 2525339 2525521 2525604 "RULECOLD" 2525679 NIL RULECOLD (NIL NIL) -8 NIL NIL NIL) (-1081 2522562 2523092 2523550 "RULE" 2525020 NIL RULE (NIL T T T) -8 NIL NIL NIL) (-1080 2522352 2522380 2522451 "RTVALUE" 2522513 T RTVALUE (NIL) -8 NIL NIL NIL) (-1079 2521823 2522069 2522163 "RSTRCAST" 2522280 T RSTRCAST (NIL) -8 NIL NIL NIL) (-1078 2516671 2517466 2518386 "RSETGCD" 2521022 NIL RSETGCD (NIL T T T T T) -7 NIL NIL NIL) (-1077 2505928 2510980 2511077 "RSETCAT" 2515196 NIL RSETCAT (NIL T T T T) -9 NIL 2516293 NIL) (-1076 2503855 2504394 2505218 "RSETCAT-" 2505223 NIL RSETCAT- (NIL T T T T T) -8 NIL NIL NIL) (-1075 2496241 2497617 2499137 "RSDCMPK" 2502454 NIL RSDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1074 2494220 2494687 2494761 "RRCC" 2495847 NIL RRCC (NIL T T) -9 NIL 2496191 NIL) (-1073 2493571 2493745 2494024 "RRCC-" 2494029 NIL RRCC- (NIL T T T) -8 NIL NIL NIL) (-1072 2493014 2493267 2493368 "RPTAST" 2493492 T RPTAST (NIL) -8 NIL NIL NIL) (-1071 2466896 2476222 2476289 "RPOLCAT" 2486953 NIL RPOLCAT (NIL T T T) -9 NIL 2490112 NIL) (-1070 2458430 2460758 2463868 "RPOLCAT-" 2463873 NIL RPOLCAT- (NIL T T T T) -8 NIL NIL NIL) (-1069 2449363 2456641 2457123 "ROUTINE" 2457970 T ROUTINE (NIL) -8 NIL NIL NIL) (-1068 2446163 2448989 2449129 "ROMAN" 2449245 T ROMAN (NIL) -8 NIL NIL NIL) (-1067 2444409 2445023 2445283 "ROIRC" 2445968 NIL ROIRC (NIL T T) -8 NIL NIL NIL) (-1066 2440645 2442925 2442955 "RNS" 2443259 T RNS (NIL) -9 NIL 2443533 NIL) (-1065 2439154 2439537 2440071 "RNS-" 2440146 NIL RNS- (NIL T) -8 NIL NIL NIL) (-1064 2438157 2438519 2438721 "RNGBIND" 2439005 NIL RNGBIND (NIL T T) -8 NIL NIL NIL) (-1063 2437560 2437968 2437998 "RNG" 2438003 T RNG (NIL) -9 NIL 2438024 NIL) (-1062 2436959 2437347 2437390 "RMODULE" 2437395 NIL RMODULE (NIL T) -9 NIL 2437422 NIL) (-1061 2435795 2435889 2436225 "RMCAT2" 2436860 NIL RMCAT2 (NIL NIL NIL T T T T T T T T) -7 NIL NIL NIL) (-1060 2432645 2435141 2435438 "RMATRIX" 2435557 NIL RMATRIX (NIL NIL NIL T) -8 NIL NIL NIL) (-1059 2425472 2427732 2427847 "RMATCAT" 2431206 NIL RMATCAT (NIL NIL NIL T T T) -9 NIL 2432188 NIL) (-1058 2424847 2424994 2425301 "RMATCAT-" 2425306 NIL RMATCAT- (NIL T NIL NIL T T T) -8 NIL NIL NIL) (-1057 2424248 2424469 2424512 "RLINSET" 2424706 NIL RLINSET (NIL T) -9 NIL 2424797 NIL) (-1056 2423815 2423890 2424018 "RINTERP" 2424167 NIL RINTERP (NIL NIL T) -7 NIL NIL NIL) (-1055 2422873 2423427 2423457 "RING" 2423513 T RING (NIL) -9 NIL 2423605 NIL) (-1054 2422665 2422709 2422806 "RING-" 2422811 NIL RING- (NIL T) -8 NIL NIL NIL) (-1053 2421506 2421743 2422001 "RIDIST" 2422429 T RIDIST (NIL) -7 NIL NIL NIL) (-1052 2412822 2420974 2421180 "RGCHAIN" 2421354 NIL RGCHAIN (NIL T NIL) -8 NIL NIL NIL) (-1051 2412172 2412578 2412619 "RGBCSPC" 2412677 NIL RGBCSPC (NIL T) -9 NIL 2412729 NIL) (-1050 2411330 2411711 2411752 "RGBCMDL" 2411984 NIL RGBCMDL (NIL T) -9 NIL 2412098 NIL) (-1049 2410976 2411039 2411142 "RFFACTOR" 2411261 NIL RFFACTOR (NIL T) -7 NIL NIL NIL) (-1048 2410701 2410736 2410833 "RFFACT" 2410935 NIL RFFACT (NIL T) -7 NIL NIL NIL) (-1047 2408818 2409182 2409564 "RFDIST" 2410341 T RFDIST (NIL) -7 NIL NIL NIL) (-1046 2405812 2406426 2407096 "RF" 2408182 NIL RF (NIL T) -7 NIL NIL NIL) (-1045 2405265 2405357 2405520 "RETSOL" 2405714 NIL RETSOL (NIL T T) -7 NIL NIL NIL) (-1044 2404901 2404981 2405024 "RETRACT" 2405157 NIL RETRACT (NIL T) -9 NIL 2405244 NIL) (-1043 2404750 2404775 2404862 "RETRACT-" 2404867 NIL RETRACT- (NIL T T) -8 NIL NIL NIL) (-1042 2404352 2404572 2404642 "RETAST" 2404702 T RETAST (NIL) -8 NIL NIL NIL) (-1041 2397092 2404005 2404132 "RESULT" 2404247 T RESULT (NIL) -8 NIL NIL NIL) (-1040 2395683 2396361 2396560 "RESRING" 2396995 NIL RESRING (NIL T T T T NIL) -8 NIL NIL NIL) (-1039 2395319 2395368 2395466 "RESLATC" 2395620 NIL RESLATC (NIL T) -7 NIL NIL NIL) (-1038 2395024 2395059 2395166 "REPSQ" 2395278 NIL REPSQ (NIL T) -7 NIL NIL NIL) (-1037 2394721 2394756 2394867 "REPDB" 2394983 NIL REPDB (NIL T) -7 NIL NIL NIL) (-1036 2388621 2390010 2391233 "REP2" 2393533 NIL REP2 (NIL T) -7 NIL NIL NIL) (-1035 2384998 2385679 2386487 "REP1" 2387848 NIL REP1 (NIL T) -7 NIL NIL NIL) (-1034 2382420 2383000 2383602 "REP" 2384418 T REP (NIL) -7 NIL NIL NIL) (-1033 2375143 2380561 2381017 "REGSET" 2382050 NIL REGSET (NIL T T T T) -8 NIL NIL NIL) (-1032 2373908 2374291 2374541 "REF" 2374928 NIL REF (NIL T) -8 NIL NIL NIL) (-1031 2373285 2373388 2373555 "REDORDER" 2373792 NIL REDORDER (NIL T T) -7 NIL NIL NIL) (-1030 2369284 2372498 2372725 "RECLOS" 2373113 NIL RECLOS (NIL T) -8 NIL NIL NIL) (-1029 2368336 2368517 2368732 "REALSOLV" 2369091 T REALSOLV (NIL) -7 NIL NIL NIL) (-1028 2364819 2365621 2366505 "REAL0Q" 2367501 NIL REAL0Q (NIL T) -7 NIL NIL NIL) (-1027 2360420 2361408 2362469 "REAL0" 2363800 NIL REAL0 (NIL T) -7 NIL NIL NIL) (-1026 2360266 2360307 2360337 "REAL" 2360342 T REAL (NIL) -9 NIL 2360377 NIL) (-1025 2359737 2359983 2360077 "RDUCEAST" 2360194 T RDUCEAST (NIL) -8 NIL NIL NIL) (-1024 2359142 2359214 2359421 "RDIV" 2359659 NIL RDIV (NIL T T T T T) -7 NIL NIL NIL) (-1023 2358210 2358384 2358597 "RDIST" 2358964 NIL RDIST (NIL T) -7 NIL NIL NIL) (-1022 2356807 2357094 2357466 "RDETRS" 2357918 NIL RDETRS (NIL T T) -7 NIL NIL NIL) (-1021 2354619 2355073 2355611 "RDETR" 2356349 NIL RDETR (NIL T T) -7 NIL NIL NIL) (-1020 2353244 2353522 2353919 "RDEEFS" 2354335 NIL RDEEFS (NIL T T) -7 NIL NIL NIL) (-1019 2351753 2352059 2352484 "RDEEF" 2352932 NIL RDEEF (NIL T T) -7 NIL NIL NIL) (-1018 2345823 2348734 2348764 "RCFIELD" 2350059 T RCFIELD (NIL) -9 NIL 2350790 NIL) (-1017 2343887 2344391 2345087 "RCFIELD-" 2345162 NIL RCFIELD- (NIL T) -8 NIL NIL NIL) (-1016 2340156 2341988 2342031 "RCAGG" 2343115 NIL RCAGG (NIL T) -9 NIL 2343580 NIL) (-1015 2339784 2339878 2340041 "RCAGG-" 2340046 NIL RCAGG- (NIL T T) -8 NIL NIL NIL) (-1014 2339119 2339231 2339396 "RATRET" 2339668 NIL RATRET (NIL T) -7 NIL NIL NIL) (-1013 2338672 2338739 2338860 "RATFACT" 2339047 NIL RATFACT (NIL T) -7 NIL NIL NIL) (-1012 2337980 2338100 2338252 "RANDSRC" 2338542 T RANDSRC (NIL) -7 NIL NIL NIL) (-1011 2337714 2337758 2337831 "RADUTIL" 2337929 T RADUTIL (NIL) -7 NIL NIL NIL) (-1010 2330851 2336547 2336857 "RADIX" 2337438 NIL RADIX (NIL NIL) -8 NIL NIL NIL) (-1009 2322481 2330693 2330823 "RADFF" 2330828 NIL RADFF (NIL T T T NIL NIL) -8 NIL NIL NIL) (-1008 2322128 2322203 2322233 "RADCAT" 2322393 T RADCAT (NIL) -9 NIL NIL NIL) (-1007 2321910 2321958 2322058 "RADCAT-" 2322063 NIL RADCAT- (NIL T) -8 NIL NIL NIL) (-1006 2320008 2321680 2321772 "QUEUE" 2321853 NIL QUEUE (NIL T) -8 NIL NIL NIL) (-1005 2319639 2319682 2319813 "QUATCT2" 2319959 NIL QUATCT2 (NIL T T T T) -7 NIL NIL NIL) (-1004 2313095 2316433 2316475 "QUATCAT" 2317266 NIL QUATCAT (NIL T) -9 NIL 2318032 NIL) (-1003 2309255 2310285 2311668 "QUATCAT-" 2311764 NIL QUATCAT- (NIL T T) -8 NIL NIL NIL) (-1002 2305799 2309188 2309236 "QUAT" 2309241 NIL QUAT (NIL T) -8 NIL NIL NIL) (-1001 2303264 2304875 2304918 "QUAGG" 2305299 NIL QUAGG (NIL T) -9 NIL 2305474 NIL) (-1000 2302866 2303086 2303156 "QQUTAST" 2303216 T QQUTAST (NIL) -8 NIL NIL NIL) (-999 2301764 2302264 2302436 "QFORM" 2302738 NIL QFORM (NIL NIL T) -8 NIL NIL NIL) (-998 2301402 2301445 2301572 "QFCAT2" 2301715 NIL QFCAT2 (NIL T T T T) -7 NIL NIL NIL) (-997 2292423 2297646 2297686 "QFCAT" 2298344 NIL QFCAT (NIL T) -9 NIL 2299345 NIL) (-996 2288031 2289220 2290799 "QFCAT-" 2290893 NIL QFCAT- (NIL T T) -8 NIL NIL NIL) (-995 2287491 2287601 2287731 "QEQUAT" 2287921 T QEQUAT (NIL) -8 NIL NIL NIL) (-994 2280637 2281710 2282894 "QCMPACK" 2286424 NIL QCMPACK (NIL T T T T T) -7 NIL NIL NIL) (-993 2279882 2280056 2280288 "QALGSET2" 2280457 NIL QALGSET2 (NIL NIL NIL) -7 NIL NIL NIL) (-992 2277437 2277883 2278309 "QALGSET" 2279539 NIL QALGSET (NIL T T T T) -8 NIL NIL NIL) (-991 2276127 2276351 2276668 "PWFFINTB" 2277210 NIL PWFFINTB (NIL T T T T) -7 NIL NIL NIL) (-990 2274326 2274494 2274848 "PUSHVAR" 2275941 NIL PUSHVAR (NIL T T T T) -7 NIL NIL NIL) (-989 2270244 2271298 2271339 "PTRANFN" 2273223 NIL PTRANFN (NIL T) -9 NIL NIL NIL) (-988 2268646 2268937 2269259 "PTPACK" 2269955 NIL PTPACK (NIL T) -7 NIL NIL NIL) (-987 2268278 2268335 2268444 "PTFUNC2" 2268583 NIL PTFUNC2 (NIL T T) -7 NIL NIL NIL) (-986 2262755 2267150 2267191 "PTCAT" 2267487 NIL PTCAT (NIL T) -9 NIL 2267640 NIL) (-985 2262413 2262448 2262572 "PSQFR" 2262714 NIL PSQFR (NIL T T T T) -7 NIL NIL NIL) (-984 2261008 2261306 2261640 "PSEUDLIN" 2262111 NIL PSEUDLIN (NIL T) -7 NIL NIL NIL) (-983 2247771 2250142 2252466 "PSETPK" 2258768 NIL PSETPK (NIL T T T T) -7 NIL NIL NIL) (-982 2240789 2243529 2243625 "PSETCAT" 2246646 NIL PSETCAT (NIL T T T T) -9 NIL 2247460 NIL) (-981 2238625 2239259 2240080 "PSETCAT-" 2240085 NIL PSETCAT- (NIL T T T T T) -8 NIL NIL NIL) (-980 2237974 2238139 2238167 "PSCURVE" 2238435 T PSCURVE (NIL) -9 NIL 2238602 NIL) (-979 2233972 2235488 2235553 "PSCAT" 2236397 NIL PSCAT (NIL T T T) -9 NIL 2236637 NIL) (-978 2233035 2233251 2233651 "PSCAT-" 2233656 NIL PSCAT- (NIL T T T T) -8 NIL NIL NIL) (-977 2231740 2232400 2232605 "PRTITION" 2232850 T PRTITION (NIL) -8 NIL NIL NIL) (-976 2231215 2231461 2231553 "PRTDAST" 2231668 T PRTDAST (NIL) -8 NIL NIL NIL) (-975 2220305 2222519 2224707 "PRS" 2229077 NIL PRS (NIL T T) -7 NIL NIL NIL) (-974 2218116 2219655 2219695 "PRQAGG" 2219878 NIL PRQAGG (NIL T) -9 NIL 2219980 NIL) (-973 2217320 2217625 2217653 "PROPLOG" 2217900 T PROPLOG (NIL) -9 NIL 2218066 NIL) (-972 2215501 2216067 2216364 "PROPFRML" 2217056 NIL PROPFRML (NIL T) -8 NIL NIL NIL) (-971 2214970 2215077 2215205 "PROPERTY" 2215393 T PROPERTY (NIL) -8 NIL NIL NIL) (-970 2209028 2213136 2213956 "PRODUCT" 2214196 NIL PRODUCT (NIL T T) -8 NIL NIL NIL) (-969 2208824 2208856 2208915 "PRINT" 2208989 T PRINT (NIL) -7 NIL NIL NIL) (-968 2208164 2208281 2208433 "PRIMES" 2208704 NIL PRIMES (NIL T) -7 NIL NIL NIL) (-967 2206229 2206630 2207096 "PRIMELT" 2207743 NIL PRIMELT (NIL T) -7 NIL NIL NIL) (-966 2205958 2206007 2206035 "PRIMCAT" 2206159 T PRIMCAT (NIL) -9 NIL NIL NIL) (-965 2204965 2205143 2205371 "PRIMARR2" 2205776 NIL PRIMARR2 (NIL T T) -7 NIL NIL NIL) (-964 2201080 2204903 2204948 "PRIMARR" 2204953 NIL PRIMARR (NIL T) -8 NIL NIL NIL) 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T PDEPROB (NIL) -8 NIL NIL NIL) (-903 2058291 2058795 2059350 "PDEPACK" 2060211 T PDEPACK (NIL) -7 NIL NIL NIL) (-902 2057203 2057393 2057644 "PDECOMP" 2058090 NIL PDECOMP (NIL T T) -7 NIL NIL NIL) (-901 2054782 2055625 2055653 "PDECAT" 2056440 T PDECAT (NIL) -9 NIL 2057153 NIL) (-900 2054533 2054566 2054656 "PCOMP" 2054743 NIL PCOMP (NIL T T) -7 NIL NIL NIL) (-899 2052711 2053334 2053631 "PBWLB" 2054262 NIL PBWLB (NIL T) -8 NIL NIL NIL) (-898 2052343 2052400 2052509 "PATTERN2" 2052648 NIL PATTERN2 (NIL T T) -7 NIL NIL NIL) (-897 2050100 2050488 2050945 "PATTERN1" 2051932 NIL PATTERN1 (NIL T T) -7 NIL NIL NIL) (-896 2042575 2044173 2045511 "PATTERN" 2048783 NIL PATTERN (NIL T) -8 NIL NIL NIL) (-895 2042139 2042206 2042338 "PATRES2" 2042502 NIL PATRES2 (NIL T T T) -7 NIL NIL NIL) (-894 2039507 2040088 2040569 "PATRES" 2041704 NIL PATRES (NIL T T) -8 NIL NIL NIL) (-893 2037390 2037795 2038202 "PATMATCH" 2039174 NIL PATMATCH (NIL T T T) -7 NIL NIL NIL) (-892 2036900 2037109 2037150 "PATMAB" 2037257 NIL PATMAB (NIL T) -9 NIL 2037340 NIL) (-891 2035418 2035754 2036012 "PATLRES" 2036705 NIL PATLRES (NIL T T T) -8 NIL NIL NIL) (-890 2034964 2035087 2035128 "PATAB" 2035133 NIL PATAB (NIL T) -9 NIL 2035305 NIL) (-889 2032445 2032977 2033550 "PARTPERM" 2034411 T PARTPERM (NIL) -7 NIL NIL NIL) (-888 2032066 2032129 2032231 "PARSURF" 2032376 NIL PARSURF (NIL T) -8 NIL NIL NIL) (-887 2031698 2031755 2031864 "PARSU2" 2032003 NIL PARSU2 (NIL T T) -7 NIL NIL NIL) (-886 2031462 2031502 2031569 "PARSER" 2031651 T PARSER (NIL) -7 NIL NIL NIL) (-885 2031083 2031146 2031248 "PARSCURV" 2031393 NIL PARSCURV (NIL T) -8 NIL NIL NIL) (-884 2030715 2030772 2030881 "PARSC2" 2031020 NIL PARSC2 (NIL T T) -7 NIL NIL NIL) (-883 2030354 2030412 2030509 "PARPCURV" 2030651 NIL PARPCURV (NIL T) -8 NIL NIL NIL) (-882 2029986 2030043 2030152 "PARPC2" 2030291 NIL PARPC2 (NIL T T) -7 NIL NIL NIL) (-881 2029047 2029359 2029541 "PARAMAST" 2029824 T PARAMAST (NIL) -8 NIL NIL NIL) (-880 2028567 2028653 2028772 "PAN2EXPR" 2028948 T PAN2EXPR (NIL) -7 NIL NIL NIL) (-879 2027344 2027688 2027916 "PALETTE" 2028359 T PALETTE (NIL) -8 NIL NIL NIL) (-878 2025737 2026349 2026709 "PAIR" 2027030 NIL PAIR (NIL T T) -8 NIL NIL NIL) (-877 2019628 2024996 2025190 "PADICRC" 2025592 NIL PADICRC (NIL NIL T) -8 NIL NIL NIL) (-876 2012878 2018974 2019158 "PADICRAT" 2019476 NIL PADICRAT (NIL NIL) -8 NIL NIL NIL) (-875 2009990 2011552 2011592 "PADICCT" 2012173 NIL PADICCT (NIL NIL) -9 NIL 2012455 NIL) (-874 2008307 2009927 2009972 "PADIC" 2009977 NIL PADIC (NIL NIL) -8 NIL NIL NIL) (-873 2007264 2007464 2007732 "PADEPAC" 2008094 NIL PADEPAC (NIL T NIL NIL) -7 NIL NIL NIL) (-872 2006476 2006609 2006815 "PADE" 2007126 NIL PADE (NIL T T T) -7 NIL NIL NIL) (-871 2004863 2005684 2005964 "OWP" 2006280 NIL OWP (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-870 2004356 2004569 2004666 "OVERSET" 2004786 T OVERSET (NIL) -8 NIL NIL NIL) (-869 2003402 2003961 2004133 "OVAR" 2004224 NIL OVAR (NIL NIL) -8 NIL NIL NIL) (-868 1992274 1994511 1996711 "OUTFORM" 2001222 T OUTFORM (NIL) -8 NIL NIL NIL) (-867 1991610 1991871 1991998 "OUTBFILE" 1992167 T OUTBFILE (NIL) -8 NIL NIL NIL) (-866 1990917 1991082 1991110 "OUTBCON" 1991428 T OUTBCON (NIL) -9 NIL 1991594 NIL) (-865 1990518 1990630 1990787 "OUTBCON-" 1990792 NIL OUTBCON- (NIL T) -8 NIL NIL NIL) (-864 1989782 1989903 1990064 "OUT" 1990377 T OUT (NIL) -7 NIL NIL NIL) (-863 1989162 1989511 1989600 "OSI" 1989713 T OSI (NIL) -8 NIL NIL NIL) (-862 1988692 1989030 1989058 "OSGROUP" 1989063 T OSGROUP (NIL) -9 NIL 1989085 NIL) (-861 1987437 1987664 1987949 "ORTHPOL" 1988439 NIL ORTHPOL (NIL T) -7 NIL NIL NIL) (-860 1985002 1987272 1987393 "OREUP" 1987398 NIL OREUP (NIL NIL T NIL NIL) -8 NIL NIL NIL) (-859 1982419 1984693 1984820 "ORESUP" 1984944 NIL ORESUP (NIL T NIL NIL) -8 NIL NIL NIL) (-858 1979947 1980447 1981008 "OREPCTO" 1981908 NIL OREPCTO (NIL T T) -7 NIL NIL NIL) (-857 1973640 1975834 1975875 "OREPCAT" 1978223 NIL OREPCAT (NIL T) -9 NIL 1979327 NIL) (-856 1970808 1971583 1972634 "OREPCAT-" 1972639 NIL OREPCAT- (NIL T T) -8 NIL NIL NIL) (-855 1969959 1970257 1970285 "ORDSET" 1970594 T ORDSET (NIL) -9 NIL 1970758 NIL) (-854 1969390 1969538 1969762 "ORDSET-" 1969767 NIL ORDSET- (NIL T) -8 NIL NIL NIL) (-853 1967955 1968746 1968774 "ORDRING" 1968976 T ORDRING (NIL) -9 NIL 1969101 NIL) (-852 1967600 1967694 1967838 "ORDRING-" 1967843 NIL ORDRING- (NIL T) -8 NIL NIL NIL) (-851 1966980 1967443 1967471 "ORDMON" 1967476 T ORDMON (NIL) -9 NIL 1967497 NIL) (-850 1966142 1966289 1966484 "ORDFUNS" 1966829 NIL ORDFUNS (NIL NIL T) -7 NIL NIL NIL) (-849 1965480 1965899 1965927 "ORDFIN" 1965992 T ORDFIN (NIL) -9 NIL 1966066 NIL) (-848 1964746 1964873 1965059 "ORDCOMP2" 1965340 NIL ORDCOMP2 (NIL T T) -7 NIL NIL NIL) (-847 1961312 1963332 1963741 "ORDCOMP" 1964370 NIL ORDCOMP (NIL T) -8 NIL NIL NIL) (-846 1957893 1958803 1959617 "OPTPROB" 1960518 T OPTPROB (NIL) -8 NIL NIL NIL) (-845 1954695 1955334 1956038 "OPTPACK" 1957209 T OPTPACK (NIL) -7 NIL NIL NIL) (-844 1952382 1953148 1953176 "OPTCAT" 1953995 T OPTCAT (NIL) -9 NIL 1954645 NIL) (-843 1951766 1952059 1952164 "OPSIG" 1952297 T OPSIG (NIL) -8 NIL NIL NIL) (-842 1951534 1951573 1951639 "OPQUERY" 1951720 T OPQUERY (NIL) -7 NIL NIL NIL) (-841 1950908 1951134 1951175 "OPERCAT" 1951387 NIL OPERCAT (NIL T) -9 NIL 1951484 NIL) (-840 1950663 1950719 1950836 "OPERCAT-" 1950841 NIL OPERCAT- (NIL T T) -8 NIL NIL NIL) (-839 1947796 1948974 1949478 "OP" 1950192 NIL OP (NIL T) -8 NIL NIL NIL) (-838 1947101 1947216 1947390 "ONECOMP2" 1947668 NIL ONECOMP2 (NIL T T) -7 NIL NIL NIL) (-837 1943921 1945898 1946267 "ONECOMP" 1946765 NIL ONECOMP (NIL T) -8 NIL NIL NIL) (-836 1943340 1943446 1943576 "OMSERVER" 1943811 T OMSERVER (NIL) -7 NIL NIL NIL) (-835 1940202 1942780 1942820 "OMSAGG" 1942881 NIL OMSAGG (NIL T) -9 NIL 1942945 NIL) (-834 1938825 1939088 1939370 "OMPKG" 1939940 T OMPKG (NIL) -7 NIL NIL NIL) (-833 1937372 1938374 1938543 "OMLO" 1938706 NIL OMLO (NIL T T) -8 NIL NIL NIL) (-832 1936332 1936479 1936699 "OMEXPR" 1937198 NIL OMEXPR (NIL T) -7 NIL NIL NIL) (-831 1935483 1935753 1935913 "OMERRK" 1936192 T OMERRK (NIL) -8 NIL NIL NIL) (-830 1934774 1935029 1935165 "OMERR" 1935367 T OMERR (NIL) -8 NIL NIL NIL) (-829 1934225 1934451 1934559 "OMENC" 1934686 T OMENC (NIL) -8 NIL NIL NIL) (-828 1928120 1929305 1930476 "OMDEV" 1933074 T OMDEV (NIL) -8 NIL NIL NIL) (-827 1927189 1927360 1927554 "OMCONN" 1927946 T OMCONN (NIL) -8 NIL NIL NIL) (-826 1926619 1926722 1926750 "OM" 1927049 T OM (NIL) -9 NIL NIL NIL) (-825 1925140 1926116 1926144 "OINTDOM" 1926149 T OINTDOM (NIL) -9 NIL 1926170 NIL) (-824 1922485 1923828 1924165 "OFMONOID" 1924835 NIL OFMONOID (NIL T) -8 NIL NIL NIL) (-823 1921896 1922422 1922467 "ODVAR" 1922472 NIL ODVAR (NIL T) -8 NIL NIL NIL) (-822 1919321 1921641 1921796 "ODR" 1921801 NIL ODR (NIL T T NIL) -8 NIL NIL NIL) (-821 1911943 1919097 1919223 "ODPOL" 1919228 NIL ODPOL (NIL T) -8 NIL NIL NIL) (-820 1905772 1911815 1911920 "ODP" 1911925 NIL ODP (NIL NIL T NIL) -8 NIL NIL NIL) (-819 1904538 1904753 1905028 "ODETOOLS" 1905546 NIL ODETOOLS (NIL T T) -7 NIL NIL NIL) (-818 1901505 1902163 1902879 "ODESYS" 1903871 NIL ODESYS (NIL T T) -7 NIL NIL NIL) (-817 1896387 1897295 1898320 "ODERTRIC" 1900580 NIL ODERTRIC (NIL T T) -7 NIL NIL NIL) (-816 1895813 1895895 1896089 "ODERED" 1896299 NIL ODERED (NIL T T T T T) -7 NIL NIL NIL) (-815 1892709 1893255 1893930 "ODERAT" 1895238 NIL ODERAT (NIL T T) -7 NIL NIL NIL) (-814 1889666 1890133 1890730 "ODEPRRIC" 1892238 NIL ODEPRRIC (NIL T T T T) -7 NIL NIL NIL) (-813 1887609 1888205 1888691 "ODEPROB" 1889200 T ODEPROB (NIL) -8 NIL NIL NIL) (-812 1884129 1884614 1885261 "ODEPRIM" 1887088 NIL ODEPRIM (NIL T T T T) -7 NIL NIL NIL) (-811 1883378 1883480 1883740 "ODEPAL" 1884021 NIL ODEPAL (NIL T T T T) -7 NIL NIL NIL) (-810 1879540 1880331 1881195 "ODEPACK" 1882534 T ODEPACK (NIL) -7 NIL NIL NIL) (-809 1878601 1878708 1878930 "ODEINT" 1879429 NIL ODEINT (NIL T T) -7 NIL NIL NIL) (-808 1872702 1874127 1875574 "ODEIFTBL" 1877174 T ODEIFTBL (NIL) -8 NIL NIL NIL) (-807 1868114 1868896 1869844 "ODEEF" 1871865 NIL ODEEF (NIL T T) -7 NIL NIL NIL) (-806 1867463 1867552 1867775 "ODECONST" 1868019 NIL ODECONST (NIL T T T) -7 NIL NIL NIL) (-805 1865588 1866249 1866277 "ODECAT" 1866882 T ODECAT (NIL) -9 NIL 1867413 NIL) (-804 1865226 1865269 1865396 "OCTCT2" 1865539 NIL OCTCT2 (NIL T T T T) -7 NIL NIL NIL) (-803 1862093 1864931 1865053 "OCT" 1865136 NIL OCT (NIL T) -8 NIL NIL NIL) (-802 1861445 1861913 1861941 "OCAMON" 1861946 T OCAMON (NIL) -9 NIL 1861967 NIL) (-801 1856101 1858529 1858569 "OC" 1859666 NIL OC (NIL T) -9 NIL 1860524 NIL) (-800 1853349 1854090 1855073 "OC-" 1855167 NIL OC- (NIL T T) -8 NIL NIL NIL) (-799 1852880 1853221 1853249 "OASGP" 1853254 T OASGP (NIL) -9 NIL 1853274 NIL) (-798 1852141 1852630 1852658 "OAMONS" 1852698 T OAMONS (NIL) -9 NIL 1852741 NIL) (-797 1851555 1851988 1852016 "OAMON" 1852021 T OAMON (NIL) -9 NIL 1852041 NIL) (-796 1850813 1851331 1851359 "OAGROUP" 1851364 T OAGROUP (NIL) -9 NIL 1851384 NIL) (-795 1850503 1850553 1850641 "NUMTUBE" 1850757 NIL NUMTUBE (NIL T) -7 NIL NIL NIL) (-794 1844076 1845594 1847130 "NUMQUAD" 1848987 T NUMQUAD (NIL) -7 NIL NIL NIL) (-793 1839832 1840820 1841845 "NUMODE" 1843071 T NUMODE (NIL) -7 NIL NIL NIL) (-792 1837187 1838067 1838095 "NUMINT" 1839018 T NUMINT (NIL) -9 NIL 1839782 NIL) (-791 1836135 1836332 1836550 "NUMFMT" 1836989 T NUMFMT (NIL) -7 NIL NIL NIL) (-790 1822494 1825439 1827971 "NUMERIC" 1833642 NIL NUMERIC (NIL T) -7 NIL NIL NIL) (-789 1816891 1821943 1822038 "NTSCAT" 1822043 NIL NTSCAT (NIL T T T T) -9 NIL 1822082 NIL) (-788 1816085 1816250 1816443 "NTPOLFN" 1816730 NIL NTPOLFN (NIL T) -7 NIL NIL NIL) (-787 1815717 1815774 1815883 "NSUP2" 1816022 NIL NSUP2 (NIL T T) -7 NIL NIL NIL) (-786 1803839 1812542 1813354 "NSUP" 1814938 NIL NSUP (NIL T) -8 NIL NIL NIL) (-785 1794115 1803613 1803746 "NSMP" 1803751 NIL NSMP (NIL T T) -8 NIL NIL NIL) (-784 1792547 1792848 1793205 "NREP" 1793803 NIL NREP (NIL T) -7 NIL NIL NIL) (-783 1791138 1791390 1791748 "NPCOEF" 1792290 NIL NPCOEF (NIL T T T T T) -7 NIL NIL NIL) (-782 1790204 1790319 1790535 "NORMRETR" 1791019 NIL NORMRETR (NIL T T T T NIL) -7 NIL NIL NIL) (-781 1788245 1788535 1788944 "NORMPK" 1789912 NIL NORMPK (NIL T T T T T) -7 NIL NIL NIL) (-780 1787930 1787958 1788082 "NORMMA" 1788211 NIL NORMMA (NIL T T T T) -7 NIL NIL NIL) (-779 1787719 1787748 1787817 "NONE1" 1787894 NIL NONE1 (NIL T) -7 NIL NIL NIL) (-778 1787519 1787676 1787705 "NONE" 1787710 T NONE (NIL) -8 NIL NIL NIL) (-777 1787016 1787078 1787257 "NODE1" 1787451 NIL NODE1 (NIL T T) -7 NIL NIL NIL) (-776 1785301 1786152 1786407 "NNI" 1786754 T NNI (NIL) -8 NIL NIL 1786989) (-775 1783721 1784034 1784398 "NLINSOL" 1784969 NIL NLINSOL (NIL T) -7 NIL NIL NIL) (-774 1779962 1780957 1781856 "NIPROB" 1782842 T NIPROB (NIL) -8 NIL NIL NIL) (-773 1778719 1778953 1779255 "NFINTBAS" 1779724 NIL NFINTBAS (NIL T T) -7 NIL NIL NIL) (-772 1777893 1778369 1778410 "NETCLT" 1778582 NIL NETCLT (NIL T) -9 NIL 1778664 NIL) (-771 1776601 1776832 1777113 "NCODIV" 1777661 NIL NCODIV (NIL T T) -7 NIL NIL NIL) (-770 1776363 1776400 1776475 "NCNTFRAC" 1776558 NIL NCNTFRAC (NIL T) -7 NIL NIL NIL) (-769 1774543 1774907 1775327 "NCEP" 1775988 NIL NCEP (NIL T) -7 NIL NIL NIL) (-768 1773401 1774167 1774195 "NASRING" 1774305 T NASRING (NIL) -9 NIL 1774385 NIL) (-767 1773196 1773240 1773334 "NASRING-" 1773339 NIL NASRING- (NIL T) -8 NIL NIL NIL) (-766 1772303 1772828 1772856 "NARNG" 1772973 T NARNG (NIL) -9 NIL 1773064 NIL) (-765 1771995 1772062 1772196 "NARNG-" 1772201 NIL NARNG- (NIL T) -8 NIL NIL NIL) (-764 1770874 1771081 1771316 "NAGSP" 1771780 T NAGSP (NIL) -7 NIL NIL NIL) (-763 1762146 1763830 1765503 "NAGS" 1769221 T NAGS (NIL) -7 NIL NIL NIL) (-762 1760694 1761002 1761333 "NAGF07" 1761835 T NAGF07 (NIL) -7 NIL NIL NIL) (-761 1755232 1756523 1757830 "NAGF04" 1759407 T NAGF04 (NIL) -7 NIL NIL NIL) (-760 1748200 1749814 1751447 "NAGF02" 1753619 T NAGF02 (NIL) -7 NIL NIL NIL) (-759 1743424 1744524 1745641 "NAGF01" 1747103 T NAGF01 (NIL) -7 NIL NIL NIL) (-758 1737052 1738618 1740203 "NAGE04" 1741859 T NAGE04 (NIL) -7 NIL NIL NIL) (-757 1728221 1730342 1732472 "NAGE02" 1734942 T NAGE02 (NIL) -7 NIL NIL NIL) (-756 1724174 1725121 1726085 "NAGE01" 1727277 T NAGE01 (NIL) -7 NIL NIL NIL) (-755 1721969 1722503 1723061 "NAGD03" 1723636 T NAGD03 (NIL) -7 NIL NIL NIL) (-754 1713719 1715647 1717601 "NAGD02" 1720035 T NAGD02 (NIL) -7 NIL NIL NIL) (-753 1707530 1708955 1710395 "NAGD01" 1712299 T NAGD01 (NIL) -7 NIL NIL NIL) (-752 1703739 1704561 1705398 "NAGC06" 1706713 T NAGC06 (NIL) -7 NIL NIL NIL) (-751 1702204 1702536 1702892 "NAGC05" 1703403 T NAGC05 (NIL) -7 NIL NIL NIL) (-750 1701580 1701699 1701843 "NAGC02" 1702080 T NAGC02 (NIL) -7 NIL NIL NIL) (-749 1700539 1701122 1701162 "NAALG" 1701241 NIL NAALG (NIL T) -9 NIL 1701302 NIL) (-748 1700374 1700403 1700493 "NAALG-" 1700498 NIL NAALG- (NIL T T) -8 NIL NIL NIL) (-747 1694324 1695432 1696619 "MULTSQFR" 1699270 NIL MULTSQFR (NIL T T T T) -7 NIL NIL NIL) (-746 1693643 1693718 1693902 "MULTFACT" 