* algebra/prtition.spad.pamphlet (powers$Partition): Return list

of pairs.
author: dos-reis <gdr@axiomatics.org> 2010-04-17 23:03:55 +0000
committer: dos-reis <gdr@axiomatics.org> 2010-04-17 23:03:55 +0000
commit: 5b24e39b978891b7a8dc22b9ab6ffa5eec341e6a (patch)
tree: 450e549494784c805ce24f0b319598a7e25cb0e8 /src/share/algebra/browse.daase
parent: 423125f1638b432ccfb944f954fd4304ddfb0183 (diff)
download: open-axiom-5b24e39b978891b7a8dc22b9ab6ffa5eec341e6a.tar.gz
1 files changed, 637 insertions, 637 deletions
diff --git a/src/share/algebra/browse.daase b/src/share/algebra/browse.daase
index dfe934e5..04329149 100644
--- a/src/share/algebra/browse.daase
+++ b/src/share/algebra/browse.daase
@@ -1,12 +1,12 @@
 
-(2267929 . 3479539532)       
+(2267801 . 3480528374)       
 (-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."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . 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}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . 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}.") (($ $) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) 
-((-4446 . T) (-4444 . T) (-4443 . T) ((-4451 "*") . T) (-4442 . T) (-4447 . T) (-4441 . T)) 
+((-4445 . T) (-4443 . T) (-4442 . T) ((-4450 "*") . T) (-4441 . T) (-4446 . T) (-4440 . T)) 
 NIL 
 (-30) 
 ((|constructor| (NIL "\\indented{1}{Plot a NON-SINGULAR plane algebraic curve \\spad{p}(\\spad{x},{}\\spad{y}) = 0.} Author: Clifton \\spad{J}. Williamson Date Created: Fall 1988 Date Last Updated: 27 April 1990 Keywords: algebraic curve,{} non-singular,{} plot Examples: References:")) (|refine| (($ $ (|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 -1673) 
+(-32 R -1674) 
 ((|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 -1047) (QUOTE (-570))))) 
 (-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 -4449))) 
+((|HasAttribute| |#1| (QUOTE -4448))) 
 (-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}."))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-37 S R) 
 ((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline"))) 
@@ -82,17 +82,17 @@ NIL
 NIL 
 (-38 R) 
 ((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . 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 -1673 UP UPUP -1771) 
+(-40 -1674 UP UPUP -4323) 
 ((|constructor| (NIL "Function field defined by \\spad{f}(\\spad{x},{} \\spad{y}) = 0.")) (|knownInfBasis| (((|Void|) (|NonNegativeInteger|)) "\\spad{knownInfBasis(n)} \\undocumented{}"))) 
-((-4442 |has| (-413 |#2|) (-368)) (-4447 |has| (-413 |#2|) (-368)) (-4441 |has| (-413 |#2|) (-368)) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-413 |#2|) (QUOTE (-146))) (|HasCategory| (-413 |#2|) (QUOTE (-148))) (|HasCategory| (-413 |#2|) (QUOTE (-354))) (-2738 (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-373))) (-2738 (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (-2738 (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-354))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -645) (QUOTE (-570)))) (-2738 (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368))))) 
-(-41 R -1673) 
+((-4441 |has| (-413 |#2|) (-368)) (-4446 |has| (-413 |#2|) (-368)) (-4440 |has| (-413 |#2|) (-368)) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-413 |#2|) (QUOTE (-146))) (|HasCategory| (-413 |#2|) (QUOTE (-148))) (|HasCategory| (-413 |#2|) (QUOTE (-354))) (-2740 (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-373))) (-2740 (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (-2740 (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-354))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -645) (QUOTE (-570)))) (-2740 (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368))))) 
+(-41 R -1674) 
 ((|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| $)) $) "\\spad{numer(x)} \\undocumented"))) 
 NIL 
 ((-12 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -436) (|devaluate| |#1|))))) 
@@ -106,23 +106,23 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-311)))) 
 (-44 R |n| |ls| |gamma|) 
 ((|constructor| (NIL "AlgebraGivenByStructuralConstants implements finite rank algebras over a commutative ring,{} given by the structural constants \\spad{gamma} with respect to a fixed basis \\spad{[a1,..,an]},{} where \\spad{gamma} is an \\spad{n}-vector of \\spad{n} by \\spad{n} matrices \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{ai * aj = gammaij1 * a1 + ... + gammaijn * an}. The symbols for the fixed basis have to be given as a list of symbols.")) (|coerce| (($ (|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."))) 
-((-4446 |has| |#1| (-562)) (-4444 . T) (-4443 . T)) 
+((-4445 |has| |#1| (-562)) (-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#1| (QUOTE (-368))) (|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."))) 
-((-4449 . T) (-4450 . T)) 
-((-2738 (-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|))))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|))))))) 
+((-4448 . T) (-4449 . T)) 
+((-2740 (-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|))))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|))))))) 
 (-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| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) 
 NIL 
 ((|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-368)))) 
 (-47 R E) 
 ((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#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| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-48) 
 ((|constructor| (NIL "Algebraic closure of the rational numbers,{} with mathematical =")) (|norm| (($ $ (|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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| $ (QUOTE (-1058))) (|HasCategory| $ (LIST (QUOTE -1047) (QUOTE (-570))))) 
 (-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|)) $) "\\spad{parameters(f)} returns the list of parameters bound by \\spad{`f'}."))) 
@@ -130,7 +130,7 @@ NIL
 NIL 
 (-50 R |lVar|) 
 ((|constructor| (NIL "The domain of antisymmetric polynomials.")) (|map| (($ (|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}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-51 S) 
 ((|constructor| (NIL "\\spadtype{AnyFunctions1} implements several utility functions for working with \\spadtype{Any}. These functions are used to go back and forth between objects of \\spadtype{Any} and objects of other types.")) (|retract| ((|#1| (|Any|)) "\\spad{retract(a)} tries to convert \\spad{a} into an object of type \\spad{S}. If possible,{} it returns the object. Error: if no such retraction is possible.")) (|retractable?| (((|Boolean|) (|Any|)) "\\spad{retractable?(a)} tests if \\spad{a} can be converted into an object of type \\spad{S}.")) (|retractIfCan| (((|Union| |#1| "failed") (|Any|)) "\\spad{retractIfCan(a)} tries change \\spad{a} into an object of type \\spad{S}. If it can,{} then such an object is returned. Otherwise,{} \"failed\" is returned.")) (|coerce| (((|Any|) |#1|) "\\spad{coerce(s)} creates an object of \\spadtype{Any} from the object \\spad{s} of type \\spad{S}."))) 
@@ -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 -1673) 
+(-54 |Base| R -1674) 
 ((|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 
@@ -158,7 +158,7 @@ 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"))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-58 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),...]}."))) 
@@ -166,65 +166,65 @@ NIL
 NIL 
 (-59 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}"))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
-(-61 -3504) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+(-61 -3503) 
 ((|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 
-(-62 -3504) 
+(-62 -3503) 
 ((|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 
-(-63 -3504) 
+(-63 -3503) 
 ((|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 
-(-64 -3504) 
+(-64 -3503) 
 ((|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 
-(-65 -3504) 
+(-65 -3503) 
 ((|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 -3504) 
+(-66 -3503) 
 ((|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 -3504) 
+(-67 -3503) 
 ((|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 -3504) 
+(-68 -3503) 
 ((|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 -3504) 
+(-69 -3503) 
 ((|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 -3504) 
+(-70 -3503) 
 ((|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 -3504) 
+(-71 -3503) 
 ((|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 -3504) 
+(-72 -3503) 
 ((|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 -3504) 
+(-73 -3503) 
 ((|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 -3504) 
+(-74 -3503) 
 ((|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 
@@ -236,55 +236,55 @@ 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 
-(-77 -3504) 
+(-77 -3503) 
 ((|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 
-(-78 -3504) 
+(-78 -3503) 
 ((|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 
-(-79 -3504) 
+(-79 -3503) 
 ((|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 -3504) 
+(-80 -3503) 
 ((|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 -3504) 
+(-81 -3503) 
 ((|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 -3504) 
+(-82 -3503) 
 ((|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 
-(-83 -3504) 
+(-83 -3503) 
 ((|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 
-(-84 -3504) 
+(-84 -3503) 
 ((|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 
-(-85 -3504) 
+(-85 -3503) 
 ((|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 
-(-86 -3504) 
+(-86 -3503) 
 ((|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 
-(-87 -3504) 
+(-87 -3503) 
 ((|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 
-(-88 -3504) 
+(-88 -3503) 
 ((|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 
-(-89 -3504) 
+(-89 -3503) 
 ((|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 
@@ -294,8 +294,8 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-368)))) 
 (-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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (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\"."))) 
-((-4449 . T)) 
+((-4448 . 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."))) 
-((-4449 . T) ((-4451 "*") . T) (-4450 . T) (-4446 . T) (-4444 . T) (-4443 . T) (-4442 . T) (-4447 . T) (-4441 . T) (-4440 . T) (-4439 . T) (-4438 . T) (-4437 . T) (-4445 . T) (-4448 . T) (|NullSquare| . T) (|JacobiIdentity| . T) (-4436 . T)) 
+((-4448 . T) ((-4450 "*") . T) (-4449 . T) (-4445 . T) (-4443 . T) (-4442 . T) (-4441 . T) (-4446 . T) (-4440 . T) (-4439 . T) (-4438 . T) (-4437 . T) (-4436 . T) (-4444 . T) (-4447 . T) (|NullSquare| . T) (|JacobiIdentity| . T) (-4435 . 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}."))) 
-((-4446 . T)) 
+((-4445 . 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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (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 (-4451 "*")))) 
+((|HasAttribute| |#1| (QUOTE (-4450 "*")))) 
 (-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"))) 
-((-4449 . T)) 
+((-4448 . 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."))) 
-((-4450 . T)) 
+((-4449 . 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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2738 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2740 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
 (-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}"))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| (-112) (QUOTE (-1109))) (|HasCategory| (-112) (LIST (QUOTE -313) (QUOTE (-112))))) (|HasCategory| (-112) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-112) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-112) (QUOTE (-1109))) (|HasCategory| (-112) (LIST (QUOTE -619) (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}"))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 NIL 
 (-112) 
 ((|constructor| (NIL "\\indented{1}{\\spadtype{Boolean} is the elementary logic with 2 values:} \\spad{true} and \\spad{false}")) (|test| (($ $) "\\spad{test(b)} returns \\spad{b} and is provided for compatibility with the new compiler.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical negation of \\spad{a} or \\spad{b}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical negation of \\spad{a} and \\spad{b}.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical exclusive {\\em or} of Boolean \\spad{a} and \\spad{b}."))) 
@@ -392,22 +392,22 @@ NIL
 ((|constructor| (NIL "A basic operator is an object that can be applied to a list of arguments from a set,{} the result being a kernel over that set.")) (|setProperties| (($ $ (|AssociationList| (|String|) (|None|))) "\\spad{setProperties(op, l)} sets the property list of \\spad{op} to \\spad{l}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|setProperty| (($ $ (|Identifier|) (|None|)) "\\spad{setProperty(op, p, v)} attaches property \\spad{p} to \\spad{op},{} and sets its value to \\spad{v}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.") (($ $ (|String|) (|None|)) "\\spad{setProperty(op, s, v)} attaches property \\spad{s} to \\spad{op},{} and sets its value to \\spad{v}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|property| (((|Maybe| (|None|)) $ (|Identifier|)) "\\spad{property(op, p)} returns the value of property \\spad{p} if it is attached to \\spad{op},{} otherwise \\spad{nothing}.") (((|Union| (|None|) "failed") $ (|String|)) "\\spad{property(op, s)} returns the value of property \\spad{s} if it is attached to \\spad{op},{} and \"failed\" otherwise.")) (|deleteProperty!| (($ $ (|Identifier|)) "\\spad{deleteProperty!(op, p)} unattaches property \\spad{p} from \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.") (($ $ (|String|)) "\\spad{deleteProperty!(op, s)} unattaches property \\spad{s} from \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|assert| (($ $ (|Identifier|)) "\\spad{assert(op, p)} attaches property \\spad{p} to \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|has?| (((|Boolean|) $ (|Identifier|)) "\\spad{has?(op,p)} tests if property \\spad{s} is attached to \\spad{op}.")) (|input| (((|Union| (|Mapping| (|InputForm|) (|List| (|InputForm|))) "failed") $) "\\spad{input(op)} returns the \"\\%input\" property of \\spad{op} if it has one attached,{} \"failed\" otherwise.") (($ $ (|Mapping| (|InputForm|) (|List| (|InputForm|)))) "\\spad{input(op, foo)} attaches foo as the \"\\%input\" property of \\spad{op}. If \\spad{op} has a \"\\%input\" property \\spad{f},{} then \\spad{op(a1,...,an)} gets converted to InputForm as \\spad{f(a1,...,an)}.")) (|display| (($ $ (|Mapping| (|OutputForm|) (|OutputForm|))) "\\spad{display(op, foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a)} gets converted to OutputForm as \\spad{f(a)}. Argument \\spad{op} must be unary.") (($ $ (|Mapping| (|OutputForm|) (|List| (|OutputForm|)))) "\\spad{display(op, foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a1,...,an)} gets converted to OutputForm as \\spad{f(a1,...,an)}.") (((|Union| (|Mapping| (|OutputForm|) (|List| (|OutputForm|))) "failed") $) "\\spad{display(op)} returns the \"\\%display\" property of \\spad{op} if it has one attached,{} and \"failed\" otherwise.")) (|comparison| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{comparison(op, foo?)} attaches foo? as the \"\\%less?\" property to \\spad{op}. If op1 and op2 have the same name,{} and one of them has a \"\\%less?\" property \\spad{f},{} then \\spad{f(op1, op2)} is called to decide whether \\spad{op1 < op2}.")) (|equality| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{equality(op, foo?)} attaches foo? as the \"\\%equal?\" property to \\spad{op}. If op1 and op2 have the same name,{} and one of them has an \"\\%equal?\" property \\spad{f},{} then \\spad{f(op1, op2)} is called to decide whether op1 and op2 should be considered equal.")) (|weight| (($ $ (|NonNegativeInteger|)) "\\spad{weight(op, n)} attaches the weight \\spad{n} to \\spad{op}.") (((|NonNegativeInteger|) $) "\\spad{weight(op)} returns the weight attached to \\spad{op}.")) (|nary?| (((|Boolean|) $) "\\spad{nary?(op)} tests if \\spad{op} has arbitrary arity.")) (|unary?| (((|Boolean|) $) "\\spad{unary?(op)} tests if \\spad{op} is unary.")) (|nullary?| (((|Boolean|) $) "\\spad{nullary?(op)} tests if \\spad{op} is nullary.")) (|operator| (($ (|Symbol|) (|Arity|)) "\\spad{operator(f, a)} makes \\spad{f} into an operator of arity \\spad{a}.") (($ (|Symbol|) (|NonNegativeInteger|)) "\\spad{operator(f, n)} makes \\spad{f} into an \\spad{n}-ary operator.") (($ (|Symbol|)) "\\spad{operator(f)} makes \\spad{f} into an operator with arbitrary arity.")) (|copy| (($ $) "\\spad{copy(op)} returns a copy of \\spad{op}.")) (|properties| (((|AssociationList| (|String|) (|None|)) $) "\\spad{properties(op)} returns the list of all the properties currently attached to \\spad{op}."))) 