1694236 NIL MULTFACT (NIL T T T T) -7 NIL NIL NIL) (-745 1686367 1690280 1690333 "MTSCAT" 1691403 NIL MTSCAT (NIL T T) -9 NIL 1691918 NIL) (-744 1686079 1686133 1686225 "MTHING" 1686307 NIL MTHING (NIL T) -7 NIL NIL NIL) (-743 1685871 1685904 1685964 "MSYSCMD" 1686039 T MSYSCMD (NIL) -7 NIL NIL NIL) (-742 1682940 1685432 1685473 "MSETAGG" 1685478 NIL MSETAGG (NIL T) -9 NIL 1685512 NIL) (-741 1679022 1681695 1682015 "MSET" 1682653 NIL MSET (NIL T) -8 NIL NIL NIL) (-740 1674865 1676401 1677146 "MRING" 1678322 NIL MRING (NIL T T) -8 NIL NIL NIL) (-739 1674431 1674498 1674629 "MRF2" 1674792 NIL MRF2 (NIL T T T) -7 NIL NIL NIL) (-738 1674049 1674084 1674228 "MRATFAC" 1674390 NIL MRATFAC (NIL T T T T) -7 NIL NIL NIL) (-737 1671661 1671956 1672387 "MPRFF" 1673754 NIL MPRFF (NIL T T T T) -7 NIL NIL NIL) (-736 1665984 1671515 1671612 "MPOLY" 1671617 NIL MPOLY (NIL NIL T) -8 NIL NIL NIL) (-735 1665474 1665509 1665717 "MPCPF" 1665943 NIL MPCPF (NIL T T T T) -7 NIL NIL NIL) (-734 1664988 1665031 1665215 "MPC3" 1665425 NIL MPC3 (NIL T T T T T T T) -7 NIL NIL NIL) (-733 1664183 1664264 1664485 "MPC2" 1664903 NIL MPC2 (NIL T T T T T T T) -7 NIL NIL NIL) (-732 1662484 1662821 1663211 "MONOTOOL" 1663843 NIL MONOTOOL (NIL T T) -7 NIL NIL NIL) (-731 1661709 1662026 1662054 "MONOID" 1662273 T MONOID (NIL) -9 NIL 1662420 NIL) (-730 1661255 1661374 1661555 "MONOID-" 1661560 NIL MONOID- (NIL T) -8 NIL NIL NIL) (-729 1651739 1657681 1657740 "MONOGEN" 1658414 NIL MONOGEN (NIL T T) -9 NIL 1658870 NIL) (-728 1648978 1649706 1650699 "MONOGEN-" 1650818 NIL MONOGEN- (NIL T T T) -8 NIL NIL NIL) (-727 1647811 1648257 1648285 "MONADWU" 1648677 T MONADWU (NIL) -9 NIL 1648915 NIL) (-726 1647183 1647342 1647590 "MONADWU-" 1647595 NIL MONADWU- (NIL T) -8 NIL NIL NIL) (-725 1646542 1646786 1646814 "MONAD" 1647021 T MONAD (NIL) -9 NIL 1647133 NIL) (-724 1646227 1646305 1646437 "MONAD-" 1646442 NIL MONAD- (NIL T) -8 NIL NIL NIL) (-723 1644516 1645140 1645419 "MOEBIUS" 1645980 NIL MOEBIUS (NIL T) -8 NIL NIL NIL) (-722 1643794 1644198 1644238 "MODULE" 1644243 NIL MODULE (NIL T) -9 NIL 1644282 NIL) (-721 1643362 1643458 1643648 "MODULE-" 1643653 NIL MODULE- (NIL T T) -8 NIL NIL NIL) (-720 1641086 1641770 1642097 "MODRING" 1643186 NIL MODRING (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-719 1638032 1639191 1639712 "MODOP" 1640615 NIL MODOP (NIL T T) -8 NIL NIL NIL) (-718 1636620 1637099 1637376 "MODMONOM" 1637895 NIL MODMONOM (NIL T T NIL) -8 NIL NIL NIL) (-717 1626702 1634911 1635325 "MODMON" 1636257 NIL MODMON (NIL T T) -8 NIL NIL NIL) (-716 1623884 1625570 1625846 "MODFIELD" 1626577 NIL MODFIELD (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-715 1622861 1623165 1623355 "MMLFORM" 1623714 T MMLFORM (NIL) -8 NIL NIL NIL) (-714 1622387 1622430 1622609 "MMAP" 1622812 NIL MMAP (NIL T T T T T T) -7 NIL NIL NIL) (-713 1620466 1621233 1621274 "MLO" 1621697 NIL MLO (NIL T) -9 NIL 1621939 NIL) (-712 1617832 1618348 1618950 "MLIFT" 1619947 NIL MLIFT (NIL T T T T) -7 NIL NIL NIL) (-711 1617223 1617307 1617461 "MKUCFUNC" 1617743 NIL MKUCFUNC (NIL T T T) -7 NIL NIL NIL) (-710 1616822 1616892 1617015 "MKRECORD" 1617146 NIL MKRECORD (NIL T T) -7 NIL NIL NIL) (-709 1615869 1616031 1616259 "MKFUNC" 1616633 NIL MKFUNC (NIL T) -7 NIL NIL NIL) (-708 1615257 1615361 1615517 "MKFLCFN" 1615752 NIL MKFLCFN (NIL T) -7 NIL NIL NIL) (-707 1614534 1614636 1614821 "MKBCFUNC" 1615150 NIL MKBCFUNC (NIL T T T T) -7 NIL NIL NIL) (-706 1611243 1614088 1614224 "MINT" 1614418 T MINT (NIL) -8 NIL NIL NIL) (-705 1610055 1610298 1610575 "MHROWRED" 1610998 NIL MHROWRED (NIL T) -7 NIL NIL NIL) (-704 1605444 1608590 1608995 "MFLOAT" 1609670 T MFLOAT (NIL) -8 NIL NIL NIL) (-703 1604801 1604877 1605048 "MFINFACT" 1605356 NIL MFINFACT (NIL T T T T) -7 NIL NIL NIL) (-702 1601136 1601979 1602858 "MESH" 1603942 T MESH (NIL) -7 NIL NIL NIL) (-701 1599526 1599838 1600191 "MDDFACT" 1600823 NIL MDDFACT (NIL T) -7 NIL NIL NIL) (-700 1596321 1598685 1598726 "MDAGG" 1598981 NIL MDAGG (NIL T) -9 NIL 1599124 NIL) (-699 1586079 1595614 1595821 "MCMPLX" 1596134 T MCMPLX (NIL) -8 NIL NIL NIL) (-698 1585220 1585366 1585566 "MCDEN" 1585928 NIL MCDEN (NIL T T) -7 NIL NIL NIL) (-697 1583110 1583380 1583760 "MCALCFN" 1584950 NIL MCALCFN (NIL T T T T) -7 NIL NIL NIL) (-696 1582035 1582275 1582508 "MAYBE" 1582916 NIL MAYBE (NIL T) -8 NIL NIL NIL) (-695 1579647 1580170 1580732 "MATSTOR" 1581506 NIL MATSTOR (NIL T) -7 NIL NIL NIL) (-694 1575603 1579019 1579267 "MATRIX" 1579432 NIL MATRIX (NIL T) -8 NIL NIL NIL) (-693 1571367 1572076 1572812 "MATLIN" 1574960 NIL MATLIN (NIL T T T T) -7 NIL NIL NIL) (-692 1569961 1570114 1570447 "MATCAT2" 1571202 NIL MATCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-691 1560061 1563250 1563327 "MATCAT" 1568210 NIL MATCAT (NIL T T T) -9 NIL 1569627 NIL) (-690 1556417 1557438 1558794 "MATCAT-" 1558799 NIL MATCAT- (NIL T T T T) -8 NIL NIL NIL) (-689 1554529 1554853 1555237 "MAPPKG3" 1556092 NIL MAPPKG3 (NIL T T T) -7 NIL NIL NIL) (-688 1553510 1553683 1553905 "MAPPKG2" 1554353 NIL MAPPKG2 (NIL T T) -7 NIL NIL NIL) (-687 1552009 1552293 1552620 "MAPPKG1" 1553216 NIL MAPPKG1 (NIL T) -7 NIL NIL NIL) (-686 1551088 1551415 1551592 "MAPPAST" 1551852 T MAPPAST (NIL) -8 NIL NIL NIL) (-685 1550699 1550757 1550880 "MAPHACK3" 1551024 NIL MAPHACK3 (NIL T T T) -7 NIL NIL NIL) (-684 1550291 1550352 1550466 "MAPHACK2" 1550631 NIL MAPHACK2 (NIL T T) -7 NIL NIL NIL) (-683 1549728 1549832 1549974 "MAPHACK1" 1550182 NIL MAPHACK1 (NIL T) -7 NIL NIL NIL) (-682 1547807 1548428 1548732 "MAGMA" 1549456 NIL MAGMA (NIL T) -8 NIL NIL NIL) (-681 1547286 1547531 1547622 "MACROAST" 1547736 T MACROAST (NIL) -8 NIL NIL NIL) (-680 1543704 1545525 1545986 "M3D" 1546858 NIL M3D (NIL T) -8 NIL NIL NIL) (-679 1537812 1542073 1542114 "LZSTAGG" 1542896 NIL LZSTAGG (NIL T) -9 NIL 1543191 NIL) (-678 1533769 1534943 1536400 "LZSTAGG-" 1536405 NIL LZSTAGG- (NIL T T) -8 NIL NIL NIL) (-677 1530856 1531660 1532147 "LWORD" 1533314 NIL LWORD (NIL T) -8 NIL NIL NIL) (-676 1530432 1530660 1530735 "LSTAST" 1530801 T LSTAST (NIL) -8 NIL NIL NIL) (-675 1523629 1530203 1530337 "LSQM" 1530342 NIL LSQM (NIL NIL T) -8 NIL NIL NIL) (-674 1522853 1522992 1523220 "LSPP" 1523484 NIL LSPP (NIL T T T T) -7 NIL NIL NIL) (-673 1519695 1520352 1521065 "LSMP1" 1522172 NIL LSMP1 (NIL T) -7 NIL NIL NIL) (-672 1517530 1517824 1518273 "LSMP" 1519391 NIL LSMP (NIL T T T T) -7 NIL NIL NIL) (-671 1511409 1516697 1516738 "LSAGG" 1516800 NIL LSAGG (NIL T) -9 NIL 1516878 NIL) (-670 1508104 1509028 1510241 "LSAGG-" 1510246 NIL LSAGG- (NIL T T) -8 NIL NIL NIL) (-669 1505703 1507248 1507497 "LPOLY" 1507899 NIL LPOLY (NIL T T) -8 NIL NIL NIL) (-668 1505285 1505370 1505493 "LPEFRAC" 1505612 NIL LPEFRAC (NIL T) -7 NIL NIL NIL) (-667 1504937 1505049 1505077 "LOGIC" 1505188 T LOGIC (NIL) -9 NIL 1505269 NIL) (-666 1504799 1504822 1504893 "LOGIC-" 1504898 NIL LOGIC- (NIL T) -8 NIL NIL NIL) (-665 1503992 1504132 1504325 "LODOOPS" 1504655 NIL LODOOPS (NIL T T) -7 NIL NIL NIL) (-664 1502530 1502765 1503118 "LODOF" 1503739 NIL LODOF (NIL T T) -7 NIL NIL NIL) (-663 1498762 1501179 1501220 "LODOCAT" 1501658 NIL LODOCAT (NIL T) -9 NIL 1501869 NIL) (-662 1498495 1498553 1498680 "LODOCAT-" 1498685 NIL LODOCAT- (NIL T T) -8 NIL NIL NIL) (-661 1495829 1498336 1498454 "LODO2" 1498459 NIL LODO2 (NIL T T) -8 NIL NIL NIL) (-660 1493278 1495766 1495811 "LODO1" 1495816 NIL LODO1 (NIL T) -8 NIL NIL NIL) (-659 1490715 1493194 1493260 "LODO" 1493265 NIL LODO (NIL T NIL) -8 NIL NIL NIL) (-658 1489596 1489761 1490066 "LODEEF" 1490538 NIL LODEEF (NIL T T T) -7 NIL NIL NIL) (-657 1487917 1488690 1488943 "LO" 1489428 NIL LO (NIL T T T) -8 NIL NIL NIL) (-656 1483156 1486047 1486088 "LNAGG" 1487035 NIL LNAGG (NIL T) -9 NIL 1487479 NIL) (-655 1482303 1482517 1482859 "LNAGG-" 1482864 NIL LNAGG- (NIL T T) -8 NIL NIL NIL) (-654 1478439 1479228 1479867 "LMOPS" 1481718 NIL LMOPS (NIL T T NIL) -8 NIL NIL NIL) (-653 1477842 1478230 1478271 "LMODULE" 1478276 NIL LMODULE (NIL T) -9 NIL 1478302 NIL) (-652 1475040 1477487 1477610 "LMDICT" 1477752 NIL LMDICT (NIL T) -8 NIL NIL NIL) (-651 1474446 1474667 1474708 "LLINSET" 1474899 NIL LLINSET (NIL T) -9 NIL 1474990 NIL) (-650 1474145 1474354 1474414 "LITERAL" 1474419 NIL LITERAL (NIL T) -8 NIL NIL NIL) (-649 1473670 1473744 1473883 "LIST3" 1474065 NIL LIST3 (NIL T T T) -7 NIL NIL NIL) (-648 1471804 1472116 1472515 "LIST2MAP" 1473317 NIL LIST2MAP (NIL T T) -7 NIL NIL NIL) (-647 1470811 1470989 1471217 "LIST2" 1471622 NIL LIST2 (NIL T T) -7 NIL NIL NIL) (-646 1463976 1469745 1470049 "LIST" 1470540 NIL LIST (NIL T) -8 NIL NIL NIL) (-645 1463572 1463809 1463850 "LINSET" 1463855 NIL LINSET (NIL T) -9 NIL 1463889 NIL) (-644 1462233 1462903 1462944 "LINEXP" 1463199 NIL LINEXP (NIL T) -9 NIL 1463348 NIL) (-643 1460880 1461140 1461437 "LINDEP" 1461985 NIL LINDEP (NIL T T) -7 NIL NIL NIL) (-642 1457718 1458418 1459176 "LIMITRF" 1460154 NIL LIMITRF (NIL T) -7 NIL NIL NIL) (-641 1456044 1456333 1456735 "LIMITPS" 1457420 NIL LIMITPS (NIL T T) -7 NIL NIL NIL) (-640 1454992 1455461 1455501 "LIECAT" 1455641 NIL LIECAT (NIL T) -9 NIL 1455792 NIL) (-639 1454833 1454860 1454948 "LIECAT-" 1454953 NIL LIECAT- (NIL T T) -8 NIL NIL NIL) (-638 1449293 1454344 1454572 "LIE" 1454654 NIL LIE (NIL T T) -8 NIL NIL NIL) (-637 1441791 1448742 1448907 "LIB" 1449148 T LIB (NIL) -8 NIL NIL NIL) (-636 1437426 1438309 1439244 "LGROBP" 1440908 NIL LGROBP (NIL NIL T) -7 NIL NIL NIL) (-635 1436266 1436958 1436986 "LFCAT" 1437193 T LFCAT (NIL) -9 NIL 1437332 NIL) (-634 1434264 1434538 1434888 "LF" 1435987 NIL LF (NIL T T) -7 NIL NIL NIL) (-633 1431166 1431796 1432484 "LEXTRIPK" 1433628 NIL LEXTRIPK (NIL T NIL) -7 NIL NIL NIL) (-632 1427910 1428736 1429239 "LEXP" 1430746 NIL LEXP (NIL T T NIL) -8 NIL NIL NIL) (-631 1427386 1427631 1427723 "LETAST" 1427838 T LETAST (NIL) -8 NIL NIL NIL) (-630 1425784 1426097 1426498 "LEADCDET" 1427068 NIL LEADCDET (NIL T T T T) -7 NIL NIL NIL) (-629 1424974 1425048 1425277 "LAZM3PK" 1425705 NIL LAZM3PK (NIL T T T T T T) -7 NIL NIL NIL) (-628 1419905 1423051 1423589 "LAUPOL" 1424486 NIL LAUPOL (NIL T T) -8 NIL NIL NIL) (-627 1419484 1419528 1419689 "LAPLACE" 1419855 NIL LAPLACE (NIL T T) -7 NIL NIL NIL) (-626 1418478 1419062 1419103 "LALG" 1419165 NIL LALG (NIL T) -9 NIL 1419224 NIL) (-625 1418192 1418251 1418387 "LALG-" 1418392 NIL LALG- (NIL T T) -8 NIL NIL NIL) (-624 1416131 1417293 1417544 "LA" 1418025 NIL LA (NIL T T T) -8 NIL NIL NIL) (-623 1415966 1415990 1416031 "KVTFROM" 1416093 NIL KVTFROM (NIL T) -9 NIL NIL NIL) (-622 1414889 1415333 1415518 "KTVLOGIC" 1415801 T KTVLOGIC (NIL) -8 NIL NIL NIL) (-621 1414724 1414748 1414789 "KRCFROM" 1414851 NIL KRCFROM (NIL T) -9 NIL NIL NIL) (-620 1413628 1413815 1414114 "KOVACIC" 1414524 NIL KOVACIC (NIL T T) -7 NIL NIL NIL) (-619 1413463 1413487 1413528 "KONVERT" 1413590 NIL KONVERT (NIL T) -9 NIL NIL NIL) (-618 1413298 1413322 1413363 "KOERCE" 1413425 NIL KOERCE (NIL T) -9 NIL NIL NIL) (-617 1412794 1412875 1413007 "KERNEL2" 1413212 NIL KERNEL2 (NIL T T) -7 NIL NIL NIL) (-616 1410624 1411387 1411764 "KERNEL" 1412450 NIL KERNEL (NIL T) -8 NIL NIL NIL) (-615 1404394 1409163 1409217 "KDAGG" 1409594 NIL KDAGG (NIL T T) -9 NIL 1409800 NIL) (-614 1403923 1404047 1404252 "KDAGG-" 1404257 NIL KDAGG- (NIL T T T) -8 NIL NIL NIL) (-613 1397073 1403584 1403739 "KAFILE" 1403801 NIL KAFILE (NIL T) -8 NIL NIL NIL) (-612 1391533 1396584 1396812 "JORDAN" 1396894 NIL JORDAN (NIL T T) -8 NIL NIL NIL) (-611 1390912 1391182 1391303 "JOINAST" 1391432 T JOINAST (NIL) -8 NIL NIL NIL) (-610 1390758 1390817 1390872 "JAVACODE" 1390877 T JAVACODE (NIL) -8 NIL NIL NIL) (-609 1387010 1388963 1389017 "IXAGG" 1389946 NIL IXAGG (NIL T T) -9 NIL 1390405 NIL) (-608 1385929 1386235 1386654 "IXAGG-" 1386659 NIL IXAGG- (NIL T T T) -8 NIL NIL NIL) (-607 1381459 1385851 1385910 "IVECTOR" 1385915 NIL IVECTOR (NIL T NIL) -8 NIL NIL NIL) (-606 1380225 1380462 1380728 "ITUPLE" 1381226 NIL ITUPLE (NIL T) -8 NIL NIL NIL) (-605 1378727 1378904 1379199 "ITRIGMNP" 1380047 NIL ITRIGMNP (NIL T T T) -7 NIL NIL NIL) (-604 1377472 1377676 1377959 "ITFUN3" 1378503 NIL ITFUN3 (NIL T T T) -7 NIL NIL NIL) (-603 1377104 1377161 1377270 "ITFUN2" 1377409 NIL ITFUN2 (NIL T T) -7 NIL NIL NIL) (-602 1377021 1377049 1377084 "ITFORM" 1377089 T ITFORM (NIL) -8 NIL NIL NIL) (-601 1374982 1376041 1376319 "ITAYLOR" 1376776 NIL ITAYLOR (NIL T) -8 NIL NIL NIL) (-600 1363927 1369119 1370282 "ISUPS" 1373852 NIL ISUPS (NIL T) -8 NIL NIL NIL) (-599 1363031 1363171 1363407 "ISUMP" 1363774 NIL ISUMP (NIL T T T T) -7 NIL NIL NIL) (-598 1358406 1362976 1363017 "ISTRING" 1363022 NIL ISTRING (NIL NIL) -8 NIL NIL NIL) (-597 1357882 1358127 1358219 "ISAST" 1358334 T ISAST (NIL) -8 NIL NIL NIL) (-596 1357091 1357173 1357389 "IRURPK" 1357796 NIL IRURPK (NIL T T T T T) -7 NIL NIL NIL) (-595 1356027 1356228 1356468 "IRSN" 1356871 T IRSN (NIL) -7 NIL NIL NIL) (-594 1354098 1354453 1354882 "IRRF2F" 1355665 NIL IRRF2F (NIL T) -7 NIL NIL NIL) (-593 1353845 1353883 1353959 "IRREDFFX" 1354054 NIL IRREDFFX (NIL T) -7 NIL NIL NIL) (-592 1352460 1352719 1353018 "IROOT" 1353578 NIL IROOT (NIL T) -7 NIL NIL NIL) (-591 1352304 1352363 1352419 "IRFORM" 1352424 T IRFORM (NIL) -8 NIL NIL NIL) (-590 1351404 1351517 1351731 "IR2F" 1352187 NIL IR2F (NIL T T) -7 NIL NIL NIL) (-589 1349017 1349512 1350078 "IR2" 1350882 NIL IR2 (NIL T T) -7 NIL NIL NIL) (-588 1345621 1346701 1347393 "IR" 1348357 NIL IR (NIL T) -8 NIL NIL NIL) (-587 1345412 1345446 1345506 "IPRNTPK" 1345581 T IPRNTPK (NIL) -7 NIL NIL NIL) (-586 1341995 1345301 1345370 "IPF" 1345375 NIL IPF (NIL NIL) -8 NIL NIL NIL) (-585 1340324 1341920 1341977 "IPADIC" 1341982 NIL IPADIC (NIL NIL NIL) -8 NIL NIL NIL) (-584 1339636 1339884 1340014 "IP4ADDR" 1340214 T IP4ADDR (NIL) -8 NIL NIL NIL) (-583 1339109 1339340 1339450 "IOMODE" 1339546 T IOMODE (NIL) -8 NIL NIL NIL) (-582 1338182 1338706 1338833 "IOBFILE" 1339002 T IOBFILE (NIL) -8 NIL NIL NIL) (-581 1337670 1338086 1338114 "IOBCON" 1338119 T IOBCON (NIL) -9 NIL 1338140 NIL) (-580 1337181 1337239 1337422 "INVLAPLA" 1337606 NIL INVLAPLA (NIL T T) -7 NIL NIL NIL) (-579 1326877 1329219 1331593 "INTTR" 1334857 NIL INTTR (NIL T T) -7 NIL NIL NIL) (-578 1323212 1323954 1324819 "INTTOOLS" 1326062 NIL INTTOOLS (NIL T T) -7 NIL NIL NIL) (-577 1322798 1322889 1323006 "INTSLPE" 1323115 T INTSLPE (NIL) -7 NIL NIL NIL) (-576 1320751 1322721 1322780 "INTRVL" 1322785 NIL INTRVL (NIL T) -8 NIL NIL NIL) (-575 1318353 1318865 1319440 "INTRF" 1320236 NIL INTRF (NIL T) -7 NIL NIL NIL) (-574 1317764 1317861 1318003 "INTRET" 1318251 NIL INTRET (NIL T) -7 NIL NIL NIL) (-573 1315761 1316150 1316620 "INTRAT" 1317372 NIL INTRAT (NIL T T) -7 NIL NIL NIL) (-572 1313024 1313607 1314226 "INTPM" 1315246 NIL INTPM (NIL T T) -7 NIL NIL NIL) (-571 1309792 1310384 1311115 "INTPAF" 1312417 NIL INTPAF (NIL T T T) -7 NIL NIL NIL) (-570 1304971 1305933 1306984 "INTPACK" 1308761 T INTPACK (NIL) -7 NIL NIL NIL) (-569 1304223 1304375 1304583 "INTHERTR" 1304813 NIL INTHERTR (NIL T T) -7 NIL NIL NIL) (-568 1303662 1303742 1303930 "INTHERAL" 1304137 NIL INTHERAL (NIL T T T T) -7 NIL NIL NIL) (-567 1301508 1301951 1302408 "INTHEORY" 1303225 T INTHEORY (NIL) -7 NIL NIL NIL) (-566 1292972 1294575 1296329 "INTG0" 1299878 NIL INTG0 (NIL T T T) -7 NIL NIL NIL) (-565 1279245 1282610 1285995 "INTFTBL" 1289607 T INTFTBL (NIL) -8 NIL NIL NIL) (-564 1278494 1278632 1278805 "INTFACT" 1279104 NIL INTFACT (NIL T) -7 NIL NIL NIL) (-563 1275927 1276371 1276926 "INTEF" 1278050 NIL INTEF (NIL T T) -7 NIL NIL NIL) (-562 1274294 1275033 1275061 "INTDOM" 1275362 T INTDOM (NIL) -9 NIL 1275569 NIL) (-561 1273663 1273837 1274079 "INTDOM-" 1274084 NIL INTDOM- (NIL T) -8 NIL NIL NIL) (-560 1270051 1271979 1272033 "INTCAT" 1272832 NIL INTCAT (NIL T) -9 NIL 1273153 NIL) (-559 1269523 1269626 1269754 "INTBIT" 1269943 T INTBIT (NIL) -7 NIL NIL NIL) (-558 1268222 1268376 1268683 "INTALG" 1269368 NIL INTALG (NIL T T T T T) -7 NIL NIL NIL) (-557 1267705 1267795 1267952 "INTAF" 1268126 NIL INTAF (NIL T T) -7 NIL NIL NIL) (-556 1261050 1267515 1267655 "INTABL" 1267660 NIL INTABL (NIL T T T) -8 NIL NIL NIL) (-555 1260391 1260857 1260922 "INT8" 1260956 T INT8 (NIL) -8 NIL NIL 1261001) (-554 1259731 1260197 1260262 "INT64" 1260296 T INT64 (NIL) -8 NIL NIL 1260341) (-553 1259071 1259537 1259602 "INT32" 1259636 T INT32 (NIL) -8 NIL NIL 1259681) (-552 1258411 1258877 1258942 "INT16" 1258976 T INT16 (NIL) -8 NIL NIL 1259021) (-551 1255361 1258208 1258317 "INT" 1258322 T INT (NIL) -8 NIL NIL NIL) (-550 1250273 1252984 1253012 "INS" 1253946 T INS (NIL) -9 NIL 1254611 NIL) (-549 1247513 1248284 1249258 "INS-" 1249331 NIL INS- (NIL T) -8 NIL NIL NIL) (-548 1246361 1246566 1246842 "INPSIGN" 1247288 NIL INPSIGN (NIL T T) -7 NIL NIL NIL) (-547 1245479 1245596 1245793 "INPRODPF" 1246241 NIL INPRODPF (NIL T T) -7 NIL NIL NIL) (-546 1244373 1244490 1244727 "INPRODFF" 1245359 NIL INPRODFF (NIL T T T T) -7 NIL NIL NIL) (-545 1243373 1243525 1243785 "INNMFACT" 1244209 NIL INNMFACT (NIL T T T T) -7 NIL NIL NIL) (-544 1242570 1242667 1242855 "INMODGCD" 1243272 NIL INMODGCD (NIL T T NIL NIL) -7 NIL NIL NIL) (-543 1241078 1241323 1241647 "INFSP" 1242315 NIL INFSP (NIL T T T) -7 NIL NIL NIL) (-542 1240262 1240379 1240562 "INFPROD0" 1240958 NIL INFPROD0 (NIL T T) -7 NIL NIL NIL) (-541 1239872 1239932 1240030 "INFORM1" 1240197 NIL INFORM1 (NIL T) -7 NIL NIL NIL) (-540 1236727 1237937 1238452 "INFORM" 1239365 T INFORM (NIL) -8 NIL NIL NIL) (-539 1236250 1236339 1236453 "INFINITY" 1236633 T INFINITY (NIL) -7 NIL NIL NIL) (-538 1235426 1235970 1236071 "INETCLTS" 1236169 T INETCLTS (NIL) -8 NIL NIL NIL) (-537 1234042 1234292 1234613 "INEP" 1235174 NIL INEP (NIL T T T) -7 NIL NIL NIL) (-536 1233291 1233939 1234004 "INDE" 1234009 NIL INDE (NIL T) -8 NIL NIL NIL) (-535 1232855 1232923 1233040 "INCRMAPS" 1233218 NIL INCRMAPS (NIL T) -7 NIL NIL NIL) (-534 1231673 1232124 1232330 "INBFILE" 1232669 T INBFILE (NIL) -8 NIL NIL NIL) (-533 1226973 1227909 1228853 "INBFF" 1230761 NIL INBFF (NIL T) -7 NIL NIL NIL) (-532 1225881 1226150 1226178 "INBCON" 1226691 T INBCON (NIL) -9 NIL 1226957 NIL) (-531 1225133 1225356 1225632 "INBCON-" 1225637 NIL INBCON- (NIL T) -8 NIL NIL NIL) (-530 1224612 1224857 1224948 "INAST" 1225062 T INAST (NIL) -8 NIL NIL NIL) (-529 1224039 1224291 1224397 "IMPTAST" 1224526 T IMPTAST (NIL) -8 NIL NIL NIL) (-528 1220484 1223883 1223987 "IMATRIX" 1223992 NIL IMATRIX (NIL T NIL NIL) -8 NIL NIL NIL) (-527 1219196 1219319 1219634 "IMATQF" 1220340 NIL IMATQF (NIL T T T T T T T T) -7 NIL NIL NIL) (-526 1217416 1217643 1217980 "IMATLIN" 1218952 NIL IMATLIN (NIL T T T T) -7 NIL NIL NIL) (-525 1211996 1217340 1217398 "ILIST" 1217403 NIL ILIST (NIL T NIL) -8 NIL NIL NIL) (-524 1209901 1211856 1211969 "IIARRAY2" 1211974 NIL IIARRAY2 (NIL T NIL NIL T T) -8 NIL NIL NIL) (-523 1205301 1209812 1209876 "IFF" 1209881 NIL IFF (NIL NIL NIL) -8 NIL NIL NIL) (-522 1204648 1204918 1205034 "IFAST" 1205205 T IFAST (NIL) -8 NIL NIL NIL) (-521 1199643 1203940 1204128 "IFARRAY" 1204505 NIL IFARRAY (NIL T NIL) -8 NIL NIL NIL) (-520 1198823 1199547 1199620 "IFAMON" 1199625 NIL IFAMON (NIL T T NIL) -8 NIL NIL NIL) (-519 1198407 1198472 1198526 "IEVALAB" 1198733 NIL IEVALAB (NIL T T) -9 NIL NIL NIL) (-518 1198082 1198150 1198310 "IEVALAB-" 1198315 NIL IEVALAB- (NIL T T T) -8 NIL NIL NIL) (-517 1197332 1197971 1198046 "IDPOAMS" 1198051 NIL IDPOAMS (NIL T T) -8 NIL NIL NIL) (-516 1196639 1197221 1197296 "IDPOAM" 1197301 NIL IDPOAM (NIL T T) -8 NIL NIL NIL) (-515 1196270 1196553 1196616 "IDPO" 1196621 NIL IDPO (NIL T T) -8 NIL NIL NIL) (-514 1195329 1195605 1195658 "IDPC" 1196071 NIL IDPC (NIL T T) -9 NIL 1196220 NIL) (-513 1194798 1195221 1195294 "IDPAM" 1195299 NIL IDPAM (NIL T T) -8 NIL NIL NIL) (-512 1194174 1194690 1194763 "IDPAG" 1194768 NIL IDPAG (NIL T T) -8 NIL NIL NIL) (-511 1193819 1194010 1194085 "IDENT" 1194119 T IDENT (NIL) -8 NIL NIL NIL) (-510 1190074 1190922 1191817 "IDECOMP" 1192976 NIL IDECOMP (NIL NIL NIL) -7 NIL NIL NIL) (-509 1182912 1183997 1185044 "IDEAL" 1189110 NIL IDEAL (NIL T T T T) -8 NIL NIL NIL) (-508 1182076 1182188 1182387 "ICDEN" 1182796 NIL ICDEN (NIL T T T T) -7 NIL NIL NIL) (-507 1181147 1181556 1181703 "ICARD" 1181949 T ICARD (NIL) -8 NIL NIL NIL) (-506 1179207 1179520 1179925 "IBPTOOLS" 1180824 NIL IBPTOOLS (NIL T T T T) -7 NIL NIL NIL) (-505 1174814 1178827 1178940 "IBITS" 1179126 NIL IBITS (NIL NIL) -8 NIL NIL NIL) (-504 1171537 1172113 1172808 "IBATOOL" 1174231 NIL IBATOOL (NIL T T T) -7 NIL NIL NIL) (-503 1169316 1169778 1170311 "IBACHIN" 1171072 NIL IBACHIN (NIL T T T) -7 NIL NIL NIL) (-502 1167145 1169162 1169265 "IARRAY2" 1169270 NIL IARRAY2 (NIL T NIL NIL) -8 NIL NIL NIL) (-501 1163251 1167071 1167128 "IARRAY1" 1167133 NIL IARRAY1 (NIL T NIL) -8 NIL NIL NIL) (-500 1157369 1161663 1162144 "IAN" 1162790 T IAN (NIL) -8 NIL NIL NIL) (-499 1156880 1156937 1157110 "IALGFACT" 1157306 NIL IALGFACT (NIL T T T T) -7 NIL NIL NIL) (-498 1156408 1156521 1156549 "HYPCAT" 1156756 T HYPCAT (NIL) -9 NIL NIL NIL) (-497 1155946 1156063 1156249 "HYPCAT-" 1156254 NIL HYPCAT- (NIL T) -8 NIL NIL NIL) (-496 1155541 1155741 1155824 "HOSTNAME" 1155883 T HOSTNAME (NIL) -8 NIL NIL NIL) (-495 1155386 1155423 1155464 "HOMOTOP" 1155469 NIL HOMOTOP (NIL T) -9 NIL 1155502 NIL) (-494 1152018 1153396 1153437 "HOAGG" 1154418 NIL HOAGG (NIL T) -9 NIL 1155097 NIL) (-493 1150612 1151011 1151537 "HOAGG-" 1151542 NIL HOAGG- (NIL T T) -8 NIL NIL NIL) (-492 1144637 1150207 1150356 "HEXADEC" 1150483 T HEXADEC (NIL) -8 NIL NIL NIL) (-491 1143385 1143607 1143870 "HEUGCD" 1144414 NIL HEUGCD (NIL T) -7 NIL NIL NIL) (-490 1142461 1143222 1143352 "HELLFDIV" 1143357 NIL HELLFDIV (NIL T T T T) -8 NIL NIL NIL) (-489 1140640 1142238 1142326 "HEAP" 1142405 NIL HEAP (NIL T) -8 NIL NIL NIL) (-488 1139903 1140192 1140326 "HEADAST" 1140526 T HEADAST (NIL) -8 NIL NIL NIL) (-487 1133776 1139818 1139880 "HDP" 1139885 NIL HDP (NIL NIL T) -8 NIL NIL NIL) (-486 1127795 1133411 1133563 "HDMP" 1133677 NIL HDMP (NIL NIL T) -8 NIL NIL NIL) (-485 1127119 1127259 1127423 "HB" 1127651 T HB (NIL) -7 NIL NIL NIL) (-484 1120507 1126965 1127069 "HASHTBL" 1127074 NIL HASHTBL (NIL T T NIL) -8 NIL NIL NIL) (-483 1119983 1120228 1120320 "HASAST" 1120435 T HASAST (NIL) -8 NIL NIL NIL) (-482 1117765 1119605 1119787 "HACKPI" 1119821 T HACKPI (NIL) -8 NIL NIL NIL) (-481 1113460 1117618 1117731 "GTSET" 1117736 NIL GTSET (NIL T T T T) -8 NIL NIL NIL) (-480 1106877 1113338 1113436 "GSTBL" 1113441 NIL GSTBL (NIL T T T NIL) -8 NIL NIL NIL) (-479 1099157 1105908 1106173 "GSERIES" 1106668 NIL GSERIES (NIL T NIL NIL) -8 NIL NIL NIL) (-478 1098298 1098715 1098743 "GROUP" 1098946 T GROUP (NIL) -9 NIL 1099080 NIL) (-477 1097664 1097823 1098074 "GROUP-" 1098079 NIL GROUP- (NIL T) -8 NIL NIL NIL) (-476 1096031 1096352 1096739 "GROEBSOL" 1097341 NIL GROEBSOL (NIL NIL T T) -7 NIL NIL NIL) (-475 1094945 1095233 1095284 "GRMOD" 1095813 NIL GRMOD (NIL T T) -9 NIL 1095981 NIL) (-474 1094713 1094749 1094877 "GRMOD-" 1094882 NIL GRMOD- (NIL T T T) -8 NIL NIL NIL) (-473 1090003 1091067 1092067 "GRIMAGE" 1093733 T GRIMAGE (NIL) -8 NIL NIL NIL) (-472 1088469 1088730 1089054 "GRDEF" 1089699 T GRDEF (NIL) -7 NIL NIL NIL) (-471 1087913 1088029 1088170 "GRAY" 1088348 T GRAY (NIL) -7 NIL NIL NIL) (-470 1087100 1087506 1087557 "GRALG" 1087710 NIL GRALG (NIL T T) -9 NIL 1087803 NIL) (-469 1086761 1086834 1086997 "GRALG-" 1087002 NIL GRALG- (NIL T T T) -8 NIL NIL NIL) (-468 1083538 1086346 1086524 "GPOLSET" 1086668 NIL GPOLSET (NIL T T T T) -8 NIL NIL NIL) (-467 1082892 1082949 1083207 "GOSPER" 1083475 NIL GOSPER (NIL T T T T T) -7 NIL NIL NIL) (-466 1078624 1079330 1079856 "GMODPOL" 1082591 NIL GMODPOL (NIL NIL T T T NIL T) -8 NIL NIL NIL) (-465 1077629 1077813 1078051 "GHENSEL" 1078436 NIL GHENSEL (NIL T T) -7 NIL NIL NIL) (-464 1071785 1072628 1073648 "GENUPS" 1076713 NIL GENUPS (NIL T T) -7 NIL NIL NIL) (-463 1071482 1071533 1071622 "GENUFACT" 1071728 NIL GENUFACT (NIL T) -7 NIL NIL NIL) (-462 1070894 1070971 1071136 "GENPGCD" 1071400 NIL GENPGCD (NIL T T T T) -7 NIL NIL NIL) (-461 1070368 1070403 1070616 "GENMFACT" 1070853 NIL GENMFACT (NIL T T T T T) -7 NIL NIL NIL) (-460 1068934 1069191 1069498 "GENEEZ" 1070111 NIL GENEEZ (NIL T T) -7 NIL NIL NIL) (-459 1063111 1068545 1068707 "GDMP" 1068857 NIL GDMP (NIL NIL T T) -8 NIL NIL NIL) (-458 1052475 1056882 1057988 "GCNAALG" 1062094 NIL GCNAALG (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-457 1050802 1051664 1051692 "GCDDOM" 1051947 T GCDDOM (NIL) -9 NIL 1052104 NIL) (-456 1050272 1050399 1050614 "GCDDOM-" 1050619 NIL GCDDOM- (NIL T) -8 NIL NIL NIL) (-455 1038888 1041218 1043610 "GBINTERN" 1047963 NIL GBINTERN (NIL T T T T) -7 NIL NIL NIL) (-454 1036725 1037017 1037438 "GBF" 1038563 NIL GBF (NIL T T T T) -7 NIL NIL NIL) (-453 1035506 1035671 1035938 "GBEUCLID" 1036541 NIL GBEUCLID (NIL T T T T) -7 NIL NIL NIL) (-452 1034178 1034363 1034667 "GB" 1035285 NIL GB (NIL T T T T) -7 NIL NIL NIL) (-451 1033527 1033652 1033801 "GAUSSFAC" 1034049 T GAUSSFAC (NIL) -7 NIL NIL NIL) (-450 1031894 1032196 1032510 "GALUTIL" 1033246 NIL GALUTIL (NIL T) -7 NIL NIL NIL) (-449 1030202 1030476 1030800 "GALPOLYU" 1031621 NIL GALPOLYU (NIL T T) -7 NIL NIL NIL) (-448 1027567 1027857 1028264 "GALFACTU" 1029899 NIL GALFACTU (NIL T T T) -7 NIL NIL NIL) (-447 1019372 1020872 1022480 "GALFACT" 1025999 NIL GALFACT (NIL T) -7 NIL NIL NIL) (-446 1016760 1017418 1017446 "FVFUN" 1018602 T FVFUN (NIL) -9 NIL 1019322 NIL) (-445 1016026 1016208 1016236 "FVC" 1016527 T FVC (NIL) -9 NIL 1016710 NIL) (-444 1015669 1015851 1015919 "FUNDESC" 1015978 T FUNDESC (NIL) -8 NIL NIL NIL) (-443 1015284 1015466 1015547 "FUNCTION" 1015621 NIL FUNCTION (NIL NIL) -8 NIL NIL NIL) (-442 1014075 1014585 1014788 "FTEM" 1015101 T FTEM (NIL) -8 NIL NIL NIL) (-441 1011831 1012406 1012869 "FT" 1013632 T FT (NIL) -8 NIL NIL NIL) (-440 1010122 1010411 1010808 "FSUPFACT" 1011522 NIL FSUPFACT (NIL T T T) -7 NIL NIL NIL) (-439 1008519 1008808 1009140 "FST" 1009810 T FST (NIL) -8 NIL NIL NIL) (-438 1007718 1007824 1008012 "FSRED" 1008401 NIL FSRED (NIL T T) -7 NIL NIL NIL) (-437 1006417 1006673 1007020 "FSPRMELT" 1007433 NIL FSPRMELT (NIL T T) -7 NIL NIL NIL) (-436 1003723 1004161 1004647 "FSPECF" 1005980 NIL FSPECF (NIL T T) -7 NIL NIL NIL) (-435 1003251 1003305 1003475 "FSINT" 1003664 NIL FSINT (NIL T T) -7 NIL NIL NIL) (-434 1001543 1002244 1002547 "FSERIES" 1003030 NIL FSERIES (NIL T T) -8 NIL NIL NIL) (-433 1000585 1000701 1000925 "FSCINT" 1001423 NIL FSCINT (NIL T T) -7 NIL NIL NIL) (-432 999627 999770 999997 "FSAGG2" 1000438 NIL FSAGG2 (NIL T T T T) -7 NIL NIL NIL) (-431 995835 998571 998612 "FSAGG" 998982 NIL FSAGG (NIL T) -9 NIL 999241 NIL) (-430 993597 994198 994994 "FSAGG-" 995089 NIL FSAGG- (NIL T T) -8 NIL NIL NIL) (-429 991279 991559 992106 "FS2UPS" 993315 NIL FS2UPS (NIL T T T T T NIL) -7 NIL NIL NIL) (-428 990157 990328 990630 "FS2EXPXP" 991104 NIL FS2EXPXP (NIL T T NIL NIL) -7 NIL NIL NIL) (-427 989791 989834 989963 "FS2" 990108 NIL FS2 (NIL T T T T) -7 NIL NIL NIL) (-426 971458 979760 979801 "FS" 983685 NIL FS (NIL T) -9 NIL 985974 NIL) (-425 960182 963148 967178 "FS-" 967478 NIL FS- (NIL T T) -8 NIL NIL NIL) (-424 959608 959723 959875 "FRUTIL" 960062 NIL FRUTIL (NIL T) -7 NIL NIL NIL) (-423 954609 957251 957291 "FRNAALG" 958687 NIL FRNAALG (NIL T) -9 NIL 959294 NIL) (-422 950333 951392 952650 "FRNAALG-" 953400 NIL FRNAALG- (NIL T T) -8 NIL NIL NIL) (-421 949971 950014 950141 "FRNAAF2" 950284 NIL FRNAAF2 (NIL T T T T) -7 NIL NIL NIL) (-420 948351 948825 949120 "FRMOD" 949783 NIL FRMOD (NIL T T T T NIL) -8 NIL NIL NIL) (-419 947546 947633 947922 "FRIDEAL2" 948258 NIL FRIDEAL2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-418 945297 945929 946246 "FRIDEAL" 947337 NIL FRIDEAL (NIL T T T T) -8 NIL NIL NIL) (-417 944437 944844 944885 "FRETRCT" 944890 NIL FRETRCT (NIL T) -9 NIL 945066 NIL) (-416 943570 943794 944138 "FRETRCT-" 944143 NIL FRETRCT- (NIL T T) -8 NIL NIL NIL) (-415 940658 941868 941927 "FRAMALG" 942809 NIL FRAMALG (NIL T T) -9 NIL 943101 NIL) (-414 938792 939247 939877 "FRAMALG-" 940100 NIL FRAMALG- (NIL T T T) -8 NIL NIL NIL) (-413 938428 938485 938592 "FRAC2" 938729 NIL FRAC2 (NIL T T) -7 NIL NIL NIL) (-412 932370 937903 938179 "FRAC" 938184 NIL FRAC (NIL T) -8 NIL NIL NIL) (-411 932006 932063 932170 "FR2" 932307 NIL FR2 (NIL T T) -7 NIL NIL NIL) (-410 923534 927582 928913 "FR" 930707 NIL FR (NIL T) -8 NIL NIL NIL) (-409 918051 920940 920968 "FPS" 922087 T FPS (NIL) -9 NIL 922644 NIL) (-408 917500 917609 917773 "FPS-" 917919 NIL FPS- (NIL T) -8 NIL NIL NIL) (-407 914804 916471 916499 "FPC" 916724 T FPC (NIL) -9 NIL 916866 NIL) (-406 914597 914637 914734 "FPC-" 914739 NIL FPC- (NIL T) -8 NIL NIL NIL) (-405 913387 914085 914126 "FPATMAB" 914131 NIL FPATMAB (NIL T) -9 NIL 914283 NIL) (-404 911060 911563 911989 "FPARFRAC" 913024 NIL FPARFRAC (NIL T T) -8 NIL NIL NIL) (-403 906493 906991 907673 "FORTRAN" 910492 NIL FORTRAN (NIL NIL NIL NIL NIL) -8 NIL NIL NIL) (-402 904169 904731 904759 "FORTFN" 905819 T FORTFN (NIL) -9 NIL 906443 NIL) (-401 903933 903983 904011 "FORTCAT" 904070 T FORTCAT (NIL) -9 NIL 904132 NIL) (-400 901649 902149 902688 "FORT" 903414 T FORT (NIL) -7 NIL NIL NIL) (-399 901437 901467 901536 "FORMULA1" 901613 NIL FORMULA1 (NIL T) -7 NIL NIL NIL) (-398 899543 900053 900443 "FORMULA" 901067 T FORMULA (NIL) -8 NIL NIL NIL) (-397 899066 899118 899291 "FORDER" 899485 NIL FORDER (NIL T T T T) -7 NIL NIL NIL) (-396 898162 898326 898519 "FOP" 898893 T FOP (NIL) -7 NIL NIL NIL) (-395 896743 897442 897616 "FNLA" 898044 NIL FNLA (NIL NIL NIL T) -8 NIL NIL NIL) (-394 895472 895887 895915 "FNCAT" 896375 T FNCAT (NIL) -9 NIL 896635 NIL) (-393 895011 895431 895459 "FNAME" 895464 T FNAME (NIL) -8 NIL NIL NIL) (-392 893574 894537 894565 "FMTC" 894570 T FMTC (NIL) -9 NIL 894606 NIL) (-391 892327 893510 893556 "FMONOID" 893561 NIL FMONOID (NIL T) -8 NIL NIL NIL) (-390 889155 890323 890364 "FMONCAT" 891581 NIL FMONCAT (NIL T) -9 NIL 892186 NIL) (-389 886579 887225 887253 "FMFUN" 888397 T FMFUN (NIL) -9 NIL 889105 NIL) (-388 883658 884518 884572 "FMCAT" 885767 NIL FMCAT (NIL T T) -9 NIL 886262 NIL) (-387 882927 883108 883136 "FMC" 883426 T FMC (NIL) -9 NIL 883608 NIL) (-386 881793 882693 882793 "FM1" 882872 NIL FM1 (NIL T T) -8 NIL NIL NIL) (-385 880985 881535 881684 "FM" 881689 NIL FM (NIL T T) -8 NIL NIL NIL) (-384 878759 879175 879669 "FLOATRP" 880536 NIL FLOATRP (NIL T) -7 NIL NIL NIL) (-383 876197 876697 877275 "FLOATCP" 878226 NIL FLOATCP (NIL T) -7 NIL NIL NIL) (-382 869775 873926 874547 "FLOAT" 875596 T FLOAT (NIL) -8 NIL NIL NIL) (-381 868515 869353 869394 "FLINEXP" 869399 NIL FLINEXP (NIL T) -9 NIL 869492 NIL) (-380 867669 867904 868232 "FLINEXP-" 868237 NIL FLINEXP- (NIL T T) -8 NIL NIL NIL) (-379 866745 866889 867113 "FLASORT" 867521 NIL FLASORT (NIL T T) -7 NIL NIL NIL) (-378 863861 864729 864781 "FLALG" 866008 NIL FLALG (NIL T T) -9 NIL 866475 NIL) (-377 862903 863046 863273 "FLAGG2" 863714 NIL FLAGG2 (NIL T T T T) -7 NIL NIL NIL) (-376 856639 860389 860430 "FLAGG" 861692 NIL FLAGG (NIL T) -9 NIL 862344 NIL) (-375 855365 855704 856194 "FLAGG-" 856199 NIL FLAGG- (NIL T T) -8 NIL NIL NIL) (-374 852216 853224 853283 "FINRALG" 854411 NIL FINRALG (NIL T T) -9 NIL 854919 NIL) (-373 851376 851605 851944 "FINRALG-" 851949 NIL FINRALG- (NIL T T T) -8 NIL NIL NIL) (-372 850756 850995 851023 "FINITE" 851219 T FINITE (NIL) -9 NIL 851326 NIL) (-371 843113 845300 845340 "FINAALG" 849007 NIL FINAALG (NIL T) -9 NIL 850460 NIL) (-370 838445 839495 840639 "FINAALG-" 842018 NIL FINAALG- (NIL T T) -8 NIL NIL NIL) (-369 837103 837441 837495 "FILECAT" 838179 NIL FILECAT (NIL T T) -9 NIL 838395 NIL) (-368 836471 836858 836961 "FILE" 837033 NIL FILE (NIL T) -8 NIL NIL NIL) (-367 834189 835715 835743 "FIELD" 835783 T FIELD (NIL) -9 NIL 835863 NIL) (-366 832809 833194 833705 "FIELD-" 833710 NIL FIELD- (NIL T) -8 NIL NIL NIL) (-365 830659 831444 831791 "FGROUP" 832495 NIL FGROUP (NIL T) -8 NIL NIL NIL) (-364 829749 829913 830133 "FGLMICPK" 830491 NIL FGLMICPK (NIL T NIL) -7 NIL NIL NIL) (-363 825583 829674 829731 "FFX" 829736 NIL FFX (NIL T NIL) -8 NIL NIL NIL) (-362 825184 825245 825380 "FFSLPE" 825516 NIL FFSLPE (NIL T T T) -7 NIL NIL NIL) (-361 824688 824724 824933 "FFPOLY2" 825142 NIL FFPOLY2 (NIL T T) -7 NIL NIL NIL) (-360 820678 821460 822256 "FFPOLY" 823924 NIL FFPOLY (NIL T) -7 NIL NIL NIL) (-359 816524 820597 820660 "FFP" 820665 NIL FFP (NIL T NIL) -8 NIL NIL NIL) (-358 811652 815867 816057 "FFNBX" 816378 NIL FFNBX (NIL T NIL) -8 NIL NIL NIL) (-357 806582 810787 811045 "FFNBP" 811506 NIL FFNBP (NIL T NIL) -8 NIL NIL NIL) (-356 801217 805866 806077 "FFNB" 806415 NIL FFNB (NIL NIL NIL) -8 NIL NIL NIL) (-355 800049 800247 800562 "FFINTBAS" 801014 NIL FFINTBAS (NIL T T T) -7 NIL NIL NIL) (-354 796120 798338 798366 "FFIELDC" 798986 T FFIELDC (NIL) -9 NIL 799362 NIL) (-353 794782 795153 795650 "FFIELDC-" 795655 NIL FFIELDC- (NIL T) -8 NIL NIL NIL) (-352 794351 794397 794521 "FFHOM" 794724 NIL FFHOM (NIL T T T) -7 NIL NIL NIL) (-351 792046 792533 793050 "FFF" 793866 NIL FFF (NIL T) -7 NIL NIL NIL) (-350 787666 791788 791889 "FFCGX" 791989 NIL FFCGX (NIL T NIL) -8 NIL NIL NIL) (-349 783290 787398 787505 "FFCGP" 787609 NIL FFCGP (NIL T NIL) -8 NIL NIL NIL) (-348 778475 783017 783125 "FFCG" 783226 NIL FFCG (NIL NIL NIL) -8 NIL NIL NIL) (-347 777886 777929 778164 "FFCAT2" 778426 NIL FFCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-346 759291 768363 768449 "FFCAT" 773614 NIL FFCAT (NIL T T T) -9 NIL 775065 NIL) (-345 754488 755536 756850 "FFCAT-" 758080 NIL FFCAT- (NIL T T T T) -8 NIL NIL NIL) (-344 749888 754399 754463 "FF" 754468 NIL FF (NIL NIL NIL) -8 NIL NIL NIL) (-343 739213 742860 744080 "FEXPR" 748740 NIL FEXPR (NIL NIL NIL T) -8 NIL NIL NIL) (-342 738213 738648 738689 "FEVALAB" 738773 NIL FEVALAB (NIL T) -9 NIL 739034 NIL) (-341 737372 737582 737920 "FEVALAB-" 737925 NIL FEVALAB- (NIL T T) -8 NIL NIL NIL) (-340 734392 735133 735248 "FDIVCAT" 736816 NIL FDIVCAT (NIL T T T T) -9 NIL 737253 NIL) (-339 734154 734181 734351 "FDIVCAT-" 734356 NIL FDIVCAT- (NIL T T T T T) -8 NIL NIL NIL) (-338 733374 733461 733738 "FDIV2" 734061 NIL FDIV2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-337 731940 732757 732960 "FDIV" 733273 NIL FDIV (NIL T T T T) -8 NIL NIL NIL) (-336 730914 731235 731437 "FCTRDATA" 731758 T FCTRDATA (NIL) -8 NIL NIL NIL) (-335 729600 729859 730148 "FCPAK1" 730645 T FCPAK1 (NIL) -7 NIL NIL NIL) (-334 728699 729100 729241 "FCOMP" 729491 NIL FCOMP (NIL T) -8 NIL NIL NIL) (-333 712404 715849 719387 "FC" 725181 T FC (NIL) -8 NIL NIL NIL) (-332 704769 708795 708835 "FAXF" 710637 NIL FAXF (NIL T) -9 NIL 711329 NIL) (-331 702045 702703 703528 "FAXF-" 703993 NIL FAXF- (NIL T T) -8 NIL NIL NIL) (-330 697097 701421 701597 "FARRAY" 701902 NIL FARRAY (NIL T) -8 NIL NIL NIL) (-329 691998 694058 694111 "FAMR" 695134 NIL FAMR (NIL T T) -9 NIL 695594 NIL) (-328 690888 691190 691625 "FAMR-" 691630 NIL FAMR- (NIL T T T) -8 NIL NIL NIL) (-327 690057 690810 690863 "FAMONOID" 690868 NIL FAMONOID (NIL T) -8 NIL NIL NIL) (-326 687843 688553 688606 "FAMONC" 689547 NIL FAMONC (NIL T T) -9 NIL 689933 NIL) (-325 686507 687597 687734 "FAGROUP" 687739 NIL FAGROUP (NIL T) -8 NIL NIL NIL) (-324 684302 684621 685024 "FACUTIL" 686188 NIL FACUTIL (NIL T T T T) -7 NIL NIL NIL) (-323 683401 683586 683808 "FACTFUNC" 684112 NIL FACTFUNC (NIL T) -7 NIL NIL NIL) (-322 675825 682704 682903 "EXPUPXS" 683257 NIL EXPUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-321 673308 673848 674434 "EXPRTUBE" 675259 T EXPRTUBE (NIL) -7 NIL NIL NIL) (-320 669579 670171 670901 "EXPRODE" 672647 NIL EXPRODE (NIL T T) -7 NIL NIL NIL) (-319 664133 664720 665526 "EXPR2UPS" 668877 NIL EXPR2UPS (NIL T T) -7 NIL NIL NIL) (-318 663765 663822 663931 "EXPR2" 664070 NIL EXPR2 (NIL T T) -7 NIL NIL NIL) (-317 649311 662414 662843 "EXPR" 663369 NIL EXPR (NIL T) -8 NIL NIL NIL) (-316 640727 648464 648754 "EXPEXPAN" 649148 NIL EXPEXPAN (NIL T T NIL NIL) -8 NIL NIL NIL) (-315 640207 640451 640542 "EXITAST" 640656 T EXITAST (NIL) -8 NIL NIL NIL) (-314 640007 640164 640193 "EXIT" 640198 T EXIT (NIL) -8 NIL NIL NIL) (-313 639634 639696 639809 "EVALCYC" 639939 NIL EVALCYC (NIL T) -7 NIL NIL NIL) (-312 639175 639293 639334 "EVALAB" 639504 NIL EVALAB (NIL T) -9 NIL 639608 NIL) (-311 638656 638778 638999 "EVALAB-" 639004 NIL EVALAB- (NIL T T) -8 NIL NIL NIL) (-310 636024 637326 637354 "EUCDOM" 637909 T EUCDOM (NIL) -9 NIL 638259 NIL) (-309 634429 634871 635461 "EUCDOM-" 635466 NIL EUCDOM- (NIL T) -8 NIL NIL NIL) (-308 634061 634118 634227 "ESTOOLS2" 634366 NIL ESTOOLS2 (NIL T T) -7 NIL NIL NIL) (-307 633812 633854 633934 "ESTOOLS1" 634013 NIL ESTOOLS1 (NIL T) -7 NIL NIL NIL) (-306 621350 624110 626860 "ESTOOLS" 631082 T ESTOOLS (NIL) -7 NIL NIL NIL) (-305 621095 621127 621209 "ESCONT1" 621312 NIL ESCONT1 (NIL NIL NIL) -7 NIL NIL NIL) (-304 617469 618230 619010 "ESCONT" 620335 T ESCONT (NIL) -7 NIL NIL NIL) (-303 617144 617194 617294 "ES2" 617413 NIL ES2 (NIL T T) -7 NIL NIL NIL) (-302 616774 616832 616941 "ES1" 617080 NIL ES1 (NIL T T) -7 NIL NIL NIL) (-301 610811 612419 612447 "ES" 615215 T ES (NIL) -9 NIL 616625 NIL) (-300 605758 607045 608862 "ES-" 609026 NIL ES- (NIL T) -8 NIL NIL NIL) (-299 604974 605103 605279 "ERROR" 605602 T ERROR (NIL) -7 NIL NIL NIL) (-298 598368 604833 604924 "EQTBL" 604929 NIL EQTBL (NIL T T) -8 NIL NIL NIL) (-297 598000 598057 598166 "EQ2" 598305 NIL EQ2 (NIL T T) -7 NIL NIL NIL) (-296 590503 593314 594763 "EQ" 596584 NIL -3968 (NIL T) -8 NIL NIL NIL) (-295 585793 586841 587934 "EP" 589442 NIL EP (NIL T) -7 NIL NIL NIL) (-294 584393 584684 584990 "ENV" 585507 T ENV (NIL) -8 NIL NIL NIL) (-293 583487 584041 584069 "ENTIRER" 584074 T ENTIRER (NIL) -9 NIL 584120 NIL) (-292 580010 581496 581866 "EMR" 583286 NIL EMR (NIL T T T NIL NIL NIL) -8 NIL NIL NIL) (-291 579154 579339 579393 "ELTAGG" 579773 NIL ELTAGG (NIL T T) -9 NIL 579984 NIL) (-290 578873 578935 579076 "ELTAGG-" 579081 NIL ELTAGG- (NIL T T T) -8 NIL NIL NIL) (-289 578662 578691 578745 "ELTAB" 578829 NIL ELTAB (NIL T T) -9 NIL NIL NIL) (-288 577788 577934 578133 "ELFUTS" 578513 NIL ELFUTS (NIL T T) -7 NIL NIL NIL) (-287 577530 577586 577614 "ELEMFUN" 577719 T ELEMFUN (NIL) -9 NIL NIL NIL) (-286 577400 577421 577489 "ELEMFUN-" 577494 NIL ELEMFUN- (NIL T) -8 NIL NIL NIL) (-285 572244 575500 575541 "ELAGG" 576481 NIL ELAGG (NIL T) -9 NIL 576944 NIL) (-284 570529 570963 571626 "ELAGG-" 571631 NIL ELAGG- (NIL T T) -8 NIL NIL NIL) (-283 569841 569978 570134 "ELABOR" 570393 T ELABOR (NIL) -8 NIL NIL NIL) (-282 568502 568781 569075 "ELABEXPR" 569567 T ELABEXPR (NIL) -8 NIL NIL NIL) (-281 561493 563169 563996 "EFUPXS" 567778 NIL EFUPXS (NIL T T T T) -8 NIL NIL NIL) (-280 555070 556744 557554 "EFULS" 560769 NIL EFULS (NIL T T T) -8 NIL NIL NIL) (-279 552555 552913 553385 "EFSTRUC" 554702 NIL EFSTRUC (NIL T T) -7 NIL NIL NIL) (-278 542346 543912 545460 "EF" 551070 NIL EF (NIL T T) -7 NIL NIL NIL) (-277 541420 541831 541980 "EAB" 542217 T EAB (NIL) -8 NIL NIL NIL) (-276 540602 541379 541407 "E04UCFA" 541412 T E04UCFA (NIL) -8 NIL NIL NIL) (-275 539784 540561 540589 "E04NAFA" 540594 T E04NAFA (NIL) -8 NIL NIL NIL) (-274 538966 539743 539771 "E04MBFA" 539776 T E04MBFA (NIL) -8 NIL NIL NIL) (-273 538148 538925 538953 "E04JAFA" 538958 T E04JAFA (NIL) -8 NIL NIL NIL) (-272 537332 538107 538135 "E04GCFA" 538140 T E04GCFA (NIL) -8 NIL NIL NIL) (-271 536516 537291 537319 "E04FDFA" 537324 T E04FDFA (NIL) -8 NIL NIL NIL) (-270 535698 536475 536503 "E04DGFA" 536508 T E04DGFA (NIL) -8 NIL NIL NIL) (-269 529871 531223 532587 "E04AGNT" 534354 T E04AGNT (NIL) -7 NIL NIL NIL) (-268 528551 529057 529097 "DVARCAT" 529572 NIL DVARCAT (NIL T) -9 NIL 529771 NIL) (-267 527755 527967 528281 "DVARCAT-" 528286 NIL DVARCAT- (NIL T T) -8 NIL NIL NIL) (-266 520933 527554 527683 "DSMP" 527688 NIL DSMP (NIL T T T) -8 NIL NIL NIL) (-265 520598 520657 520755 "DROPT1" 520868 NIL DROPT1 (NIL T) -7 NIL NIL NIL) (-264 515713 516839 517976 "DROPT0" 519481 T DROPT0 (NIL) -7 NIL NIL NIL) (-263 510494 511658 512726 "DROPT" 514665 T DROPT (NIL) -8 NIL NIL NIL) (-262 508839 509164 509550 "DRAWPT" 510128 T DRAWPT (NIL) -7 NIL NIL NIL) (-261 508472 508525 508643 "DRAWHACK" 508780 NIL DRAWHACK (NIL T) -7 NIL NIL NIL) (-260 507203 507472 507763 "DRAWCX" 508201 T DRAWCX (NIL) -7 NIL NIL NIL) (-259 506718 506787 506938 "DRAWCURV" 507129 NIL DRAWCURV (NIL T T) -7 NIL NIL NIL) (-258 497186 499148 501263 "DRAWCFUN" 504623 T DRAWCFUN (NIL) -7 NIL NIL NIL) (-257 491773 492696 493775 "DRAW" 496160 NIL DRAW (NIL T) -7 NIL NIL NIL) (-256 488537 490466 490507 "DQAGG" 491136 NIL DQAGG (NIL T) -9 NIL 491410 NIL) (-255 476697 483130 483213 "DPOLCAT" 485065 NIL DPOLCAT (NIL T T T T) -9 NIL 485610 NIL) (-254 471584 472916 474857 "DPOLCAT-" 474862 NIL DPOLCAT- (NIL T T T T T) -8 NIL NIL NIL) (-253 464713 471445 471543 "DPMO" 471548 NIL DPMO (NIL NIL T T) -8 NIL NIL NIL) (-252 457745 464493 464660 "DPMM" 464665 NIL DPMM (NIL NIL T T T) -8 NIL NIL NIL) (-251 457223 457437 457535 "DOMTMPLT" 457667 T DOMTMPLT (NIL) -8 NIL NIL NIL) (-250 456656 457025 457105 "DOMCTOR" 457163 T DOMCTOR (NIL) -8 NIL NIL NIL) (-249 455868 456136 456287 "DOMAIN" 456525 T DOMAIN (NIL) -8 NIL NIL NIL) (-248 449887 455503 455655 "DMP" 455769 NIL DMP (NIL NIL T) -8 NIL NIL NIL) (-247 449487 449543 449687 "DLP" 449825 NIL DLP (NIL T) -7 NIL NIL NIL) (-246 443311 448814 449004 "DLIST" 449329 NIL DLIST (NIL T) -8 NIL NIL NIL) (-245 440109 442164 442205 "DLAGG" 442755 NIL DLAGG (NIL T) -9 NIL 442985 NIL) (-244 438785 439449 439477 "DIVRING" 439569 T DIVRING (NIL) -9 NIL 439652 NIL) (-243 438022 438212 438512 "DIVRING-" 438517 NIL DIVRING- (NIL T) -8 NIL NIL NIL) (-242 436124 436481 436887 "DISPLAY" 437636 T DISPLAY (NIL) -7 NIL NIL NIL) (-241 434972 435175 435440 "DIRPROD2" 435917 NIL DIRPROD2 (NIL NIL T T) -7 NIL NIL NIL) (-240 428867 434886 434949 "DIRPROD" 434954 NIL DIRPROD (NIL NIL T) -8 NIL NIL NIL) (-239 417649 423648 423701 "DIRPCAT" 424111 NIL DIRPCAT (NIL NIL T) -9 NIL 424951 NIL) (-238 414975 415617 416498 "DIRPCAT-" 416835 NIL DIRPCAT- (NIL T NIL T) -8 NIL NIL NIL) (-237 414262 414422 414608 "DIOSP" 414809 T DIOSP (NIL) -7 NIL NIL NIL) (-236 410917 413174 413215 "DIOPS" 413649 NIL DIOPS (NIL T) -9 NIL 413878 NIL) (-235 410466 410580 410771 "DIOPS-" 410776 NIL DIOPS- (NIL T T) -8 NIL NIL NIL) (-234 409289 409917 409945 "DIFRING" 410132 T DIFRING (NIL) -9 NIL 410242 NIL) (-233 408935 409012 409164 "DIFRING-" 409169 NIL DIFRING- (NIL T) -8 NIL NIL NIL) (-232 406671 407943 407984 "DIFEXT" 408347 NIL DIFEXT (NIL T) -9 NIL 408641 NIL) (-231 404956 405384 406050 "DIFEXT-" 406055 NIL DIFEXT- (NIL T T) -8 NIL NIL NIL) (-230 402231 404488 404529 "DIAGG" 404534 NIL DIAGG (NIL T) -9 NIL 404554 NIL) (-229 401615 401772 402024 "DIAGG-" 402029 NIL DIAGG- (NIL T T) -8 NIL NIL NIL) (-228 397031 400574 400851 "DHMATRIX" 401384 NIL DHMATRIX (NIL T) -8 NIL NIL NIL) (-227 392643 393552 394562 "DFSFUN" 396041 T DFSFUN (NIL) -7 NIL NIL NIL) (-226 387726 391574 391886 "DFLOAT" 392351 T DFLOAT (NIL) -8 NIL NIL NIL) (-225 385989 386270 386659 "DFINTTLS" 387434 NIL DFINTTLS (NIL T T) -7 NIL NIL NIL) (-224 383018 384010 384410 "DERHAM" 385655 NIL DERHAM (NIL T NIL) -8 NIL NIL NIL) (-223 380819 382793 382882 "DEQUEUE" 382962 NIL DEQUEUE (NIL T) -8 NIL NIL NIL) (-222 380073 380206 380389 "DEGRED" 380681 NIL DEGRED (NIL T T) -7 NIL NIL NIL) (-221 376683 377383 378184 "DEFINTRF" 379346 NIL DEFINTRF (NIL T) -7 NIL NIL NIL) (-220 374350 374791 375355 "DEFINTEF" 376230 NIL DEFINTEF (NIL T T) -7 NIL NIL NIL) (-219 373700 373970 374085 "DEFAST" 374255 T DEFAST (NIL) -8 NIL NIL NIL) (-218 367725 373295 373444 "DECIMAL" 373571 T DECIMAL (NIL) -8 NIL NIL NIL) (-217 365237 365695 366201 "DDFACT" 367269 NIL DDFACT (NIL T T) -7 NIL NIL NIL) (-216 364833 364876 365027 "DBLRESP" 365188 NIL DBLRESP (NIL T T T T) -7 NIL NIL NIL) (-215 362705 363066 363426 "DBASE" 364600 NIL DBASE (NIL T) -8 NIL NIL NIL) (-214 361947 362185 362331 "DATAARY" 362604 NIL DATAARY (NIL NIL T) -8 NIL NIL NIL) (-213 361053 361906 361934 "D03FAFA" 361939 T D03FAFA (NIL) -8 NIL NIL NIL) (-212 360160 361012 361040 "D03EEFA" 361045 T D03EEFA (NIL) -8 NIL NIL NIL) (-211 358110 358576 359065 "D03AGNT" 359691 T D03AGNT (NIL) -7 NIL NIL NIL) (-210 357399 358069 358097 "D02EJFA" 358102 T D02EJFA (NIL) -8 NIL NIL NIL) (-209 356688 357358 357386 "D02CJFA" 357391 T D02CJFA (NIL) -8 NIL NIL NIL) (-208 355977 356647 356675 "D02BHFA" 356680 T D02BHFA (NIL) -8 NIL NIL NIL) (-207 355266 355936 355964 "D02BBFA" 355969 T D02BBFA (NIL) -8 NIL NIL NIL) (-206 348463 350052 351658 "D02AGNT" 353680 T D02AGNT (NIL) -7 NIL NIL NIL) (-205 346231 346754 347300 "D01WGTS" 347937 T D01WGTS (NIL) -7 NIL NIL NIL) (-204 345298 346190 346218 "D01TRNS" 346223 T D01TRNS (NIL) -8 NIL NIL NIL) (-203 344366 345257 345285 "D01GBFA" 345290 T D01GBFA (NIL) -8 NIL NIL NIL) (-202 343434 344325 344353 "D01FCFA" 344358 T D01FCFA (NIL) -8 NIL NIL NIL) (-201 342502 343393 343421 "D01ASFA" 343426 T D01ASFA (NIL) -8 NIL NIL NIL) (-200 341570 342461 342489 "D01AQFA" 342494 T D01AQFA (NIL) -8 NIL NIL NIL) (-199 340638 341529 341557 "D01APFA" 341562 T D01APFA (NIL) -8 NIL NIL NIL) (-198 339706 340597 340625 "D01ANFA" 340630 T D01ANFA (NIL) -8 NIL NIL NIL) (-197 338774 339665 339693 "D01AMFA" 339698 T D01AMFA (NIL) -8 NIL NIL NIL) (-196 337842 338733 338761 "D01ALFA" 338766 T D01ALFA (NIL) -8 NIL NIL NIL) (-195 336910 337801 337829 "D01AKFA" 337834 T D01AKFA (NIL) -8 NIL NIL NIL) (-194 335978 336869 336897 "D01AJFA" 336902 T D01AJFA (NIL) -8 NIL NIL NIL) (-193 329273 330826 332387 "D01AGNT" 334437 T D01AGNT (NIL) -7 NIL NIL NIL) (-192 328610 328738 328890 "CYCLOTOM" 329141 T CYCLOTOM (NIL) -7 NIL NIL NIL) (-191 325345 326058 326785 "CYCLES" 327903 T CYCLES (NIL) -7 NIL NIL NIL) (-190 324657 324791 324962 "CVMP" 325206 NIL CVMP (NIL T) -7 NIL NIL NIL) (-189 322498 322756 323125 "CTRIGMNP" 324385 NIL CTRIGMNP (NIL T T) -7 NIL NIL NIL) (-188 322007 322229 322330 "CTORKIND" 322417 T CTORKIND (NIL) -8 NIL NIL NIL) (-187 321298 321614 321642 "CTORCAT" 321824 T CTORCAT (NIL) -9 NIL 321937 NIL) (-186 320896 321007 321166 "CTORCAT-" 321171 NIL CTORCAT- (NIL T) -8 NIL NIL NIL) (-185 320358 320570 320678 "CTORCALL" 320820 NIL CTORCALL (NIL T) -8 NIL NIL NIL) (-184 319794 320152 320225 "CTOR" 320305 T CTOR (NIL) -8 NIL NIL NIL) (-183 319168 319267 319420 "CSTTOOLS" 319691 NIL CSTTOOLS (NIL T T) -7 NIL NIL NIL) (-182 314967 315624 316382 "CRFP" 318480 NIL CRFP (NIL T T) -7 NIL NIL NIL) (-181 314442 314688 314780 "CRCEAST" 314895 T CRCEAST (NIL) -8 NIL NIL NIL) (-180 313489 313674 313902 "CRAPACK" 314246 NIL CRAPACK (NIL T) -7 NIL NIL NIL) (-179 312873 312974 313178 "CPMATCH" 313365 NIL CPMATCH (NIL T T T) -7 NIL NIL NIL) (-178 312598 312626 312732 "CPIMA" 312839 NIL CPIMA (NIL T T T) -7 NIL NIL NIL) (-177 308946 309618 310337 "COORDSYS" 311933 NIL COORDSYS (NIL T) -7 NIL NIL NIL) (-176 308358 308479 308621 "CONTOUR" 308824 T CONTOUR (NIL) -8 NIL NIL NIL) (-175 304251 306361 306853 "CONTFRAC" 307898 NIL CONTFRAC (NIL T) -8 NIL NIL NIL) (-174 304131 304152 304180 "CONDUIT" 304217 T CONDUIT (NIL) -9 NIL NIL NIL) (-173 303219 303773 303801 "COMRING" 303806 T COMRING (NIL) -9 NIL 303858 NIL) (-172 302273 302577 302761 "COMPPROP" 303055 T COMPPROP (NIL) -8 NIL NIL NIL) (-171 301934 301969 302097 "COMPLPAT" 302232 NIL COMPLPAT (NIL T T T) -7 NIL NIL NIL) (-170 301570 301627 301734 "COMPLEX2" 301871 NIL COMPLEX2 (NIL T T) -7 NIL NIL NIL) (-169 291879 301379 301488 "COMPLEX" 301493 NIL COMPLEX (NIL T) -8 NIL NIL NIL) (-168 291468 291537 291645 "COMPILER" 291791 T COMPILER (NIL) -8 NIL NIL NIL) (-167 291186 291221 291319 "COMPFACT" 291427 NIL COMPFACT (NIL T T) -7 NIL NIL NIL) (-166 275275 285260 285300 "COMPCAT" 286304 NIL COMPCAT (NIL T) -9 NIL 287652 NIL) (-165 264808 267728 271348 "COMPCAT-" 271704 NIL COMPCAT- (NIL T T) -8 NIL NIL NIL) (-164 264537 264565 264668 "COMMUPC" 264774 NIL COMMUPC (NIL T T T) -7 NIL NIL NIL) (-163 264331 264365 264424 "COMMONOP" 264498 T COMMONOP (NIL) -7 NIL NIL NIL) (-162 263907 264135 264210 "COMMAAST" 264276 T COMMAAST (NIL) -8 NIL NIL NIL) (-161 263463 263658 263745 "COMM" 263840 T COMM (NIL) -8 NIL NIL NIL) (-160 262712 262906 262934 "COMBOPC" 263272 T COMBOPC (NIL) -9 NIL 263447 NIL) (-159 261608 261818 262060 "COMBINAT" 262502 NIL COMBINAT (NIL T) -7 NIL NIL NIL) (-158 258065 258639 259266 "COMBF" 261030 NIL COMBF (NIL T T) -7 NIL NIL NIL) (-157 256823 257181 257416 "COLOR" 257850 T COLOR (NIL) -8 NIL NIL NIL) (-156 256299 256544 256636 "COLONAST" 256751 T COLONAST (NIL) -8 NIL NIL NIL) (-155 255939 255986 256111 "CMPLXRT" 256246 NIL CMPLXRT (NIL T T) -7 NIL NIL NIL) (-154 255387 255639 255738 "CLLCTAST" 255860 T CLLCTAST (NIL) -8 NIL NIL NIL) (-153 250886 251917 252997 "CLIP" 254327 T CLIP (NIL) -7 NIL NIL NIL) (-152 249232 249992 250231 "CLIF" 250713 NIL CLIF (NIL NIL T NIL) -8 NIL NIL NIL) (-151 245407 247378 247419 "CLAGG" 248348 NIL CLAGG (NIL T) -9 NIL 248884 NIL) (-150 243829 244286 244869 "CLAGG-" 244874 NIL CLAGG- (NIL T T) -8 NIL NIL NIL) (-149 243373 243458 243598 "CINTSLPE" 243738 NIL CINTSLPE (NIL T T) -7 NIL NIL NIL) (-148 240874 241345 241893 "CHVAR" 242901 NIL CHVAR (NIL T T T) -7 NIL NIL NIL) (-147 240048 240602 240630 "CHARZ" 240635 T CHARZ (NIL) -9 NIL 240650 NIL) (-146 239802 239842 239920 "CHARPOL" 240002 NIL CHARPOL (NIL T) -7 NIL NIL NIL) (-145 238860 239447 239475 "CHARNZ" 239522 T CHARNZ (NIL) -9 NIL 239578 NIL) (-144 236766 237514 237867 "CHAR" 238527 T CHAR (NIL) -8 NIL NIL NIL) (-143 236492 236553 236581 "CFCAT" 236692 T CFCAT (NIL) -9 NIL NIL NIL) (-142 235737 235848 236030 "CDEN" 236376 NIL CDEN (NIL T T T) -7 NIL NIL NIL) (-141 231702 234890 235170 "CCLASS" 235477 T CCLASS (NIL) -8 NIL NIL NIL) (-140 230953 231110 231287 "CATEGORY" 231545 T -10 (NIL) -8 NIL NIL NIL) (-139 230526 230872 230920 "CATCTOR" 230925 T CATCTOR (NIL) -8 NIL NIL NIL) (-138 229977 230229 230327 "CATAST" 230448 T CATAST (NIL) -8 NIL NIL NIL) (-137 229453 229698 229790 "CASEAST" 229905 T CASEAST (NIL) -8 NIL NIL NIL) (-136 228561 228709 228930 "CARTEN2" 229300 NIL CARTEN2 (NIL NIL NIL T T) -7 NIL NIL NIL) (-135 223570 224590 225343 "CARTEN" 227864 NIL CARTEN (NIL NIL NIL T) -8 NIL NIL NIL) (-134 221886 222720 222977 "CARD" 223333 T CARD (NIL) -8 NIL NIL NIL) (-133 221462 221690 221765 "CAPSLAST" 221831 T CAPSLAST (NIL) -8 NIL NIL NIL) (-132 220966 221174 221202 "CACHSET" 221334 T CACHSET (NIL) -9 NIL 221412 NIL) (-131 220436 220758 220786 "CABMON" 220836 T CABMON (NIL) -9 NIL 220892 NIL) (-130 219909 220140 220250 "BYTEORD" 220346 T BYTEORD (NIL) -8 NIL NIL NIL) (-129 215259 219414 219586 "BYTEBUF" 219757 T BYTEBUF (NIL) -8 NIL NIL NIL) (-128 214241 214793 214935 "BYTE" 215098 T BYTE (NIL) -8 NIL NIL 215220) (-127 211752 213933 214040 "BTREE" 214167 NIL BTREE (NIL T) -8 NIL NIL NIL) (-126 209203 211400 211522 "BTOURN" 211662 NIL BTOURN (NIL T) -8 NIL NIL NIL) (-125 206575 208673 208714 "BTCAT" 208782 NIL BTCAT (NIL T) -9 NIL 208859 NIL) (-124 206242 206322 206471 "BTCAT-" 206476 NIL BTCAT- (NIL T T) -8 NIL NIL NIL) (-123 201507 205385 205413 "BTAGG" 205635 T BTAGG (NIL) -9 NIL 205796 NIL) (-122 200997 201122 201328 "BTAGG-" 201333 NIL BTAGG- (NIL T) -8 NIL NIL NIL) (-121 197994 200275 200490 "BSTREE" 200814 NIL BSTREE (NIL T) -8 NIL NIL NIL) (-120 197132 197258 197442 "BRILL" 197850 NIL BRILL (NIL T) -7 NIL NIL NIL) (-119 193785 195858 195899 "BRAGG" 196548 NIL BRAGG (NIL T) -9 NIL 196806 NIL) (-118 192317 192722 193276 "BRAGG-" 193281 NIL BRAGG- (NIL T T) -8 NIL NIL NIL) (-117 185567 191663 191847 "BPADICRT" 192165 NIL BPADICRT (NIL NIL) -8 NIL NIL NIL) (-116 183884 185504 185549 "BPADIC" 185554 NIL BPADIC (NIL NIL) -8 NIL NIL NIL) (-115 183582 183612 183726 "BOUNDZRO" 183848 NIL BOUNDZRO (NIL T T) -7 NIL NIL NIL) (-114 181363 181767 182242 "BOP1" 183140 NIL BOP1 (NIL T) -7 NIL NIL NIL) (-113 176591 177789 178701 "BOP" 180471 T BOP (NIL) -8 NIL NIL NIL) (-112 175416 176165 176314 "BOOLEAN" 176462 T BOOLEAN (NIL) -8 NIL NIL NIL) (-111 174695 175099 175153 "BMODULE" 175158 NIL BMODULE (NIL T T) -9 NIL 175223 NIL) (-110 170496 174493 174566 "BITS" 174642 T BITS (NIL) -8 NIL NIL NIL) (-109 169917 170036 170176 "BINDING" 170376 T BINDING (NIL) -8 NIL NIL NIL) (-108 163945 169514 169662 "BINARY" 169789 T BINARY (NIL) -8 NIL NIL NIL) (-107 161725 163200 163241 "BGAGG" 163501 NIL BGAGG (NIL T) -9 NIL 163638 NIL) (-106 161556 161588 161679 "BGAGG-" 161684 NIL BGAGG- (NIL T T) -8 NIL NIL NIL) (-105 160627 160940 161145 "BFUNCT" 161371 T BFUNCT (NIL) -8 NIL NIL NIL) (-104 159311 159492 159780 "BEZOUT" 160451 NIL BEZOUT (NIL T T T T T) -7 NIL NIL NIL) (-103 155782 158163 158493 "BBTREE" 159014 NIL BBTREE (NIL T) -8 NIL NIL NIL) (-102 155516 155569 155597 "BASTYPE" 155716 T BASTYPE (NIL) -9 NIL NIL NIL) (-101 155368 155397 155470 "BASTYPE-" 155475 NIL BASTYPE- (NIL T) -8 NIL NIL NIL) (-100 154802 154878 155030 "BALFACT" 155279 NIL BALFACT (NIL T T) -7 NIL NIL NIL) (-99 153658 154217 154403 "AUTOMOR" 154647 NIL AUTOMOR (NIL T) -8 NIL NIL NIL) (-98 153384 153389 153415 "ATTREG" 153420 T ATTREG (NIL) -9 NIL NIL NIL) (-97 151636 152081 152433 "ATTRBUT" 153050 T ATTRBUT (NIL) -8 NIL NIL NIL) (-96 151244 151464 151530 "ATTRAST" 151588 T ATTRAST (NIL) -8 NIL NIL NIL) (-95 150780 150893 150919 "ATRIG" 151120 T ATRIG (NIL) -9 NIL NIL NIL) (-94 150589 150630 150717 "ATRIG-" 150722 NIL ATRIG- (NIL T) -8 NIL NIL NIL) (-93 150234 150420 150446 "ASTCAT" 150451 T ASTCAT (NIL) -9 NIL 150481 NIL) (-92 149961 150020 150139 "ASTCAT-" 150144 NIL ASTCAT- (NIL T) -8 NIL NIL NIL) (-91 148110 149737 149825 "ASTACK" 149904 NIL ASTACK (NIL T) -8 NIL NIL NIL) (-90 146615 146912 147277 "ASSOCEQ" 147792 NIL ASSOCEQ (NIL T T) -7 NIL NIL NIL) (-89 145669 146274 146398 "ASP9" 146522 NIL ASP9 (NIL NIL) -8 NIL NIL NIL) (-88 144559 145274 145416 "ASP80" 145558 NIL ASP80 (NIL NIL) -8 NIL NIL NIL) (-87 144322 144507 144546 "ASP8" 144551 NIL ASP8 (NIL NIL) -8 NIL NIL NIL) (-86 143298 143999 144117 "ASP78" 144235 NIL ASP78 (NIL NIL) -8 NIL NIL NIL) (-85 142289 142978 143095 "ASP77" 143212 NIL ASP77 (NIL NIL) -8 NIL NIL NIL) (-84 141223 141927 142058 "ASP74" 142189 NIL ASP74 (NIL NIL) -8 NIL NIL NIL) (-83 140145 140858 140990 "ASP73" 141122 NIL ASP73 (NIL NIL) -8 NIL NIL NIL) (-82 139065 139780 139912 "ASP7" 140044 NIL ASP7 (NIL NIL) -8 NIL NIL NIL) (-81 138191 138891 138991 "ASP6" 138996 NIL ASP6 (NIL NIL) -8 NIL NIL NIL) (-80 137158 137868 137986 "ASP55" 138104 NIL ASP55 (NIL NIL) -8 NIL NIL NIL) (-79 136129 136832 136951 "ASP50" 137070 NIL ASP50 (NIL NIL) -8 NIL NIL NIL) (-78 135239 135830 135940 "ASP49" 136050 NIL ASP49 (NIL NIL) -8 NIL NIL NIL) (-77 134045 134778 134946 "ASP42" 135128 NIL ASP42 (NIL NIL NIL NIL) -8 NIL NIL NIL) (-76 132843 133578 133748 "ASP41" 133932 NIL ASP41 (NIL NIL NIL NIL) -8 NIL NIL NIL) (-75 131953 132544 132654 "ASP4" 132764 NIL ASP4 (NIL NIL) -8 NIL NIL NIL) (-74 130925 131630 131748 "ASP35" 131866 NIL ASP35 (NIL NIL) -8 NIL NIL NIL) (-73 130690 130873 130912 "ASP34" 130917 NIL ASP34 (NIL NIL) -8 NIL NIL NIL) (-72 130427 130494 130570 "ASP33" 130645 NIL ASP33 (NIL NIL) -8 NIL NIL NIL) (-71 129342 130062 130194 "ASP31" 130326 NIL ASP31 (NIL NIL) -8 NIL NIL NIL) (-70 129107 129290 129329 "ASP30" 129334 NIL ASP30 (NIL NIL) -8 NIL NIL NIL) (-69 128842 128911 128987 "ASP29" 129062 NIL ASP29 (NIL NIL) -8 NIL NIL NIL) (-68 128607 128790 128829 "ASP28" 128834 NIL ASP28 (NIL NIL) -8 NIL NIL NIL) (-67 128372 128555 128594 "ASP27" 128599 NIL ASP27 (NIL NIL) -8 NIL NIL NIL) (-66 127478 128070 128181 "ASP24" 128292 NIL ASP24 (NIL NIL) -8 NIL NIL NIL) (-65 126576 127280 127392 "ASP20" 127397 NIL ASP20 (NIL NIL) -8 NIL NIL NIL) (-64 125540 126250 126369 "ASP19" 126488 NIL ASP19 (NIL NIL) -8 NIL NIL NIL) (-63 125277 125344 125420 "ASP12" 125495 NIL ASP12 (NIL NIL) -8 NIL NIL NIL) (-62 124151 124876 125020 "ASP10" 125164 NIL ASP10 (NIL NIL) -8 NIL NIL NIL) (-61 123261 123852 123962 "ASP1" 124072 NIL ASP1 (NIL NIL) -8 NIL NIL NIL) (-60 121112 123105 123196 "ARRAY2" 123201 NIL ARRAY2 (NIL T) -8 NIL NIL NIL) (-59 120144 120317 120538 "ARRAY12" 120935 NIL ARRAY12 (NIL T T) -7 NIL NIL NIL) (-58 115909 119792 119906 "ARRAY1" 120061 NIL ARRAY1 (NIL T) -8 NIL NIL NIL) (-57 110221 112139 112214 "ARR2CAT" 114844 NIL ARR2CAT (NIL T T T) -9 NIL 115602 NIL) (-56 107655 108399 109353 "ARR2CAT-" 109358 NIL ARR2CAT- (NIL T T T T) -8 NIL NIL NIL) (-55 106972 107282 107407 "ARITY" 107548 T ARITY (NIL) -8 NIL NIL NIL) (-54 105748 105900 106199 "APPRULE" 106808 NIL APPRULE (NIL T T T) -7 NIL NIL NIL) (-53 105399 105447 105566 "APPLYORE" 105694 NIL APPLYORE (NIL T T T) -7 NIL NIL NIL) (-52 104677 104800 104957 "ANY1" 105273 NIL ANY1 (NIL T) -7 NIL NIL NIL) (-51 104031 104270 104390 "ANY" 104575 T ANY (NIL) -8 NIL NIL NIL) (-50 101561 102468 102795 "ANTISYM" 103755 NIL ANTISYM (NIL T NIL) -8 NIL NIL NIL) (-49 101053 101268 101364 "ANON" 101483 T ANON (NIL) -8 NIL NIL NIL) (-48 95311 99592 100046 "AN" 100617 T AN (NIL) -8 NIL NIL NIL) (-47 91209 92597 92648 "AMR" 93396 NIL AMR (NIL T T) -9 NIL 93996 NIL) (-46 90321 90542 90905 "AMR-" 90910 NIL AMR- (NIL T T T) -8 NIL NIL NIL) (-45 74766 90238 90299 "ALIST" 90304 NIL ALIST (NIL T T) -8 NIL NIL NIL) (-44 71601 74360 74529 "ALGSC" 74684 NIL ALGSC (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-43 68156 68711 69318 "ALGPKG" 71041 NIL ALGPKG (NIL T T) -7 NIL NIL NIL) (-42 67433 67534 67718 "ALGMFACT" 68042 NIL ALGMFACT (NIL T T T) -7 NIL NIL NIL) (-41 63468 64047 64641 "ALGMANIP" 67017 NIL ALGMANIP (NIL T T) -7 NIL NIL NIL) (-40 54849 63094 63244 "ALGFF" 63401 NIL ALGFF (NIL T T T NIL) -8 NIL NIL NIL) (-39 54045 54176 54355 "ALGFACT" 54707 NIL ALGFACT (NIL T) -7 NIL NIL NIL) (-38 52986 53586 53624 "ALGEBRA" 53629 NIL ALGEBRA (NIL T) -9 NIL 53670 NIL) (-37 52704 52763 52895 "ALGEBRA-" 52900 NIL ALGEBRA- (NIL T T) -8 NIL NIL NIL) (-36 34803 50706 50758 "ALAGG" 50894 NIL ALAGG (NIL T T) -9 NIL 51055 NIL) (-35 34339 34452 34478 "AHYP" 34679 T AHYP (NIL) -9 NIL NIL NIL) (-34 33270 33518 33544 "AGG" 34043 T AGG (NIL) -9 NIL 34322 NIL) (-33 32704 32866 33080 "AGG-" 33085 NIL AGG- (NIL T) -8 NIL NIL NIL) (-32 30510 30933 31338 "AF" 32346 NIL AF (NIL T T) -7 NIL NIL NIL) (-31 29990 30235 30325 "ADDAST" 30438 T ADDAST (NIL) -8 NIL NIL NIL) (-30 29258 29517 29673 "ACPLOT" 29852 T ACPLOT (NIL) -8 NIL NIL NIL) (-29 18637 26385 26423 "ACFS" 27030 NIL ACFS (NIL T) -9 NIL 27269 NIL) (-28 16664 17154 17916 "ACFS-" 17921 NIL ACFS- (NIL T T) -8 NIL NIL NIL) (-27 12784 14711 14737 "ACF" 15616 T ACF (NIL) -9 NIL 16029 NIL) (-26 11488 11822 12315 "ACF-" 12320 NIL ACF- (NIL T) -8 NIL NIL NIL) (-25 11060 11255 11281 "ABELSG" 11373 T ABELSG (NIL) -9 NIL 11438 NIL) (-24 10927 10952 11018 "ABELSG-" 11023 NIL ABELSG- (NIL T) -8 NIL NIL NIL) (-23 10270 10557 10583 "ABELMON" 10753 T ABELMON (NIL) -9 NIL 10865 NIL) (-22 9934 10018 10156 "ABELMON-" 10161 NIL ABELMON- (NIL T) -8 NIL NIL NIL) (-21 9282 9654 9680 "ABELGRP" 9752 T ABELGRP (NIL) -9 NIL 9827 NIL) (-20 8745 8874 9090 "ABELGRP-" 9095 NIL ABELGRP- (NIL T) -8 NIL NIL NIL) (-19 4334 8084 8123 "A1AGG" 8128 NIL A1AGG (NIL T) -9 NIL 8168 NIL) (-18 30 1252 2814 "A1AGG-" 2819 NIL A1AGG- (NIL T T) -8 NIL NIL NIL)) 
\ No newline at end of file
diff --git a/src/share/algebra/operation.daase b/src/share/algebra/operation.daase
index f836d3f8..0f3e38cc 100644
--- a/src/share/algebra/operation.daase
+++ b/src/share/algebra/operation.daase
@@ -1,3804 +1,3804 @@
 
-(724049 . 3477420788)        
-(((*1 *2 *3 *4)
-  (|partial| -12 (-5 *3 (-1270 *4)) (-4 *4 (-642 (-550)))
-   (-5 *2 (-1270 (-411 (-550)))) (-5 *1 (-1298 *4))))) 
-(((*1 *2 *3)
-  (|partial| -12 (-5 *3 (-1270 *4)) (-4 *4 (-642 (-550)))
-   (-5 *2 (-1270 (-550))) (-5 *1 (-1298 *4))))) 
-(((*1 *2 *3)
-  (-12 (-5 *3 (-1270 *4)) (-4 *4 (-642 (-550))) (-5 *2 (-112))
-   (-5 *1 (-1298 *4))))) 
-(((*1 *2 *3)
-  (-12 (-4 *5 (-13 (-617 *2) (-173))) (-5 *2 (-894 *4)) (-5 *1 (-171 *4 *5 *3))
-   (-4 *4 (-1105)) (-4 *3 (-166 *5))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-644 (-1093 (-845 (-381)))))
-   (-5 *2 (-644 (-1093 (-845 (-226))))) (-5 *1 (-305))))
- ((*1 *1 *2 *3) (-12 (-5 *2 (-866)) (-5 *3 (-550)) (-5 *1 (-397))))
- ((*1 *1 *2)
-  (-12 (-5 *2 (-1270 *3)) (-4 *3 (-173)) (-4 *1 (-414 *3 *4))
-   (-4 *4 (-1246 *3))))
- ((*1 *2 *1)
-  (-12 (-4 *1 (-414 *3 *4)) (-4 *3 (-173)) (-4 *4 (-1246 *3))
-   (-5 *2 (-1270 *3))))
- ((*1 *1 *2) (-12 (-5 *2 (-1270 *3)) (-4 *3 (-173)) (-4 *1 (-422 *3))))
- ((*1 *2 *1) (-12 (-4 *1 (-422 *3)) (-4 *3 (-173)) (-5 *2 (-1270 *3))))
- ((*1 *1 *2)
-  (-12 (-5 *2 (-409 *1)) (-4 *1 (-425 *3)) (-4 *3 (-561)) (-4 *3 (-1105))))
- ((*1 *1 *2)
-  (-12 (-5 *2 (-644 *6)) (-4 *6 (-1069 *3 *4 *5)) (-4 *3 (-1053))
-   (-4 *4 (-796)) (-4 *5 (-853)) (-5 *1 (-467 *3 *4 *5 *6))))
- ((*1 *1 *2) (-12 (-5 *2 (-1107)) (-5 *1 (-539))))
- ((*1 *2 *1) (-12 (-4 *1 (-617 *2)) (-4 *2 (-1220))))
- ((*1 *1 *2) (-12 (-4 *1 (-621 *2)) (-4 *2 (-1220))))
- ((*1 *1 *2) (-12 (-4 *3 (-173)) (-4 *1 (-727 *3 *2)) (-4 *2 (-1246 *3))))
- ((*1 *1 *2) (-12 (-5 *2 (-644 (-894 *3))) (-5 *1 (-894 *3)) (-4 *3 (-1105))))
- ((*1 *1 *2)
-  (-12 (-5 *2 (-950 *3)) (-4 *3 (-1053)) (-4 *1 (-1069 *3 *4 *5))
-   (-4 *5 (-617 (-1181))) (-4 *4 (-796)) (-4 *5 (-853))))
- ((*1 *1 *2)
-  (-3962
-   (-12 (-5 *2 (-950 (-550))) (-4 *1 (-1069 *3 *4 *5))
-    (-12 (-3748 (-4 *3 (-38 (-411 (-550))))) (-4 *3 (-38 (-550)))
-     (-4 *5 (-617 (-1181))))
-    (-4 *3 (-1053)) (-4 *4 (-796)) (-4 *5 (-853)))
-   (-12 (-5 *2 (-950 (-550))) (-4 *1 (-1069 *3 *4 *5))
-    (-12 (-4 *3 (-38 (-411 (-550)))) (-4 *5 (-617 (-1181)))) (-4 *3 (-1053))
-    (-4 *4 (-796)) (-4 *5 (-853)))))
- ((*1 *1 *2)
-  (-12 (-5 *2 (-950 (-411 (-550)))) (-4 *1 (-1069 *3 *4 *5))
-   (-4 *3 (-38 (-411 (-550)))) (-4 *5 (-617 (-1181))) (-4 *3 (-1053))
-   (-4 *4 (-796)) (-4 *5 (-853))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-2 (|:| |val| (-644 *7)) (|:| -1710 *8)))
-   (-4 *7 (-1069 *4 *5 *6)) (-4 *8 (-1075 *4 *5 *6 *7)) (-4 *4 (-456))
-   (-4 *5 (-796)) (-4 *6 (-853)) (-5 *2 (-1163))
-   (-5 *1 (-1073 *4 *5 *6 *7 *8))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-2 (|:| |val| (-644 *7)) (|:| -1710 *8)))
-   (-4 *7 (-1069 *4 *5 *6)) (-4 *8 (-1113 *4 *5 *6 *7)) (-4 *4 (-456))
-   (-4 *5 (-796)) (-4 *6 (-853)) (-5 *2 (-1163))
-   (-5 *1 (-1149 *4 *5 *6 *7 *8))))
- ((*1 *1 *2) (-12 (-5 *2 (-1107)) (-5 *1 (-1186))))
- ((*1 *2 *1) (-12 (-5 *2 (-1107)) (-5 *1 (-1186))))
- ((*1 *1 *2 *3 *2) (-12 (-5 *2 (-866)) (-5 *3 (-550)) (-5 *1 (-1200))))
- ((*1 *1 *2 *3) (-12 (-5 *2 (-866)) (-5 *3 (-550)) (-5 *1 (-1200))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-783 *4 (-867 *5))) (-4 *4 (-13 (-851) (-309) (-147) (-1024)))
-   (-14 *5 (-644 (-1181))) (-5 *2 (-783 *4 (-867 *6))) (-5 *1 (-1297 *4 *5 *6))
-   (-14 *6 (-644 (-1181)))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-950 *4)) (-4 *4 (-13 (-851) (-309) (-147) (-1024)))
-   (-5 *2 (-950 (-1028 (-411 *4)))) (-5 *1 (-1297 *4 *5 *6))
-   (-14 *5 (-644 (-1181))) (-14 *6 (-644 (-1181)))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-783 *4 (-867 *6))) (-4 *4 (-13 (-851) (-309) (-147) (-1024)))
-   (-14 *6 (-644 (-1181))) (-5 *2 (-950 (-1028 (-411 *4))))
-   (-5 *1 (-1297 *4 *5 *6)) (-14 *5 (-644 (-1181)))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-1175 *4)) (-4 *4 (-13 (-851) (-309) (-147) (-1024)))
-   (-5 *2 (-1175 (-1028 (-411 *4)))) (-5 *1 (-1297 *4 *5 *6))
-   (-14 *5 (-644 (-1181))) (-14 *6 (-644 (-1181)))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-1150 *4 (-535 (-867 *6)) (-867 *6) (-783 *4 (-867 *6))))
-   (-4 *4 (-13 (-851) (-309) (-147) (-1024))) (-14 *6 (-644 (-1181)))
-   (-5 *2 (-644 (-783 *4 (-867 *6)))) (-5 *1 (-1297 *4 *5 *6))
-   (-14 *5 (-644 (-1181)))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-409 *3)) (-5 *1 (-563 *3)) (-4 *3 (-549))))
- ((*1 *2 *3)
-  (-12 (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-309)) (-5 *2 (-409 *3))
-   (-5 *1 (-745 *4 *5 *6 *3)) (-4 *3 (-954 *6 *4 *5))))
- ((*1 *2 *3)
-  (-12 (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-309)) (-4 *7 (-954 *6 *4 *5))
-   (-5 *2 (-409 (-1175 *7))) (-5 *1 (-745 *4 *5 *6 *7)) (-5 *3 (-1175 *7))))
- ((*1 *2 *1)
-  (-12 (-4 *3 (-456)) (-4 *3 (-1053)) (-4 *4 (-796)) (-4 *5 (-853))
-   (-5 *2 (-409 *1)) (-4 *1 (-954 *3 *4 *5))))
- ((*1 *2 *3)
-  (-12 (-4 *4 (-853)) (-4 *5 (-796)) (-4 *6 (-456)) (-5 *2 (-409 *3))
-   (-5 *1 (-983 *4 *5 *6 *3)) (-4 *3 (-954 *6 *5 *4))))
- ((*1 *2 *3)
-  (-12 (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-456)) (-4 *7 (-954 *6 *4 *5))
-   (-5 *2 (-409 (-1175 (-411 *7)))) (-5 *1 (-1177 *4 *5 *6 *7))
-   (-5 *3 (-1175 (-411 *7)))))
- ((*1 *2 *1) (-12 (-5 *2 (-409 *1)) (-4 *1 (-1225))))
- ((*1 *2 *3)
-  (-12 (-4 *4 (-561)) (-5 *2 (-409 *3)) (-5 *1 (-1250 *4 *3))
-   (-4 *3 (-13 (-1246 *4) (-561) (-10 -8 (-15 -3566 ($ $ $)))))))
- ((*1 *2 *3)
-  (-12 (-5 *3 (-1050 *4 *5)) (-4 *4 (-13 (-851) (-309) (-147) (-1024)))
-   (-14 *5 (-644 (-1181)))
-   (-5 *2 (-644 (-1150 *4 (-535 (-867 *6)) (-867 *6) (-783 *4 (-867 *6)))))
-   (-5 *1 (-1297 *4 *5 *6)) (-14 *6 (-644 (-1181)))))) 
-(((*1 *2 *3)
-  (-12 (-5 *3 (-1050 *4 *5)) (-4 *4 (-13 (-851) (-309) (-147) (-1024)))
-   (-14 *5 (-644 (-1181))) (-5 *2 (-644 (-644 (-1028 (-411 *4)))))
-   (-5 *1 (-1297 *4 *5 *6)) (-14 *6 (-644 (-1181)))))
+(724368 . 3477425186)        
+(((*1 *2 *3 *4)
+  (|partial| -12 (-5 *3 (-1272 *4)) (-4 *4 (-644 (-551)))
+   (-5 *2 (-1272 (-412 (-551)))) (-5 *1 (-1300 *4))))) 
+(((*1 *2 *3)
+  (|partial| -12 (-5 *3 (-1272 *4)) (-4 *4 (-644 (-551)))
+   (-5 *2 (-1272 (-551))) (-5 *1 (-1300 *4))))) 
+(((*1 *2 *3)
+  (-12 (-5 *3 (-1272 *4)) (-4 *4 (-644 (-551))) (-5 *2 (-112))
+   (-5 *1 (-1300 *4))))) 
+(((*1 *2 *3)
+  (-12 (-4 *5 (-13 (-619 *2) (-173))) (-5 *2 (-896 *4)) (-5 *1 (-171 *4 *5 *3))
+   (-4 *4 (-1107)) (-4 *3 (-166 *5))))
+ ((*1 *2 *3)
+  (-12 (-5 *3 (-646 (-1095 (-847 (-382)))))
+   (-5 *2 (-646 (-1095 (-847 (-226))))) (-5 *1 (-306))))
+ ((*1 *1 *2 *3) (-12 (-5 *2 (-868)) (-5 *3 (-551)) (-5 *1 (-398))))
+ ((*1 *1 *2)
+  (-12 (-5 *2 (-1272 *3)) (-4 *3 (-173)) (-4 *1 (-415 *3 *4))
+   (-4 *4 (-1248 *3))))
+ ((*1 *2 *1)
+  (-12 (-4 *1 (-415 *3 *4)) (-4 *3 (-173)) (-4 *4 (-1248 *3))
+   (-5 *2 (-1272 *3))))
+ ((*1 *1 *2) (-12 (-5 *2 (-1272 *3)) (-4 *3 (-173)) (-4 *1 (-423 *3))))
+ ((*1 *2 *1) (-12 (-4 *1 (-423 *3)) (-4 *3 (-173)) (-5 *2 (-1272 *3))))
+ ((*1 *1 *2)
+  (-12 (-5 *2 (-410 *1)) (-4 *1 (-426 *3)) (-4 *3 (-562)) (-4 *3 (-1107))))
+ ((*1 *1 *2)
+  (-12 (-5 *2 (-646 *6)) (-4 *6 (-1071 *3 *4 *5)) (-4 *3 (-1055))
+   (-4 *4 (-798)) (-4 *5 (-855)) (-5 *1 (-468 *3 *4 *5 *6))))
+ ((*1 *1 *2) (-12 (-5 *2 (-1109)) (-5 *1 (-540))))
+ ((*1 *2 *1) (-12 (-4 *1 (-619 *2)) (-4 *2 (-1222))))
+ ((*1 *1 *2) (-12 (-4 *1 (-623 *2)) (-4 *2 (-1222))))
+ ((*1 *1 *2) (-12 (-4 *3 (-173)) (-4 *1 (-729 *3 *2)) (-4 *2 (-1248 *3))))
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    (-5 *2
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 (((*1 *1 *1 *1) (-4 *1 (-21))) ((*1 *1 *1) (-4 *1 (-21)))
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-   (-4 *3 (-978))))
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 (((*1 *2 *1) (-12 (-5 *2 (-112)) (-5 *1 (-172))))
- ((*1 *2 *1) (-12 (-5 *2 (-112)) (-5 *1 (-1216 *3)) (-4 *3 (-978))))) 
+ ((*1 *2 *1) (-12 (-5 *2 (-112)) (-5 *1 (-1218 *3)) (-4 *3 (-980))))) 
 (((*1 *2 *1)
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 (((*1 *2 *3 *4 *5)
   (|partial| -12 (-5 *4 (-1 (-112) *9)) (-5 *5 (-1 (-112) *9 *9))
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-   (-4 *1 (-1215 *5 *6 *7 *8))))) 
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 (((*1 *2 *1 *3)
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    (-5 *2 (-112))))
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 (((*1 *2 *1)
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-   (-4 *6 (-1069 *3 *4 *5)) (-5 *2 (-112))))
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 (((*1 *2 *1 *3)
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    (-5 *2 (-112))))) 
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-   (-4 *8 (-1069 *5 *6 *7))))) 
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-   (-4 *2 (-1069 *3 *4 *5))))) 
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-   (-5 *1 (-1150 *5 *6 *7 *8))))
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-  (-12 (-5 *2 (-774)) (-4 *3 (-1220)) (-4 *1 (-57 *3 *4 *5)) (-4 *4 (-375 *3))
-   (-4 *5 (-375 *3))))
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+   (-4 *5 (-376 *3))))
  ((*1 *1) (-5 *1 (-172)))
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- ((*1 *1 *2 *3) (-12 (-5 *2 (-774)) (-4 *1 (-654 *3)) (-4 *3 (-1220))))
+ ((*1 *1) (-12 (-5 *1 (-214 *2 *3)) (-14 *2 (-925)) (-4 *3 (-1107))))
+ ((*1 *1 *2 *2 *2) (-12 (-5 *2 (-1165)) (-4 *1 (-394))))
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+  (-12 (-4 *3 (-1107)) (-5 *1 (-891 *2 *3 *4)) (-4 *2 (-1107))
+   (-4 *4 (-671 *3))))
+ ((*1 *1) (-12 (-5 *1 (-894 *2 *3)) (-4 *2 (-1107)) (-4 *3 (-1107))))
+ ((*1 *1 *2) (-12 (-5 *1 (-1148 *3 *2)) (-14 *3 (-776)) (-4 *2 (-1055))))
+ ((*1 *1) (-12 (-5 *1 (-1171 *2 *3)) (-14 *2 (-925)) (-4 *3 (-1055))))
+ ((*1 *1 *1) (-5 *1 (-1183))) ((*1 *1) (-5 *1 (-1183)))
+ ((*1 *1) (-5 *1 (-1202)))) 
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+ ((*1 *2 *1) (-12 (-5 *2 (-646 (-1165))) (-5 *1 (-1202))))) 
+(((*1 *2 *1 *2) (-12 (-5 *2 (-646 (-1165))) (-5 *1 (-398))))
+ ((*1 *2 *1 *2) (-12 (-5 *2 (-646 (-1165))) (-5 *1 (-1202))))) 
+(((*1 *2 *1 *2) (-12 (-5 *2 (-646 (-1165))) (-5 *1 (-1202))))) 
+(((*1 *2 *1 *2) (-12 (-5 *2 (-646 (-1165))) (-5 *1 (-398))))
+ ((*1 *2 *1 *2) (-12 (-5 *2 (-646 (-1165))) (-5 *1 (-1202))))) 
+(((*1 *2 *2) (-12 (-5 *2 (-1165)) (-5 *1 (-1201))))) 
+(((*1 *2 *3) (-12 (-5 *3 (-1165)) (-5 *2 (-51)) (-5 *1 (-1201))))) 
+(((*1 *1 *2 *1) (-12 (-4 *1 (-107 *2)) (-4 *2 (-1222))))
+ ((*1 *1 *2 *1) (-12 (-5 *1 (-121 *2)) (-4 *2 (-855))))
+ ((*1 *1 *2 *1) (-12 (-5 *1 (-126 *2)) (-4 *2 (-855))))
+ ((*1 *1 *1 *1 *2) (-12 (-5 *2 (-551)) (-4 *1 (-285 *3)) (-4 *3 (-1222))))