 NIL 
 NIL 
-(-116 -1673 UP) 
+(-116 -1674 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 
 (-117 |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}."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-118 |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}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-117 |#1|) (QUOTE (-916))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-148))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-117 |#1|) (QUOTE (-1031))) (|HasCategory| (-117 |#1|) (QUOTE (-826))) (-2738 (|HasCategory| (-117 |#1|) (QUOTE (-826))) (|HasCategory| (-117 |#1|) (QUOTE (-856)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-117 |#1|) (QUOTE (-1161))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| (-117 |#1|) (QUOTE (-235))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -520) (QUOTE (-1186)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -313) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -290) (LIST (QUOTE -117) (|devaluate| |#1|)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (QUOTE (-311))) (|HasCategory| (-117 |#1|) (QUOTE (-551))) (|HasCategory| (-117 |#1|) (QUOTE (-856))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-916)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-117 |#1|) (QUOTE (-916))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-148))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-117 |#1|) (QUOTE (-1031))) (|HasCategory| (-117 |#1|) (QUOTE (-826))) (-2740 (|HasCategory| (-117 |#1|) (QUOTE (-826))) (|HasCategory| (-117 |#1|) (QUOTE (-856)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-117 |#1|) (QUOTE (-1161))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| (-117 |#1|) (QUOTE (-235))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -520) (QUOTE (-1186)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -313) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (LIST (QUOTE -290) (LIST (QUOTE -117) (|devaluate| |#1|)) (LIST (QUOTE -117) (|devaluate| |#1|)))) (|HasCategory| (-117 |#1|) (QUOTE (-311))) (|HasCategory| (-117 |#1|) (QUOTE (-551))) (|HasCategory| (-117 |#1|) (QUOTE (-856))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-117 |#1|) (QUOTE (-916)))) (|HasCategory| (-117 |#1|) (QUOTE (-146))))) 
 (-119 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| (($ $) "\\spad{left(u)} returns the left child."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4450))) 
+((|HasAttribute| |#1| (QUOTE -4449))) 
 (-120 S) 
 ((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "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 
@@ -418,15 +418,15 @@ NIL
 NIL 
 (-122 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"))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-123 S) 
 ((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical {\\em exclusive-or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical {\\em nor} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical {\\em nand} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}."))) 
 NIL 
 NIL 
 (-124) 
 ((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical {\\em exclusive-or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical {\\em nor} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical {\\em nand} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-125 A S) 
 ((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#2| $) "\\spad{node(left,v,right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}.")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) 
@@ -434,20 +434,20 @@ NIL
 NIL 
 (-126 S) 
 ((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#1| $) "\\spad{node(left,v,right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}.")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-127 S) 
 ((|constructor| (NIL "\\spadtype{BinaryTournament(S)} is the domain of binary trees where elements are ordered down the tree. A binary search tree is either empty or is a node containing a \\spadfun{value} of type \\spad{S},{} and a \\spadfun{right} and a \\spadfun{left} which are both \\spadtype{BinaryTree(S)}")) (|insert!| (($ |#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."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-128 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."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-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."))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| (-130) (QUOTE (-856))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130))))) (-12 (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130)))))) (-2738 (-12 (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130))))) (|HasCategory| (-130) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-130) (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| (-130) (QUOTE (-856))) (|HasCategory| (-130) (QUOTE (-1109)))) (|HasCategory| (-130) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130)))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| (-130) (QUOTE (-856))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130))))) (-12 (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130)))))) (-2740 (-12 (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130))))) (|HasCategory| (-130) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-130) (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| (-130) (QUOTE (-856))) (|HasCategory| (-130) (QUOTE (-1109)))) (|HasCategory| (-130) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-130) (QUOTE (-1109))) (|HasCategory| (-130) (LIST (QUOTE -313) (QUOTE (-130)))))) 
 (-130) 
 ((|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 
@@ -470,13 +470,13 @@ NIL
 NIL 
 (-135) 
 ((|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") $ $) "\\spad{x - y} returns an element \\spad{z} such that \\spad{z+y=x} or \"failed\" if no such element exists.")) (|commutative| ((|attribute| "*") "a domain \\spad{D} has \\spad{commutative(\"*\")} if it has an operation \\spad{\"*\": (D,D) -> D} which is commutative."))) 
-(((-4451 "*") . T)) 
+(((-4450 "*") . T)) 
 NIL 
-(-136 |minix| -2407 S T$) 
+(-136 |minix| -2408 S T$) 
 ((|constructor| (NIL "This package provides functions to enable conversion of tensors given conversion of the components.")) (|map| (((|CartesianTensor| |#1| |#2| |#4|) (|Mapping| |#4| |#3|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{map(f,ts)} does a componentwise conversion of the tensor \\spad{ts} to a tensor with components of type \\spad{T}.")) (|reshape| (((|CartesianTensor| |#1| |#2| |#4|) (|List| |#4|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{reshape(lt,ts)} organizes the list of components \\spad{lt} into a tensor with the same shape as \\spad{ts}."))) 
 NIL 
 NIL 
-(-137 |minix| -2407 R) 
+(-137 |minix| -2408 R) 
 ((|constructor| (NIL "CartesianTensor(minix,{}dim,{}\\spad{R}) provides Cartesian tensors with components belonging to a commutative ring \\spad{R}. These tensors can have any number of indices. Each index takes values from \\spad{minix} to \\spad{minix + dim - 1}.")) (|sample| (($) "\\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 
@@ -498,8 +498,8 @@ NIL
 NIL 
 (-142) 
 ((|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}."))) 
-((-4449 . T) (-4439 . T) (-4450 . T)) 
-((-2738 (-12 (|HasCategory| (-145) (QUOTE (-373))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-145) (QUOTE (-373))) (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) 
+((-4448 . T) (-4438 . T) (-4449 . T)) 
+((-2740 (-12 (|HasCategory| (-145) (QUOTE (-373))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-145) (QUOTE (-373))) (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) 
 (-143 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 
@@ -514,7 +514,7 @@ NIL
 NIL 
 (-146) 
 ((|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."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-147 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}."))) 
@@ -522,9 +522,9 @@ NIL
 NIL 
 (-148) 
 ((|constructor| (NIL "Rings of Characteristic Zero."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
-(-149 -1673 UP UPUP) 
+(-149 -1674 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 
@@ -535,14 +535,14 @@ NIL
 (-151 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 -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasAttribute| |#1| (QUOTE -4449))) 
+((|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasAttribute| |#1| (QUOTE -4448))) 
 (-152 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 
 (-153 |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."))) 
-((-4444 . T) (-4443 . T) (-4446 . T)) 
+((-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
 (-154) 
 ((|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."))) 
@@ -564,7 +564,7 @@ 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 
-(-159 R -1673) 
+(-159 R -1674) 
 ((|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 
@@ -595,10 +595,10 @@ NIL
 (-166 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 (-916))) (|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (QUOTE (-1011))) (|HasCategory| |#2| (QUOTE (-1211))) (|HasCategory| |#2| (QUOTE (-1069))) (|HasCategory| |#2| (QUOTE (-1031))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4445)) (|HasAttribute| |#2| (QUOTE -4448)) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-562)))) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (QUOTE (-1011))) (|HasCategory| |#2| (QUOTE (-1211))) (|HasCategory| |#2| (QUOTE (-1069))) (|HasCategory| |#2| (QUOTE (-1031))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4444)) (|HasAttribute| |#2| (QUOTE -4447)) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-562)))) 
 (-167 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})"))) 
-((-4442 -2738 (|has| |#1| (-562)) (-12 (|has| |#1| (-311)) (|has| |#1| (-916)))) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4445 |has| |#1| (-6 -4445)) (-4448 |has| |#1| (-6 -4448)) (-3035 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 -2740 (|has| |#1| (-562)) (-12 (|has| |#1| (-311)) (|has| |#1| (-916)))) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4444 |has| |#1| (-6 -4444)) (-4447 |has| |#1| (-6 -4447)) (-3035 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-168 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."))) 
@@ -614,8 +614,8 @@ NIL
 NIL 
 (-171 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}."))) 
-((-4442 -2738 (|has| |#1| (-562)) (-12 (|has| |#1| (-311)) (|has| |#1| (-916)))) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4445 |has| |#1| (-6 -4445)) (-4448 |has| |#1| (-6 -4448)) (-3035 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-354))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-2738 (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (QUOTE (-235))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-834)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-1031)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-1211)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-368))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-916))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-916)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-916))))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1211)))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (QUOTE (-1031))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-562)))) (-2738 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#1| (QUOTE (-1069))) (-12 (|HasCategory| |#1| (QUOTE (-1069))) (|HasCategory| |#1| (QUOTE (-1211)))) (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-368)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-235))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasAttribute| |#1| (QUOTE -4445)) (|HasAttribute| |#1| (QUOTE -4448)) (-12 (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (QUOTE (-368)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-354))))) 
+((-4441 -2740 (|has| |#1| (-562)) (-12 (|has| |#1| (-311)) (|has| |#1| (-916)))) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4444 |has| |#1| (-6 -4444)) (-4447 |has| |#1| (-6 -4447)) (-3035 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-354))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-2740 (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (QUOTE (-235))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-373)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-834)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-1031)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-1211)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-368))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-916))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-916)))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-916))))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1211)))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (QUOTE (-1031))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-562)))) (-2740 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| |#1| (QUOTE (-1069))) (-12 (|HasCategory| |#1| (QUOTE (-1069))) (|HasCategory| |#1| (QUOTE (-1211)))) (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-368)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-235))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasAttribute| |#1| (QUOTE -4444)) (|HasAttribute| |#1| (QUOTE -4447)) (-12 (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (QUOTE (-368)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-354))))) 
 (-172 R S CS) 
 ((|constructor| (NIL "This package supports converting complex expressions to patterns")) (|convert| (((|Pattern| |#1|) |#3|) "\\spad{convert(cs)} converts the complex expression \\spad{cs} to a pattern"))) 
 NIL 
@@ -626,7 +626,7 @@ NIL
 NIL 
 (-174) 
 ((|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."))) 
-(((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-175) 
 ((|constructor| (NIL "This category is the root of the I/O conduits.")) (|close!| (($ $) "\\spad{close!(c)} closes the conduit \\spad{c},{} changing its state to one that is invalid for future read or write operations."))) 
@@ -634,7 +634,7 @@ NIL
 NIL 
 (-176 R) 
 ((|constructor| (NIL "\\spadtype{ContinuedFraction} implements general \\indented{1}{continued fractions.\\space{2}This version is not restricted to simple,{}} \\indented{1}{finite fractions and uses the \\spadtype{Stream} as a} \\indented{1}{representation.\\space{2}The arithmetic functions assume that the} \\indented{1}{approximants alternate below/above the convergence point.} \\indented{1}{This is enforced by ensuring the partial numerators and partial} \\indented{1}{denominators are greater than 0 in the Euclidean domain view of \\spad{R}} \\indented{1}{(\\spadignore{i.e.} \\spad{sizeLess?(0, x)}).}")) (|complete| (($ $) "\\spad{complete(x)} causes all entries in \\spadvar{\\spad{x}} to be computed. Normally entries are only computed as needed. If \\spadvar{\\spad{x}} is an infinite continued fraction,{} a user-initiated interrupt is necessary to stop the computation.")) (|extend| (($ $ (|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}."))) 
-(((-4451 "*") . T) (-4442 . T) (-4447 . T) (-4441 . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") . T) (-4441 . T) (-4446 . T) (-4440 . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-177) 
 ((|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}."))) 
@@ -688,7 +688,7 @@ NIL
 ((|constructor| (NIL "This domain provides implementations for constructors.")) (|findConstructor| (((|Maybe| $) (|Identifier|)) "\\spad{findConstructor(s)} attempts to find a constructor named \\spad{s}. If successful,{} returns that constructor; otherwise,{} returns \\spad{nothing}."))) 
 NIL 
 NIL 
-(-190 R -1673) 
+(-190 R -1674) 
 ((|constructor| (NIL "\\spadtype{ComplexTrigonometricManipulations} provides function that compute the real and imaginary parts of complex functions.")) (|complexForm| (((|Complex| (|Expression| |#1|)) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f, imag f]}.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| (((|Expression| |#1|) |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| (((|Expression| |#1|) |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f, x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f, x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) 
 NIL 
 NIL 
@@ -796,23 +796,23 @@ NIL
 ((|constructor| (NIL "\\indented{1}{This domain implements a simple view of a database whose fields are} indexed by symbols")) (- (($ $ $) "\\spad{db1-db2} returns the difference of databases \\spad{db1} and \\spad{db2} \\spadignore{i.e.} consisting of elements in \\spad{db1} but not in \\spad{db2}")) (+ (($ $ $) "\\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 
-(-217 -1673 UP UPUP R) 
+(-217 -1674 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 
-(-218 -1673 FP) 
+(-218 -1674 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 
 (-219) 
 ((|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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2738 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2740 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
 (-220) 
 ((|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|) $) "\\spad{head(d)} returns the head of the definition \\spad{`d'}. This is a list of identifiers starting with the name of the operation followed by the name of the parameters,{} if any."))) 
 NIL 
 NIL 
-(-221 R -1673) 
+(-221 R -1674) 
 ((|constructor| (NIL "\\spadtype{ElementaryFunctionDefiniteIntegration} provides functions to compute definite integrals of elementary functions.")) (|innerint| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{innerint(f, x, a, b, ignore?)} should be local but conditional")) (|integrate| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|)) (|String|)) "\\spad{integrate(f, x = a..b, \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| "failed") (|:| |pole| "potentialPole")) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|))) "\\spad{integrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}."))) 
 NIL 
 NIL 
@@ -826,19 +826,19 @@ NIL
 NIL 
 (-224 S) 
 ((|constructor| (NIL "Linked list implementation of a Dequeue")) (|dequeue| (($ (|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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-225 |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}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
-(-226 R -1673) 
+(-226 R -1674) 
 ((|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 
 (-227) 
 ((|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}."))) 
-((-3026 . T) (-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-3026 . T) (-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-228) 
 ((|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)}.}"))) 
@@ -846,15 +846,15 @@ NIL
 NIL 
 (-229 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}"))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4451 "*"))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4450 "*"))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-230 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 
 (-231 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."))) 
-((-4450 . T)) 
+((-4449 . T)) 
 NIL 
 (-232 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}."))) 
@@ -862,7 +862,7 @@ NIL
 ((|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235)))) 
 (-233 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}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-234 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}.") (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified."))) 
@@ -870,36 +870,36 @@ NIL
 NIL 
 (-235) 
 ((|constructor| (NIL "An ordinary differential ring,{} that is,{} a ring with an operation \\spadfun{differentiate}. \\blankline")) (D (($ $ (|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}.") (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}. This function is a simple differential operator where no variable needs to be specified."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-236 A S) 
 ((|constructor| (NIL "This category is a collection of operations common to both categories \\spadtype{Dictionary} and \\spadtype{MultiDictionary}")) (|select!| (($ (|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 -4449))) 
+((|HasAttribute| |#1| (QUOTE -4448))) 
 (-237 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}."))) 
-((-4450 . T)) 
+((-4449 . T)) 
 NIL 
 (-238) 
 ((|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 
-(-239 S -2407 R) 
+(-239 S -2408 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 (-368))) (|HasCategory| |#3| (QUOTE (-799))) (|HasCategory| |#3| (QUOTE (-854))) (|HasAttribute| |#3| (QUOTE -4446)) (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (QUOTE (-732))) (|HasCategory| |#3| (QUOTE (-132))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-1058))) (|HasCategory| |#3| (QUOTE (-1109)))) 
-(-240 -2407 R) 
+((|HasCategory| |#3| (QUOTE (-368))) (|HasCategory| |#3| (QUOTE (-799))) (|HasCategory| |#3| (QUOTE (-854))) (|HasAttribute| |#3| (QUOTE -4445)) (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (QUOTE (-732))) (|HasCategory| |#3| (QUOTE (-132))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-1058))) (|HasCategory| |#3| (QUOTE (-1109)))) 
+(-240 -2408 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"))) 
-((-4443 |has| |#2| (-1058)) (-4444 |has| |#2| (-1058)) (-4446 |has| |#2| (-6 -4446)) ((-4451 "*") |has| |#2| (-174)) (-4449 . T)) 
+((-4442 |has| |#2| (-1058)) (-4443 |has| |#2| (-1058)) (-4445 |has| |#2| (-6 -4445)) ((-4450 "*") |has| |#2| (-174)) (-4448 . T)) 
 NIL 
-(-241 -2407 A B) 
+(-241 -2408 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 
-(-242 -2407 R) 
+(-242 -2408 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."))) 
-((-4443 |has| |#2| (-1058)) (-4444 |has| |#2| (-1058)) (-4446 |has| |#2| (-6 -4446)) ((-4451 "*") |has| |#2| (-174)) (-4449 . T)) 
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+((-4442 |has| |#2| (-1058)) (-4443 |has| |#2| (-1058)) (-4445 |has| |#2| (-6 -4445)) ((-4450 "*") |has| |#2| (-174)) (-4448 . T)) 
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 (-243) 
 ((|constructor| (NIL "DisplayPackage allows one to print strings in a nice manner,{} including highlighting substrings.")) (|sayLength| (((|Integer|) (|List| (|String|))) "\\spad{sayLength(l)} returns the length of a list of strings \\spad{l} as an integer.") (((|Integer|) (|String|)) "\\spad{sayLength(s)} returns the length of a string \\spad{s} as an integer.")) (|say| (((|Void|) (|List| (|String|))) "\\spad{say(l)} sends a list of strings \\spad{l} to output.") (((|Void|) (|String|)) "\\spad{say(s)} sends a string \\spad{s} to output.")) (|center| (((|List| (|String|)) (|List| (|String|)) (|Integer|) (|String|)) "\\spad{center(l,i,s)} takes a list of strings \\spad{l},{} and centers them within a list of strings which is \\spad{i} characters long,{} in which the remaining spaces are filled with strings composed of as many repetitions as possible of the last string parameter \\spad{s}.") (((|String|) (|String|) (|Integer|) (|String|)) "\\spad{center(s,i,s)} takes the first string \\spad{s},{} and centers it within a string of length \\spad{i},{} in which the other elements of the string are composed of as many replications as possible of the second indicated string,{} \\spad{s} which must have a length greater than that of an empty string.")) (|copies| (((|String|) (|Integer|) (|String|)) "\\spad{copies(i,s)} will take a string \\spad{s} and create a new string composed of \\spad{i} copies of \\spad{s}.")) (|newLine| (((|String|)) "\\spad{newLine()} sends a new line command to output.")) (|bright| (((|List| (|String|)) (|List| (|String|))) "\\spad{bright(l)} sets the font property of a list of strings,{} \\spad{l},{} to bold-face type.") (((|List| (|String|)) (|String|)) "\\spad{bright(s)} sets the font property of the string \\spad{s} to bold-face type."))) 
 NIL 
@@ -910,7 +910,7 @@ NIL
 NIL 
 (-245) 
 ((|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}."))) 
-((-4442 . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-246 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."))) 
@@ -918,16 +918,16 @@ NIL
 NIL 
 (-247 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}"))) 
-((-4450 . T) (-4449 . T)) 
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+((-4449 . T) (-4448 . T)) 
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 (-248 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 
 (-249 |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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 ((|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 
@@ -942,23 +942,23 @@ NIL
 NIL 
 (-253 |n| R M S) 
 ((|constructor| (NIL "This constructor provides a direct product type with a left matrix-module view."))) 
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(QUOTE (-570))))) (-12 (|HasCategory| |#3| (QUOTE (-1058))) (|HasCategory| |#3| (LIST (QUOTE -907) (QUOTE (-1186)))))) (|HasCategory| |#3| (QUOTE (-132))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|))))) 
 (-255 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 (-235)))) 
 (-256 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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
 (-257 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}."))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-258) 
 ((|constructor| (NIL "TopLevelDrawFunctionsForCompiledFunctions provides top level functions for drawing graphics of expressions.")) (|recolor| (((|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{recolor()},{} uninteresting to top level user; exported in order to compile package.")) (|makeObject| (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(surface(f,g,h),a..b,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)}.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(surface(f,g,h),a..b,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)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(f,a..b,c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{f(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)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f,a..b,c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{f(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)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(f,a..b,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)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f,a..b,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)},{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{makeObject(sp,curve(f,g,h),a..b)} returns the space \\spad{sp} of the domain \\spadtype{ThreeSpace} with the addition of the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f,g,h),a..b,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{makeObject(sp,curve(f,g,h),a..b)} returns the space \\spad{sp} of the domain \\spadtype{ThreeSpace} with the addition of the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f,g,h),a..b,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.")) (|draw| (((|ThreeDimensionalViewport|) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(surface(f,g,h),a..b,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)}.") (((|ThreeDimensionalViewport|) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(surface(f,g,h),a..b,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)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b,c..d)} draws the graph of the parametric surface \\spad{f(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)} The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,a..b,c..d)} draws the graph of the parametric surface \\spad{f(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)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b,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)}.") (((|ThreeDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,a..b,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)}. and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b,l)} draws the graph of the parametric curve \\spad{f} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,a..b,l)} draws the graph of the parametric curve \\spad{f} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{draw(curve(f,g,h),a..b,l)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f,g,h),a..b,l)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{draw(curve(f,g),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)}.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f,g),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)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|TwoDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,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)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied."))) 
@@ -998,8 +998,8 @@ NIL
 NIL 
 (-267 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"))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
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 (-268 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 
@@ -1044,11 +1044,11 @@ 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 
-(-279 R -1673) 
+(-279 R -1674) 
 ((|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 
-(-280 R -1673) 
+(-280 R -1674) 
 ((|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 
@@ -1074,7 +1074,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109)))) 
 (-286 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}."))) 
-((-4450 . T)) 
+((-4449 . T)) 
 NIL 
 (-287 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}."))) 
@@ -1095,18 +1095,18 @@ NIL
 (-291 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 -4450))) 
+((|HasAttribute| |#1| (QUOTE -4449))) 
 (-292 |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 
-(-293 S R |Mod| -2942 -1585 |exactQuo|) 
+(-293 S R |Mod| -4052 -3068 |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| $) "\\spad{modulus(x)} \\undocumented"))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-294) 
 ((|constructor| (NIL "Entire Rings (non-commutative Integral Domains),{} \\spadignore{i.e.} a ring not necessarily commutative which has no zero divisors. \\blankline")) (|noZeroDivisors| ((|attribute|) "if a product is zero then one of the factors must be zero."))) 
-((-4442 . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-295) 
 ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: March 18,{} 2010. An `Environment' is a stack of scope.")) (|categoryFrame| (($) "the current category environment in the interpreter.")) (|interactiveEnv| (($) "the current interactive environment in effect.")) (|currentEnv| (($) "the current normal environment in effect.")) (|putProperties| (($ (|Identifier|) (|List| (|Property|)) $) "\\spad{putProperties(n,props,e)} set the list of properties of \\spad{n} to \\spad{props} in \\spad{e}.")) (|getProperties| (((|List| (|Property|)) (|Identifier|) $) "\\spad{getBinding(n,e)} returns the list of properties of \\spad{n} in \\spad{e}.")) (|putProperty| (($ (|Identifier|) (|Identifier|) (|SExpression|) $) "\\spad{putProperty(n,p,v,e)} binds the property \\spad{(p,v)} to \\spad{n} in the topmost scope of \\spad{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 \\spad{e}. Otherwise,{} \\spad{nothing}.")) (|scopes| (((|List| (|Scope|)) $) "\\spad{scopes(e)} returns the stack of scopes in environment \\spad{e}.")) (|empty| (($) "\\spad{empty()} constructs an empty environment"))) 
@@ -1122,21 +1122,21 @@ NIL
 NIL 
 (-298 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."))) 
-((-4446 -2738 (|has| |#1| (-1058)) (|has| |#1| (-479))) (-4443 |has| |#1| (-1058)) (-4444 |has| |#1| (-1058))) 
-((|HasCategory| |#1| (QUOTE (-368))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2738 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-732)))) (|HasCategory| |#1| (QUOTE (-479))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-1109)))) (-2738 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1121)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-306))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-479)))) (-2738 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732)))) (-2738 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-732)))) 
+((-4445 -2740 (|has| |#1| (-1058)) (|has| |#1| (-479))) (-4442 |has| |#1| (-1058)) (-4443 |has| |#1| (-1058))) 
+((|HasCategory| |#1| (QUOTE (-368))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-1058)))) (-2740 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-732)))) (|HasCategory| |#1| (QUOTE (-479))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-1109)))) (-2740 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1121)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-306))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-479)))) (-2740 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732)))) (-2740 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-732)))) 
 (-299 |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."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-300) 
 ((|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 
-(-301 -1673 S) 
+(-301 -1674 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 -1673) 
+(-302 E -1674) 
 ((|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 
@@ -1174,7 +1174,7 @@ NIL
 NIL 
 (-311) 
 ((|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|) $ $) "\\spad{sizeLess?(x,y)} tests whether \\spad{x} is strictly smaller than \\spad{y} with respect to the \\spadfunFrom{euclideanSize}{EuclideanDomain}."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-312 S R) 
 ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\\spad{''} substitutions.")) (|eval| (($ $ (|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}."))) 
@@ -1184,7 +1184,7 @@ NIL
 ((|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 
-(-314 -1673) 
+(-314 -1674) 
 ((|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 
@@ -1198,8 +1198,8 @@ NIL
 NIL 
 (-317 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))}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
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+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-916))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-146))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-148))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-1031))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-826))) (-2740 (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-826))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-856)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-1161))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-235))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -520) (QUOTE (-1186)) (LIST (QUOTE -1262) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -313) (LIST (QUOTE -1262) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (LIST (QUOTE -290) (LIST (QUOTE -1262) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)) (LIST (QUOTE -1262) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-311))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-551))) (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-856))) (-12 (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-916))) (|HasCategory| $ (QUOTE (-146)))) (-2740 (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-146))) (-12 (|HasCategory| (-1262 |#1| |#2| |#3| |#4|) (QUOTE (-916))) (|HasCategory| $ (QUOTE (-146)))))) 
 (-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 
@@ -1210,9 +1210,9 @@ NIL
 NIL 
 (-320 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."))) 
-((-4446 -2738 (-1764 (|has| |#1| (-1058)) (|has| |#1| (-645 (-570)))) (-12 (|has| |#1| (-562)) (-2738 (-1764 (|has| |#1| (-1058)) (|has| |#1| (-645 (-570)))) (|has| |#1| (-1058)) (|has| |#1| (-479)))) (|has| |#1| (-1058)) (|has| |#1| (-479))) (-4444 |has| |#1| (-174)) (-4443 |has| |#1| (-174)) ((-4451 "*") |has| |#1| (-562)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-562)) (-4441 |has| |#1| (-562))) 
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-(-321 R -1673) 
+((-4445 -2740 (-1765 (|has| |#1| (-1058)) (|has| |#1| (-645 (-570)))) (-12 (|has| |#1| (-562)) (-2740 (-1765 (|has| |#1| (-1058)) (|has| |#1| (-645 (-570)))) (|has| |#1| (-1058)) (|has| |#1| (-479)))) (|has| |#1| (-1058)) (|has| |#1| (-479))) (-4443 |has| |#1| (-174)) (-4442 |has| |#1| (-174)) ((-4450 "*") |has| |#1| (-562)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-562)) (-4440 |has| |#1| (-562))) 
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+(-321 R -1674) 
 ((|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 
@@ -1222,8 +1222,8 @@ NIL
 NIL 
 (-323 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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2738 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3665) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2740 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3555) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
 (-324 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 
@@ -1234,7 +1234,7 @@ NIL
 NIL 
 (-326 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."))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-798)))) 
 (-327 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| $) "\\spad{s + x} returns the sum of \\spad{s} and \\spad{x}."))) 
@@ -1250,19 +1250,19 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174)))) 
 (-330 R E) 
 ((|constructor| (NIL "This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.")) (|primitivePart| (($ $) "\\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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-331 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."))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
-(-332 S -1673) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+(-332 S -1674) 
 ((|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 (-373)))) 
-(-333 -1673) 
+(-333 -1674) 
 ((|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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-334) 
 ((|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."))) 
@@ -1284,15 +1284,15 @@ NIL
 ((|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 
-(-339 S -1673 UP UPUP R) 
+(-339 S -1674 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 
-(-340 -1673 UP UPUP R) 
+(-340 -1674 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 
-(-341 -1673 UP UPUP R) 
+(-341 -1674 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 
@@ -1306,32 +1306,32 @@ NIL
 NIL 
 (-344 |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.}"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#3| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#3| (LIST (QUOTE -1047) (QUOTE (-384)))) (|HasCategory| $ (QUOTE (-1058))) (|HasCategory| $ (LIST (QUOTE -1047) (QUOTE (-570))))) 
 (-345 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 
-(-346 S -1673 UP UPUP) 
+(-346 S -1674 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 (-373))) (|HasCategory| |#2| (QUOTE (-368)))) 
-(-347 -1673 UP UPUP) 
+(-347 -1674 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."))) 