+ ((*1 *1 *2 *1 *3) (-12 (-5 *3 (-551)) (-4 *1 (-285 *2)) (-4 *2 (-1222))))
  ((*1 *1 *2)
   (-12
    (-5 *2
     (-2
-     (|:| -4294
-          (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-           (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+     (|:| -4301
+          (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+           (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
            (|:| |relerr| (-226))))
-     (|:| -2256
+     (|:| -2263
           (-2
            (|:| |endPointContinuity|
                 (-3 (|:| |continuous| "Continuous at the end points")
@@ -3811,7978 +3811,7978 @@
                  (|:| |notEvaluated|
                       "End point continuity not yet evaluated")))
            (|:| |singularitiesStream|
-                (-3 (|:| |str| (-1158 (-226)))
+                (-3 (|:| |str| (-1160 (-226)))
                  (|:| |notEvaluated|
                       "Internal singularities not yet evaluated")))
-           (|:| -1609
+           (|:| -1612
                 (-3 (|:| |finite| "The range is finite")
                  (|:| |lowerInfinite| "The bottom of range is infinite")
                  (|:| |upperInfinite| "The top of range is infinite")
                  (|:| |bothInfinite| "Both top and bottom points are infinite")
                  (|:| |notEvaluated| "Range not yet evaluated")))))))
-   (-5 *1 (-564))))
- ((*1 *1 *2 *1 *3) (-12 (-5 *3 (-774)) (-4 *1 (-698 *2)) (-4 *2 (-1105))))
+   (-5 *1 (-565))))
+ ((*1 *1 *2 *1 *3) (-12 (-5 *3 (-776)) (-4 *1 (-700 *2)) (-4 *2 (-1107))))
  ((*1 *1 *2)
   (-12
    (-5 *2
     (-2
-     (|:| -4294
+     (|:| -4301
           (-2 (|:| |xinit| (-226)) (|:| |xend| (-226))
-           (|:| |fn| (-1270 (-316 (-226)))) (|:| |yinit| (-644 (-226)))
-           (|:| |intvals| (-644 (-226))) (|:| |g| (-316 (-226)))
+           (|:| |fn| (-1272 (-317 (-226)))) (|:| |yinit| (-646 (-226)))
+           (|:| |intvals| (-646 (-226))) (|:| |g| (-317 (-226)))
            (|:| |abserr| (-226)) (|:| |relerr| (-226))))
-     (|:| -2256
-          (-2 (|:| |stiffness| (-381)) (|:| |stability| (-381))
-           (|:| |expense| (-381)) (|:| |accuracy| (-381))
-           (|:| |intermediateResults| (-381))))))
-   (-5 *1 (-806))))
+     (|:| -2263
+          (-2 (|:| |stiffness| (-382)) (|:| |stability| (-382))
+           (|:| |expense| (-382)) (|:| |accuracy| (-382))
+           (|:| |intermediateResults| (-382))))))
+   (-5 *1 (-808))))
  ((*1 *2 *3 *4)
-  (-12 (-5 *2 (-1276)) (-5 *1 (-1198 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
+  (-12 (-5 *2 (-1278)) (-5 *1 (-1200 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))))) 
 (((*1 *2 *3)
-  (|partial| -12 (-4 *2 (-1105)) (-5 *1 (-1198 *3 *2)) (-4 *3 (-1105))))) 
+  (|partial| -12 (-4 *2 (-1107)) (-5 *1 (-1200 *3 *2)) (-4 *3 (-1107))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-112)) (-5 *1 (-1198 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
+  (-12 (-5 *2 (-112)) (-5 *1 (-1200 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-112)) (-5 *1 (-1198 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
+  (-12 (-5 *2 (-112)) (-5 *1 (-1200 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-112)) (-5 *1 (-1198 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
+  (-12 (-5 *2 (-112)) (-5 *1 (-1200 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-1276)) (-5 *1 (-1198 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
+  (-12 (-5 *2 (-1278)) (-5 *1 (-1200 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-1276)) (-5 *1 (-1198 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
+  (-12 (-5 *2 (-1278)) (-5 *1 (-1200 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))))) 
 (((*1 *2 *3)
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-   (-4 *5 (-1105))))) 
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+   (-4 *5 (-1107))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *3 (-1163)) (-5 *2 (-1276)) (-5 *1 (-1198 *4 *5)) (-4 *4 (-1105))
-   (-4 *5 (-1105))))) 
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+   (-4 *5 (-1107))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-1276)) (-5 *1 (-1198 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
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 (((*1 *1 *2)
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 (((*1 *2 *3 *4)
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-(((*1 *2 *3) (-12 (-5 *3 (-644 *2)) (-5 *1 (-1195 *2)) (-4 *2 (-366))))) 
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 (((*1 *2 *1)
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-   (-4 *5 (-375 *3)) (-5 *2 (-644 (-644 *3)))))
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+   (-4 *7 (-239 *3 *5)) (-5 *2 (-646 (-646 *5)))))
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 (((*1 *2 *3)
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+  (-12 (-4 *4 (-855))
    (-5 *2
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-   (-5 *1 (-1192 *4)) (-5 *3 (-644 (-644 (-644 *4))))))) 
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 (((*1 *2 *3 *4 *5 *4 *4 *4)
-  (-12 (-4 *6 (-853)) (-5 *3 (-644 *6)) (-5 *5 (-644 *3))
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    (-5 *2
-    (-2 (|:| |f1| *3) (|:| |f2| (-644 *5)) (|:| |f3| *5) (|:| |f4| (-644 *5))))
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+   (-5 *1 (-1194 *6)) (-5 *4 (-646 *5))))) 
 (((*1 *2 *2)
-  (|partial| -12 (-4 *3 (-366)) (-4 *4 (-375 *3)) (-4 *5 (-375 *3))
-   (-5 *1 (-525 *3 *4 *5 *2)) (-4 *2 (-689 *3 *4 *5))))
+  (|partial| -12 (-4 *3 (-367)) (-4 *4 (-376 *3)) (-4 *5 (-376 *3))
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  ((*1 *2 *3)
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  ((*1 *1 *1)
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-   (-4 *4 (-375 *2)) (-4 *2 (-366))))
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+   (-4 *4 (-376 *2)) (-4 *2 (-367))))
  ((*1 *2 *2)
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- ((*1 *1 *1) (|partial| -12 (-5 *1 (-692 *2)) (-4 *2 (-366)) (-4 *2 (-1053))))
+  (|partial| -12 (-4 *3 (-367)) (-4 *3 (-173)) (-4 *4 (-376 *3))
+   (-4 *5 (-376 *3)) (-5 *1 (-693 *3 *4 *5 *2)) (-4 *2 (-691 *3 *4 *5))))
+ ((*1 *1 *1) (|partial| -12 (-5 *1 (-694 *2)) (-4 *2 (-367)) (-4 *2 (-1055))))
  ((*1 *1 *1)
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- ((*1 *2 *2) (-12 (-5 *2 (-644 *3)) (-4 *3 (-853)) (-5 *1 (-1192 *3))))) 
+  (|partial| -12 (-4 *1 (-1129 *2 *3 *4 *5)) (-4 *3 (-1055))
+   (-4 *4 (-239 *2 *3)) (-4 *5 (-239 *2 *3)) (-4 *3 (-367))))
+ ((*1 *2 *2) (-12 (-5 *2 (-646 *3)) (-4 *3 (-855)) (-5 *1 (-1194 *3))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-853)) (-5 *2 (-644 (-644 *4))) (-5 *1 (-1192 *4))
-   (-5 *3 (-644 *4))))) 
-(((*1 *2 *2) (-12 (-5 *2 (-644 (-644 *3))) (-4 *3 (-853)) (-5 *1 (-1192 *3))))) 
+  (-12 (-4 *4 (-855)) (-5 *2 (-646 (-646 *4))) (-5 *1 (-1194 *4))
+   (-5 *3 (-646 *4))))) 
+(((*1 *2 *2) (-12 (-5 *2 (-646 (-646 *3))) (-4 *3 (-855)) (-5 *1 (-1194 *3))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-853)) (-5 *2 (-1194 (-644 *4))) (-5 *1 (-1192 *4))
-   (-5 *3 (-644 *4))))) 
+  (-12 (-4 *4 (-855)) (-5 *2 (-1196 (-646 *4))) (-5 *1 (-1194 *4))
+   (-5 *3 (-646 *4))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-853)) (-5 *2 (-644 (-644 (-644 *4)))) (-5 *1 (-1192 *4))
-   (-5 *3 (-644 (-644 *4)))))) 
+  (-12 (-4 *4 (-855)) (-5 *2 (-646 (-646 (-646 *4)))) (-5 *1 (-1194 *4))
+   (-5 *3 (-646 (-646 *4)))))) 
 (((*1 *2 *3)
-  (-12 (-5 *3 (-1194 (-644 *4))) (-4 *4 (-853)) (-5 *2 (-644 (-644 *4)))
-   (-5 *1 (-1192 *4))))) 
+  (-12 (-5 *3 (-1196 (-646 *4))) (-4 *4 (-855)) (-5 *2 (-646 (-646 *4)))
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 (((*1 *2 *3)
-  (-12 (-5 *3 (-644 (-644 (-644 *4)))) (-5 *2 (-644 (-644 *4)))
-   (-5 *1 (-1192 *4)) (-4 *4 (-853))))) 
+  (-12 (-5 *3 (-646 (-646 (-646 *4)))) (-5 *2 (-646 (-646 *4)))
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 (((*1 *2 *2 *3)
-  (-12 (-5 *3 (-644 (-644 (-644 *4)))) (-5 *2 (-644 (-644 *4))) (-4 *4 (-853))
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+  (-12 (-5 *3 (-646 (-646 (-646 *4)))) (-5 *2 (-646 (-646 *4))) (-4 *4 (-855))
+   (-5 *1 (-1194 *4))))) 
 (((*1 *2 *3 *2)
-  (-12 (-5 *2 (-644 (-644 (-644 *4)))) (-5 *3 (-644 *4)) (-4 *4 (-853))
-   (-5 *1 (-1192 *4))))) 
+  (-12 (-5 *2 (-646 (-646 (-646 *4)))) (-5 *3 (-646 *4)) (-4 *4 (-855))
+   (-5 *1 (-1194 *4))))) 
 (((*1 *2 *3 *4 *2)
-  (-12 (-5 *2 (-644 (-644 (-644 *5)))) (-5 *3 (-1 (-112) *5 *5))
-   (-5 *4 (-644 *5)) (-4 *5 (-853)) (-5 *1 (-1192 *5))))) 
+  (-12 (-5 *2 (-646 (-646 (-646 *5)))) (-5 *3 (-1 (-112) *5 *5))
+   (-5 *4 (-646 *5)) (-4 *5 (-855)) (-5 *1 (-1194 *5))))) 
 (((*1 *2 *3 *4 *5)
-  (-12 (-5 *3 (-1 (-112) *6 *6)) (-4 *6 (-853)) (-5 *4 (-644 *6))
-   (-5 *2 (-2 (|:| |fs| (-112)) (|:| |sd| *4) (|:| |td| (-644 *4))))
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-(((*1 *2 *3) (-12 (-5 *3 (-1163)) (-5 *2 (-1276)) (-5 *1 (-1191))))) 
-(((*1 *2 *3 *4)
-  (-12 (-5 *3 (-644 (-411 (-950 *5)))) (-5 *4 (-644 (-1181))) (-4 *5 (-561))
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-(((*1 *2 *3 *4)
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-   (-5 *2 (-644 (-644 (-295 (-950 *4))))) (-5 *1 (-383 *4))
-   (-4 *4 (-13 (-851) (-366)))))
- ((*1 *2 *3 *4)
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- ((*1 *2 *3 *4)
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- ((*1 *2 *3 *4)
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+(((*1 *2 *3 *4)
+  (-12 (-5 *3 (-646 (-412 (-952 *5)))) (-5 *4 (-646 (-1183))) (-4 *5 (-562))
+   (-5 *2 (-646 (-646 (-952 *5)))) (-5 *1 (-1192 *5))))) 
+(((*1 *2 *3 *4)
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    (-5 *2
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-  (-12 (-5 *3 (-1239 *5 *4)) (-4 *4 (-823)) (-14 *5 (-1181)) (-5 *2 (-550))
-   (-5 *1 (-1118 *4 *5))))) 
+  (-12 (-5 *3 (-1241 *5 *4)) (-4 *4 (-825)) (-14 *5 (-1183)) (-5 *2 (-551))
+   (-5 *1 (-1120 *4 *5))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *3 (-1239 *5 *4)) (-4 *4 (-823)) (-14 *5 (-1181)) (-5 *2 (-550))
-   (-5 *1 (-1118 *4 *5))))) 
+  (-12 (-5 *3 (-1241 *5 *4)) (-4 *4 (-825)) (-14 *5 (-1183)) (-5 *2 (-551))
+   (-5 *1 (-1120 *4 *5))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *3 (-1239 *5 *4)) (-4 *4 (-823)) (-14 *5 (-1181)) (-5 *2 (-644 *4))
-   (-5 *1 (-1118 *4 *5))))) 
+  (-12 (-5 *3 (-1241 *5 *4)) (-4 *4 (-825)) (-14 *5 (-1183)) (-5 *2 (-646 *4))
+   (-5 *1 (-1120 *4 *5))))) 
 (((*1 *2 *3 *3)
-  (-12 (-4 *4 (-823)) (-14 *5 (-1181)) (-5 *2 (-644 (-1239 *5 *4)))
-   (-5 *1 (-1118 *4 *5)) (-5 *3 (-1239 *5 *4))))) 
+  (-12 (-4 *4 (-825)) (-14 *5 (-1183)) (-5 *2 (-646 (-1241 *5 *4)))
+   (-5 *1 (-1120 *4 *5)) (-5 *3 (-1241 *5 *4))))) 
 (((*1 *2 *3 *3)
-  (-12 (-4 *4 (-823)) (-14 *5 (-1181)) (-5 *2 (-644 (-1239 *5 *4)))
-   (-5 *1 (-1118 *4 *5)) (-5 *3 (-1239 *5 *4))))) 
-(((*1 *2 *3 *3 *3) (-12 (-5 *2 (-644 (-550))) (-5 *1 (-1114)) (-5 *3 (-550))))) 
-(((*1 *2 *3 *3 *3) (-12 (-5 *2 (-644 (-550))) (-5 *1 (-1114)) (-5 *3 (-550))))) 
-(((*1 *2 *3 *3 *3) (-12 (-5 *2 (-644 (-550))) (-5 *1 (-1114)) (-5 *3 (-550))))) 
-(((*1 *2 *2 *2) (-12 (-5 *2 (-550)) (-5 *1 (-1114))))) 
-(((*1 *2 *2 *2 *3) (-12 (-5 *2 (-1270 (-550))) (-5 *3 (-550)) (-5 *1 (-1114))))
+  (-12 (-4 *4 (-825)) (-14 *5 (-1183)) (-5 *2 (-646 (-1241 *5 *4)))
+   (-5 *1 (-1120 *4 *5)) (-5 *3 (-1241 *5 *4))))) 
+(((*1 *2 *3 *3 *3) (-12 (-5 *2 (-646 (-551))) (-5 *1 (-1116)) (-5 *3 (-551))))) 
+(((*1 *2 *3 *3 *3) (-12 (-5 *2 (-646 (-551))) (-5 *1 (-1116)) (-5 *3 (-551))))) 
+(((*1 *2 *3 *3 *3) (-12 (-5 *2 (-646 (-551))) (-5 *1 (-1116)) (-5 *3 (-551))))) 
+(((*1 *2 *2 *2) (-12 (-5 *2 (-551)) (-5 *1 (-1116))))) 
+(((*1 *2 *2 *2 *3) (-12 (-5 *2 (-1272 (-551))) (-5 *3 (-551)) (-5 *1 (-1116))))
  ((*1 *2 *3 *2 *4)
-  (-12 (-5 *2 (-1270 (-550))) (-5 *3 (-644 (-550))) (-5 *4 (-550))
-   (-5 *1 (-1114))))) 
-(((*1 *2 *2 *2 *3) (-12 (-5 *2 (-644 (-550))) (-5 *3 (-112)) (-5 *1 (-1114))))) 
+  (-12 (-5 *2 (-1272 (-551))) (-5 *3 (-646 (-551))) (-5 *4 (-551))
+   (-5 *1 (-1116))))) 
+(((*1 *2 *2 *2 *3) (-12 (-5 *2 (-646 (-551))) (-5 *3 (-112)) (-5 *1 (-1116))))) 
 (((*1 *2 *3 *3 *2)
-  (-12 (-5 *2 (-692 (-550))) (-5 *3 (-644 (-550))) (-5 *1 (-1114))))) 
+  (-12 (-5 *2 (-694 (-551))) (-5 *3 (-646 (-551))) (-5 *1 (-1116))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *3 (-644 (-550))) (-5 *2 (-692 (-550))) (-5 *1 (-1114))))) 
+  (-12 (-5 *3 (-646 (-551))) (-5 *2 (-694 (-551))) (-5 *1 (-1116))))) 
 (((*1 *2 *3)
-  (-12 (-5 *3 (-644 (-550))) (-5 *2 (-644 (-692 (-550)))) (-5 *1 (-1114))))) 
+  (-12 (-5 *3 (-646 (-551))) (-5 *2 (-646 (-694 (-551)))) (-5 *1 (-1116))))) 
 (((*1 *2 *2 *2 *3)
-  (-12 (-5 *2 (-644 (-550))) (-5 *3 (-692 (-550))) (-5 *1 (-1114))))) 
+  (-12 (-5 *2 (-646 (-551))) (-5 *3 (-694 (-551))) (-5 *1 (-1116))))) 
 (((*1 *2 *3 *3 *3)
-  (-12 (-5 *3 (-644 (-550))) (-5 *2 (-692 (-550))) (-5 *1 (-1114))))) 
+  (-12 (-5 *3 (-646 (-551))) (-5 *2 (-694 (-551))) (-5 *1 (-1116))))) 
 (((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 (-2 (|:| |val| *3) (|:| -1710 *4))))
-   (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 (-2 (|:| |val| *3) (|:| -1717 *4))))
+   (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 *4)) (-5 *1 (-1112 *5 *6 *7 *3 *4))
-   (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 *4)) (-5 *1 (-1114 *5 *6 *7 *3 *4))
+   (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-112)) (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-112)) (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))
  ((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 (-2 (|:| |val| (-112)) (|:| -1710 *4))))
-   (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 (-2 (|:| |val| (-112)) (|:| -1717 *4))))
+   (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 *4)) (-5 *1 (-1112 *5 *6 *7 *3 *4))
-   (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 *4)) (-5 *1 (-1114 *5 *6 *7 *3 *4))
+   (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 (-2 (|:| |val| (-112)) (|:| -1710 *4))))
-   (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 (-2 (|:| |val| (-112)) (|:| -1717 *4))))
+   (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 *4)) (-5 *1 (-1112 *5 *6 *7 *3 *4))
-   (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 *4)) (-5 *1 (-1114 *5 *6 *7 *3 *4))
+   (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 (-2 (|:| |val| (-112)) (|:| -1710 *4))))
-   (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 (-2 (|:| |val| (-112)) (|:| -1717 *4))))
+   (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 (-2 (|:| |val| *3) (|:| -1710 *4))))
-   (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 (-2 (|:| |val| *3) (|:| -1717 *4))))
+   (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 (-2 (|:| |val| *3) (|:| -1710 *4))))
-   (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 (-2 (|:| |val| *3) (|:| -1717 *4))))
+   (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *3 *4 *5 *5)
-  (-12 (-5 *5 (-112)) (-4 *6 (-456)) (-4 *7 (-796)) (-4 *8 (-853))
-   (-4 *3 (-1069 *6 *7 *8)) (-5 *2 (-644 (-2 (|:| |val| *3) (|:| -1710 *4))))
-   (-5 *1 (-1112 *6 *7 *8 *3 *4)) (-4 *4 (-1075 *6 *7 *8 *3))))
+  (-12 (-5 *5 (-112)) (-4 *6 (-457)) (-4 *7 (-798)) (-4 *8 (-855))
+   (-4 *3 (-1071 *6 *7 *8)) (-5 *2 (-646 (-2 (|:| |val| *3) (|:| -1717 *4))))
+   (-5 *1 (-1114 *6 *7 *8 *3 *4)) (-4 *4 (-1077 *6 *7 *8 *3))))
  ((*1 *2 *3 *4 *5)
-  (-12 (-5 *3 (-644 (-2 (|:| |val| (-644 *8)) (|:| -1710 *9)))) (-5 *5 (-112))
-   (-4 *8 (-1069 *6 *7 *4)) (-4 *9 (-1075 *6 *7 *4 *8)) (-4 *6 (-456))
-   (-4 *7 (-796)) (-4 *4 (-853))
-   (-5 *2 (-644 (-2 (|:| |val| *8) (|:| -1710 *9))))
-   (-5 *1 (-1112 *6 *7 *4 *8 *9))))) 
+  (-12 (-5 *3 (-646 (-2 (|:| |val| (-646 *8)) (|:| -1717 *9)))) (-5 *5 (-112))
+   (-4 *8 (-1071 *6 *7 *4)) (-4 *9 (-1077 *6 *7 *4 *8)) (-4 *6 (-457))
+   (-4 *7 (-798)) (-4 *4 (-855))
+   (-5 *2 (-646 (-2 (|:| |val| *8) (|:| -1717 *9))))
+   (-5 *1 (-1114 *6 *7 *4 *8 *9))))) 
 (((*1 *2 *3 *3 *4)
-  (-12 (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853)) (-4 *3 (-1069 *5 *6 *7))
-   (-5 *2 (-644 (-2 (|:| |val| (-644 *3)) (|:| -1710 *4))))
-   (-5 *1 (-1112 *5 *6 *7 *3 *4)) (-4 *4 (-1075 *5 *6 *7 *3))))) 
+  (-12 (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855)) (-4 *3 (-1071 *5 *6 *7))
+   (-5 *2 (-646 (-2 (|:| |val| (-646 *3)) (|:| -1717 *4))))
+   (-5 *1 (-1114 *5 *6 *7 *3 *4)) (-4 *4 (-1077 *5 *6 *7 *3))))) 
 (((*1 *2)
-  (-12 (-4 *3 (-456)) (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-1069 *3 *4 *5))
-   (-5 *2 (-1276)) (-5 *1 (-1076 *3 *4 *5 *6 *7)) (-4 *7 (-1075 *3 *4 *5 *6))))
+  (-12 (-4 *3 (-457)) (-4 *4 (-798)) (-4 *5 (-855)) (-4 *6 (-1071 *3 *4 *5))
+   (-5 *2 (-1278)) (-5 *1 (-1078 *3 *4 *5 *6 *7)) (-4 *7 (-1077 *3 *4 *5 *6))))
  ((*1 *2)
-  (-12 (-4 *3 (-456)) (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-1069 *3 *4 *5))
-   (-5 *2 (-1276)) (-5 *1 (-1112 *3 *4 *5 *6 *7)) (-4 *7 (-1075 *3 *4 *5 *6))))) 
+  (-12 (-4 *3 (-457)) (-4 *4 (-798)) (-4 *5 (-855)) (-4 *6 (-1071 *3 *4 *5))
+   (-5 *2 (-1278)) (-5 *1 (-1114 *3 *4 *5 *6 *7)) (-4 *7 (-1077 *3 *4 *5 *6))))) 
 (((*1 *2 *3 *3 *3)
-  (-12 (-5 *3 (-1163)) (-4 *4 (-456)) (-4 *5 (-796)) (-4 *6 (-853))
-   (-4 *7 (-1069 *4 *5 *6)) (-5 *2 (-1276)) (-5 *1 (-1076 *4 *5 *6 *7 *8))
-   (-4 *8 (-1075 *4 *5 *6 *7))))
+  (-12 (-5 *3 (-1165)) (-4 *4 (-457)) (-4 *5 (-798)) (-4 *6 (-855))
+   (-4 *7 (-1071 *4 *5 *6)) (-5 *2 (-1278)) (-5 *1 (-1078 *4 *5 *6 *7 *8))
+   (-4 *8 (-1077 *4 *5 *6 *7))))
  ((*1 *2 *3 *3 *3)
-  (-12 (-5 *3 (-1163)) (-4 *4 (-456)) (-4 *5 (-796)) (-4 *6 (-853))
-   (-4 *7 (-1069 *4 *5 *6)) (-5 *2 (-1276)) (-5 *1 (-1112 *4 *5 *6 *7 *8))
-   (-4 *8 (-1075 *4 *5 *6 *7))))) 
+  (-12 (-5 *3 (-1165)) (-4 *4 (-457)) (-4 *5 (-798)) (-4 *6 (-855))
+   (-4 *7 (-1071 *4 *5 *6)) (-5 *2 (-1278)) (-5 *1 (-1114 *4 *5 *6 *7 *8))
+   (-4 *8 (-1077 *4 *5 *6 *7))))) 
 (((*1 *2)
-  (-12 (-4 *3 (-456)) (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-1069 *3 *4 *5))
-   (-5 *2 (-1276)) (-5 *1 (-1076 *3 *4 *5 *6 *7)) (-4 *7 (-1075 *3 *4 *5 *6))))
+  (-12 (-4 *3 (-457)) (-4 *4 (-798)) (-4 *5 (-855)) (-4 *6 (-1071 *3 *4 *5))
+   (-5 *2 (-1278)) (-5 *1 (-1078 *3 *4 *5 *6 *7)) (-4 *7 (-1077 *3 *4 *5 *6))))
  ((*1 *2)
-  (-12 (-4 *3 (-456)) (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-1069 *3 *4 *5))
-   (-5 *2 (-1276)) (-5 *1 (-1112 *3 *4 *5 *6 *7)) (-4 *7 (-1075 *3 *4 *5 *6))))) 
+  (-12 (-4 *3 (-457)) (-4 *4 (-798)) (-4 *5 (-855)) (-4 *6 (-1071 *3 *4 *5))
+   (-5 *2 (-1278)) (-5 *1 (-1114 *3 *4 *5 *6 *7)) (-4 *7 (-1077 *3 *4 *5 *6))))) 
 (((*1 *2 *3 *3 *3)
-  (-12 (-5 *3 (-1163)) (-4 *4 (-456)) (-4 *5 (-796)) (-4 *6 (-853))
-   (-4 *7 (-1069 *4 *5 *6)) (-5 *2 (-1276)) (-5 *1 (-1076 *4 *5 *6 *7 *8))
-   (-4 *8 (-1075 *4 *5 *6 *7))))
+  (-12 (-5 *3 (-1165)) (-4 *4 (-457)) (-4 *5 (-798)) (-4 *6 (-855))
+   (-4 *7 (-1071 *4 *5 *6)) (-5 *2 (-1278)) (-5 *1 (-1078 *4 *5 *6 *7 *8))
+   (-4 *8 (-1077 *4 *5 *6 *7))))
  ((*1 *2 *3 *3 *3)
-  (-12 (-5 *3 (-1163)) (-4 *4 (-456)) (-4 *5 (-796)) (-4 *6 (-853))
-   (-4 *7 (-1069 *4 *5 *6)) (-5 *2 (-1276)) (-5 *1 (-1112 *4 *5 *6 *7 *8))
-   (-4 *8 (-1075 *4 *5 *6 *7))))) 
+  (-12 (-5 *3 (-1165)) (-4 *4 (-457)) (-4 *5 (-798)) (-4 *6 (-855))
+   (-4 *7 (-1071 *4 *5 *6)) (-5 *2 (-1278)) (-5 *1 (-1114 *4 *5 *6 *7 *8))
+   (-4 *8 (-1077 *4 *5 *6 *7))))) 
 (((*1 *2 *3 *4 *3 *5 *5 *5 *5 *5)
-  (|partial| -12 (-5 *5 (-112)) (-4 *6 (-456)) (-4 *7 (-796)) (-4 *8 (-853))
-   (-4 *9 (-1069 *6 *7 *8))
-   (-5 *2 (-2 (|:| -3689 (-644 *9)) (|:| -1710 *4) (|:| |ineq| (-644 *9))))
-   (-5 *1 (-992 *6 *7 *8 *9 *4)) (-5 *3 (-644 *9))
-   (-4 *4 (-1075 *6 *7 *8 *9))))
+  (|partial| -12 (-5 *5 (-112)) (-4 *6 (-457)) (-4 *7 (-798)) (-4 *8 (-855))
+   (-4 *9 (-1071 *6 *7 *8))
+   (-5 *2 (-2 (|:| -3696 (-646 *9)) (|:| -1717 *4) (|:| |ineq| (-646 *9))))
+   (-5 *1 (-994 *6 *7 *8 *9 *4)) (-5 *3 (-646 *9))
+   (-4 *4 (-1077 *6 *7 *8 *9))))
  ((*1 *2 *3 *4 *3 *5 *5 *5 *5 *5)
-  (|partial| -12 (-5 *5 (-112)) (-4 *6 (-456)) (-4 *7 (-796)) (-4 *8 (-853))
-   (-4 *9 (-1069 *6 *7 *8))
-   (-5 *2 (-2 (|:| -3689 (-644 *9)) (|:| -1710 *4) (|:| |ineq| (-644 *9))))
-   (-5 *1 (-1111 *6 *7 *8 *9 *4)) (-5 *3 (-644 *9))
-   (-4 *4 (-1075 *6 *7 *8 *9))))) 
+  (|partial| -12 (-5 *5 (-112)) (-4 *6 (-457)) (-4 *7 (-798)) (-4 *8 (-855))
+   (-4 *9 (-1071 *6 *7 *8))
+   (-5 *2 (-2 (|:| -3696 (-646 *9)) (|:| -1717 *4) (|:| |ineq| (-646 *9))))
+   (-5 *1 (-1113 *6 *7 *8 *9 *4)) (-5 *3 (-646 *9))
+   (-4 *4 (-1077 *6 *7 *8 *9))))) 
 (((*1 *2 *3 *4 *5 *5)
-  (-12 (-5 *4 (-644 *10)) (-5 *5 (-112)) (-4 *10 (-1075 *6 *7 *8 *9))
-   (-4 *6 (-456)) (-4 *7 (-796)) (-4 *8 (-853)) (-4 *9 (-1069 *6 *7 *8))
+  (-12 (-5 *4 (-646 *10)) (-5 *5 (-112)) (-4 *10 (-1077 *6 *7 *8 *9))
+   (-4 *6 (-457)) (-4 *7 (-798)) (-4 *8 (-855)) (-4 *9 (-1071 *6 *7 *8))
    (-5 *2
-    (-644 (-2 (|:| -3689 (-644 *9)) (|:| -1710 *10) (|:| |ineq| (-644 *9)))))
-   (-5 *1 (-992 *6 *7 *8 *9 *10)) (-5 *3 (-644 *9))))
+    (-646 (-2 (|:| -3696 (-646 *9)) (|:| -1717 *10) (|:| |ineq| (-646 *9)))))
+   (-5 *1 (-994 *6 *7 *8 *9 *10)) (-5 *3 (-646 *9))))
  ((*1 *2 *3 *4 *5 *5)
-  (-12 (-5 *4 (-644 *10)) (-5 *5 (-112)) (-4 *10 (-1075 *6 *7 *8 *9))
-   (-4 *6 (-456)) (-4 *7 (-796)) (-4 *8 (-853)) (-4 *9 (-1069 *6 *7 *8))
+  (-12 (-5 *4 (-646 *10)) (-5 *5 (-112)) (-4 *10 (-1077 *6 *7 *8 *9))
+   (-4 *6 (-457)) (-4 *7 (-798)) (-4 *8 (-855)) (-4 *9 (-1071 *6 *7 *8))
    (-5 *2
-    (-644 (-2 (|:| -3689 (-644 *9)) (|:| -1710 *10) (|:| |ineq| (-644 *9)))))
-   (-5 *1 (-1111 *6 *7 *8 *9 *10)) (-5 *3 (-644 *9))))) 
+    (-646 (-2 (|:| -3696 (-646 *9)) (|:| -1717 *10) (|:| |ineq| (-646 *9)))))
+   (-5 *1 (-1113 *6 *7 *8 *9 *10)) (-5 *3 (-646 *9))))) 
 (((*1 *2 *2)
-  (-12 (-5 *2 (-644 (-2 (|:| |val| (-644 *6)) (|:| -1710 *7))))
-   (-4 *6 (-1069 *3 *4 *5)) (-4 *7 (-1075 *3 *4 *5 *6)) (-4 *3 (-456))
-   (-4 *4 (-796)) (-4 *5 (-853)) (-5 *1 (-992 *3 *4 *5 *6 *7))))
+  (-12 (-5 *2 (-646 (-2 (|:| |val| (-646 *6)) (|:| -1717 *7))))
+   (-4 *6 (-1071 *3 *4 *5)) (-4 *7 (-1077 *3 *4 *5 *6)) (-4 *3 (-457))
+   (-4 *4 (-798)) (-4 *5 (-855)) (-5 *1 (-994 *3 *4 *5 *6 *7))))
  ((*1 *2 *2)
-  (-12 (-5 *2 (-644 (-2 (|:| |val| (-644 *6)) (|:| -1710 *7))))
-   (-4 *6 (-1069 *3 *4 *5)) (-4 *7 (-1075 *3 *4 *5 *6)) (-4 *3 (-456))
-   (-4 *4 (-796)) (-4 *5 (-853)) (-5 *1 (-1111 *3 *4 *5 *6 *7))))) 
+  (-12 (-5 *2 (-646 (-2 (|:| |val| (-646 *6)) (|:| -1717 *7))))
+   (-4 *6 (-1071 *3 *4 *5)) (-4 *7 (-1077 *3 *4 *5 *6)) (-4 *3 (-457))
+   (-4 *4 (-798)) (-4 *5 (-855)) (-5 *1 (-1113 *3 *4 *5 *6 *7))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *3 (-2 (|:| |val| (-644 *7)) (|:| -1710 *8)))
-   (-4 *7 (-1069 *4 *5 *6)) (-4 *8 (-1075 *4 *5 *6 *7)) (-4 *4 (-456))
-   (-4 *5 (-796)) (-4 *6 (-853)) (-5 *2 (-112)) (-5 *1 (-992 *4 *5 *6 *7 *8))))
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-   (-4 *5 (-796)) (-4 *6 (-853)) (-5 *2 (-112))
-   (-5 *1 (-1111 *4 *5 *6 *7 *8))))) 
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- ((*1 *2 *2)
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-   (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-1069 *3 *4 *5))
-   (-5 *1 (-1111 *3 *4 *5 *6 *7))))) 
+  (-12 (-5 *3 (-2 (|:| |val| (-646 *7)) (|:| -1717 *8)))