-((-4442 |has| (-413 |#2|) (-368)) (-4447 |has| (-413 |#2|) (-368)) (-4441 |has| (-413 |#2|) (-368)) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 |has| (-413 |#2|) (-368)) (-4446 |has| (-413 |#2|) (-368)) (-4440 |has| (-413 |#2|) (-368)) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| (-917 |#1|) (QUOTE (-146))) (|HasCategory| (-917 |#1|) (QUOTE (-373)))) (|HasCategory| (-917 |#1|) (QUOTE (-148))) (|HasCategory| (-917 |#1|) (QUOTE (-373))) (|HasCategory| (-917 |#1|) (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| (-917 |#1|) (QUOTE (-146))) (|HasCategory| (-917 |#1|) (QUOTE (-373)))) (|HasCategory| (-917 |#1|) (QUOTE (-148))) (|HasCategory| (-917 |#1|) (QUOTE (-373))) (|HasCategory| (-917 |#1|) (QUOTE (-146)))) 
 (-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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
 (-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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
 (-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 
@@ -1346,33 +1346,33 @@ NIL
 NIL 
 (-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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
-(-355 R UP -1673) 
+(-355 R UP -1674) 
 ((|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 
 (-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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| (-917 |#1|) (QUOTE (-146))) (|HasCategory| (-917 |#1|) (QUOTE (-373)))) (|HasCategory| (-917 |#1|) (QUOTE (-148))) (|HasCategory| (-917 |#1|) (QUOTE (-373))) (|HasCategory| (-917 |#1|) (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| (-917 |#1|) (QUOTE (-146))) (|HasCategory| (-917 |#1|) (QUOTE (-373)))) (|HasCategory| (-917 |#1|) (QUOTE (-148))) (|HasCategory| (-917 |#1|) (QUOTE (-373))) (|HasCategory| (-917 |#1|) (QUOTE (-146)))) 
 (-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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
 (-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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
 (-359 |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}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| (-917 |#1|) (QUOTE (-146))) (|HasCategory| (-917 |#1|) (QUOTE (-373)))) (|HasCategory| (-917 |#1|) (QUOTE (-148))) (|HasCategory| (-917 |#1|) (QUOTE (-373))) (|HasCategory| (-917 |#1|) (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| (-917 |#1|) (QUOTE (-146))) (|HasCategory| (-917 |#1|) (QUOTE (-373)))) (|HasCategory| (-917 |#1|) (QUOTE (-148))) (|HasCategory| (-917 |#1|) (QUOTE (-373))) (|HasCategory| (-917 |#1|) (QUOTE (-146)))) 
 (-360 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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
-(-361 -1673 GF) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
+(-361 -1674 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 
@@ -1380,21 +1380,21 @@ NIL
 ((|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 
-(-363 -1673 FP FPP) 
+(-363 -1674 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 
 (-364 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}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-146)))) 
 (-365 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 
 (-366 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| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-367 S) 
 ((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) 
@@ -1402,7 +1402,7 @@ NIL
 NIL 
 (-368) 
 ((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-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."))) 
@@ -1418,7 +1418,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-562)))) 
 (-372 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| $)) "\\spad{someBasis()} returns some \\spad{R}-module basis."))) 
-((-4446 |has| |#1| (-562)) (-4444 . T) (-4443 . T)) 
+((-4445 |has| |#1| (-562)) (-4443 . T) (-4442 . T)) 
 NIL 
 (-373) 
 ((|constructor| (NIL "The category of domains composed of a finite set of elements. We include the functions \\spadfun{lookup} and \\spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \\blankline")) (|random| (($) "\\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."))) 
@@ -1430,7 +1430,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-368)))) 
 (-375 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| $)) "\\spad{regularRepresentation(a,basis)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-376 S A R B) 
 ((|constructor| (NIL "FiniteLinearAggregateFunctions2 provides functions involving two FiniteLinearAggregates where the underlying domains might be different. An example of this might be creating a list of rational numbers by mapping a function across a list of integers where the function divides each integer by 1000.")) (|scan| ((|#4| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-aggregates \\spad{x} of aggregrate \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}.")) (|reduce| ((|#3| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the aggregate \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$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."))) 
@@ -1439,14 +1439,14 @@ NIL
 (-377 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|) $ $) "\\spad{merge(p,a,b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4450)) (|HasCategory| |#2| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109)))) 
+((|HasAttribute| |#1| (QUOTE -4449)) (|HasCategory| |#2| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109)))) 
 (-378 S) 
 ((|constructor| (NIL "A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \\spadtype{LinearAggregate} such as \\spadfun{reverse},{} \\spadfun{sort},{} and so on.")) (|sort!| (($ $) "\\spad{sort!(u)} returns \\spad{u} with its elements in ascending order.") (($ (|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|) $ $) "\\spad{merge(p,a,b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) 
-((-4449 . T)) 
+((-4448 . T)) 
 NIL 
 (-379 |VarSet| R) 
 ((|constructor| (NIL "The category of free Lie algebras. It is used by domains of non-commutative algebra: \\spadtype{LiePolynomial} and \\spadtype{XPBWPolynomial}. \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr})")) (|eval| (($ $ (|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) (-4444 . T) (-4443 . T)) 
+((|JacobiIdentity| . T) (|NullSquare| . T) (-4443 . T) (-4442 . T)) 
 NIL 
 (-380 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."))) 
@@ -1458,7 +1458,7 @@ NIL
 ((|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570))))) 
 (-382 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}"))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-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}."))) 
@@ -1466,7 +1466,7 @@ NIL
 NIL 
 (-384) 
 ((|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}."))) 
-((-4432 . T) (-4440 . T) (-3026 . T) (-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4431 . T) (-4439 . T) (-3026 . T) (-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-385 |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}."))) 
@@ -1474,11 +1474,11 @@ NIL
 NIL 
 (-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}"))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174)))) 
 (-387 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}."))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 NIL 
 (-388) 
 ((|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}."))) 
@@ -1490,7 +1490,7 @@ NIL
 NIL 
 (-390 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."))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174)))) 
 (-391 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| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) 
@@ -1502,7 +1502,7 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-856)))) 
 (-393) 
 ((|constructor| (NIL "A category of domains which model machine arithmetic used by machines in the AXIOM-NAG link."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-394) 
 ((|constructor| (NIL "This domain provides an interface to names in the file system."))) 
@@ -1514,13 +1514,13 @@ NIL
 NIL 
 (-396 |n| |class| R) 
 ((|constructor| (NIL "Generate the Free Lie Algebra over a ring \\spad{R} with identity; A \\spad{P}. Hall basis is generated by a package call to HallBasis.")) (|generator| (($ (|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"))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 NIL 
 (-397) 
 ((|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 
-(-398 -1673 UP UPUP R) 
+(-398 -1674 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 
@@ -1544,11 +1544,11 @@ NIL
 ((|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 
-(-404 -3504 |returnType| -3890 |symbols|) 
+(-404 -3503 |returnType| -3890 |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 
-(-405 -1673 UP) 
+(-405 -1674 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| $) "\\spad{p + x} returns the sum of \\spad{p} and \\spad{x}"))) 
 NIL 
 NIL 
@@ -1562,15 +1562,15 @@ NIL
 NIL 
 (-408) 
 ((|constructor| (NIL "FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \\spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.")) (|primeFrobenius| (($ $ (|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|)) $) "\\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \\spad{a} is 0."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-409 S) 
 ((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline \\indented{2}{1: \\spadfunFrom{base}{FloatingPointSystem} of the \\spadfunFrom{exponent}{FloatingPointSystem}.} \\indented{9}{(actual implemenations are usually binary or decimal)} \\indented{2}{2: \\spadfunFrom{precision}{FloatingPointSystem} of the \\spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \\indented{2}{3: rounding error for operations} \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\\spad{\"+\"}) does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\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 -4432)) (|HasAttribute| |#1| (QUOTE -4440))) 
+((|HasAttribute| |#1| (QUOTE -4431)) (|HasAttribute| |#1| (QUOTE -4439))) 
 (-410) 
 ((|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\"."))) 
-((-3026 . T) (-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-3026 . T) (-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-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."))) 
@@ -1582,15 +1582,15 @@ NIL
 NIL 
 (-413 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."))) 
-((-4436 -12 (|has| |#1| (-6 -4447)) (|has| |#1| (-458)) (|has| |#1| (-6 -4436))) (-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (QUOTE (-1031))) (|HasCategory| |#1| (QUOTE (-826))) (-2738 (|HasCategory| |#1| (QUOTE (-826))) (|HasCategory| |#1| (QUOTE (-856)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-1161))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834))))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834))))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-551))) (-12 (|HasAttribute| |#1| (QUOTE -4447)) (|HasAttribute| |#1| (QUOTE -4436)) (|HasCategory| |#1| (QUOTE (-458)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
+((-4435 -12 (|has| |#1| (-6 -4446)) (|has| |#1| (-458)) (|has| |#1| (-6 -4435))) (-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (QUOTE (-1031))) (|HasCategory| |#1| (QUOTE (-826))) (-2740 (|HasCategory| |#1| (QUOTE (-826))) (|HasCategory| |#1| (QUOTE (-856)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-1161))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834))))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834))))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-834)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-551))) (-12 (|HasAttribute| |#1| (QUOTE -4446)) (|HasAttribute| |#1| (QUOTE -4435)) (|HasCategory| |#1| (QUOTE (-458)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
 (-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 
 (-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| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-416 A S) 
 ((|constructor| (NIL "\\indented{2}{A is fully retractable to \\spad{B} means that A is retractable to \\spad{B},{} and,{}} \\indented{2}{in addition,{} if \\spad{B} is retractable to the integers or rational} \\indented{2}{numbers then so is A.} \\indented{2}{In particular,{} what we are asserting is that there are no integers} \\indented{2}{(rationals) in A which don\\spad{'t} retract into \\spad{B}.} Date Created: March 1990 Date Last Updated: 9 April 1991"))) 
@@ -1604,11 +1604,11 @@ NIL
 ((|constructor| (NIL "\\indented{1}{Lifting of morphisms to fractional ideals.} Author: Manuel Bronstein Date Created: 1 Feb 1989 Date Last Updated: 27 Feb 1990 Keywords: ideal,{} algebra,{} module.")) (|map| (((|FractionalIdeal| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FractionalIdeal| |#1| |#2| |#3| |#4|)) "\\spad{map(f,i)} \\undocumented{}"))) 
 NIL 
 NIL 
-(-419 R -1673 UP A) 
+(-419 R -1674 UP A) 
 ((|constructor| (NIL "Fractional ideals in a framed algebra.")) (|randomLC| ((|#4| (|NonNegativeInteger|) (|Vector| |#4|)) "\\spad{randomLC(n,x)} should be local but conditional.")) (|minimize| (($ $) "\\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)}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
-(-420 R -1673 UP A |ibasis|) 
+(-420 R -1674 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|) $) "\\spad{basis((f1,...,fn))} = the vector \\spad{[f1,...,fn]}."))) 
 NIL 
 ((|HasCategory| |#4| (LIST (QUOTE -1047) (|devaluate| |#2|)))) 
@@ -1622,12 +1622,12 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-368)))) 
 (-423 R) 
 ((|constructor| (NIL "FramedNonAssociativeAlgebra(\\spad{R}) is a \\spadtype{FiniteRankNonAssociativeAlgebra} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank) over a commutative ring \\spad{R} together with a fixed \\spad{R}-module basis.")) (|apply| (($ (|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."))) 
-((-4446 |has| |#1| (-562)) (-4444 . T) (-4443 . T)) 
+((-4445 |has| |#1| (-562)) (-4443 . T) (-4442 . T)) 
 NIL 
 (-424 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| "nil" "sqfr" "irred" "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| "nil" "sqfr" "irred" "prime") $ (|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| "nil" "sqfr" "irred" "prime")) (|:| |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| "nil" "sqfr" "irred" "prime")) (|:| |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."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -313) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -290) (QUOTE $) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-1230))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-1230)))) (|HasCategory| |#1| (QUOTE (-1031))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-458)))) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -313) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -290) (QUOTE $) (QUOTE $))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-1230))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-1230)))) (|HasCategory| |#1| (QUOTE (-1031))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-458)))) 
 (-425 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 
@@ -1654,17 +1654,17 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-373)))) 
 (-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|) $) "\\spad{cardinality(u)} returns the number of elements of \\spad{u}. Note: \\axiom{cardinality(\\spad{u}) = \\#u}."))) 
-((-4449 . T) (-4439 . T) (-4450 . T)) 
+((-4448 . T) (-4438 . T) (-4449 . T)) 
 NIL 
-(-432 R -1673) 
+(-432 R -1674) 
 ((|constructor| (NIL "\\spadtype{FunctionSpaceComplexIntegration} provides functions for the indefinite integration of complex-valued functions.")) (|complexIntegrate| ((|#2| |#2| (|Symbol|)) "\\spad{complexIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable.")) (|internalIntegrate0| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate0 should} be a local function,{} but is conditional.")) (|internalIntegrate| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable."))) 
 NIL 
 NIL 
 (-433 R E) 
 ((|constructor| (NIL "\\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:")) (|makeCos| (($ |#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"))) 
-((-4436 -12 (|has| |#1| (-6 -4436)) (|has| |#2| (-6 -4436))) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-12 (|HasAttribute| |#1| (QUOTE -4436)) (|HasAttribute| |#2| (QUOTE -4436)))) 
-(-434 R -1673) 
+((-4435 -12 (|has| |#1| (-6 -4435)) (|has| |#2| (-6 -4435))) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-12 (|HasAttribute| |#1| (QUOTE -4435)) (|HasAttribute| |#2| (QUOTE -4435)))) 
+(-434 R -1674) 
 ((|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 
@@ -1674,17 +1674,17 @@ NIL
 ((|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-1058))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-1121))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) 
 (-436 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}."))) 
-((-4446 -2738 (|has| |#1| (-1058)) (|has| |#1| (-479))) (-4444 |has| |#1| (-174)) (-4443 |has| |#1| (-174)) ((-4451 "*") |has| |#1| (-562)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-562)) (-4441 |has| |#1| (-562))) 
+((-4445 -2740 (|has| |#1| (-1058)) (|has| |#1| (-479))) (-4443 |has| |#1| (-174)) (-4442 |has| |#1| (-174)) ((-4450 "*") |has| |#1| (-562)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-562)) (-4440 |has| |#1| (-562))) 
 NIL 
-(-437 R -1673) 
+(-437 R -1674) 
 ((|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 
-(-438 R -1673) 
+(-438 R -1674) 
 ((|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)))) 
-(-439 R -1673) 
+(-439 R -1674) 
 ((|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 
@@ -1692,7 +1692,7 @@ NIL
 ((|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 
-(-441 R -1673 UP) 
+(-441 R -1674 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 -1047) (QUOTE (-48))))) 
@@ -1724,7 +1724,7 @@ NIL
 ((|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 
-(-449 R UP -1673) 
+(-449 R UP -1674) 
 ((|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 
@@ -1762,16 +1762,16 @@ NIL
 NIL 
 (-458) 
 ((|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}."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-459 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"))) 
-((-4446 |has| (-413 (-959 |#1|)) (-562)) (-4444 . T) (-4443 . T)) 
+((-4445 |has| (-413 (-959 |#1|)) (-562)) (-4443 . T) (-4442 . T)) 
 ((|HasCategory| (-413 (-959 |#1|)) (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| (-413 (-959 |#1|)) (QUOTE (-562)))) 
 (-460 |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"))) 
-(((-4451 "*") |has| |#2| (-174)) (-4442 |has| |#2| (-562)) (-4447 |has| |#2| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#2| (QUOTE (-916))) (-2738 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (-2738 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4447)) (|HasCategory| |#2| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
+(((-4450 "*") |has| |#2| (-174)) (-4441 |has| |#2| (-562)) (-4446 |has| |#2| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4446)) (|HasCategory| |#2| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
 (-461 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 
@@ -1798,7 +1798,7 @@ NIL
 NIL 
 (-467 |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| $) "\\spad{leadingCoefficient(x)} \\undocumented"))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 NIL 
 (-468 E V R P Q) 
 ((|constructor| (NIL "Gosper\\spad{'s} summation algorithm.")) (|GospersMethod| (((|Union| |#5| "failed") |#5| |#2| (|Mapping| |#2|)) "\\spad{GospersMethod(b, n, new)} returns a rational function \\spad{rf(n)} such that \\spad{a(n) * rf(n)} is the indefinite sum of \\spad{a(n)} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{a(n+1) * rf(n+1) - a(n) * rf(n) = a(n)},{} where \\spad{b(n) = a(n)/a(n-1)} is a rational function. Returns \"failed\" if no such rational function \\spad{rf(n)} exists. Note: \\spad{new} is a nullary function returning a new \\spad{V} every time. The condition on \\spad{a(n)} is that \\spad{a(n)/a(n-1)} is a rational function of \\spad{n}."))) 
@@ -1806,7 +1806,7 @@ NIL
 NIL 
 (-469 R E |VarSet| P) 
 ((|constructor| (NIL "A domain for polynomial sets.")) (|convert| (($ (|List| |#4|)) "\\axiom{convert(\\spad{lp})} returns the polynomial set whose members are the polynomials of \\axiom{\\spad{lp}}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#4| (LIST (QUOTE -313) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-470 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}."))) 
@@ -1836,7 +1836,7 @@ NIL
 ((|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| $) "\\spad{degree(g)} names the degree of \\spad{g}. The set of all elements of a given degree form an \\spad{R}-module."))) 
 NIL 
 NIL 
-(-477 |lv| -1673 R) 
+(-477 |lv| -1674 R) 
 ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni,{} Summer \\spad{'88},{} revised November \\spad{'89}} Solve systems of polynomial equations using Groebner bases Total order Groebner bases are computed and then converted to lex ones This package is mostly intended for internal use.")) (|genericPosition| (((|Record| (|:| |dpolys| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |coords| (|List| (|Integer|)))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{genericPosition(lp,lv)} puts a radical zero dimensional ideal in general position,{} for system \\spad{lp} in variables \\spad{lv}.")) (|testDim| (((|Union| (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "failed") (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{testDim(lp,lv)} tests if the polynomial system \\spad{lp} in variables \\spad{lv} is zero dimensional.")) (|groebSolve| (((|List| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{groebSolve(lp,lv)} reduces the polynomial system \\spad{lp} in variables \\spad{lv} to triangular form. Algorithm based on groebner bases algorithm with linear algebra for change of ordering. Preprocessing for the general solver. The polynomials in input are of type \\spadtype{DMP}."))) 
 NIL 
 NIL 
@@ -1846,23 +1846,23 @@ NIL
 NIL 
 (-479) 
 ((|constructor| (NIL "The class of multiplicative groups,{} \\spadignore{i.e.} monoids with multiplicative inverses. \\blankline")) (|commutator| (($ $ $) "\\spad{commutator(p,q)} computes \\spad{inv(p) * inv(q) * p * q}.")) (|conjugate| (($ $ $) "\\spad{conjugate(p,q)} computes \\spad{inv(q) * p * q}; this is 'right action by conjugation'.")) (|unitsKnown| ((|attribute|) "unitsKnown asserts that recip only returns \"failed\" for non-units.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}.")) (/ (($ $ $) "\\spad{x/y} is the same as \\spad{x} times the inverse of \\spad{y}.")) (|inv| (($ $) "\\spad{inv(x)} returns the inverse of \\spad{x}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-480 |Coef| |var| |cen|) 
 ((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x\\^r)}.")) (|integrate| (($ $ (|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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2738 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3665) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2740 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3555) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
 (-481 |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."))) 
-((-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-856))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109)))) 
+((-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-856))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109)))) 
 (-482 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)}"))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#4| (LIST (QUOTE -313) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-483) 
 ((|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}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-484) 
 ((|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'."))) 
@@ -1870,29 +1870,29 @@ NIL
 NIL 
 (-485 |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."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-486) 
 ((|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 
 (-487 |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"))) 
-(((-4451 "*") |has| |#2| (-174)) (-4442 |has| |#2| (-562)) (-4447 |has| |#2| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#2| (QUOTE (-916))) (-2738 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (-2738 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4447)) (|HasCategory| |#2| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
-(-488 -2407 S) 
+(((-4450 "*") |has| |#2| (-174)) (-4441 |has| |#2| (-562)) (-4446 |has| |#2| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4446)) (|HasCategory| |#2| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
+(-488 -2408 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}."))) 
-((-4443 |has| |#2| (-1058)) (-4444 |has| |#2| (-1058)) (-4446 |has| |#2| (-6 -4446)) ((-4451 "*") |has| |#2| (-174)) (-4449 . T)) 
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(LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-732))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-799))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-854))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-1058))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))))) (|HasCategory| (-570) (QUOTE (-856))) (-12 (|HasCategory| |#2| (QUOTE (-1058))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (QUOTE (-235))) (|HasCategory| |#2| (QUOTE (-1058)))) (-12 (|HasCategory| |#2| (QUOTE (-1058))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186))))) (-2740 (|HasCategory| |#2| (QUOTE (-1058))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-1109)))) (|HasAttribute| |#2| (QUOTE -4445)) (|HasCategory| |#2| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))))) 
 (-489) 
 ((|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 
 (-490 S) 
 ((|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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
-(-491 -1673 UP UPUP R) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+(-491 -1674 UP UPUP R) 
 ((|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 
@@ -1902,12 +1902,12 @@ NIL
 NIL 
 (-493) 
 ((|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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2738 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2740 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
 (-494 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 -4449)) (|HasAttribute| |#1| (QUOTE -4450)) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((|HasAttribute| |#1| (QUOTE -4448)) (|HasAttribute| |#1| (QUOTE -4449)) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-495 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|) $) "\\spad{map(f,u)} returns a copy of \\spad{u} with each element \\spad{x} replaced by \\spad{f}(\\spad{x}). For collections,{} \\axiom{map(\\spad{f},{}\\spad{u}) = [\\spad{f}(\\spad{x}) for \\spad{x} in \\spad{u}]}."))) 
 NIL 
@@ -1928,33 +1928,33 @@ NIL
 ((|constructor| (NIL "Category for the hyperbolic trigonometric functions.")) (|tanh| (($ $) "\\spad{tanh(x)} returns the hyperbolic tangent of \\spad{x}.")) (|sinh| (($ $) "\\spad{sinh(x)} returns the hyperbolic sine of \\spad{x}.")) (|sech| (($ $) "\\spad{sech(x)} returns the hyperbolic secant of \\spad{x}.")) (|csch| (($ $) "\\spad{csch(x)} returns the hyperbolic cosecant of \\spad{x}.")) (|coth| (($ $) "\\spad{coth(x)} returns the hyperbolic cotangent of \\spad{x}.")) (|cosh| (($ $) "\\spad{cosh(x)} returns the hyperbolic cosine of \\spad{x}."))) 
 NIL 
 NIL 
-(-500 -1673 UP |AlExt| |AlPol|) 
+(-500 -1674 UP |AlExt| |AlPol|) 
 ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of a field over which we can factor UP\\spad{'s}.")) (|factor| (((|Factored| |#4|) |#4| (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{factor(p, f)} returns a prime factorisation of \\spad{p}; \\spad{f} is a factorisation map for elements of UP."))) 
 NIL 
 NIL 
 (-501) 
 ((|constructor| (NIL "Algebraic closure of the rational numbers.")) (|norm| (($ $ (|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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| $ (QUOTE (-1058))) (|HasCategory| $ (LIST (QUOTE -1047) (QUOTE (-570))))) 
 (-502 S |mn|) 
 ((|constructor| (NIL "\\indented{1}{Author Micheal Monagan Aug/87} This is the basic one dimensional array data type."))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-503 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."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-504 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 
-(-505 R UP -1673) 
+(-505 R UP -1674) 
 ((|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| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) 
 NIL 
 NIL 
 (-506 |mn|) 
 ((|constructor| (NIL "\\spadtype{IndexedBits} is a domain to compactly represent large quantities of Boolean data.")) (|And| (($ $ $) "\\spad{And(n,m)} returns the bit-by-bit logical {\\em And} of \\spad{n} and \\spad{m}.")) (|Or| (($ $ $) "\\spad{Or(n,m)} returns the bit-by-bit logical {\\em Or} of \\spad{n} and \\spad{m}.")) (|Not| (($ $) "\\spad{Not(n)} returns the bit-by-bit logical {\\em Not} of \\spad{n}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| (-112) (QUOTE (-1109))) (|HasCategory| (-112) (LIST (QUOTE -313) (QUOTE (-112))))) (|HasCategory| (-112) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-112) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-112) (QUOTE (-1109))) (|HasCategory| (-112) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-507 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)}."))) 
@@ -1968,7 +1968,7 @@ NIL
 ((|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 
-(-510 -1673 |Expon| |VarSet| |DPoly|) 
+(-510 -1674 |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}.")) (* (($ $ $) "\\spad{I*J} computes the product of the ideal \\spad{I} and \\spad{J}."))) 
 NIL 
 ((|HasCategory| |#3| (LIST (QUOTE -620) (QUOTE (-1186))))) 
@@ -2018,36 +2018,36 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-798)))) 
 (-522 S |mn|) 
 ((|constructor| (NIL "\\indented{1}{Author: Michael Monagan July/87,{} modified \\spad{SMW} June/91} A FlexibleArray is the notion of an array intended to allow for growth at the end only. Hence the following efficient operations \\indented{2}{\\spad{append(x,a)} meaning append item \\spad{x} at the end of the array \\spad{a}} \\indented{2}{\\spad{delete(a,n)} meaning delete the last item from the array \\spad{a}} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets.")) (|shrinkable| (((|Boolean|) (|Boolean|)) "\\spad{shrinkable(b)} sets the shrinkable attribute of flexible arrays to \\spad{b} and returns the previous value")) (|physicalLength!| (($ $ (|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}"))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-523) 
 ((|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 
 (-524 |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}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| (-587 |#1|) (QUOTE (-146))) (|HasCategory| (-587 |#1|) (QUOTE (-373)))) (|HasCategory| (-587 |#1|) (QUOTE (-148))) (|HasCategory| (-587 |#1|) (QUOTE (-373))) (|HasCategory| (-587 |#1|) (QUOTE (-146)))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((-2740 (|HasCategory| (-587 |#1|) (QUOTE (-146))) (|HasCategory| (-587 |#1|) (QUOTE (-373)))) (|HasCategory| (-587 |#1|) (QUOTE (-148))) (|HasCategory| (-587 |#1|) (QUOTE (-373))) (|HasCategory| (-587 |#1|) (QUOTE (-146)))) 
 (-525 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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-526 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."))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-527 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 -4450))) 
+((|HasAttribute| |#3| (QUOTE -4449))) 
 (-528 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 -4450))) 
+((|HasAttribute| |#7| (QUOTE -4449))) 
 (-529 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."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4451 "*"))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4450 "*"))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-530) 
 ((|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 
@@ -2080,7 +2080,7 @@ NIL
 ((|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 
-(-538 K -1673 |Par|) 
+(-538 K -1674 |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 
@@ -2104,7 +2104,7 @@ NIL
 ((|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 
-(-544 K -1673 |Par|) 
+(-544 K -1674 |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 
@@ -2134,7 +2134,7 @@ NIL
 NIL 
 (-551) 
 ((|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."))) 
-((-4447 . T) (-4448 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4446 . T) (-4447 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-552) 
 ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 16 bits."))) 
@@ -2154,13 +2154,13 @@ NIL
 NIL 
 (-556 |Key| |Entry| |addDom|) 
 ((|constructor| (NIL "This domain is used to provide a conditional \"add\" domain for the implementation of \\spadtype{Table}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
-(-557 R -1673) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
+(-557 R -1674) 
 ((|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 
-(-558 R0 -1673 UP UPUP R) 
+(-558 R0 -1674 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 
@@ -2170,7 +2170,7 @@ NIL
 NIL 
 (-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."))) 
-((-3026 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-3026 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-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."))) 
@@ -2178,9 +2178,9 @@ NIL
 NIL 
 (-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."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
-(-563 R -1673) 
+(-563 R -1674) 
 ((|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|)) "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|)) "failed") |#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 
@@ -2192,7 +2192,7 @@ NIL
 ((|constructor| (NIL "\\blankline")) (|entry| (((|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")))) (|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| "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"))))))) $) "\\spad{entries(x)} \\undocumented{}")) (|showAttributes| (((|Union| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "Continuous at the end points") (|:| |lowerSingular| "There is a singularity at the lower end point") (|:| |upperSingular| "There is a singularity at the upper end point") (|:| |bothSingular| "There are singularities at both end points") (|:| |notEvaluated| "End point continuity not yet evaluated"))) (|:| |singularitiesStream| (|Union| (|:| |str| (|Stream| (|DoubleFloat|))) (|:| |notEvaluated| "Internal singularities not yet evaluated"))) (|:| |range| (|Union| (|:| |finite| "The range is finite") (|:| |lowerInfinite| "The bottom of range is infinite") (|:| |upperInfinite| "The top of range is infinite") (|:| |bothInfinite| "Both top and bottom points are infinite") (|:| |notEvaluated| "Range not yet evaluated")))) "failed") (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) "\\spad{showAttributes(x)} \\undocumented{}")) (|insert!| (($ (|Record| (|:| |key| (|Record| (|:| |var| (|Symbol|)) (|:| |fn| (|Expression| (|DoubleFloat|))) (|:| |range| (|Segment| (|OrderedCompletion| (|DoubleFloat|)))) (|:| |abserr| (|DoubleFloat|)) (|:| |relerr| (|DoubleFloat|)))) (|:| |entry| (|Record| (|:| |endPointContinuity| (|Union| (|:| |continuous| "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{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| "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{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 
-(-566 R -1673 L) 
+(-566 R -1674 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| "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| "failed")) (|:| |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| "failed") |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| "failed") |#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| "failed") |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| "failed") |#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|))))) "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|))))) "failed") |#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|)) "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|)) "failed") |#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 -662) (|devaluate| |#2|)))) 
@@ -2200,31 +2200,31 @@ NIL
 ((|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 
-(-568 -1673 UP UPUP R) 
+(-568 -1674 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 
-(-569 -1673 UP) 
+(-569 -1674 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 
 (-570) 
 ((|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."))) 
-((-4431 . T) (-4437 . T) (-4441 . T) (-4436 . T) (-4447 . T) (-4448 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4430 . T) (-4436 . T) (-4440 . T) (-4435 . T) (-4446 . T) (-4447 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-571) 
 ((|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 
-(-572 R -1673 L) 
+(-572 R -1674 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| "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| "failed") |#2| |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| "failed") |#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 -662) (|devaluate| |#2|)))) 
-(-573 R -1673) 
+(-573 R -1674) 
 ((|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 -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-1148)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-635))))) 
-(-574 -1673 UP) 
+(-574 -1674 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 
@@ -2232,27 +2232,27 @@ NIL
 ((|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 
-(-576 -1673) 
+(-576 -1674) 
 ((|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 
 (-577 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."))) 