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+(((*1 *2 *2)
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+ ((*1 *2 *2)
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+   (-4 *4 (-798)) (-4 *5 (-855)) (-4 *6 (-1071 *3 *4 *5))
+   (-5 *1 (-1113 *3 *4 *5 *6 *7))))) 
 (((*1 *2 *3 *3)
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-   (-5 *1 (-992 *5 *6 *7 *8 *3))))
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 (((*1 *2 *3 *3)
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-   (-4 *3 (-1075 *4 *5 *6 *7))))
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-   (-4 *3 (-1075 *4 *5 *6 *7))))) 
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-   (-4 *8 (-1075 *4 *5 *6 *7))))
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-   (-4 *8 (-1075 *4 *5 *6 *7))))) 
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-   (-4 *8 (-1075 *4 *5 *6 *7))))
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-   (-4 *8 (-1075 *4 *5 *6 *7))))) 
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 (((*1 *2 *3 *3)
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-   (-4 *8 (-1075 *4 *5 *6 *7))))
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-   (-4 *8 (-1075 *4 *5 *6 *7))))) 
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 (((*1 *2 *3 *3)
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 (((*1 *2 *3 *3)
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 (((*1 *2 *2)
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-   (-5 *1 (-992 *3 *4 *5 *6 *7))))
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-   (-5 *1 (-1111 *3 *4 *5 *6 *7))))) 
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 (((*1 *2 *3 *3)
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 (((*1 *2)
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 (((*1 *2 *3 *3 *3)
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-   (-4 *8 (-1075 *4 *5 *6 *7))))
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 (((*1 *2 *1)
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 (((*1 *2 *1)
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 (((*1 *2 *3 *3 *4)
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 (((*1 *2 *3 *3 *4)
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 (((*1 *2 *1)
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 (((*1 *2 *3 *1)
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 (((*1 *2 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *3 *4)
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 (((*1 *2 *3)
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 (((*1 *2 *1 *1) (-12 (-4 *1 (-102)) (-5 *2 (-112))))
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 (((*1 *1 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *3 *3 *3)
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 (((*1 *2 *3 *3 *3 *4 *5)
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-   (-4 *6 (-13 (-366) (-147) (-1042 *4))) (-5 *4 (-550))
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    (-5 *2
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+     (|:| -3696
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            (|:| |beta| *3)))))
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 (((*1 *2 *3 *3)
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 (((*1 *2 *3 *4 *4 *4 *5 *6 *7)
-  (|partial| -12 (-5 *5 (-1181))
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     (-1
      (-3
       (-2 (|:| |mainpart| *4)
-       (|:| |limitedlogs| (-644 (-2 (|:| |coeff| *4) (|:| |logand| *4)))))
+       (|:| |limitedlogs| (-646 (-2 (|:| |coeff| *4) (|:| |logand| *4)))))
       "failed")
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 (((*1 *2 *3 *4 *4 *5 *6 *7)
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-       (|:| |limitedlogs| (-644 (-2 (|:| |coeff| *4) (|:| |logand| *4)))))
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       "failed")
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 (((*1 *1 *1)
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 (((*1 *2 *3 *4 *5 *6 *7)
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             (|:| |bsoln|
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       (|:| |bsoln|
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 (((*1 *2 *3 *4)
   (-12
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       (|:| |bsoln|
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+(((*1 *2 *3 *4)
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       (|:| |bsoln|
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 (((*1 *2 *3 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *2 *3)
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 (((*1 *2 *2 *3)
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-  (-12 (-5 *2 (-644 *7)) (-5 *3 (-1163)) (-4 *7 (-954 *4 *5 *6)) (-4 *4 (-456))
-   (-4 *5 (-796)) (-4 *6 (-853)) (-5 *1 (-451 *4 *5 *6 *7))))
+  (-12 (-5 *2 (-646 *7)) (-5 *3 (-1165)) (-4 *7 (-956 *4 *5 *6)) (-4 *4 (-457))
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  ((*1 *1 *1)
-  (-12 (-4 *2 (-366)) (-4 *3 (-796)) (-4 *4 (-853)) (-5 *1 (-508 *2 *3 *4 *5))
-   (-4 *5 (-954 *2 *3 *4))))
+  (-12 (-4 *2 (-367)) (-4 *3 (-798)) (-4 *4 (-855)) (-5 *1 (-509 *2 *3 *4 *5))
+   (-4 *5 (-956 *2 *3 *4))))
  ((*1 *2 *2)
-  (-12 (-5 *2 (-644 (-783 *3 (-867 *4)))) (-4 *3 (-456))
-   (-14 *4 (-644 (-1181))) (-5 *1 (-631 *3 *4))))) 
+  (-12 (-5 *2 (-646 (-785 *3 (-869 *4)))) (-4 *3 (-457))
+   (-14 *4 (-646 (-1183))) (-5 *1 (-633 *3 *4))))) 
 (((*1 *2 *2)
-  (|partial| -12 (-5 *2 (-644 (-950 *3))) (-4 *3 (-456)) (-5 *1 (-363 *3 *4))
-   (-14 *4 (-644 (-1181)))))
+  (|partial| -12 (-5 *2 (-646 (-952 *3))) (-4 *3 (-457)) (-5 *1 (-364 *3 *4))
+   (-14 *4 (-646 (-1183)))))
  ((*1 *2 *2)
-  (|partial| -12 (-5 *2 (-644 (-783 *3 (-867 *4)))) (-4 *3 (-456))
-   (-14 *4 (-644 (-1181))) (-5 *1 (-631 *3 *4))))) 
+  (|partial| -12 (-5 *2 (-646 (-785 *3 (-869 *4)))) (-4 *3 (-457))
+   (-14 *4 (-646 (-1183))) (-5 *1 (-633 *3 *4))))) 
 (((*1 *2 *3)
-  (-12 (-5 *3 (-644 (-950 *4))) (-4 *4 (-456)) (-5 *2 (-112))
-   (-5 *1 (-363 *4 *5)) (-14 *5 (-644 (-1181)))))
+  (-12 (-5 *3 (-646 (-952 *4))) (-4 *4 (-457)) (-5 *2 (-112))
+   (-5 *1 (-364 *4 *5)) (-14 *5 (-646 (-1183)))))
  ((*1 *2 *3)
-  (-12 (-5 *3 (-644 (-783 *4 (-867 *5)))) (-4 *4 (-456))
-   (-14 *5 (-644 (-1181))) (-5 *2 (-112)) (-5 *1 (-631 *4 *5))))) 
+  (-12 (-5 *3 (-646 (-785 *4 (-869 *5)))) (-4 *4 (-457))
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 (((*1 *2 *3)
-  (-12 (-5 *3 (-644 *4)) (-4 *4 (-853)) (-5 *2 (-644 (-667 *4 *5)))
-   (-5 *1 (-630 *4 *5 *6)) (-4 *5 (-13 (-173) (-720 (-411 (-550)))))
-   (-14 *6 (-923))))) 
+  (-12 (-5 *3 (-646 *4)) (-4 *4 (-855)) (-5 *2 (-646 (-669 *4 *5)))
+   (-5 *1 (-632 *4 *5 *6)) (-4 *5 (-13 (-173) (-722 (-412 (-551)))))
+   (-14 *6 (-925))))) 
 (((*1 *2 *1)
-  (-12 (-5 *2 (-644 (-2 (|:| |k| (-675 *3)) (|:| |c| *4))))
-   (-5 *1 (-630 *3 *4 *5)) (-4 *3 (-853))
-   (-4 *4 (-13 (-173) (-720 (-411 (-550))))) (-14 *5 (-923))))) 
+  (-12 (-5 *2 (-646 (-2 (|:| |k| (-677 *3)) (|:| |c| *4))))
+   (-5 *1 (-632 *3 *4 *5)) (-4 *3 (-855))
+   (-4 *4 (-13 (-173) (-722 (-412 (-551))))) (-14 *5 (-925))))) 
 (((*1 *2 *1 *1)
-  (-12 (-5 *2 (-644 (-295 *4))) (-5 *1 (-630 *3 *4 *5)) (-4 *3 (-853))
-   (-4 *4 (-13 (-173) (-720 (-411 (-550))))) (-14 *5 (-923))))) 
+  (-12 (-5 *2 (-646 (-296 *4))) (-5 *1 (-632 *3 *4 *5)) (-4 *3 (-855))
+   (-4 *4 (-13 (-173) (-722 (-412 (-551))))) (-14 *5 (-925))))) 
 (((*1 *2 *3 *4 *5 *6 *7 *6)
   (|partial| -12
    (-5 *5
     (-2 (|:| |contp| *3)
-     (|:| -1956 (-644 (-2 (|:| |irr| *10) (|:| -2560 (-550)))))))
-   (-5 *6 (-644 *3)) (-5 *7 (-644 *8)) (-4 *8 (-853)) (-4 *3 (-309))
-   (-4 *10 (-954 *3 *9 *8)) (-4 *9 (-796))
+     (|:| -1963 (-646 (-2 (|:| |irr| *10) (|:| -2567 (-551)))))))
+   (-5 *6 (-646 *3)) (-5 *7 (-646 *8)) (-4 *8 (-855)) (-4 *3 (-310))
+   (-4 *10 (-956 *3 *9 *8)) (-4 *9 (-798))
    (-5 *2
-    (-2 (|:| |polfac| (-644 *10)) (|:| |correct| *3)
-     (|:| |corrfact| (-644 (-1175 *3)))))
-   (-5 *1 (-628 *8 *9 *3 *10)) (-5 *4 (-644 (-1175 *3)))))) 
+    (-2 (|:| |polfac| (-646 *10)) (|:| |correct| *3)
+     (|:| |corrfact| (-646 (-1177 *3)))))
+   (-5 *1 (-630 *8 *9 *3 *10)) (-5 *4 (-646 (-1177 *3)))))) 
 (((*1 *2 *3 *4 *5)
-  (-12 (-5 *4 (-774)) (-5 *5 (-644 *3)) (-4 *3 (-309)) (-4 *6 (-853))
-   (-4 *7 (-796)) (-5 *2 (-112)) (-5 *1 (-628 *6 *7 *3 *8))
-   (-4 *8 (-954 *3 *7 *6))))) 
-(((*1 *2 *2)
-  (-12 (-4 *3 (-456)) (-4 *4 (-796)) (-4 *5 (-853)) (-4 *6 (-1069 *3 *4 *5))
-   (-5 *1 (-627 *3 *4 *5 *6 *7 *2)) (-4 *7 (-1075 *3 *4 *5 *6))
-   (-4 *2 (-1113 *3 *4 *5 *6))))) 
-(((*1 *2 *1) (-12 (-4 *2 (-561)) (-5 *1 (-626 *2 *3)) (-4 *3 (-1246 *2))))) 
+  (-12 (-5 *4 (-776)) (-5 *5 (-646 *3)) (-4 *3 (-310)) (-4 *6 (-855))
+   (-4 *7 (-798)) (-5 *2 (-112)) (-5 *1 (-630 *6 *7 *3 *8))
+   (-4 *8 (-956 *3 *7 *6))))) 
+(((*1 *2 *2)
+  (-12 (-4 *3 (-457)) (-4 *4 (-798)) (-4 *5 (-855)) (-4 *6 (-1071 *3 *4 *5))
+   (-5 *1 (-629 *3 *4 *5 *6 *7 *2)) (-4 *7 (-1077 *3 *4 *5 *6))
+   (-4 *2 (-1115 *3 *4 *5 *6))))) 
+(((*1 *2 *1) (-12 (-4 *2 (-562)) (-5 *1 (-628 *2 *3)) (-4 *3 (-1248 *2))))) 
 (((*1 *2 *2 *3 *3)
-  (-12 (-5 *3 (-1181)) (-4 *4 (-13 (-309) (-147) (-1042 (-550)) (-642 (-550))))
-   (-5 *1 (-625 *4 *2)) (-4 *2 (-13 (-1206) (-964) (-29 *4)))))) 
-(((*1 *1) (-5 *1 (-620)))) 
+  (-12 (-5 *3 (-1183)) (-4 *4 (-13 (-310) (-147) (-1044 (-551)) (-644 (-551))))
+   (-5 *1 (-627 *4 *2)) (-4 *2 (-13 (-1208) (-966) (-29 *4)))))) 
+(((*1 *1) (-5 *1 (-622)))) 
 (((*1 *2 *3 *3 *3)
   (|partial| -12
-   (-4 *4 (-13 (-147) (-27) (-1042 (-550)) (-1042 (-411 (-550)))))
-   (-4 *5 (-1246 *4)) (-5 *2 (-1175 (-411 *5))) (-5 *1 (-618 *4 *5))
-   (-5 *3 (-411 *5))))
+   (-4 *4 (-13 (-147) (-27) (-1044 (-551)) (-1044 (-412 (-551)))))
+   (-4 *5 (-1248 *4)) (-5 *2 (-1177 (-412 *5))) (-5 *1 (-620 *4 *5))
+   (-5 *3 (-412 *5))))
  ((*1 *2 *3 *3 *3 *4)
-  (|partial| -12 (-5 *4 (-1 (-409 *6) *6)) (-4 *6 (-1246 *5))
-   (-4 *5 (-13 (-147) (-27) (-1042 (-550)) (-1042 (-411 (-550)))))
-   (-5 *2 (-1175 (-411 *6))) (-5 *1 (-618 *5 *6)) (-5 *3 (-411 *6))))) 
-(((*1 *2 *3)
-  (|partial| -12 (-5 *3 (-614 *4)) (-4 *4 (-1105)) (-4 *2 (-1105))
-   (-5 *1 (-615 *2 *4))))) 
-(((*1 *2 *3)
-  (-12 (-5 *2 (-614 *4)) (-5 *1 (-615 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105))))) 
-(((*1 *2 *1) (-12 (-4 *1 (-166 *2)) (-4 *2 (-173)) (-4 *2 (-1206))))
- ((*1 *2 *1) (-12 (-5 *1 (-333 *2)) (-4 *2 (-853))))
- ((*1 *2 *1) (-12 (-5 *2 (-644 *3)) (-5 *1 (-614 *3)) (-4 *3 (-1105))))) 
-(((*1 *1 *2 *3) (-12 (-5 *2 (-113)) (-5 *3 (-644 *1)) (-4 *1 (-300))))
- ((*1 *1 *2 *1) (-12 (-4 *1 (-300)) (-5 *2 (-113))))
- ((*1 *1 *2) (-12 (-5 *2 (-1181)) (-5 *1 (-614 *3)) (-4 *3 (-1105))))
+  (|partial| -12 (-5 *4 (-1 (-410 *6) *6)) (-4 *6 (-1248 *5))
+   (-4 *5 (-13 (-147) (-27) (-1044 (-551)) (-1044 (-412 (-551)))))
+   (-5 *2 (-1177 (-412 *6))) (-5 *1 (-620 *5 *6)) (-5 *3 (-412 *6))))) 
+(((*1 *2 *3)
+  (|partial| -12 (-5 *3 (-616 *4)) (-4 *4 (-1107)) (-4 *2 (-1107))
+   (-5 *1 (-617 *2 *4))))) 
+(((*1 *2 *3)
+  (-12 (-5 *2 (-616 *4)) (-5 *1 (-617 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))))) 
+(((*1 *2 *1) (-12 (-4 *1 (-166 *2)) (-4 *2 (-173)) (-4 *2 (-1208))))
+ ((*1 *2 *1) (-12 (-5 *1 (-334 *2)) (-4 *2 (-855))))
+ ((*1 *2 *1) (-12 (-5 *2 (-646 *3)) (-5 *1 (-616 *3)) (-4 *3 (-1107))))) 
+(((*1 *1 *2 *3) (-12 (-5 *2 (-113)) (-5 *3 (-646 *1)) (-4 *1 (-301))))
+ ((*1 *1 *2 *1) (-12 (-4 *1 (-301)) (-5 *2 (-113))))
+ ((*1 *1 *2) (-12 (-5 *2 (-1183)) (-5 *1 (-616 *3)) (-4 *3 (-1107))))
  ((*1 *1 *2 *3 *4)
-  (-12 (-5 *2 (-113)) (-5 *3 (-644 *5)) (-5 *4 (-774)) (-4 *5 (-1105))
-   (-5 *1 (-614 *5))))) 
-(((*1 *2 *1) (|partial| -12 (-5 *2 (-1181)) (-5 *1 (-614 *3)) (-4 *3 (-1105))))) 
+  (-12 (-5 *2 (-113)) (-5 *3 (-646 *5)) (-5 *4 (-776)) (-4 *5 (-1107))
+   (-5 *1 (-616 *5))))) 
+(((*1 *2 *1) (|partial| -12 (-5 *2 (-1183)) (-5 *1 (-616 *3)) (-4 *3 (-1107))))) 
 (((*1 *2 *3 *1)
-  (-12 (-4 *1 (-613 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1105)) (-5 *2 (-112))))) 
+  (-12 (-4 *1 (-615 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107)) (-5 *2 (-112))))) 
 (((*1 *2 *3 *1)
-  (|partial| -12 (-4 *1 (-613 *3 *2)) (-4 *3 (-1105)) (-4 *2 (-1105))))) 
-(((*1 *2 *1) (-12 (-5 *2 (-962 (-185 (-139)))) (-5 *1 (-335))))
- ((*1 *2 *1) (-12 (-5 *2 (-644 (-1221))) (-5 *1 (-609))))) 
+  (|partial| -12 (-4 *1 (-615 *3 *2)) (-4 *3 (-1107)) (-4 *2 (-1107))))) 
+(((*1 *2 *1) (-12 (-5 *2 (-964 (-185 (-139)))) (-5 *1 (-336))))
+ ((*1 *2 *1) (-12 (-5 *2 (-646 (-1223))) (-5 *1 (-611))))) 
 (((*1 *2 *1)
   (-12
    (-5 *2
-    (-644
+    (-646
      (-2
-      (|:| -4294
-           (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-            (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+      (|:| -4301
+           (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+            (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
             (|:| |relerr| (-226))))
-      (|:| -2256
+      (|:| -2263
            (-2
             (|:| |endPointContinuity|
                  (-3 (|:| |continuous| "Continuous at the end points")
@@ -11795,499 +11795,499 @@
                   (|:| |notEvaluated|
                        "End point continuity not yet evaluated")))
             (|:| |singularitiesStream|
-                 (-3 (|:| |str| (-1158 (-226)))
+                 (-3 (|:| |str| (-1160 (-226)))
                   (|:| |notEvaluated|
                        "Internal singularities not yet evaluated")))
-            (|:| -1609
+            (|:| -1612
                  (-3 (|:| |finite| "The range is finite")
                   (|:| |lowerInfinite| "The bottom of range is infinite")
                   (|:| |upperInfinite| "The top of range is infinite")
                   (|:| |bothInfinite|
                        "Both top and bottom points are infinite")
                   (|:| |notEvaluated| "Range not yet evaluated"))))))))
-   (-5 *1 (-564))))
+   (-5 *1 (-565))))
  ((*1 *2 *1)
-  (-12 (-4 *1 (-607 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1220)) (-5 *2 (-644 *4))))) 
+  (-12 (-4 *1 (-609 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1222)) (-5 *2 (-646 *4))))) 
 (((*1 *2 *3 *1)
-  (-12 (-4 *1 (-607 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1220)) (-5 *2 (-112))))) 
+  (-12 (-4 *1 (-609 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1222)) (-5 *2 (-112))))) 
 (((*1 *2 *1)
-  (-12 (-4 *1 (-607 *3 *4)) (-4 *3 (-1105)) (-4 *4 (-1220)) (-5 *2 (-644 *3))))) 
+  (-12 (-4 *1 (-609 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1222)) (-5 *2 (-646 *3))))) 
 (((*1 *2 *3 *1)
-  (-12 (|has| *1 (-6 -4427)) (-4 *1 (-607 *4 *3)) (-4 *4 (-1105))
-   (-4 *3 (-1220)) (-4 *3 (-1105)) (-5 *2 (-112))))) 
+  (-12 (|has| *1 (-6 -4434)) (-4 *1 (-609 *4 *3)) (-4 *4 (-1107))
+   (-4 *3 (-1222)) (-4 *3 (-1107)) (-5 *2 (-112))))) 
 (((*1 *2 *1)
-  (-12 (-4 *1 (-607 *2 *3)) (-4 *3 (-1220)) (-4 *2 (-1105)) (-4 *2 (-853))))) 
+  (-12 (-4 *1 (-609 *2 *3)) (-4 *3 (-1222)) (-4 *2 (-1107)) (-4 *2 (-855))))) 
 (((*1 *2 *1)
-  (-12 (-4 *1 (-607 *2 *3)) (-4 *3 (-1220)) (-4 *2 (-1105)) (-4 *2 (-853))))) 
+  (-12 (-4 *1 (-609 *2 *3)) (-4 *3 (-1222)) (-4 *2 (-1107)) (-4 *2 (-855))))) 
 (((*1 *1 *1 *2)
-  (-12 (-4 *1 (-57 *2 *3 *4)) (-4 *2 (-1220)) (-4 *3 (-375 *2))
-   (-4 *4 (-375 *2))))
+  (-12 (-4 *1 (-57 *2 *3 *4)) (-4 *2 (-1222)) (-4 *3 (-376 *2))
+   (-4 *4 (-376 *2))))
  ((*1 *1 *1 *2)
-  (-12 (|has| *1 (-6 -4428)) (-4 *1 (-607 *3 *2)) (-4 *3 (-1105))
-   (-4 *2 (-1220))))) 
+  (-12 (|has| *1 (-6 -4435)) (-4 *1 (-609 *3 *2)) (-4 *3 (-1107))
+   (-4 *2 (-1222))))) 
 (((*1 *2 *1 *3 *3)
-  (-12 (|has| *1 (-6 -4428)) (-4 *1 (-607 *3 *4)) (-4 *3 (-1105))
-   (-4 *4 (-1220)) (-5 *2 (-1276))))) 
+  (-12 (|has| *1 (-6 -4435)) (-4 *1 (-609 *3 *4)) (-4 *3 (-1107))
+   (-4 *4 (-1222)) (-5 *2 (-1278))))) 
 (((*1 *2 *2 *3 *4)
-  (-12 (-5 *3 (-644 (-614 *2))) (-5 *4 (-644 (-1181)))
-   (-4 *2 (-13 (-425 (-169 *5)) (-1006) (-1206))) (-4 *5 (-561))
-   (-5 *1 (-603 *5 *6 *2)) (-4 *6 (-13 (-425 *5) (-1006) (-1206)))))) 
+  (-12 (-5 *3 (-646 (-616 *2))) (-5 *4 (-646 (-1183)))
+   (-4 *2 (-13 (-426 (-169 *5)) (-1008) (-1208))) (-4 *5 (-562))
+   (-5 *1 (-605 *5 *6 *2)) (-4 *6 (-13 (-426 *5) (-1008) (-1208)))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-561)) (-5 *2 (-169 *5)) (-5 *1 (-603 *4 *5 *3))
-   (-4 *5 (-13 (-425 *4) (-1006) (-1206)))
-   (-4 *3 (-13 (-425 (-169 *4)) (-1006) (-1206)))))) 
+  (-12 (-4 *4 (-562)) (-5 *2 (-169 *5)) (-5 *1 (-605 *4 *5 *3))
+   (-4 *5 (-13 (-426 *4) (-1008) (-1208)))
+   (-4 *3 (-13 (-426 (-169 *4)) (-1008) (-1208)))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-561)) (-4 *2 (-13 (-425 (-169 *4)) (-1006) (-1206)))
-   (-5 *1 (-603 *4 *3 *2)) (-4 *3 (-13 (-425 *4) (-1006) (-1206)))))) 
+  (-12 (-4 *4 (-562)) (-4 *2 (-13 (-426 (-169 *4)) (-1008) (-1208)))
+   (-5 *1 (-605 *4 *3 *2)) (-4 *3 (-13 (-426 *4) (-1008) (-1208)))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-561)) (-4 *2 (-13 (-425 *4) (-1006) (-1206)))
-   (-5 *1 (-603 *4 *2 *3)) (-4 *3 (-13 (-425 (-169 *4)) (-1006) (-1206)))))) 
+  (-12 (-4 *4 (-562)) (-4 *2 (-13 (-426 *4) (-1008) (-1208)))
+   (-5 *1 (-605 *4 *2 *3)) (-4 *3 (-13 (-426 (-169 *4)) (-1008) (-1208)))))) 
 (((*1 *2 *3)
-  (-12 (-5 *3 (-169 *5)) (-4 *5 (-13 (-425 *4) (-1006) (-1206))) (-4 *4 (-561))
-   (-4 *2 (-13 (-425 (-169 *4)) (-1006) (-1206))) (-5 *1 (-603 *4 *5 *2))))) 
+  (-12 (-5 *3 (-169 *5)) (-4 *5 (-13 (-426 *4) (-1008) (-1208))) (-4 *4 (-562))
+   (-4 *2 (-13 (-426 (-169 *4)) (-1008) (-1208))) (-5 *1 (-605 *4 *5 *2))))) 
 (((*1 *1 *2 *3)
-  (-12 (-5 *2 (-1030 (-845 (-550))))
-   (-5 *3 (-1158 (-2 (|:| |k| (-550)) (|:| |c| *4)))) (-4 *4 (-1053))
-   (-5 *1 (-599 *4))))) 
+  (-12 (-5 *2 (-1032 (-847 (-551))))
+   (-5 *3 (-1160 (-2 (|:| |k| (-551)) (|:| |c| *4)))) (-4 *4 (-1055))
+   (-5 *1 (-600 *4))))) 
 (((*1 *2 *1)
-  (-12 (-5 *2 (-1030 (-845 (-550)))) (-5 *1 (-599 *3)) (-4 *3 (-1053))))) 
+  (-12 (-5 *2 (-1032 (-847 (-551)))) (-5 *1 (-600 *3)) (-4 *3 (-1055))))) 
 (((*1 *2 *1)
-  (-12 (-5 *2 (-1158 (-2 (|:| |k| (-550)) (|:| |c| *3)))) (-5 *1 (-599 *3))
-   (-4 *3 (-1053))))) 
+  (-12 (-5 *2 (-1160 (-2 (|:| |k| (-551)) (|:| |c| *3)))) (-5 *1 (-600 *3))
+   (-4 *3 (-1055))))) 
 (((*1 *1 *1 *1 *2)
-  (|partial| -12 (-5 *2 (-112)) (-5 *1 (-599 *3)) (-4 *3 (-1053))))) 
-(((*1 *1 *1) (-12 (-5 *1 (-599 *2)) (-4 *2 (-1053))))) 
-(((*1 *1 *1 *1) (-12 (-5 *1 (-599 *2)) (-4 *2 (-1053))))) 
+  (|partial| -12 (-5 *2 (-112)) (-5 *1 (-600 *3)) (-4 *3 (-1055))))) 
+(((*1 *1 *1) (-12 (-5 *1 (-600 *2)) (-4 *2 (-1055))))) 
+(((*1 *1 *1 *1) (-12 (-5 *1 (-600 *2)) (-4 *2 (-1055))))) 
 (((*1 *2 *3 *4 *5 *6 *7)
-  (-12 (-5 *3 (-1158 (-2 (|:| |k| (-550)) (|:| |c| *6))))
-   (-5 *4 (-1030 (-845 (-550)))) (-5 *5 (-1181)) (-5 *7 (-411 (-550)))
-   (-4 *6 (-1053)) (-5 *2 (-866)) (-5 *1 (-599 *6))))) 
+  (-12 (-5 *3 (-1160 (-2 (|:| |k| (-551)) (|:| |c| *6))))
+   (-5 *4 (-1032 (-847 (-551)))) (-5 *5 (-1183)) (-5 *7 (-412 (-551)))
+   (-4 *6 (-1055)) (-5 *2 (-868)) (-5 *1 (-600 *6))))) 
 (((*1 *1 *1 *2)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1 *2)
-  (-12 (-5 *2 (-411 (-550))) (-5 *1 (-599 *3)) (-4 *3 (-38 *2))
-   (-4 *3 (-1053))))) 
+  (-12 (-5 *2 (-412 (-551))) (-5 *1 (-600 *3)) (-4 *3 (-38 *2))
+   (-4 *3 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
 (((*1 *1 *1)
-  (-12 (-5 *1 (-599 *2)) (-4 *2 (-38 (-411 (-550)))) (-4 *2 (-1053))))) 
+  (-12 (-5 *1 (-600 *2)) (-4 *2 (-38 (-412 (-551)))) (-4 *2 (-1055))))) 
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 (((*1 *2 *3 *4 *4 *5 *3 *6)
-  (|partial| -12 (-5 *4 (-614 *3)) (-5 *5 (-644 *3)) (-5 *6 (-1175 *3))
-   (-4 *3 (-13 (-425 *7) (-27) (-1206)))
-   (-4 *7 (-13 (-456) (-1042 (-550)) (-147) (-642 (-550))))
+  (|partial| -12 (-5 *4 (-616 *3)) (-5 *5 (-646 *3)) (-5 *6 (-1177 *3))
+   (-4 *3 (-13 (-426 *7) (-27) (-1208)))
+   (-4 *7 (-13 (-457) (-1044 (-551)) (-147) (-644 (-551))))
    (-5 *2
     (-2 (|:| |mainpart| *3)
-     (|:| |limitedlogs| (-644 (-2 (|:| |coeff| *3) (|:| |logand| *3))))))
-   (-5 *1 (-565 *7 *3 *8)) (-4 *8 (-1105))))
+     (|:| |limitedlogs| (-646 (-2 (|:| |coeff| *3) (|:| |logand| *3))))))
+   (-5 *1 (-566 *7 *3 *8)) (-4 *8 (-1107))))
  ((*1 *2 *3 *4 *4 *5 *4 *3 *6)
-  (|partial| -12 (-5 *4 (-614 *3)) (-5 *5 (-644 *3)) (-5 *6 (-411 (-1175 *3)))
-   (-4 *3 (-13 (-425 *7) (-27) (-1206)))
-   (-4 *7 (-13 (-456) (-1042 (-550)) (-147) (-642 (-550))))
+  (|partial| -12 (-5 *4 (-616 *3)) (-5 *5 (-646 *3)) (-5 *6 (-412 (-1177 *3)))
+   (-4 *3 (-13 (-426 *7) (-27) (-1208)))
+   (-4 *7 (-13 (-457) (-1044 (-551)) (-147) (-644 (-551))))
    (-5 *2
     (-2 (|:| |mainpart| *3)
-     (|:| |limitedlogs| (-644 (-2 (|:| |coeff| *3) (|:| |logand| *3))))))
-   (-5 *1 (-565 *7 *3 *8)) (-4 *8 (-1105))))) 
+     (|:| |limitedlogs| (-646 (-2 (|:| |coeff| *3) (|:| |logand| *3))))))
+   (-5 *1 (-566 *7 *3 *8)) (-4 *8 (-1107))))) 
 (((*1 *2 *3 *4 *4 *3 *3 *5)
-  (|partial| -12 (-5 *4 (-614 *3)) (-5 *5 (-1175 *3))
-   (-4 *3 (-13 (-425 *6) (-27) (-1206)))
-   (-4 *6 (-13 (-456) (-1042 (-550)) (-147) (-642 (-550))))
-   (-5 *2 (-2 (|:| -2320 *3) (|:| |coeff| *3))) (-5 *1 (-565 *6 *3 *7))
-   (-4 *7 (-1105))))
+  (|partial| -12 (-5 *4 (-616 *3)) (-5 *5 (-1177 *3))
+   (-4 *3 (-13 (-426 *6) (-27) (-1208)))
+   (-4 *6 (-13 (-457) (-1044 (-551)) (-147) (-644 (-551))))
+   (-5 *2 (-2 (|:| -2327 *3) (|:| |coeff| *3))) (-5 *1 (-566 *6 *3 *7))
+   (-4 *7 (-1107))))
  ((*1 *2 *3 *4 *4 *3 *4 *3 *5)
-  (|partial| -12 (-5 *4 (-614 *3)) (-5 *5 (-411 (-1175 *3)))
-   (-4 *3 (-13 (-425 *6) (-27) (-1206)))
-   (-4 *6 (-13 (-456) (-1042 (-550)) (-147) (-642 (-550))))
-   (-5 *2 (-2 (|:| -2320 *3) (|:| |coeff| *3))) (-5 *1 (-565 *6 *3 *7))
-   (-4 *7 (-1105))))) 
+  (|partial| -12 (-5 *4 (-616 *3)) (-5 *5 (-412 (-1177 *3)))
+   (-4 *3 (-13 (-426 *6) (-27) (-1208)))
+   (-4 *6 (-13 (-457) (-1044 (-551)) (-147) (-644 (-551))))
+   (-5 *2 (-2 (|:| -2327 *3) (|:| |coeff| *3))) (-5 *1 (-566 *6 *3 *7))
+   (-4 *7 (-1107))))) 
 (((*1 *2 *3 *4 *4 *3 *5)
-  (-12 (-5 *4 (-614 *3)) (-5 *5 (-1175 *3))
-   (-4 *3 (-13 (-425 *6) (-27) (-1206)))
-   (-4 *6 (-13 (-456) (-1042 (-550)) (-147) (-642 (-550)))) (-5 *2 (-587 *3))
-   (-5 *1 (-565 *6 *3 *7)) (-4 *7 (-1105))))
+  (-12 (-5 *4 (-616 *3)) (-5 *5 (-1177 *3))
+   (-4 *3 (-13 (-426 *6) (-27) (-1208)))
+   (-4 *6 (-13 (-457) (-1044 (-551)) (-147) (-644 (-551)))) (-5 *2 (-588 *3))
+   (-5 *1 (-566 *6 *3 *7)) (-4 *7 (-1107))))
  ((*1 *2 *3 *4 *4 *4 *3 *5)
-  (-12 (-5 *4 (-614 *3)) (-5 *5 (-411 (-1175 *3)))
-   (-4 *3 (-13 (-425 *6) (-27) (-1206)))
-   (-4 *6 (-13 (-456) (-1042 (-550)) (-147) (-642 (-550)))) (-5 *2 (-587 *3))
-   (-5 *1 (-565 *6 *3 *7)) (-4 *7 (-1105))))) 
+  (-12 (-5 *4 (-616 *3)) (-5 *5 (-412 (-1177 *3)))
+   (-4 *3 (-13 (-426 *6) (-27) (-1208)))
+   (-4 *6 (-13 (-457) (-1044 (-551)) (-147) (-644 (-551)))) (-5 *2 (-588 *3))
+   (-5 *1 (-566 *6 *3 *7)) (-4 *7 (-1107))))) 
 (((*1 *2 *3)
   (-12
    (-5 *3
-    (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-     (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+    (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+     (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
      (|:| |relerr| (-226))))
    (-5 *2
     (-2
@@ -12300,20 +12300,20 @@