-((-3026 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-3026 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-578) 
 ((|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 
-(-579 R -1673) 
+(-579 R -1674) 
 ((|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| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-288))) (|HasCategory| |#2| (QUOTE (-635))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-1186))))) (-12 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-288)))) (|HasCategory| |#1| (QUOTE (-562)))) 
-(-580 -1673 UP) 
+(-580 -1674 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|)) "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|)) "failed") |#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|)) "failed") |#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|)) "failed") |#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 
-(-581 R -1673) 
+(-581 R -1674) 
 ((|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 
@@ -2274,21 +2274,21 @@ NIL
 NIL 
 (-586 |p| |unBalanced?|) 
 ((|constructor| (NIL "This domain implements \\spad{Zp},{} the \\spad{p}-adic completion of the integers. This is an internal domain."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-587 |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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| $ (QUOTE (-148))) (|HasCategory| $ (QUOTE (-146))) (|HasCategory| $ (QUOTE (-373)))) 
 (-588) 
 ((|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 
-(-589 R -1673) 
+(-589 R -1674) 
 ((|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 E -1673) 
+(-590 E -1674) 
 ((|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 
@@ -2296,9 +2296,9 @@ NIL
 ((|constructor| (NIL "This domain provides representations for the intermediate form data structure used by the Spad elaborator.")) (|irDef| (($ (|Identifier|) (|InternalTypeForm|) $) "\\spad{irDef(f,ts,e)} returns an IR representation for a definition of a function named \\spad{f},{} with signature \\spad{ts} and body \\spad{e}.")) (|irCtor| (($ (|Identifier|) (|InternalTypeForm|)) "\\spad{irCtor(n,t)} returns an IR for a constructor reference of type designated by the type form \\spad{t}")) (|irVar| (($ (|Identifier|) (|InternalTypeForm|)) "\\spad{irVar(x,t)} returns an IR for a variable reference of type designated by the type form \\spad{t}"))) 
 NIL 
 NIL 
-(-592 -1673) 
+(-592 -1674) 
 ((|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}."))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-1186))))) 
 (-593 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"))) 
@@ -2326,19 +2326,19 @@ NIL
 NIL 
 (-599 |mn|) 
 ((|constructor| (NIL "This domain implements low-level strings"))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (-2738 (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109)))) (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (-2740 (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109)))) (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) 
 (-600 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 
 (-601 |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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-570)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-570)) (|devaluate| |#1|)))) (|HasCategory| (-570) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-570)))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-570)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-570)) (|devaluate| |#1|)))) (|HasCategory| (-570) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-570)))))) 
 (-602 |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|) $) "\\spad{coefficients(x)} returns a stream of ring elements. \\indented{1}{When \\spad{x} is a univariate series,{} this is a stream of Taylor} \\indented{1}{coefficients. When \\spad{x} is a multivariate series,{} the} \\indented{1}{\\spad{n}th element of the stream is a form of} \\indented{1}{degree \\spad{n} in the power series variables.}"))) 
-(((-4451 "*") |has| |#1| (-562)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-562)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-562)))) 
 (-603) 
 ((|constructor| (NIL "This domain provides representations for internal type form.")) (|mappingMode| (($ $ (|List| $)) "\\spad{mappingMode(r,ts)} returns a mapping mode with return mode \\spad{r},{} and parameter modes \\spad{ts}.")) (|categoryMode| (($) "\\spad{categoryMode} is a constant mode denoting Category.")) (|voidMode| (($) "\\spad{voidMode} is a constant mode denoting Void.")) (|noValueMode| (($) "\\spad{noValueMode} is a constant mode that indicates that the value of an expression is to be ignored.")) (|jokerMode| (($) "\\spad{jokerMode} is a constant that stands for any mode in a type inference context"))) 
@@ -2352,7 +2352,7 @@ NIL
 ((|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 
-(-606 R -1673 FG) 
+(-606 R -1674 FG) 
 ((|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 
@@ -2362,12 +2362,12 @@ NIL
 NIL 
 (-608 R |mn|) 
 ((|constructor| (NIL "\\indented{2}{This type represents vector like objects with varying lengths} and a user-specified initial index."))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-609 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 -4450)) (|HasCategory| |#2| (QUOTE (-856))) (|HasAttribute| |#1| (QUOTE -4449)) (|HasCategory| |#3| (QUOTE (-1109)))) 
+((|HasAttribute| |#1| (QUOTE -4449)) (|HasCategory| |#2| (QUOTE (-856))) (|HasAttribute| |#1| (QUOTE -4448)) (|HasCategory| |#3| (QUOTE (-1109)))) 
 (-610 |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|) $) "\\spad{entries(u)} returns a list of all the entries of aggregate \\spad{u} in no assumed order."))) 
 NIL 
@@ -2382,11 +2382,11 @@ NIL
 NIL 
 (-613 R A) 
 ((|constructor| (NIL "\\indented{1}{AssociatedJordanAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$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)."))) 
-((-4446 -2738 (-1764 (|has| |#2| (-372 |#1|)) (|has| |#1| (-562))) (-12 (|has| |#2| (-423 |#1|)) (|has| |#1| (-562)))) (-4444 . T) (-4443 . T)) 
-((-2738 (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) 
+((-4445 -2740 (-1765 (|has| |#2| (-372 |#1|)) (|has| |#1| (-562))) (-12 (|has| |#2| (-423 |#1|)) (|has| |#1| (-562)))) (-4443 . T) (-4442 . T)) 
+((-2740 (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) 
 (-614 |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."))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 ((-12 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1168))) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#1|)))))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| (-1168) (QUOTE (-856))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-615 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}."))) 
@@ -2394,7 +2394,7 @@ NIL
 NIL 
 (-616 |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}."))) 
-((-4450 . T)) 
+((-4449 . T)) 
 NIL 
 (-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"))) 
@@ -2412,7 +2412,7 @@ NIL
 ((|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| $) "\\spad{convert(a)} transforms a into an element of \\spad{S}."))) 
 NIL 
 NIL 
-(-621 -1673 UP) 
+(-621 -1674 UP) 
 ((|constructor| (NIL "\\spadtype{Kovacic} provides a modified Kovacic\\spad{'s} algorithm for solving explicitely irreducible 2nd order linear ordinary differential equations.")) (|kovacic| (((|Union| (|SparseUnivariatePolynomial| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{kovacic(a_0,a_1,a_2,ezfactor)} returns either \"failed\" or \\spad{P}(\\spad{u}) such that \\spad{\\$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 
@@ -2434,19 +2434,19 @@ NIL
 NIL 
 (-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."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-627 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}."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-854)))) 
-(-628 R -1673) 
+(-628 R -1674) 
 ((|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 
 (-629 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"))) 
-((-4444 . T) (-4443 . T) ((-4451 "*") . T) (-4442 . T) (-4446 . T)) 
+((-4443 . T) (-4442 . T) ((-4450 "*") . T) (-4441 . T) (-4445 . T)) 
 ((|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) 
 (-630 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."))) 
@@ -2462,7 +2462,7 @@ NIL
 NIL 
 (-633 |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}}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-634 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}}."))) 
@@ -2472,30 +2472,30 @@ NIL
 ((|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 
-(-636 R -1673) 
+(-636 R -1674) 
 ((|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 
-(-637 |lv| -1673) 
+(-637 |lv| -1674) 
 ((|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 
 (-638) 
 ((|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."))) 
-((-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1168))) (LIST (QUOTE |:|) (QUOTE -2223) (QUOTE (-52))))))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-52) (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -313) (QUOTE (-52))))) (|HasCategory| (-1168) (QUOTE (-856))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109)))) 
+((-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1168))) (LIST (QUOTE |:|) (QUOTE -2223) (QUOTE (-52))))))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-52) (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -313) (QUOTE (-52))))) (|HasCategory| (-1168) (QUOTE (-856))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 (-52))) (QUOTE (-1109)))) 
 (-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| (($ $ $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket of \\axiom{\\spad{x}} and \\axiom{\\spad{y}}."))) 
 NIL 
 ((|HasCategory| |#2| (QUOTE (-368)))) 
 (-640 R) 
 ((|constructor| (NIL "\\axiom{JacobiIdentity} means that \\axiom{[\\spad{x},{}[\\spad{y},{}\\spad{z}]]+[\\spad{y},{}[\\spad{z},{}\\spad{x}]]+[\\spad{z},{}[\\spad{x},{}\\spad{y}]] = 0} holds.")) (/ (($ $ |#1|) "\\axiom{\\spad{x/r}} returns the division of \\axiom{\\spad{x}} by \\axiom{\\spad{r}}.")) (|construct| (($ $ $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket of \\axiom{\\spad{x}} and \\axiom{\\spad{y}}."))) 
-((|JacobiIdentity| . T) (|NullSquare| . T) (-4444 . T) (-4443 . T)) 
+((|JacobiIdentity| . T) (|NullSquare| . T) (-4443 . T) (-4442 . T)) 
 NIL 
 (-641 R A) 
 ((|constructor| (NIL "AssociatedLieAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$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)."))) 
-((-4446 -2738 (-1764 (|has| |#2| (-372 |#1|)) (|has| |#1| (-562))) (-12 (|has| |#2| (-423 |#1|)) (|has| |#1| (-562)))) (-4444 . T) (-4443 . T)) 
-((-2738 (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) 
+((-4445 -2740 (-1765 (|has| |#2| (-372 |#1|)) (|has| |#1| (-562))) (-12 (|has| |#2| (-423 |#1|)) (|has| |#1| (-562)))) (-4443 . T) (-4442 . T)) 
+((-2740 (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -423) (|devaluate| |#1|))))) (|HasCategory| |#2| (LIST (QUOTE -372) (|devaluate| |#1|)))) 
 (-642 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|) "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|) "failed")) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| |#2|) "failed"))) "failed") |#2| (|Equation| (|OrderedCompletion| |#2|))) "\\spad{limit(f(x),x = a)} computes the real limit \\spad{lim(x -> a,f(x))}."))) 
 NIL 
@@ -2510,7 +2510,7 @@ NIL
 ((-1754 (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-645 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| $)) "\\spad{reducedSystem(A)} returns a matrix \\spad{B} such that \\spad{A x = 0} and \\spad{B x = 0} have the same solutions in \\spad{R}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-646 R) 
 ((|constructor| (NIL "\\indented{2}{A set is an \\spad{R}-linear set if it is stable by dilation} \\indented{2}{by elements in the ring \\spad{R}.\\space{2}This category differs from} \\indented{2}{\\spad{Module} in that no other assumption (such as addition)} \\indented{2}{is made about the underlying set.} See Also: LeftLinearSet,{} RightLinearSet."))) 
@@ -2530,8 +2530,8 @@ NIL
 NIL 
 (-650 S) 
 ((|constructor| (NIL "\\spadtype{List} implements singly-linked lists that are addressable by indices; the index of the first element is 1. In addition to the operations provided by \\spadtype{IndexedList},{} this constructor provides some LISP-like functions such as \\spadfun{null} and \\spadfun{cons}.")) (|setDifference| (($ $ $) "\\spad{setDifference(u1,u2)} returns a list of the elements of \\spad{u1} that are not also in \\spad{u2}. The order of elements in the resulting list is unspecified.")) (|setIntersection| (($ $ $) "\\spad{setIntersection(u1,u2)} returns a list of the elements that lists \\spad{u1} and \\spad{u2} have in common. The order of elements in the resulting list is unspecified.")) (|setUnion| (($ $ $) "\\spad{setUnion(u1,u2)} appends the two lists \\spad{u1} and \\spad{u2},{} then removes all duplicates. The order of elements in the resulting list is unspecified.")) (|append| (($ $ $) "\\spad{append(u1,u2)} appends the elements of list \\spad{u1} onto the front of list \\spad{u2}. This new list and \\spad{u2} will share some structure.")) (|cons| (($ |#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."))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-834))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-651 T$) 
 ((|constructor| (NIL "This domain represents AST for Spad literals."))) 
 NIL 
@@ -2542,8 +2542,8 @@ NIL
 NIL 
 (-653 S) 
 ((|substitute| (($ |#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."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-654 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 
@@ -2555,22 +2555,22 @@ NIL
 (-656 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 -4450))) 
+((|HasAttribute| |#1| (QUOTE -4449))) 
 (-657 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 
-(-658 R -1673 L) 
+(-658 R -1674 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 
 (-659 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}}"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-660 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| (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-661 S A) 
 ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|directSum| (($ $ $) "\\spad{directSum(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}.")) (|symmetricSquare| (($ $) "\\spad{symmetricSquare(a)} computes \\spad{symmetricProduct(a,a)} using a more efficient method.")) (|symmetricPower| (($ $ (|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}."))) 
@@ -2578,15 +2578,15 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-368)))) 
 (-662 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}."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
-(-663 -1673 UP) 
+(-663 -1674 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)))) 
-(-664 A -3710) 
+(-664 A -2737) 
 ((|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}}"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-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."))) 
@@ -2602,7 +2602,7 @@ NIL
 NIL 
 (-668 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}."))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#1| (QUOTE (-797)))) 
 (-669 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."))) 
@@ -2610,7 +2610,7 @@ NIL
 NIL 
 (-670 |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) (-4444 . T) (-4443 . T)) 
+((|JacobiIdentity| . T) (|NullSquare| . T) (-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-174)))) 
 (-671 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}."))) 
@@ -2618,13 +2618,13 @@ NIL
 NIL 
 (-672 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}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
-(-673 -1673) 
+(-673 -1674) 
 ((|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|) "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|) "failed")) (|:| |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|) "failed")) (|:| |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|) "failed")) (|:| |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|) "failed")) (|:| |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 
-(-674 -1673 |Row| |Col| M) 
+(-674 -1674 |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| "failed") |#4| |#3|) "\\spad{particularSolution(A,B)} finds a particular solution of the linear system \\spad{AX = B}.")) (|solve| (((|List| (|Record| (|:| |particular| (|Union| |#3| "failed")) (|:| |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| "failed")) (|:| |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 
@@ -2634,8 +2634,8 @@ NIL
 NIL 
 (-676 |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."))) 
-((-4446 . T) (-4449 . T) (-4443 . T) (-4444 . T)) 
-((|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235))) (|HasAttribute| |#2| (QUOTE (-4451 "*"))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (-12 (|HasCategory| |#2| (QUOTE (-235))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-562))) (-2738 (|HasAttribute| |#2| (QUOTE (-4451 "*"))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-174)))) 
+((-4445 . T) (-4448 . T) (-4442 . T) (-4443 . T)) 
+((|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235))) (|HasAttribute| |#2| (QUOTE (-4450 "*"))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (-12 (|HasCategory| |#2| (QUOTE (-235))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-562))) (-2740 (|HasAttribute| |#2| (QUOTE (-4450 "*"))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-174)))) 
 (-677) 
 ((|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 
@@ -2655,7 +2655,7 @@ NIL
 (-681 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 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (QUOTE (-1058))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-682) 
 ((|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 
@@ -2699,10 +2699,10 @@ NIL
 (-692 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 (-4451 "*"))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-562)))) 
+((|HasAttribute| |#2| (QUOTE (-4450 "*"))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-562)))) 
 (-693 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"))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-694 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."))) 
@@ -2710,8 +2710,8 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-562)))) 
 (-695 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."))) 
-((-4449 . T) (-4450 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4451 "*"))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4448 . T) (-4449 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-562))) (|HasAttribute| |#1| (QUOTE (-4450 "*"))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-696 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 
@@ -2720,7 +2720,7 @@ NIL
 ((|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 
-(-698 S -1673 FLAF FLAS) 
+(-698 S -1674 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 
@@ -2730,11 +2730,11 @@ NIL
 NIL 
 (-700) 
 ((|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"))) 
-((-4442 . T) (-4447 |has| (-705) (-368)) (-4441 |has| (-705) (-368)) (-3035 . T) (-4448 |has| (-705) (-6 -4448)) (-4445 |has| (-705) (-6 -4445)) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-705) (QUOTE (-148))) (|HasCategory| (-705) (QUOTE (-146))) (|HasCategory| (-705) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-705) (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| (-705) (QUOTE (-373))) (|HasCategory| (-705) (QUOTE (-368))) (-2738 (|HasCategory| (-705) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-705) (QUOTE (-368)))) (|HasCategory| (-705) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-705) (QUOTE (-235))) (-2738 (|HasCategory| (-705) (QUOTE (-368))) (|HasCategory| (-705) (QUOTE (-354)))) (|HasCategory| (-705) (QUOTE (-354))) (|HasCategory| (-705) (LIST (QUOTE -290) (QUOTE (-705)) (QUOTE (-705)))) (|HasCategory| (-705) (LIST (QUOTE -313) (QUOTE (-705)))) (|HasCategory| (-705) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-705)))) (|HasCategory| (-705) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-705) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-705) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-705) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (-2738 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-368))) (|HasCategory| (-705) (QUOTE (-354)))) (|HasCategory| (-705) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-705) (QUOTE (-1031))) (|HasCategory| (-705) (QUOTE (-1211))) (-12 (|HasCategory| (-705) (QUOTE (-1011))) (|HasCategory| (-705) (QUOTE (-1211)))) (-2738 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-368))) (-12 (|HasCategory| (-705) (QUOTE (-354))) (|HasCategory| (-705) (QUOTE (-916))))) (-2738 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (-12 (|HasCategory| (-705) (QUOTE (-368))) (|HasCategory| (-705) (QUOTE (-916)))) (-12 (|HasCategory| (-705) (QUOTE (-354))) (|HasCategory| (-705) (QUOTE (-916))))) (|HasCategory| (-705) (QUOTE (-551))) (-12 (|HasCategory| (-705) (QUOTE (-1069))) (|HasCategory| (-705) (QUOTE (-1211)))) (|HasCategory| (-705) (QUOTE (-1069))) (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916))) (-2738 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-368)))) (-2738 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-562)))) (-12 (|HasCategory| (-705) (QUOTE (-235))) (|HasCategory| (-705) (QUOTE (-368)))) (-12 (|HasCategory| (-705) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-705) (QUOTE (-368)))) (|HasCategory| (-705) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-705) (QUOTE (-562))) (|HasAttribute| (-705) (QUOTE -4448)) (|HasAttribute| (-705) (QUOTE -4445)) (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-146)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-354))))) 
+((-4441 . T) (-4446 |has| (-705) (-368)) (-4440 |has| (-705) (-368)) (-3035 . T) (-4447 |has| (-705) (-6 -4447)) (-4444 |has| (-705) (-6 -4444)) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-705) (QUOTE (-148))) (|HasCategory| (-705) (QUOTE (-146))) (|HasCategory| (-705) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-705) (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| (-705) (QUOTE (-373))) (|HasCategory| (-705) (QUOTE (-368))) (-2740 (|HasCategory| (-705) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-705) (QUOTE (-368)))) (|HasCategory| (-705) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-705) (QUOTE (-235))) (-2740 (|HasCategory| (-705) (QUOTE (-368))) (|HasCategory| (-705) (QUOTE (-354)))) (|HasCategory| (-705) (QUOTE (-354))) (|HasCategory| (-705) (LIST (QUOTE -290) (QUOTE (-705)) (QUOTE (-705)))) (|HasCategory| (-705) (LIST (QUOTE -313) (QUOTE (-705)))) (|HasCategory| (-705) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-705)))) (|HasCategory| (-705) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-705) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-705) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-705) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (-2740 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-368))) (|HasCategory| (-705) (QUOTE (-354)))) (|HasCategory| (-705) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-705) (QUOTE (-1031))) (|HasCategory| (-705) (QUOTE (-1211))) (-12 (|HasCategory| (-705) (QUOTE (-1011))) (|HasCategory| (-705) (QUOTE (-1211)))) (-2740 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-368))) (-12 (|HasCategory| (-705) (QUOTE (-354))) (|HasCategory| (-705) (QUOTE (-916))))) (-2740 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (-12 (|HasCategory| (-705) (QUOTE (-368))) (|HasCategory| (-705) (QUOTE (-916)))) (-12 (|HasCategory| (-705) (QUOTE (-354))) (|HasCategory| (-705) (QUOTE (-916))))) (|HasCategory| (-705) (QUOTE (-551))) (-12 (|HasCategory| (-705) (QUOTE (-1069))) (|HasCategory| (-705) (QUOTE (-1211)))) (|HasCategory| (-705) (QUOTE (-1069))) (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916))) (-2740 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-368)))) (-2740 (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-562)))) (-12 (|HasCategory| (-705) (QUOTE (-235))) (|HasCategory| (-705) (QUOTE (-368)))) (-12 (|HasCategory| (-705) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-705) (QUOTE (-368)))) (|HasCategory| (-705) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-705) (QUOTE (-562))) (|HasAttribute| (-705) (QUOTE -4447)) (|HasAttribute| (-705) (QUOTE -4444)) (-12 (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-146)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-705) (QUOTE (-311))) (|HasCategory| (-705) (QUOTE (-916)))) (|HasCategory| (-705) (QUOTE (-354))))) 
 (-701 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}."))) 
-((-4450 . T)) 
+((-4449 . T)) 
 NIL 
 (-702 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}."))) 
@@ -2744,13 +2744,13 @@ NIL
 ((|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|)) "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|)) "undefined") (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(f,g,h,j,s1,s2,l)} \\undocumented"))) 
 NIL 
 NIL 
-(-704 OV E -1673 PG) 
+(-704 OV E -1674 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 
 (-705) 
 ((|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}"))) 
-((-3026 . T) (-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-3026 . T) (-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-706 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."))) 
@@ -2758,7 +2758,7 @@ NIL
 NIL 
 (-707) 
 ((|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}"))) 
-((-4448 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4447 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-708 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"))) 
@@ -2776,7 +2776,7 @@ NIL
 ((|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 
-(-712 S -2790 I) 
+(-712 S -2791 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 
@@ -2786,7 +2786,7 @@ NIL
 NIL 
 (-714 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)}.}"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-715 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}."))) 
@@ -2796,25 +2796,25 @@ NIL
 ((|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 
-(-717 R |Mod| -2942 -1585 |exactQuo|) 
+(-717 R |Mod| -4052 -3068 |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| $) "\\spad{modulus(x)} \\undocumented"))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-718 R |Rep|) 
 ((|constructor| (NIL "This package \\undocumented")) (|frobenius| (($ $) "\\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"))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4445 |has| |#1| (-368)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1161))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-235))) (|HasAttribute| |#1| (QUOTE -4447)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4444 |has| |#1| (-368)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1161))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (QUOTE (-235))) (|HasAttribute| |#1| (QUOTE -4446)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
 (-719 IS E |ff|) 
 ((|constructor| (NIL "This package \\undocumented")) (|construct| (($ |#1| |#2|) "\\spad{construct(i,e)} \\undocumented")) (|index| ((|#1| $) "\\spad{index(x)} \\undocumented")) (|exponent| ((|#2| $) "\\spad{exponent(x)} \\undocumented"))) 
 NIL 
 NIL 
 (-720 R M) 
 ((|constructor| (NIL "Algebra of ADDITIVE operators on a module.")) (|makeop| (($ |#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| (($ $ $) "\\spad{adjoint(op1, op2)} sets the adjoint of \\spad{op1} to be op2. \\spad{op1} must be a basic operator") (($ $) "\\spad{adjoint(op)} returns the adjoint of the operator \\spad{op}."))) 
-((-4444 |has| |#1| (-174)) (-4443 |has| |#1| (-174)) (-4446 . T)) 
+((-4443 |has| |#1| (-174)) (-4442 |has| |#1| (-174)) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148)))) 
-(-721 R |Mod| -2942 -1585 |exactQuo|) 
+(-721 R |Mod| -4052 -3068 |exactQuo|) 
 ((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{EuclideanModularRing} ,{}\\spadtype{ModularField}")) (|inv| (($ $) "\\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"))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-722 S R) 
 ((|constructor| (NIL "The category of modules over a commutative ring. \\blankline"))) 
@@ -2822,11 +2822,11 @@ NIL
 NIL 
 (-723 R) 
 ((|constructor| (NIL "The category of modules over a commutative ring. \\blankline"))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 NIL 
-(-724 -1673) 
+(-724 -1674) 
 ((|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]]}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-725 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."))) 
@@ -2850,7 +2850,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-354))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-373)))) 
 (-730 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."))) 
-((-4442 |has| |#1| (-368)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 |has| |#1| (-368)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-731 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."))) 
@@ -2860,7 +2860,7 @@ NIL
 ((|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 
-(-733 -1673 UP) 
+(-733 -1674 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 
@@ -2878,8 +2878,8 @@ NIL
 NIL 
 (-737 |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."))) 
-(((-4451 "*") |has| |#2| (-174)) (-4442 |has| |#2| (-562)) (-4447 |has| |#2| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#2| (QUOTE (-916))) (-2738 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (-2738 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4447)) (|HasCategory| |#2| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
+(((-4450 "*") |has| |#2| (-174)) (-4441 |has| |#2| (-562)) (-4446 |has| |#2| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-870 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368))) (|HasAttribute| |#2| (QUOTE -4446)) (|HasCategory| |#2| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
 (-738 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|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) 
 NIL 
@@ -2894,15 +2894,15 @@ NIL
 NIL 
 (-741 R M) 
 ((|constructor| (NIL "\\spadtype{MonoidRing}(\\spad{R},{}\\spad{M}),{} implements the algebra of all maps from the monoid \\spad{M} to the commutative ring \\spad{R} with finite support. Multiplication of two maps \\spad{f} and \\spad{g} is defined to map an element \\spad{c} of \\spad{M} to the (convolution) sum over {\\em f(a)g(b)} such that {\\em ab = c}. Thus \\spad{M} can be identified with a canonical basis and the maps can also be considered as formal linear combinations of the elements in \\spad{M}. Scalar multiples of a basis element are called monomials. A prominent example is the class of polynomials where the monoid is a direct product of the natural numbers with pointwise addition. When \\spad{M} is \\spadtype{FreeMonoid Symbol},{} one gets polynomials in infinitely many non-commuting variables. Another application area is representation theory of finite groups \\spad{G},{} where modules over \\spadtype{MonoidRing}(\\spad{R},{}\\spad{G}) are studied.")) (|reductum| (($ $) "\\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}."))) 
-((-4444 |has| |#1| (-174)) (-4443 |has| |#1| (-174)) (-4446 . T)) 
+((-4443 |has| |#1| (-174)) (-4442 |has| |#1| (-174)) (-4445 . T)) 
 ((-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-373)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-856)))) 
 (-742 S) 
 ((|constructor| (NIL "A multi-set aggregate is a set which keeps track of the multiplicity of its elements."))) 
-((-4439 . T) (-4450 . T)) 
+((-4438 . T) (-4449 . T)) 
 NIL 
 (-743 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}."))) 
-((-4449 . T) (-4439 . T) (-4450 . T)) 
+((-4448 . T) (-4438 . T) (-4449 . T)) 
 ((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-744) 
 ((|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."))) 
@@ -2914,7 +2914,7 @@ NIL
 NIL 
 (-746 |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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4444 . T) (-4443 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
 (-747 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"))) 
@@ -2930,7 +2930,7 @@ NIL
 NIL 
 (-750 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}."))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 NIL 
 (-751) 
 ((|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}."))) 
@@ -3012,11 +3012,11 @@ NIL
 ((|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 
-(-771 -1673) 
+(-771 -1674) 
 ((|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 
-(-772 P -1673) 
+(-772 P -1674) 
 ((|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 
@@ -3024,7 +3024,7 @@ NIL
 NIL 
 NIL 
 NIL 
-(-774 UP -1673) 
+(-774 UP -1674) 
 ((|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 
@@ -3038,9 +3038,9 @@ NIL
 NIL 
 (-777) 
 ((|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."))) 
-(((-4451 "*") . T)) 
+(((-4450 "*") . T)) 
 NIL 
-(-778 R -1673) 
+(-778 R -1674) 
 ((|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 
@@ -3060,7 +3060,7 @@ NIL
 ((|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 
-(-783 -1673 |ExtF| |SUEx| |ExtP| |n|) 
+(-783 -1674 |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 
@@ -3074,23 +3074,23 @@ NIL
 NIL 
 (-786 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."))) 
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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 
 (-788 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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+((|HasCategory| |#1| (QUOTE (-916))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-1161))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-235))) (|HasAttribute| |#1| (QUOTE -4446)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
 (-789 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 -413) (QUOTE (-570)))))) 
 (-790 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.}"))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-791 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}."))) 
@@ -3142,25 +3142,25 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (QUOTE (-1069))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-373)))) 
 (-803 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| $) "\\spad{real(o)} extracts real part of octonion \\spad{o}.")) (|conjugate| (($ $) "\\spad{conjugate(o)} negates the imaginary parts \\spad{i},{}\\spad{j},{}\\spad{k},{}\\spad{E},{}\\spad{I},{}\\spad{J},{}\\spad{K} of octonian \\spad{o}."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
-(-804 -2738 R OS S) 
+(-804 -2740 R OS S) 
 ((|constructor| (NIL "OctonionCategoryFunctions2 implements functions between two octonion domains defined over different rings. The function map is used to coerce between octonion types.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the component parts of the octonion \\spad{u}."))) 
 NIL 
 NIL 
 (-805 R) 
 ((|constructor| (NIL "Octonion implements octonions (Cayley-Dixon algebra) over a commutative ring,{} an eight-dimensional non-associative algebra,{} doubling the quaternions in the same way as doubling the complex numbers to get the quaternions the main constructor function is {\\em octon} which takes 8 arguments: the real part,{} the \\spad{i} imaginary part,{} the \\spad{j} imaginary part,{} the \\spad{k} imaginary part,{} (as with quaternions) and in addition the imaginary parts \\spad{E},{} \\spad{I},{} \\spad{J},{} \\spad{K}.")) (|octon| (($ (|Quaternion| |#1|) (|Quaternion| |#1|)) "\\spad{octon(qe,qE)} constructs an octonion from two quaternions using the relation {\\em O = Q + QE}."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (-2738 (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2738 (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-1069))) (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (-2740 (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2740 (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-1069))) (|HasCategory| |#1| (QUOTE (-551))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1008 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) 
 (-806) 
 ((|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."))) 
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-(-807 R -1673 L) 
+(-807 R -1674 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}."))) 
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-(-808 R -1673) 
+(-808 R -1674) 
 ((|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| "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| "failed") (|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| "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| "failed") (|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."))) 
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@@ -3168,7 +3168,7 @@ NIL
 ((|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."))) 
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-(-810 R -1673) 
+(-810 R -1674) 
 ((|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 
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@@ -3176,11 +3176,11 @@ NIL
 ((|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 
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-(-812 -1673 UP UPUP R) 
+(-812 -1674 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 
-(-813 -1673 UP L LQ) 
+(-813 -1674 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 
@@ -3188,41 +3188,41 @@ NIL
 ((|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 
-(-815 -1673 UP L LQ) 
+(-815 -1674 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 
-(-816 -1673 UP) 
+(-816 -1674 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|) "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|) "failed")) (|:| |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 
-(-817 -1673 L UP A LO) 
+(-817 -1674 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 
-(-818 -1673 UP) 
+(-818 -1674 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)))) 
-(-819 -1673 LO) 
+(-819 -1674 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 
-(-820 -1673 LODO) 
+(-820 -1674 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 
-(-821 -2407 S |f|) 
+(-821 -2408 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}."))) 