            (|:| |bothSingular| "There are singularities at both end points")
            (|:| |notEvaluated| "End point continuity not yet evaluated")))
      (|:| |singularitiesStream|
-          (-3 (|:| |str| (-1158 (-226)))
+          (-3 (|:| |str| (-1160 (-226)))
            (|:| |notEvaluated| "Internal singularities not yet evaluated")))
-     (|:| -1609
+     (|:| -1612
           (-3 (|:| |finite| "The range is finite")
            (|:| |lowerInfinite| "The bottom of range is infinite")
            (|:| |upperInfinite| "The top of range is infinite")
            (|:| |bothInfinite| "Both top and bottom points are infinite")
            (|:| |notEvaluated| "Range not yet evaluated")))))
-   (-5 *1 (-564))))) 
+   (-5 *1 (-565))))) 
 (((*1 *2 *3)
   (|partial| -12
    (-5 *3
-    (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-     (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+    (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+     (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
      (|:| |relerr| (-226))))
    (-5 *2
     (-2
@@ -12326,25 +12326,25 @@
            (|:| |bothSingular| "There are singularities at both end points")
            (|:| |notEvaluated| "End point continuity not yet evaluated")))
      (|:| |singularitiesStream|
-          (-3 (|:| |str| (-1158 (-226)))
+          (-3 (|:| |str| (-1160 (-226)))
            (|:| |notEvaluated| "Internal singularities not yet evaluated")))
-     (|:| -1609
+     (|:| -1612
           (-3 (|:| |finite| "The range is finite")
            (|:| |lowerInfinite| "The bottom of range is infinite")
            (|:| |upperInfinite| "The top of range is infinite")
            (|:| |bothInfinite| "Both top and bottom points are infinite")
            (|:| |notEvaluated| "Range not yet evaluated")))))
-   (-5 *1 (-564))))) 
+   (-5 *1 (-565))))) 
 (((*1 *1 *2)
   (-12
    (-5 *2
-    (-644
+    (-646
      (-2
-      (|:| -4294
-           (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-            (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+      (|:| -4301
+           (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+            (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
             (|:| |relerr| (-226))))
-      (|:| -2256
+      (|:| -2263
            (-2
             (|:| |endPointContinuity|
                  (-3 (|:| |continuous| "Continuous at the end points")
@@ -12357,1672 +12357,1672 @@
                   (|:| |notEvaluated|
                        "End point continuity not yet evaluated")))
             (|:| |singularitiesStream|
-                 (-3 (|:| |str| (-1158 (-226)))
+                 (-3 (|:| |str| (-1160 (-226)))
                   (|:| |notEvaluated|
                        "Internal singularities not yet evaluated")))
-            (|:| -1609
+            (|:| -1612
                  (-3 (|:| |finite| "The range is finite")
                   (|:| |lowerInfinite| "The bottom of range is infinite")
                   (|:| |upperInfinite| "The top of range is infinite")
                   (|:| |bothInfinite|
                        "Both top and bottom points are infinite")
                   (|:| |notEvaluated| "Range not yet evaluated"))))))))
-   (-5 *1 (-564))))) 
-(((*1 *2) (-12 (-5 *2 (-1276)) (-5 *1 (-564))))) 
-(((*1 *1) (-5 *1 (-564)))) 
-(((*1 *2 *2) (|partial| -12 (-5 *1 (-563 *2)) (-4 *2 (-549))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-409 *3)) (-5 *1 (-563 *3)) (-4 *3 (-549))))) 
+   (-5 *1 (-565))))) 
+(((*1 *2) (-12 (-5 *2 (-1278)) (-5 *1 (-565))))) 
+(((*1 *1) (-5 *1 (-565)))) 
+(((*1 *2 *2) (|partial| -12 (-5 *1 (-564 *2)) (-4 *2 (-550))))) 
+(((*1 *2 *3) (-12 (-5 *2 (-410 *3)) (-5 *1 (-564 *3)) (-4 *3 (-550))))) 
 (((*1 *2 *3 *4 *5 *6)
-  (|partial| -12 (-5 *4 (-1181)) (-5 *6 (-644 (-614 *3))) (-5 *5 (-614 *3))
-   (-4 *3 (-13 (-27) (-1206) (-425 *7)))
-   (-4 *7 (-13 (-456) (-147) (-1042 (-550)) (-642 (-550))))
-   (-5 *2 (-2 (|:| -2320 *3) (|:| |coeff| *3))) (-5 *1 (-562 *7 *3))))) 
-(((*1 *2 *3 *4)
-  (-12 (-5 *4 (-1181)) (-4 *5 (-13 (-456) (-147) (-1042 (-550)) (-642 (-550))))
-   (-5 *2 (-587 *3)) (-5 *1 (-562 *5 *3))
-   (-4 *3 (-13 (-27) (-1206) (-425 *5)))))) 
+  (|partial| -12 (-5 *4 (-1183)) (-5 *6 (-646 (-616 *3))) (-5 *5 (-616 *3))
+   (-4 *3 (-13 (-27) (-1208) (-426 *7)))
+   (-4 *7 (-13 (-457) (-147) (-1044 (-551)) (-644 (-551))))
+   (-5 *2 (-2 (|:| -2327 *3) (|:| |coeff| *3))) (-5 *1 (-563 *7 *3))))) 
+(((*1 *2 *3 *4)
+  (-12 (-5 *4 (-1183)) (-4 *5 (-13 (-457) (-147) (-1044 (-551)) (-644 (-551))))
+   (-5 *2 (-588 *3)) (-5 *1 (-563 *5 *3))
+   (-4 *3 (-13 (-27) (-1208) (-426 *5)))))) 
 (((*1 *2 *2 *3)
-  (|partial| -12 (-5 *3 (-1181))
-   (-4 *4 (-13 (-456) (-147) (-1042 (-550)) (-642 (-550))))
-   (-5 *1 (-562 *4 *2)) (-4 *2 (-13 (-27) (-1206) (-425 *4)))))) 
+  (|partial| -12 (-5 *3 (-1183))
+   (-4 *4 (-13 (-457) (-147) (-1044 (-551)) (-644 (-551))))
+   (-5 *1 (-563 *4 *2)) (-4 *2 (-13 (-27) (-1208) (-426 *4)))))) 
 (((*1 *2 *3 *4 *5)
-  (|partial| -12 (-5 *4 (-1181)) (-5 *5 (-644 *3))
-   (-4 *3 (-13 (-27) (-1206) (-425 *6)))
-   (-4 *6 (-13 (-456) (-147) (-1042 (-550)) (-642 (-550))))
+  (|partial| -12 (-5 *4 (-1183)) (-5 *5 (-646 *3))
+   (-4 *3 (-13 (-27) (-1208) (-426 *6)))
+   (-4 *6 (-13 (-457) (-147) (-1044 (-551)) (-644 (-551))))
    (-5 *2
     (-2 (|:| |mainpart| *3)
-     (|:| |limitedlogs| (-644 (-2 (|:| |coeff| *3) (|:| |logand| *3))))))
-   (-5 *1 (-562 *6 *3))))) 
+     (|:| |limitedlogs| (-646 (-2 (|:| |coeff| *3) (|:| |logand| *3))))))
+   (-5 *1 (-563 *6 *3))))) 
 (((*1 *2 *3 *4 *3)
-  (|partial| -12 (-5 *4 (-1181))
-   (-4 *5 (-13 (-456) (-147) (-1042 (-550)) (-642 (-550))))
-   (-5 *2 (-2 (|:| -2320 *3) (|:| |coeff| *3))) (-5 *1 (-562 *5 *3))
-   (-4 *3 (-13 (-27) (-1206) (-425 *5)))))) 
-(((*1 *2 *1)
-  (-12 (-5 *2 (-2 (|:| -1949 *1) (|:| -4414 *1) (|:| |associate| *1)))
-   (-4 *1 (-561))))) 
-(((*1 *1 *1) (-4 *1 (-561)))) 
-(((*1 *2 *1 *1) (-12 (-4 *1 (-561)) (-5 *2 (-112))))) 
-(((*1 *2 *1) (-12 (-4 *1 (-561)) (-5 *2 (-112))))) 
+  (|partial| -12 (-5 *4 (-1183))
+   (-4 *5 (-13 (-457) (-147) (-1044 (-551)) (-644 (-551))))
+   (-5 *2 (-2 (|:| -2327 *3) (|:| |coeff| *3))) (-5 *1 (-563 *5 *3))
+   (-4 *3 (-13 (-27) (-1208) (-426 *5)))))) 
+(((*1 *2 *1)
+  (-12 (-5 *2 (-2 (|:| -1956 *1) (|:| -4421 *1) (|:| |associate| *1)))
+   (-4 *1 (-562))))) 
+(((*1 *1 *1) (-4 *1 (-562)))) 
+(((*1 *2 *1 *1) (-12 (-4 *1 (-562)) (-5 *2 (-112))))) 
+(((*1 *2 *1) (-12 (-4 *1 (-562)) (-5 *2 (-112))))) 
 (((*1 *1 *2)
-  (-12 (-5 *2 (-411 (-550))) (-4 *1 (-559 *3)) (-4 *3 (-13 (-408) (-1206)))))
- ((*1 *1 *2) (-12 (-4 *1 (-559 *2)) (-4 *2 (-13 (-408) (-1206)))))
- ((*1 *1 *2 *2) (-12 (-4 *1 (-559 *2)) (-4 *2 (-13 (-408) (-1206)))))) 
-(((*1 *1 *2 *2) (-12 (-4 *1 (-559 *2)) (-4 *2 (-13 (-408) (-1206)))))) 
-(((*1 *2 *1) (-12 (-4 *1 (-559 *2)) (-4 *2 (-13 (-408) (-1206)))))) 
+  (-12 (-5 *2 (-412 (-551))) (-4 *1 (-560 *3)) (-4 *3 (-13 (-409) (-1208)))))
+ ((*1 *1 *2) (-12 (-4 *1 (-560 *2)) (-4 *2 (-13 (-409) (-1208)))))
+ ((*1 *1 *2 *2) (-12 (-4 *1 (-560 *2)) (-4 *2 (-13 (-409) (-1208)))))) 
+(((*1 *1 *2 *2) (-12 (-4 *1 (-560 *2)) (-4 *2 (-13 (-409) (-1208)))))) 
+(((*1 *2 *1) (-12 (-4 *1 (-560 *2)) (-4 *2 (-13 (-409) (-1208)))))) 
 (((*1 *2 *1 *3)
-  (-12 (-4 *1 (-559 *3)) (-4 *3 (-13 (-408) (-1206))) (-5 *2 (-112))))) 
-(((*1 *2 *3 *3) (-12 (-5 *3 (-550)) (-5 *2 (-112)) (-5 *1 (-558))))) 
-(((*1 *2 *2 *2) (-12 (-5 *2 (-550)) (-5 *1 (-558))))) 
-(((*1 *2 *2) (-12 (-5 *2 (-550)) (-5 *1 (-558))))) 
+  (-12 (-4 *1 (-560 *3)) (-4 *3 (-13 (-409) (-1208))) (-5 *2 (-112))))) 
+(((*1 *2 *3 *3) (-12 (-5 *3 (-551)) (-5 *2 (-112)) (-5 *1 (-559))))) 
+(((*1 *2 *2 *2) (-12 (-5 *2 (-551)) (-5 *1 (-559))))) 
+(((*1 *2 *2) (-12 (-5 *2 (-551)) (-5 *1 (-559))))) 
 (((*1 *2 *2 *3)
-  (|partial| -12 (-5 *3 (-1 *6 *6)) (-4 *6 (-1246 *5))
-   (-4 *5 (-13 (-27) (-425 *4))) (-4 *4 (-13 (-561) (-1042 (-550))))
-   (-4 *7 (-1246 (-411 *6))) (-5 *1 (-557 *4 *5 *6 *7 *2))
-   (-4 *2 (-345 *5 *6 *7))))) 
-(((*1 *2 *3 *4)
-  (-12 (-5 *4 (-1 *7 *7)) (-4 *7 (-1246 *6)) (-4 *6 (-13 (-27) (-425 *5)))
-   (-4 *5 (-13 (-561) (-1042 (-550)))) (-4 *8 (-1246 (-411 *7)))
-   (-5 *2 (-587 *3)) (-5 *1 (-557 *5 *6 *7 *8 *3)) (-4 *3 (-345 *6 *7 *8))))) 
-(((*1 *2 *3 *4)
-  (-12 (-5 *4 (-1 *7 *7)) (-4 *7 (-1246 *6)) (-4 *6 (-13 (-27) (-425 *5)))
-   (-4 *5 (-13 (-561) (-1042 (-550)))) (-4 *8 (-1246 (-411 *7)))
-   (-5 *2 (-587 *3)) (-5 *1 (-557 *5 *6 *7 *8 *3)) (-4 *3 (-345 *6 *7 *8))))) 
+  (|partial| -12 (-5 *3 (-1 *6 *6)) (-4 *6 (-1248 *5))
+   (-4 *5 (-13 (-27) (-426 *4))) (-4 *4 (-13 (-562) (-1044 (-551))))
+   (-4 *7 (-1248 (-412 *6))) (-5 *1 (-558 *4 *5 *6 *7 *2))
+   (-4 *2 (-346 *5 *6 *7))))) 
+(((*1 *2 *3 *4)
+  (-12 (-5 *4 (-1 *7 *7)) (-4 *7 (-1248 *6)) (-4 *6 (-13 (-27) (-426 *5)))
+   (-4 *5 (-13 (-562) (-1044 (-551)))) (-4 *8 (-1248 (-412 *7)))
+   (-5 *2 (-588 *3)) (-5 *1 (-558 *5 *6 *7 *8 *3)) (-4 *3 (-346 *6 *7 *8))))) 
+(((*1 *2 *3 *4)
+  (-12 (-5 *4 (-1 *7 *7)) (-4 *7 (-1248 *6)) (-4 *6 (-13 (-27) (-426 *5)))
+   (-4 *5 (-13 (-562) (-1044 (-551)))) (-4 *8 (-1248 (-412 *7)))
+   (-5 *2 (-588 *3)) (-5 *1 (-558 *5 *6 *7 *8 *3)) (-4 *3 (-346 *6 *7 *8))))) 
 (((*1 *2 *3 *4 *4 *5)
-  (-12 (-5 *4 (-614 *3)) (-5 *5 (-1 (-1175 *3) (-1175 *3)))
-   (-4 *3 (-13 (-27) (-425 *6))) (-4 *6 (-561)) (-5 *2 (-587 *3))
-   (-5 *1 (-556 *6 *3))))) 
-(((*1 *2 *1 *1) (-12 (-4 *1 (-549)) (-5 *2 (-112))))) 
-(((*1 *1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1 *1 *1) (-4 *1 (-549)))) 
-(((*1 *1 *1 *1) (-4 *1 (-549)))) 
+  (-12 (-5 *4 (-616 *3)) (-5 *5 (-1 (-1177 *3) (-1177 *3)))
+   (-4 *3 (-13 (-27) (-426 *6))) (-4 *6 (-562)) (-5 *2 (-588 *3))
+   (-5 *1 (-557 *6 *3))))) 
+(((*1 *2 *1 *1) (-12 (-4 *1 (-550)) (-5 *2 (-112))))) 
+(((*1 *1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1 *1 *1) (-4 *1 (-550)))) 
+(((*1 *1 *1 *1) (-4 *1 (-550)))) 
 (((*1 *2 *3 *2 *4)
-  (|partial| -12 (-5 *4 (-1 (-3 (-550) #1="failed") *5)) (-4 *5 (-1053))
-   (-5 *2 (-550)) (-5 *1 (-547 *5 *3)) (-4 *3 (-1246 *5))))
+  (|partial| -12 (-5 *4 (-1 (-3 (-551) #1="failed") *5)) (-4 *5 (-1055))
+   (-5 *2 (-551)) (-5 *1 (-548 *5 *3)) (-4 *3 (-1248 *5))))
  ((*1 *2 *3 *4 *2 *5)
-  (|partial| -12 (-5 *5 (-1 (-3 (-550) #1#) *4)) (-4 *4 (-1053)) (-5 *2 (-550))
-   (-5 *1 (-547 *4 *3)) (-4 *3 (-1246 *4))))
+  (|partial| -12 (-5 *5 (-1 (-3 (-551) #1#) *4)) (-4 *4 (-1055)) (-5 *2 (-551))
+   (-5 *1 (-548 *4 *3)) (-4 *3 (-1248 *4))))
  ((*1 *2 *3 *4 *5)
-  (|partial| -12 (-5 *5 (-1 (-3 (-550) #1#) *4)) (-4 *4 (-1053)) (-5 *2 (-550))
-   (-5 *1 (-547 *4 *3)) (-4 *3 (-1246 *4))))) 
-(((*1 *2 *2 *3) (-12 (-4 *3 (-309)) (-5 *1 (-459 *3 *2)) (-4 *2 (-1246 *3))))
- ((*1 *2 *2 *3) (-12 (-4 *3 (-309)) (-5 *1 (-464 *3 *2)) (-4 *2 (-1246 *3))))
+  (|partial| -12 (-5 *5 (-1 (-3 (-551) #1#) *4)) (-4 *4 (-1055)) (-5 *2 (-551))
+   (-5 *1 (-548 *4 *3)) (-4 *3 (-1248 *4))))) 
+(((*1 *2 *2 *3) (-12 (-4 *3 (-310)) (-5 *1 (-460 *3 *2)) (-4 *2 (-1248 *3))))
+ ((*1 *2 *2 *3) (-12 (-4 *3 (-310)) (-5 *1 (-465 *3 *2)) (-4 *2 (-1248 *3))))
  ((*1 *2 *2 *3)
-  (-12 (-4 *3 (-309)) (-14 *4 *3) (-14 *5 (-1 *3 *3 (-774)))
-   (-5 *1 (-543 *3 *2 *4 *5)) (-4 *2 (-1246 *3))))) 
+  (-12 (-4 *3 (-310)) (-14 *4 *3) (-14 *5 (-1 *3 *3 (-776)))
+   (-5 *1 (-544 *3 *2 *4 *5)) (-4 *2 (-1248 *3))))) 
 (((*1 *2 *3)
-  (-12 (-5 *3 (-644 *2)) (-4 *2 (-1246 *4)) (-5 *1 (-543 *4 *2 *5 *6))
-   (-4 *4 (-309)) (-14 *5 *4) (-14 *6 (-1 *4 *4 (-774)))))) 
+  (-12 (-5 *3 (-646 *2)) (-4 *2 (-1248 *4)) (-5 *1 (-544 *4 *2 *5 *6))
+   (-4 *4 (-310)) (-14 *5 *4) (-14 *6 (-1 *4 *4 (-776)))))) 
 (((*1 *2 *3)
-  (-12 (-5 *3 (-644 *2)) (-4 *2 (-1246 *4)) (-5 *1 (-543 *4 *2 *5 *6))
-   (-4 *4 (-309)) (-14 *5 *4) (-14 *6 (-1 *4 *4 (-774)))))) 
+  (-12 (-5 *3 (-646 *2)) (-4 *2 (-1248 *4)) (-5 *1 (-544 *4 *2 *5 *6))
+   (-4 *4 (-310)) (-14 *5 *4) (-14 *6 (-1 *4 *4 (-776)))))) 
 (((*1 *2 *3 *4)
-  (-12 (-5 *3 (-644 *6)) (-5 *4 (-644 (-1181))) (-4 *6 (-366))
-   (-5 *2 (-644 (-295 (-950 *6)))) (-5 *1 (-542 *5 *6 *7)) (-4 *5 (-456))
-   (-4 *7 (-13 (-366) (-851)))))) 
+  (-12 (-5 *3 (-646 *6)) (-5 *4 (-646 (-1183))) (-4 *6 (-367))
+   (-5 *2 (-646 (-296 (-952 *6)))) (-5 *1 (-543 *5 *6 *7)) (-4 *5 (-457))
+   (-4 *7 (-13 (-367) (-853)))))) 
 (((*1 *2 *3 *3 *4 *5)
-  (-12 (-5 *3 (-644 (-950 *6))) (-5 *4 (-644 (-1181))) (-4 *6 (-456))
-   (-5 *2 (-644 (-644 *7))) (-5 *1 (-542 *6 *7 *5)) (-4 *7 (-366))
-   (-4 *5 (-13 (-366) (-851)))))) 
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-(((*1 *2 *2 *2) (-12 (-5 *2 (-550)) (-5 *1 (-484))))) 
-(((*1 *2 *3 *4)
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+(((*1 *2 *3 *4)
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+   (-5 *1 (-476 *5 *6 *7)) (-5 *3 (-646 (-248 *5 *6))) (-4 *7 (-457))))) 
 (((*1 *2 *2 *3)
-  (|partial| -12 (-5 *2 (-644 (-485 *4 *5))) (-5 *3 (-644 (-867 *4)))
-   (-14 *4 (-644 (-1181))) (-4 *5 (-456)) (-5 *1 (-475 *4 *5 *6))
-   (-4 *6 (-456))))) 
-(((*1 *2 *3 *4)
-  (-12 (-5 *4 (-644 (-867 *5))) (-14 *5 (-644 (-1181))) (-4 *6 (-456))
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-   (-5 *3 (-644 (-248 *5 *6))) (-4 *7 (-456))))) 
-(((*1 *1) (-5 *1 (-472)))) 
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+   (-14 *4 (-646 (-1183))) (-4 *5 (-457)) (-5 *1 (-476 *4 *5 *6))
+   (-4 *6 (-457))))) 
+(((*1 *2 *3 *4)
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 (((*1 *1 *2 *3 *3 *4 *5)
-  (-12 (-5 *2 (-644 (-644 (-947 (-226))))) (-5 *3 (-644 (-877)))
-   (-5 *4 (-644 (-923))) (-5 *5 (-644 (-263))) (-5 *1 (-472))))
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- ((*1 *1 *1) (-5 *1 (-472)))) 
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-(((*1 *1 *2) (-12 (-5 *2 (-644 (-1093 (-381)))) (-5 *1 (-263))))
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+ ((*1 *2 *1) (-12 (-5 *2 (-646 (-1095 (-382)))) (-5 *1 (-473))))) 
 (((*1 *2 *1 *3 *4 *4 *5)
-  (-12 (-5 *3 (-947 (-226))) (-5 *4 (-877)) (-5 *5 (-923)) (-5 *2 (-1276))
-   (-5 *1 (-472))))
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 (((*1 *2 *2 *3)
-  (-12 (-5 *2 (-644 (-644 (-947 (-226))))) (-5 *3 (-644 (-877)))
-   (-5 *1 (-472))))) 
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 (((*1 *2 *3)
-  (-12 (-5 *3 (-644 (-644 (-947 (-226))))) (-5 *2 (-644 (-226)))
-   (-5 *1 (-472))))) 
+  (-12 (-5 *3 (-646 (-646 (-949 (-226))))) (-5 *2 (-646 (-226)))
+   (-5 *1 (-473))))) 
 (((*1 *1 *2) (-12 (-5 *2 (-112)) (-5 *1 (-263))))
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-(((*1 *2 *3)
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 (((*1 *2 *2 *3)
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 (((*1 *2 *2 *3 *4)
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-   (-4 *7 (-954 *6 *5 *3)) (-5 *1 (-466 *5 *3 *6 *7 *2))
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    (-4 *2
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-     (-10 -8 (-15 -4380 ($ *7)) (-15 -3401 (*7 $)) (-15 -3400 (*7 $)))))))) 
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+     (-10 -8 (-15 -4387 ($ *7)) (-15 -3408 (*7 $)) (-15 -3407 (*7 $)))))))) 
 (((*1 *2 *1)
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-   (-4 *5 (-853)) (-4 *7 (-954 *4 *2 (-867 *3)))))) 
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 (((*1 *2 *1)
-  (-12 (-14 *3 (-644 (-1181))) (-4 *4 (-173)) (-4 *5 (-239 (-4391 *3) (-774)))
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-   (-4 *7 (-954 *4 *5 (-867 *3)))))) 
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+   (-4 *2 (-855)) (-5 *1 (-466 *3 *4 *2 *5 *6 *7))
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 (((*1 *1 *2 *3 *4)
-  (-12 (-14 *5 (-644 (-1181))) (-4 *2 (-173)) (-4 *4 (-239 (-4391 *5) (-774)))
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-   (-4 *7 (-954 *2 *4 (-867 *5)))))) 
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 (((*1 *1 *2 *3 *1)
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-   (-4 *7 (-954 *2 *3 (-867 *4)))))) 
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 (((*1 *2 *3 *2 *4 *5)
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-   (-5 *1 (-464 *4 *3))))) 
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 (((*1 *2 *3 *4 *5 *6)
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-   (-5 *4 (-644 *3))))) 
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+   (-5 *2 (-2 (|:| |plist| (-646 *3)) (|:| |modulo| *5))) (-5 *1 (-465 *5 *3))
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 (((*1 *2 *3 *4)
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-   (-5 *1 (-459 *3 *5))))) 
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 (((*1 *2 *3 *4 *5)
-  (|partial| -12 (-5 *5 (-1270 (-644 *3))) (-4 *4 (-309)) (-5 *2 (-644 *3))
-   (-5 *1 (-459 *4 *3)) (-4 *3 (-1246 *4))))) 
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 (((*1 *2 *3 *4 *5)
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-   (-5 *2 (-1270 (-644 *6))) (-5 *1 (-459 *4 *6)) (-5 *5 (-644 *6))))) 
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 (((*1 *2 *3 *4)
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-   (-5 *1 (-459 *5 *3))))) 
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 (((*1 *2)
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-   (-14 *5 (-644 (-1181))) (-14 *6 (-1270 (-692 *3)))))) 
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 (((*1 *2)
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-   (-14 *5 (-644 (-1181))) (-14 *6 (-1270 (-692 *3)))))) 
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 (((*1 *1 *2 *3)
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-   (-14 *7 (-1270 (-692 *4)))))
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-   (-4 *3 (-422 *4))))
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 (((*1 *2 *1)
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 (((*1 *2 *1)
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 (((*1 *2)
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-   (-4 *3 (-561)) (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
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+   (-4 *3 (-562)) (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2 *1)
-  (-12 (-5 *2 (-1175 (-411 (-950 *3)))) (-5 *1 (-457 *3 *4 *5 *6))
-   (-4 *3 (-561)) (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
+  (-12 (-5 *2 (-1177 (-412 (-952 *3)))) (-5 *1 (-458 *3 *4 *5 *6))
+   (-4 *3 (-562)) (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2 *1)
-  (-12 (-5 *2 (-411 (-950 *3))) (-5 *1 (-457 *3 *4 *5 *6)) (-4 *3 (-561))
-   (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
+  (-12 (-5 *2 (-412 (-952 *3))) (-5 *1 (-458 *3 *4 *5 *6)) (-4 *3 (-562))
+   (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2 *1)
-  (-12 (-5 *2 (-411 (-950 *3))) (-5 *1 (-457 *3 *4 *5 *6)) (-4 *3 (-561))
-   (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
+  (-12 (-5 *2 (-412 (-952 *3))) (-5 *1 (-458 *3 *4 *5 *6)) (-4 *3 (-562))
+   (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2 *1 *1)
-  (-12 (-5 *2 (-411 (-950 *3))) (-5 *1 (-457 *3 *4 *5 *6)) (-4 *3 (-561))
-   (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
+  (-12 (-5 *2 (-412 (-952 *3))) (-5 *1 (-458 *3 *4 *5 *6)) (-4 *3 (-562))
+   (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-411 (-950 *3))) (-5 *1 (-457 *3 *4 *5 *6)) (-4 *3 (-561))
-   (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
+  (-12 (-5 *2 (-412 (-952 *3))) (-5 *1 (-458 *3 *4 *5 *6)) (-4 *3 (-562))
+   (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2 *1 *1)
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-   (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
+  (-12 (-5 *2 (-412 (-952 *3))) (-5 *1 (-458 *3 *4 *5 *6)) (-4 *3 (-562))
+   (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2)
-  (-12 (-5 *2 (-411 (-950 *3))) (-5 *1 (-457 *3 *4 *5 *6)) (-4 *3 (-561))
-   (-4 *3 (-173)) (-14 *4 (-923)) (-14 *5 (-644 (-1181)))
-   (-14 *6 (-1270 (-692 *3)))))) 
+  (-12 (-5 *2 (-412 (-952 *3))) (-5 *1 (-458 *3 *4 *5 *6)) (-4 *3 (-562))
+   (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))) 
 (((*1 *2 *3)
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-   (-5 *2 (-644 (-950 *4)))))
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+   (-5 *2 (-646 (-952 *4)))))
  ((*1 *2)
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-   (-4 *3 (-422 *4))))
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+  (-12 (-4 *4 (-173)) (-5 *2 (-646 (-952 *4))) (-5 *1 (-422 *3 *4))
+   (-4 *3 (-423 *4))))
+ ((*1 *2) (-12 (-4 *1 (-423 *3)) (-4 *3 (-173)) (-5 *2 (-646 (-952 *3)))))
  ((*1 *2)
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-   (-14 *6 (-1270 (-692 *3)))))
- ((*1 *2 *3)
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- ((*1 *1 *1 *1) (-4 *1 (-456)))) 
-(((*1 *2 *3)
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-   (-5 *1 (-454 *4 *5 *6 *3)) (-4 *3 (-954 *4 *5 *6))))) 
+  (-12 (-5 *2 (-646 (-952 *3))) (-5 *1 (-458 *3 *4 *5 *6)) (-4 *3 (-562))
+   (-4 *3 (-173)) (-14 *4 (-925)) (-14 *5 (-646 (-1183)))
+   (-14 *6 (-1272 (-694 *3)))))
+ ((*1 *2 *3)
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+   (-14 *6 (-646 (-1183))) (-14 *7 (-1272 (-694 *4)))))) 
+(((*1 *1 *2) (-12 (-5 *2 (-646 *1)) (-4 *1 (-457))))
+ ((*1 *1 *1 *1) (-4 *1 (-457)))) 
+(((*1 *2 *3)
+  (-12 (-4 *4 (-457)) (-4 *5 (-798)) (-4 *6 (-855)) (-5 *2 (-776))
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 (((*1 *2 *3 *4 *5)
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    (-5 *2
     (-2 (|:| |lcmfij| *7) (|:| |totdeg| *5) (|:| |poli| *4) (|:| |polj| *4)))
-   (-5 *1 (-454 *6 *7 *8 *4))))) 
+   (-5 *1 (-455 *6 *7 *8 *4))))) 
 (((*1 *2 *3 *3)
   (-12
    (-5 *3
-    (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-774)) (|:| |poli| *7)
+    (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-776)) (|:| |poli| *7)
      (|:| |polj| *7)))
-   (-4 *5 (-796)) (-4 *7 (-954 *4 *5 *6)) (-4 *4 (-456)) (-4 *6 (-853))
-   (-5 *2 (-112)) (-5 *1 (-454 *4 *5 *6 *7))))) 
+   (-4 *5 (-798)) (-4 *7 (-956 *4 *5 *6)) (-4 *4 (-457)) (-4 *6 (-855))
+   (-5 *2 (-112)) (-5 *1 (-455 *4 *5 *6 *7))))) 
 (((*1 *2 *3)
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-   (-5 *2 (-1276)) (-5 *1 (-454 *4 *5 *6 *7)) (-4 *7 (-954 *4 *5 *6))))) 
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+   (-5 *2 (-1278)) (-5 *1 (-455 *4 *5 *6 *7)) (-4 *7 (-956 *4 *5 *6))))) 
 (((*1 *2 *3)
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-   (-4 *6 (-853)) (-5 *2 (-1276)) (-5 *1 (-454 *4 *5 *6 *7))))) 
+  (-12 (-5 *3 (-646 *7)) (-4 *7 (-956 *4 *5 *6)) (-4 *4 (-457)) (-4 *5 (-798))
+   (-4 *6 (-855)) (-5 *2 (-1278)) (-5 *1 (-455 *4 *5 *6 *7))))) 
 (((*1 *2 *3 *4 *4 *2 *2 *2 *2)
-  (-12 (-5 *2 (-550))
+  (-12 (-5 *2 (-551))
    (-5 *3
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+    (-2 (|:| |lcmfij| *6) (|:| |totdeg| (-776)) (|:| |poli| *4)
      (|:| |polj| *4)))
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-   (-5 *1 (-454 *5 *6 *7 *4))))) 
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+   (-5 *1 (-455 *5 *6 *7 *4))))) 
 (((*1 *2 *3 *4 *4 *2 *2 *2)
-  (-12 (-5 *2 (-550))
+  (-12 (-5 *2 (-551))
    (-5 *3
-    (-2 (|:| |lcmfij| *6) (|:| |totdeg| (-774)) (|:| |poli| *4)
+    (-2 (|:| |lcmfij| *6) (|:| |totdeg| (-776)) (|:| |poli| *4)
      (|:| |polj| *4)))
-   (-4 *6 (-796)) (-4 *4 (-954 *5 *6 *7)) (-4 *5 (-456)) (-4 *7 (-853))
-   (-5 *1 (-454 *5 *6 *7 *4))))) 
+   (-4 *6 (-798)) (-4 *4 (-956 *5 *6 *7)) (-4 *5 (-457)) (-4 *7 (-855))
+   (-5 *1 (-455 *5 *6 *7 *4))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-456)) (-4 *5 (-796)) (-4 *6 (-853)) (-5 *2 (-1276))
-   (-5 *1 (-454 *4 *5 *6 *3)) (-4 *3 (-954 *4 *5 *6))))) 
+  (-12 (-4 *4 (-457)) (-4 *5 (-798)) (-4 *6 (-855)) (-5 *2 (-1278))
+   (-5 *1 (-455 *4 *5 *6 *3)) (-4 *3 (-956 *4 *5 *6))))) 
 (((*1 *2 *3)
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-   (-5 *1 (-454 *4 *5 *6 *3)) (-4 *3 (-954 *4 *5 *6))))) 
+  (-12 (-4 *4 (-457)) (-4 *5 (-798)) (-4 *6 (-855)) (-5 *2 (-551))
+   (-5 *1 (-455 *4 *5 *6 *3)) (-4 *3 (-956 *4 *5 *6))))) 
 (((*1 *2 *2)
-  (-12 (-5 *2 (-644 *6)) (-4 *6 (-954 *3 *4 *5)) (-4 *3 (-456)) (-4 *4 (-796))
-   (-4 *5 (-853)) (-5 *1 (-454 *3 *4 *5 *6))))) 
+  (-12 (-5 *2 (-646 *6)) (-4 *6 (-956 *3 *4 *5)) (-4 *3 (-457)) (-4 *4 (-798))
+   (-4 *5 (-855)) (-5 *1 (-455 *3 *4 *5 *6))))) 
 (((*1 *2 *2 *2)
   (-12
    (-5 *2
-    (-644
-     (-2 (|:| |lcmfij| *4) (|:| |totdeg| (-774)) (|:| |poli| *6)
+    (-646
+     (-2 (|:| |lcmfij| *4) (|:| |totdeg| (-776)) (|:| |poli| *6)
       (|:| |polj| *6))))
-   (-4 *4 (-796)) (-4 *6 (-954 *3 *4 *5)) (-4 *3 (-456)) (-4 *5 (-853))
-   (-5 *1 (-454 *3 *4 *5 *6))))) 
+   (-4 *4 (-798)) (-4 *6 (-956 *3 *4 *5)) (-4 *3 (-457)) (-4 *5 (-855))
+   (-5 *1 (-455 *3 *4 *5 *6))))) 
 (((*1 *2 *3)
   (-12
    (-5 *3
-    (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-774)) (|:| |poli| *2)
+    (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-776)) (|:| |poli| *2)
      (|:| |polj| *2)))
-   (-4 *5 (-796)) (-4 *2 (-954 *4 *5 *6)) (-5 *1 (-454 *4 *5 *6 *2))
-   (-4 *4 (-456)) (-4 *6 (-853))))) 
+   (-4 *5 (-798)) (-4 *2 (-956 *4 *5 *6)) (-5 *1 (-455 *4 *5 *6 *2))
+   (-4 *4 (-457)) (-4 *6 (-855))))) 
 (((*1 *2 *3 *4 *2)
-  (-12 (-5 *2 (-644 (-2 (|:| |totdeg| (-774)) (|:| -2184 *3)))) (-5 *4 (-774))
-   (-4 *3 (-954 *5 *6 *7)) (-4 *5 (-456)) (-4 *6 (-796)) (-4 *7 (-853))
-   (-5 *1 (-454 *5 *6 *7 *3))))) 
+  (-12 (-5 *2 (-646 (-2 (|:| |totdeg| (-776)) (|:| -2191 *3)))) (-5 *4 (-776))
+   (-4 *3 (-956 *5 *6 *7)) (-4 *5 (-457)) (-4 *6 (-798)) (-4 *7 (-855))
+   (-5 *1 (-455 *5 *6 *7 *3))))) 
 (((*1 *2 *2)
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-   (-4 *2 (-954 *3 *4 *5))))) 
+  (-12 (-4 *3 (-457)) (-4 *4 (-798)) (-4 *5 (-855)) (-5 *1 (-455 *3 *4 *5 *2))
+   (-4 *2 (-956 *3 *4 *5))))) 
 (((*1 *2 *3 *4)
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-   (-4 *7 (-853)) (-5 *2 (-2 (|:| |poly| *3) (|:| |mult| *5)))
-   (-5 *1 (-454 *5 *6 *7 *3))))) 
+  (-12 (-5 *4 (-646 *3)) (-4 *3 (-956 *5 *6 *7)) (-4 *5 (-457)) (-4 *6 (-798))
+   (-4 *7 (-855)) (-5 *2 (-2 (|:| |poly| *3) (|:| |mult| *5)))
+   (-5 *1 (-455 *5 *6 *7 *3))))) 
 (((*1 *2 *3 *2)
   (-12
    (-5 *2
-    (-644
-     (-2 (|:| |lcmfij| *3) (|:| |totdeg| (-774)) (|:| |poli| *6)
+    (-646
+     (-2 (|:| |lcmfij| *3) (|:| |totdeg| (-776)) (|:| |poli| *6)
       (|:| |polj| *6))))
-   (-4 *3 (-796)) (-4 *6 (-954 *4 *3 *5)) (-4 *4 (-456)) (-4 *5 (-853))
-   (-5 *1 (-454 *4 *3 *5 *6))))) 
+   (-4 *3 (-798)) (-4 *6 (-956 *4 *3 *5)) (-4 *4 (-457)) (-4 *5 (-855))
+   (-5 *1 (-455 *4 *3 *5 *6))))) 
 (((*1 *2 *2)
   (-12
    (-5 *2
-    (-644
-     (-2 (|:| |lcmfij| *4) (|:| |totdeg| (-774)) (|:| |poli| *6)
+    (-646
+     (-2 (|:| |lcmfij| *4) (|:| |totdeg| (-776)) (|:| |poli| *6)
       (|:| |polj| *6))))
-   (-4 *4 (-796)) (-4 *6 (-954 *3 *4 *5)) (-4 *3 (-456)) (-4 *5 (-853))
-   (-5 *1 (-454 *3 *4 *5 *6))))) 
+   (-4 *4 (-798)) (-4 *6 (-956 *3 *4 *5)) (-4 *3 (-457)) (-4 *5 (-855))
+   (-5 *1 (-455 *3 *4 *5 *6))))) 
 (((*1 *2 *3 *2)
   (-12
    (-5 *2
-    (-644
-     (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-774)) (|:| |poli| *3)
+    (-646
+     (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-776)) (|:| |poli| *3)
       (|:| |polj| *3))))
-   (-4 *5 (-796)) (-4 *3 (-954 *4 *5 *6)) (-4 *4 (-456)) (-4 *6 (-853))
-   (-5 *1 (-454 *4 *5 *6 *3))))) 
+   (-4 *5 (-798)) (-4 *3 (-956 *4 *5 *6)) (-4 *4 (-457)) (-4 *6 (-855))
+   (-5 *1 (-455 *4 *5 *6 *3))))) 
 (((*1 *2 *3 *3 *3 *3)
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-   (-5 *1 (-454 *4 *3 *5 *6)) (-4 *6 (-954 *4 *3 *5))))) 
+  (-12 (-4 *4 (-457)) (-4 *3 (-798)) (-4 *5 (-855)) (-5 *2 (-112))
+   (-5 *1 (-455 *4 *3 *5 *6)) (-4 *6 (-956 *4 *3 *5))))) 
 (((*1 *2 *3 *3)
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-   (-5 *1 (-454 *4 *3 *5 *6)) (-4 *6 (-954 *4 *3 *5))))) 
+  (-12 (-4 *4 (-457)) (-4 *3 (-798)) (-4 *5 (-855)) (-5 *2 (-112))
+   (-5 *1 (-455 *4 *3 *5 *6)) (-4 *6 (-956 *4 *3 *5))))) 
 (((*1 *2 *3)
   (-12
    (-5 *3
-    (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-774)) (|:| |poli| *7)
+    (-2 (|:| |lcmfij| *5) (|:| |totdeg| (-776)) (|:| |poli| *7)
      (|:| |polj| *7)))
-   (-4 *5 (-796)) (-4 *7 (-954 *4 *5 *6)) (-4 *4 (-456)) (-4 *6 (-853))
-   (-5 *2 (-112)) (-5 *1 (-454 *4 *5 *6 *7))))) 
+   (-4 *5 (-798)) (-4 *7 (-956 *4 *5 *6)) (-4 *4 (-457)) (-4 *6 (-855))
+   (-5 *2 (-112)) (-5 *1 (-455 *4 *5 *6 *7))))) 
 (((*1 *2 *2 *3 *3)
-  (-12 (-5 *2 (-644 *7)) (-5 *3 (-550)) (-4 *7 (-954 *4 *5 *6)) (-4 *4 (-456))
-   (-4 *5 (-796)) (-4 *6 (-853)) (-5 *1 (-454 *4 *5 *6 *7))))) 
+  (-12 (-5 *2 (-646 *7)) (-5 *3 (-551)) (-4 *7 (-956 *4 *5 *6)) (-4 *4 (-457))
+   (-4 *5 (-798)) (-4 *6 (-855)) (-5 *1 (-455 *4 *5 *6 *7))))) 
 (((*1 *2 *2 *3)
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-   (-4 *6 (-853)) (-5 *1 (-454 *4 *5 *6 *2))))) 
+  (-12 (-5 *3 (-646 *2)) (-4 *2 (-956 *4 *5 *6)) (-4 *4 (-457)) (-4 *5 (-798))
+   (-4 *6 (-855)) (-5 *1 (-455 *4 *5 *6 *2))))) 
 (((*1 *2 *2 *3)
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-   (-4 *6 (-853)) (-5 *1 (-454 *4 *5 *6 *2))))) 
+  (-12 (-5 *3 (-646 *2)) (-4 *2 (-956 *4 *5 *6)) (-4 *4 (-457)) (-4 *5 (-798))
+   (-4 *6 (-855)) (-5 *1 (-455 *4 *5 *6 *2))))) 
 (((*1 *2 *3 *3)
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-   (-5 *3 (-644 *7))))
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+   (-4 *7 (-956 *4 *5 *6)) (-5 *2 (-646 (-646 *7))) (-5 *1 (-454 *4 *5 *6 *7))
+   (-5 *3 (-646 *7))))
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-   (-5 *3 (-644 *8))))
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+   (-4 *8 (-956 *5 *6 *7)) (-5 *2 (-646 (-646 *8))) (-5 *1 (-454 *5 *6 *7 *8))
+   (-5 *3 (-646 *8))))
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-   (-5 *3 (-644 *7))))
+  (-12 (-4 *4 (-13 (-310) (-147))) (-4 *5 (-798)) (-4 *6 (-855))
+   (-4 *7 (-956 *4 *5 *6)) (-5 *2 (-646 (-646 *7))) (-5 *1 (-454 *4 *5 *6 *7))
+   (-5 *3 (-646 *7))))
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-   (-5 *3 (-644 *8))))) 
+  (-12 (-5 *4 (-112)) (-4 *5 (-13 (-310) (-147))) (-4 *6 (-798)) (-4 *7 (-855))
+   (-4 *8 (-956 *5 *6 *7)) (-5 *2 (-646 (-646 *8))) (-5 *1 (-454 *5 *6 *7 *8))
+   (-5 *3 (-646 *8))))) 
 (((*1 *2 *3)
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-   (-4 *7 (-954 *4 *5 *6)) (-5 *2 (-644 (-644 *7))) (-5 *1 (-453 *4 *5 *6 *7))
-   (-5 *3 (-644 *7))))
+  (-12 (-4 *4 (-13 (-310) (-147))) (-4 *5 (-798)) (-4 *6 (-855))
+   (-4 *7 (-956 *4 *5 *6)) (-5 *2 (-646 (-646 *7))) (-5 *1 (-454 *4 *5 *6 *7))
+   (-5 *3 (-646 *7))))
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-   (-5 *3 (-644 *8))))) 
+  (-12 (-5 *4 (-112)) (-4 *5 (-13 (-310) (-147))) (-4 *6 (-798)) (-4 *7 (-855))
+   (-4 *8 (-956 *5 *6 *7)) (-5 *2 (-646 (-646 *8))) (-5 *1 (-454 *5 *6 *7 *8))
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 (((*1 *2 *1)
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 (((*1 *2 *3 *4)
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 (((*1 *2 *3 *4)
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 (((*1 *2 *3 *4)
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 (((*1 *1 *2 *3 *1)
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+ ((*1 *1 *2) (-12 (-5 *2 (-317 (-551))) (-5 *1 (-333))))
+ ((*1 *1 *2) (-12 (-5 *2 (-317 (-382))) (-5 *1 (-333))))
+ ((*1 *1 *2) (-12 (-5 *2 (-317 (-699))) (-5 *1 (-333))))
+ ((*1 *1 *2) (-12 (-5 *2 (-317 (-706))) (-5 *1 (-333))))
+ ((*1 *1 *2) (-12 (-5 *2 (-317 (-704))) (-5 *1 (-333))))
+ ((*1 *1) (-5 *1 (-333)))) 
+(((*1 *1 *2 *1 *1) (-12 (-5 *2 (-1182)) (-5 *1 (-333))))
+ ((*1 *1 *2 *1) (-12 (-5 *2 (-1182)) (-5 *1 (-333))))) 
+(((*1 *1 *2) (-12 (-5 *2 (-646 (-333))) (-5 *1 (-333))))) 
+(((*1 *1) (-5 *1 (-333)))) 
+(((*1 *1) (-5 *1 (-333)))) 
+(((*1 *1 *2) (-12 (-5 *2 (-646 (-868))) (-5 *1 (-333))))) 
+(((*1 *1 *2 *3) (-12 (-5 *3 (-646 (-1183))) (-5 *2 (-1183)) (-5 *1 (-333))))) 
 (((*1 *2 *1)
   (-12
    (-5 *2
@@ -14033,155 +14033,155 @@
      (|:| |Goto| "goto") (|:| |Continue| "continue")
      (|:| |ArrayAssignment| "arrayAssignment") (|:| |Save| "save")
      (|:| |Stop| "stop") (|:| |Common| "common") (|:| |Print| "print")))
-   (-5 *1 (-332))))) 
+   (-5 *1 (-333))))) 
 (((*1 *2 *1)
   (-12
    (-5 *2
     (-3 (|:| |nullBranch| "null")
      (|:| |assignmentBranch|
-          (-2 (|:| |var| (-1181)) (|:| |arrayIndex| (-644 (-950 (-550))))
-           (|:| |rand| (-2 (|:| |ints2Floats?| (-112)) (|:| -3676 (-866))))))
+          (-2 (|:| |var| (-1183)) (|:| |arrayIndex| (-646 (-952 (-551))))
+           (|:| |rand| (-2 (|:| |ints2Floats?| (-112)) (|:| -3683 (-868))))))
      (|:| |arrayAssignmentBranch|
-          (-2 (|:| |var| (-1181)) (|:| |rand| (-866))
+          (-2 (|:| |var| (-1183)) (|:| |rand| (-868))
            (|:| |ints2Floats?| (-112))))
      (|:| |conditionalBranch|
-          (-2 (|:| |switch| (-1180)) (|:| |thenClause| (-332))
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 (((*1 *2 *3)
   (-12
    (-5 *3
-    (-2 (|:| |stiffness| (-381)) (|:| |stability| (-381))
-     (|:| |expense| (-381)) (|:| |accuracy| (-381))
-     (|:| |intermediateResults| (-381))))
-   (-5 *2 (-1039)) (-5 *1 (-305))))) 
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 (((*1 *2 *3)
   (-12
    (-5 *3
@@ -14195,787 +14195,792 @@
            (|:| |bothSingular| "There are singularities at both end points")
            (|:| |notEvaluated| "End point continuity not yet evaluated")))
      (|:| |singularitiesStream|
-          (-3 (|:| |str| (-1158 (-226)))
+          (-3 (|:| |str| (-1160 (-226)))
            (|:| |notEvaluated| "Internal singularities not yet evaluated")))
-     (|:| -1609
+     (|:| -1612
           (-3 (|:| |finite| "The range is finite")
            (|:| |lowerInfinite| "The bottom of range is infinite")
            (|:| |upperInfinite| "The top of range is infinite")
            (|:| |bothInfinite| "Both top and bottom points are infinite")
            (|:| |notEvaluated| "Range not yet evaluated")))))
-   (-5 *2 (-1039)) (-5 *1 (-305))))) 
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 (((*1 *2 *3)
   (-12
    (-5 *3
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-     (|:| |explanations| (-644 (-1163)))))
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  ((*1 *2 *3)
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-(((*1 *2 *3)
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 (((*1 *2 *3)
   (-12
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      (|:| |relerr| (-226))))
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 (((*1 *2 *3 *4)
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 (((*1 *2 *3 *4 *5)
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 (((*1 *2 *3)
   (-12
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 (((*1 *2 *3 *4 *5 *5)
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 (((*1 *2 *1 *3 *3 *2)
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+     (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
      (|:| |relerr| (-226))))
-   (-5 *2 (-2 (|:| -2904 (-113)) (|:| |w| (-226)))) (-5 *1 (-205))))) 
-(((*1 *2 *3 *3 *2) (-12 (-5 *2 (-1039)) (-5 *3 (-1181)) (-5 *1 (-193))))) 
+   (-5 *2 (-2 (|:| -2911 (-113)) (|:| |w| (-226)))) (-5 *1 (-205))))) 
+(((*1 *2 *3 *3 *2) (-12 (-5 *2 (-1041)) (-5 *3 (-1183)) (-5 *1 (-193))))) 
 (((*1 *2 *3)
   (-12
    (-5 *3
-    (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-     (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+    (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+     (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
      (|:| |relerr| (-226))))
-   (-5 *2 (-381)) (-5 *1 (-193))))) 
+   (-5 *2 (-382)) (-5 *1 (-193))))) 
 (((*1 *2 *3)
   (-12
    (-5 *3
-    (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-     (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+    (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+     (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
      (|:| |relerr| (-226))))
    (-5 *2
     (-3 (|:| |continuous| "Continuous at the end points")
@@ -14987,8 +14992,8 @@
 (((*1 *2 *3)
   (-12
    (-5 *3
-    (-2 (|:| |var| (-1181)) (|:| |fn| (-316 (-226)))
-     (|:| -1609 (-1093 (-845 (-226)))) (|:| |abserr| (-226))
+    (-2 (|:| |var| (-1183)) (|:| |fn| (-317 (-226)))
+     (|:| -1612 (-1095 (-847 (-226)))) (|:| |abserr| (-226))
      (|:| |relerr| (-226))))
    (-5 *2
     (-3 (|:| |finite| "The range is finite")
@@ -14997,240 +15002,241 @@
      (|:| |bothInfinite| "Both top and bottom points are infinite")
      (|:| |notEvaluated| "Range not yet evaluated")))
    (-5 *1 (-193))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-409 (-1175 (-550)))) (-5 *1 (-192)) (-5 *3 (-550))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-644 (-1175 (-550)))) (-5 *1 (-192)) (-5 *3 (-550))))) 
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+(((*1 *2 *3) (-12 (-5 *2 (-646 (-1177 (-551)))) (-5 *1 (-192)) (-5 *3 (-551))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *3 (-644 (-550))) (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191))))) 
+  (-12 (-5 *3 (-646 (-551))) (-5 *2 (-1185 (-412 (-551)))) (-5 *1 (-191))))) 
 (((*1 *2 *3)
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-(((*1 *2 *2 *2) (-12 (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191))))) 
-(((*1 *2 *2 *2) (-12 (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191))))) 
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+(((*1 *2 *2 *2) (-12 (-5 *2 (-1185 (-412 (-551)))) (-5 *1 (-191))))) 
+(((*1 *2 *2 *2) (-12 (-5 *2 (-1185 (-412 (-551)))) (-5 *1 (-191))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *3 (-1183 (-411 (-550)))) (-5 *2 (-411 (-550))) (-5 *1 (-191))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191)) (-5 *3 (-550))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191)) (-5 *3 (-550))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191)) (-5 *3 (-550))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191)) (-5 *3 (-550))))) 
-(((*1 *2 *3) (-12 (-5 *2 (-1183 (-411 (-550)))) (-5 *1 (-191)) (-5 *3 (-550))))) 
-(((*1 *2 *3)
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-   (-5 *2 (-1270 (-692 (-950 *4)))) (-5 *1 (-190 *4))))) 
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+(((*1 *2 *3) (-12 (-5 *2 (-1185 (-412 (-551)))) (-5 *1 (-191)) (-5 *3 (-551))))) 
+(((*1 *2 *3) (-12 (-5 *2 (-1185 (-412 (-551)))) (-5 *1 (-191)) (-5 *3 (-551))))) 
+(((*1 *2 *3) (-12 (-5 *2 (-1185 (-412 (-551)))) (-5 *1 (-191)) (-5 *3 (-551))))) 
+(((*1 *2 *3)
+  (-12 (-5 *3 (-1272 (-694 *4))) (-4 *4 (-173))
+   (-5 *2 (-1272 (-694 (-952 *4)))) (-5 *1 (-190 *4))))) 
 (((*1 *1) (-5 *1 (-188)))) 
 (((*1 *1) (-5 *1 (-188)))) 
 (((*1 *1) (-5 *1 (-188)))) 
 (((*1 *2 *1) (-12 (-5 *2 (-188)) (-5 *1 (-138))))
  ((*1 *2 *1) (-12 (-4 *1 (-187)) (-5 *2 (-188))))) 
-(((*1 *2 *1) (-12 (-4 *1 (-187)) (-5 *2 (-644 (-112)))))) 
-(((*1 *2 *1) (-12 (-4 *1 (-187)) (-5 *2 (-644 (-868)))))) 
-(((*1 *2 *1) (-12 (-5 *2 (-644 (-1186))) (-5 *1 (-185 *3)) (-4 *3 (-187))))) 
-(((*1 *2 *3) (-12 (-5 *3 (-510)) (-5 *2 (-694 (-184))) (-5 *1 (-184))))) 
-(((*1 *2 *2 *2) (-12 (-4 *3 (-1220)) (-5 *1 (-183 *3 *2)) (-4 *2 (-677 *3))))) 
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+(((*1 *2 *1) (-12 (-4 *1 (-187)) (-5 *2 (-646 (-870)))))) 
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+(((*1 *2 *2 *2) (-12 (-4 *3 (-1222)) (-5 *1 (-183 *3 *2)) (-4 *2 (-679 *3))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-1220)) (-5 *2 (-774)) (-5 *1 (-183 *4 *3)) (-4 *3 (-677 *4))))) 
+  (-12 (-4 *4 (-1222)) (-5 *2 (-776)) (-5 *1 (-183 *4 *3)) (-4 *3 (-679 *4))))) 
 (((*1 *2 *2)
-  (|partial| -12 (-4 *3 (-1220)) (-5 *1 (-183 *3 *2)) (-4 *2 (-677 *3))))) 
+  (|partial| -12 (-4 *3 (-1222)) (-5 *1 (-183 *3 *2)) (-4 *2 (-679 *3))))) 
 (((*1 *2 *3)
-  (-12 (-4 *4 (-13 (-366) (-851)))
-   (-5 *2 (-2 (|:| |start| *3) (|:| -1956 (-409 *3)))) (-5 *1 (-182 *4 *3))
-   (-4 *3 (-1246 (-169 *4)))))) 
+  (-12 (-4 *4 (-13 (-367) (-853)))
+   (-5 *2 (-2 (|:| |start| *3) (|:| -1963 (-410 *3)))) (-5 *1 (-182 *4 *3))
+   (-4 *3 (-1248 (-169 *4)))))) 
 (((*1 *2 *2)
-  (-12 (-4 *2 (-13 (-366) (-851))) (-5 *1 (-182 *2 *3))
-   (-4 *3 (-1246 (-169 *2)))))) 
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 (((*1 *2 *3)
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-   (-4 *3 (-1246 *2))))) 
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+   (-4 *3 (-1248 *2))))) 
 (((*1 *2 *3 *2)
-  (-12 (-4 *2 (-13 (-366) (-851))) (-5 *1 (-182 *2 *3))
-   (-4 *3 (-1246 (-169 *2)))))
+  (-12 (-4 *2 (-13 (-367) (-853))) (-5 *1 (-182 *2 *3))
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  ((*1 *2 *3)
-  (-12 (-4 *2 (-13 (-366) (-851))) (-5 *1 (-182 *2 *3))
-   (-4 *3 (-1246 (-169 *2)))))) 
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 (((*1 *2 *2)
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 (((*1 *2 *3 *4 *5)
-  (-12 (-5 *5 (-112)) (-4 *4 (-13 (-366) (-851))) (-5 *2 (-409 *3))
-   (-5 *1 (-182 *4 *3)) (-4 *3 (-1246 (-169 *4)))))
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  ((*1 *2 *3 *4)
-  (-12 (-4 *4 (-13 (-366) (-851))) (-5 *2 (-409 *3)) (-5 *1 (-182 *4 *3))
-   (-4 *3 (-1246 (-169 *4)))))) 
+  (-12 (-4 *4 (-13 (-367) (-853))) (-5 *2 (-410 *3)) (-5 *1 (-182 *4 *3))
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 (((*1 *2 *2)
-  (-12 (-4 *3 (-13 (-366) (-851))) (-5 *1 (-182 *3 *2))
-   (-4 *2 (-1246 (-169 *3)))))) 
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 (((*1 *2 *3 *3 *4)
-  (-12 (-5 *4 (-112)) (-4 *5 (-13 (-366) (-851)))
-   (-5 *2 (-644 (-2 (|:| -1956 (-644 *3)) (|:| -1706 *5))))
-   (-5 *1 (-182 *5 *3)) (-4 *3 (-1246 (-169 *5)))))
+  (-12 (-5 *4 (-112)) (-4 *5 (-13 (-367) (-853)))
+   (-5 *2 (-646 (-2 (|:| -1963 (-646 *3)) (|:| -1713 *5))))
+   (-5 *1 (-182 *5 *3)) (-4 *3 (-1248 (-169 *5)))))
  ((*1 *2 *3 *3)
-  (-12 (-4 *4 (-13 (-366) (-851)))
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-   (-5 *1 (-182 *4 *3)) (-4 *3 (-1246 (-169 *4)))))) 
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+   (-5 *2 (-646 (-2 (|:| -1963 (-646 *3)) (|:| -1713 *4))))
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 (((*1 *2 *3 *4)
-  (-12 (-5 *2 (-644 (-169 *4))) (-5 *1 (-155 *3 *4))
-   (-4 *3 (-1246 (-169 (-550)))) (-4 *4 (-13 (-366) (-851)))))
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  ((*1 *2 *3)
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-   (-5 *1 (-182 *4 *3)) (-4 *3 (-1246 (-169 *4)))))
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  ((*1 *2 *3 *4)
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-(((*1 *2 *3 *2) (-12 (-5 *2 (-644 *3)) (-4 *3 (-309)) (-5 *1 (-180 *3))))) 
+  (-12 (-4 *4 (-13 (-367) (-853))) (-5 *2 (-646 (-169 *4)))
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+(((*1 *2 *3 *2) (-12 (-5 *2 (-646 *3)) (-4 *3 (-310)) (-5 *1 (-180 *3))))) 
 (((*1 *2 *3 *3)
-  (-12 (-5 *2 (-1 (-947 *3) (-947 *3))) (-5 *1 (-177 *3))
-   (-4 *3 (-13 (-366) (-1206) (-1006)))))) 
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 (((*1 *2 *3)
-  (-12 (-5 *2 (-1 (-947 *3) (-947 *3))) (-5 *1 (-177 *3))
-   (-4 *3 (-13 (-366) (-1206) (-1006)))))) 
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 (((*1 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *3)
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 (((*1 *2 *2)
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 (((*1 *2 *2)
-  (-12 (-5 *2 (-947 *3)) (-4 *3 (-13 (-366) (-1206) (-1006)))
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 (((*1 *2 *2)
-  (-12 (-5 *2 (-947 *3)) (-4 *3 (-13 (-366) (-1206) (-1006)))
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 (((*1 *2 *2)
-  (-12 (-5 *2 (-947 *3)) (-4 *3 (-13 (-366) (-1206) (-1006)))
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 (((*1 *2 *2)
-  (-12 (-5 *2 (-947 *3)) (-4 *3 (-13 (-366) (-1206) (-1006)))
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 (((*1 *2 *2)
-  (-12 (-5 *2 (-947 *3)) (-4 *3 (-13 (-366) (-1206) (-1006)))
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 (((*1 *2 *2)
-  (-12 (-5 *2 (-947 *3)) (-4 *3 (-13 (-366) (-1206) (-1006)))
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-(((*1 *2 *1) (-12 (-5 *2 (-644 (-109))) (-5 *1 (-176))))) 
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 (((*1 *1 *2 *1) (-12 (-5 *2 (-109)) (-5 *1 (-176))))) 
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 (((*1 *2 *1) (-12 (-5 *2 (-112)) (-5 *1 (-172))))) 
 (((*1 *2 *1 *2) (-12 (-5 *2 (-112)) (-5 *1 (-172))))) 
-(((*1 *2 *2 *3) (-12 (-5 *2 (-1186)) (-5 *3 (-293)) (-5 *1 (-168))))) 
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 (((*1 *1) (-12 (-4 *1 (-166 *2)) (-4 *2 (-173))))) 
 (((*1 *1 *2 *2) (-12 (-4 *1 (-166 *2)) (-4 *2 (-173))))) 
 (((*1 *2 *1)
-  (-12 (-4 *1 (-166 *3)) (-4 *3 (-173)) (-4 *3 (-1064)) (-4 *3 (-1206))
+  (-12 (-4 *1 (-166 *3)) (-4 *3 (-173)) (-4 *3 (-1066)) (-4 *3 (-1208))
    (-5 *2 (-2 (|:| |r| *3) (|:| |phi| *3)))))) 
 (((*1 *1 *1 *1) (-5 *1 (-161)))
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-(((*1 *2 *2) (-12 (-4 *3 (-561)) (-5 *1 (-158 *3 *2)) (-4 *2 (-425 *3))))
+ ((*1 *1 *2) (-12 (-5 *2 (-551)) (-5 *1 (-161))))) 
+(((*1 *2 *2) (-12 (-4 *3 (-562)) (-5 *1 (-158 *3 *2)) (-4 *2 (-426 *3))))
  ((*1 *2 *2 *3)
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+ ((*1 *1 *1 *2) (-12 (-4 *1 (-160)) (-5 *2 (-1183))))
  ((*1 *1 *1) (-4 *1 (-160)))) 
 (((*1 *2 *2 *3)
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  ((*1 *2 *2 *3)
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 (((*1 *1 *1 *1) (-4 *1 (-143)))
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+ ((*1 *2 *2 *2) (-12 (-4 *3 (-562)) (-5 *1 (-158 *3 *2)) (-4 *2 (-426 *3))))
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 (((*1 *2 *3 *1)
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-   (-5 *2 (-2 (|:| -4294 *3) (|:| -2256 *4)))))) 
+  (|partial| -12 (-4 *1 (-36 *3 *4)) (-4 *3 (-1107)) (-4 *4 (-1107))
+   (-5 *2 (-2 (|:| -4301 *3) (|:| -2263 *4)))))) 
 (((*1 *2 *1 *1) (-12 (-4 *1 (-34)) (-5 *2 (-112))))) 
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 (((*1 *2 *3 *4)
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-   (-4 *3 (-561))))) 
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+   (-4 *3 (-562))))) 
 (((*1 *2 *3)
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 (((*1 *2 *3 *2)
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 (((*1 *1 *2 *3 *3 *4 *4)
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\ No newline at end of file