-((-4443 |has| |#2| (-1058)) (-4444 |has| |#2| (-1058)) (-4446 |has| |#2| (-6 -4446)) ((-4451 "*") |has| |#2| (-174)) (-4449 . T)) 
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(|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186))))) (-2740 (|HasCategory| |#2| (QUOTE (-1058))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-1109)))) (|HasAttribute| |#2| (QUOTE -4445)) (|HasCategory| |#2| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))))) 
 (-822 R) 
 ((|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"))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-916))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4447)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-824 (-1186)) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4446)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
 (-823 |Kernels| R |var|) 
 ((|constructor| (NIL "This constructor produces an ordinary differential ring from a partial differential ring by specifying a variable."))) 
-(((-4451 "*") |has| |#2| (-368)) (-4442 |has| |#2| (-368)) (-4447 |has| |#2| (-368)) (-4441 |has| |#2| (-368)) (-4446 . T) (-4444 . T) (-4443 . T)) 
+(((-4450 "*") |has| |#2| (-368)) (-4441 |has| |#2| (-368)) (-4446 |has| |#2| (-368)) (-4440 |has| |#2| (-368)) (-4445 . T) (-4443 . T) (-4442 . T)) 
 ((|HasCategory| |#2| (QUOTE (-368)))) 
 (-824 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}))."))) 
@@ -3234,7 +3234,7 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-856)))) 
 (-826) 
 ((|constructor| (NIL "The category of ordered commutative integral domains,{} where ordering and the arithmetic operations are compatible \\blankline"))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-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}"))) 
@@ -3262,7 +3262,7 @@ NIL
 NIL 
 (-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}."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-235)))) 
 (-834) 
 ((|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."))) 
@@ -3274,7 +3274,7 @@ NIL
 NIL 
 (-836 S) 
 ((|constructor| (NIL "to become an in order iterator")) (|min| ((|#1| $) "\\spad{min(u)} returns the smallest entry in the multiset aggregate \\spad{u}."))) 
-((-4449 . T) (-4439 . T) (-4450 . T)) 
+((-4448 . T) (-4438 . T) (-4449 . T)) 
 NIL 
 (-837) 
 ((|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."))) 
@@ -3286,8 +3286,8 @@ NIL
 NIL 
 (-839 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."))) 
-((-4446 |has| |#1| (-854))) 
-((|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (QUOTE (-21))) (-2738 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-854)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2738 (|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-551)))) 
+((-4445 |has| |#1| (-854))) 
+((|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (QUOTE (-21))) (-2740 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-854)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2740 (|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-551)))) 
 (-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 
@@ -3298,7 +3298,7 @@ NIL
 NIL 
 (-842 R) 
 ((|constructor| (NIL "Algebra of ADDITIVE operators over a ring."))) 
-((-4444 |has| |#1| (-174)) (-4443 |has| |#1| (-174)) (-4446 . T)) 
+((-4443 |has| |#1| (-174)) (-4442 |has| |#1| (-174)) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148)))) 
 (-843) 
 ((|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)."))) 
@@ -3326,13 +3326,13 @@ NIL
 NIL 
 (-849 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| (($) "\\spad{plusInfinity()} returns +infinity."))) 
-((-4446 |has| |#1| (-854))) 
-((|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (QUOTE (-21))) (-2738 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-854)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2738 (|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-551)))) 
+((-4445 |has| |#1| (-854))) 
+((|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (QUOTE (-21))) (-2740 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-854)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2740 (|HasCategory| |#1| (QUOTE (-854))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-551)))) 
 (-850) 
 ((|constructor| (NIL "Ordered finite sets.")) (|max| (($) "\\spad{max} is the maximum value of \\%.")) (|min| (($) "\\spad{min} is the minimum value of \\%."))) 
 NIL 
 NIL 
-(-851 -2407 S) 
+(-851 -2408 S) 
 ((|constructor| (NIL "\\indented{3}{This package provides ordering functions on vectors which} are suitable parameters for OrderedDirectProduct.")) (|reverseLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{reverseLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by the reverse lexicographic ordering.")) (|totalLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{totalLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by lexicographic ordering.")) (|pureLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{pureLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the lexicographic ordering."))) 
 NIL 
 NIL 
@@ -3346,7 +3346,7 @@ NIL
 NIL 
 (-854) 
 ((|constructor| (NIL "Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \\blankline")) (|abs| (($ $) "\\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|) $) "\\spad{positive?(x)} tests whether \\spad{x} is strictly greater than 0."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-855 S) 
 ((|constructor| (NIL "The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \\spad{(a,b)} exactly one of the following relations holds \\spad{a<b or a=b or b<a} and the relation is transitive,{} \\spadignore{i.e.} \\spad{a<b and b<c => a<c}.")) (|min| (($ $ $) "\\spad{min(x,y)} returns the minimum of \\spad{x} and \\spad{y} relative to \\spad{\"<\"}.")) (|max| (($ $ $) "\\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|) $ $) "\\spad{x < y} is a strict total ordering on the elements of the set."))) 
@@ -3362,7 +3362,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174)))) 
 (-858 R) 
 ((|constructor| (NIL "This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}. This category is an evolution of the types \\indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \\indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \\spad{M}. Watt.")) (|leftLcm| (($ $ $) "\\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)}.}"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-859 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."))) 
@@ -3370,11 +3370,11 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) 
 (-860 R |sigma| -3048) 
 ((|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."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-861 |x| R |sigma| -3048) 
 ((|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}."))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-368)))) 
 (-862 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..)}."))) 
@@ -3418,7 +3418,7 @@ NIL
 NIL 
 (-872 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") $ $) "\\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \\spad{R} is a Field)"))) 
-((-4444 |has| |#1| (-174)) (-4443 |has| |#1| (-174)) (-4446 . T)) 
+((-4443 |has| |#1| (-174)) (-4442 |has| |#1| (-174)) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-873 R PS UP) 
 ((|constructor| (NIL "\\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \\indented{2}{\"Pade Approximants,{} Part I: Basic Theory\",{} Baker & Graves-Morris.}")) (|padecf| (((|Union| (|ContinuedFraction| |#3|) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) |#2| |#2|) "\\spad{padecf(nd,dd,ns,ds)} computes the approximant as a continued fraction of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function).")) (|pade| (((|Union| (|Fraction| |#3|) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) |#2| |#2|) "\\spad{pade(nd,dd,ns,ds)} computes the approximant as a quotient of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function)."))) 
@@ -3430,24 +3430,24 @@ NIL
 NIL 
 (-875 |p|) 
 ((|constructor| (NIL "This is the catefory of stream-based representations of \\indented{2}{the \\spad{p}-adic integers.}")) (|root| (($ (|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}."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-876 |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)."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-877 |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)."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-876 |#1|) (QUOTE (-916))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-876 |#1|) (QUOTE (-146))) (|HasCategory| (-876 |#1|) (QUOTE (-148))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-876 |#1|) (QUOTE (-1031))) (|HasCategory| (-876 |#1|) (QUOTE (-826))) (-2738 (|HasCategory| (-876 |#1|) (QUOTE (-826))) (|HasCategory| (-876 |#1|) (QUOTE (-856)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-876 |#1|) (QUOTE (-1161))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| (-876 |#1|) (QUOTE (-235))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -520) (QUOTE (-1186)) (LIST (QUOTE -876) (|devaluate| |#1|)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -313) (LIST (QUOTE -876) (|devaluate| |#1|)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -290) (LIST (QUOTE -876) (|devaluate| |#1|)) (LIST (QUOTE -876) (|devaluate| |#1|)))) (|HasCategory| (-876 |#1|) (QUOTE (-311))) (|HasCategory| (-876 |#1|) (QUOTE (-551))) (|HasCategory| (-876 |#1|) (QUOTE (-856))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-876 |#1|) (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-876 |#1|) (QUOTE (-916)))) (|HasCategory| (-876 |#1|) (QUOTE (-146))))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-876 |#1|) (QUOTE (-916))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-876 |#1|) (QUOTE (-146))) (|HasCategory| (-876 |#1|) (QUOTE (-148))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-876 |#1|) (QUOTE (-1031))) (|HasCategory| (-876 |#1|) (QUOTE (-826))) (-2740 (|HasCategory| (-876 |#1|) (QUOTE (-826))) (|HasCategory| (-876 |#1|) (QUOTE (-856)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-876 |#1|) (QUOTE (-1161))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| (-876 |#1|) (QUOTE (-235))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -520) (QUOTE (-1186)) (LIST (QUOTE -876) (|devaluate| |#1|)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -313) (LIST (QUOTE -876) (|devaluate| |#1|)))) (|HasCategory| (-876 |#1|) (LIST (QUOTE -290) (LIST (QUOTE -876) (|devaluate| |#1|)) (LIST (QUOTE -876) (|devaluate| |#1|)))) (|HasCategory| (-876 |#1|) (QUOTE (-311))) (|HasCategory| (-876 |#1|) (QUOTE (-551))) (|HasCategory| (-876 |#1|) (QUOTE (-856))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-876 |#1|) (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-876 |#1|) (QUOTE (-916)))) (|HasCategory| (-876 |#1|) (QUOTE (-146))))) 
 (-878 |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)}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-1031))) (|HasCategory| |#2| (QUOTE (-826))) (-2738 (|HasCategory| |#2| (QUOTE (-826))) (|HasCategory| |#2| (QUOTE (-856)))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-1161))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-235))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -290) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (QUOTE (-856))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-1031))) (|HasCategory| |#2| (QUOTE (-826))) (-2740 (|HasCategory| |#2| (QUOTE (-826))) (|HasCategory| |#2| (QUOTE (-856)))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-1161))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-235))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -290) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (QUOTE (-856))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
 (-879 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 (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))))) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))))) 
 (-880) 
 ((|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 
@@ -3516,7 +3516,7 @@ NIL
 ((|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 
-(-897 R -2790) 
+(-897 R -2791) 
 ((|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 
@@ -3540,7 +3540,7 @@ NIL
 ((|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 
-(-903 UP -1673) 
+(-903 UP -1674) 
 ((|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 
@@ -3558,19 +3558,19 @@ NIL
 NIL 
 (-907 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}."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-908 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 (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-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 
 (-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."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
 (-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|)) $) "\\spad{coerce(gp)} returns the generators of the group {\\em gp}."))) 
@@ -3578,8 +3578,8 @@ NIL
 NIL 
 (-912 S) 
 ((|constructor| (NIL "Permutation(\\spad{S}) implements the group of all bijections \\indented{2}{on a set \\spad{S},{} which move only a finite number of points.} \\indented{2}{A permutation is considered as a map from \\spad{S} into \\spad{S}. In particular} \\indented{2}{multiplication is defined as composition of maps:} \\indented{2}{{\\em pi1 * pi2 = pi1 o pi2}.} \\indented{2}{The internal representation of permuatations are two lists} \\indented{2}{of equal length representing preimages and images.}")) (|coerceImages| (($ (|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."))) 
-((-4446 . T)) 
-((-2738 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-856)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-856)))) 
+((-4445 . T)) 
+((-2740 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-856)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-856)))) 
 (-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 
@@ -3594,13 +3594,13 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-146)))) 
 (-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}."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-917 |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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 ((|HasCategory| $ (QUOTE (-148))) (|HasCategory| $ (QUOTE (-146))) (|HasCategory| $ (QUOTE (-373)))) 
-(-918 R0 -1673 UP UPUP R) 
+(-918 R0 -1674 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 
@@ -3614,7 +3614,7 @@ NIL
 NIL 
 (-921 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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-922 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."))) 
@@ -3628,7 +3628,7 @@ NIL
 ((|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 
-(-925 -1673) 
+(-925 -1674) 
 ((|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 
@@ -3638,17 +3638,17 @@ NIL
 NIL 
 (-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| $)) "\\spad{principalIdeal([f1,...,fn])} returns a record whose generator component is a generator of the ideal generated by \\spad{[f1,...,fn]} whose coef component satisfies \\spad{generator = sum (input.i * coef.i)}"))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-928) 
 ((|constructor| (NIL "\\spadtype{PositiveInteger} provides functions for \\indented{2}{positive integers.}")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} means multiplication is commutative : x*y = \\spad{y*x}")) (|gcd| (($ $ $) "\\spad{gcd(a,b)} computes the greatest common divisor of two positive integers \\spad{a} and \\spad{b}."))) 
-(((-4451 "*") . T)) 
+(((-4450 "*") . T)) 
 NIL 
-(-929 -1673 P) 
+(-929 -1674 P) 
 ((|constructor| (NIL "This package exports interpolation algorithms")) (|LagrangeInterpolation| ((|#2| (|List| |#1|) (|List| |#1|)) "\\spad{LagrangeInterpolation(l1,l2)} \\undocumented"))) 
 NIL 
 NIL 
-(-930 |xx| -1673) 
+(-930 |xx| -1674) 
 ((|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 
@@ -3672,7 +3672,7 @@ NIL
 ((|constructor| (NIL "This package exports plotting tools")) (|calcRanges| (((|List| (|Segment| (|DoubleFloat|))) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{calcRanges(l)} \\undocumented"))) 
 NIL 
 NIL 
-(-936 R -1673) 
+(-936 R -1674) 
 ((|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 
@@ -3684,7 +3684,7 @@ NIL
 ((|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 
-(-939 S R -1673) 
+(-939 S R -1674) 
 ((|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 
@@ -3704,11 +3704,11 @@ NIL
 ((|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 -893) (|devaluate| |#1|)))) 
-(-944 R -1673 -2790) 
+(-944 R -1674 -2791) 
 ((|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 
-(-945 -2790) 
+(-945 -2791) 
 ((|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 
@@ -3730,8 +3730,8 @@ NIL
 NIL 
 (-950 R) 
 ((|constructor| (NIL "This domain implements points in coordinate space"))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-951 |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 
@@ -3751,12 +3751,12 @@ NIL
 (-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 (-916))) (|HasAttribute| |#2| (QUOTE -4447)) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#4| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#4| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#4| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#4| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) 
+((|HasCategory| |#2| (QUOTE (-916))) (|HasAttribute| |#2| (QUOTE -4446)) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#4| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#4| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#4| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#4| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) 
 (-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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
-(-957 E V R P -1673) 
+(-957 E V R P -1674) 
 ((|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 
@@ -3766,9 +3766,9 @@ NIL
 NIL 
 (-959 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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-916))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4447)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
-(-960 E V R P -1673) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1186) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4446)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
+(-960 E V R P -1674) 
 ((|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 (-458)))) 
@@ -3790,13 +3790,13 @@ NIL
 NIL 
 (-965 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"))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-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 
-(-967 -1673) 
+(-967 -1674) 
 ((|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 
@@ -3810,26 +3810,26 @@ NIL
 NIL 
 (-970 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}"))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-132)))) (|HasAttribute| |#1| (QUOTE -4447))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-132)))) (|HasAttribute| |#1| (QUOTE -4446))) 
 (-971 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"))) 
-((-4446 -12 (|has| |#2| (-479)) (|has| |#1| (-479)))) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799)))) (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-856))))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799))))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799))))) (-12 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-479)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-479)))) (-12 (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#2| (QUOTE (-732))))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-373)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-479)))) (-12 (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#2| (QUOTE (-732)))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799))))) (-12 (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#2| (QUOTE (-732)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-856))))) 
+((-4445 -12 (|has| |#2| (-479)) (|has| |#1| (-479)))) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799)))) (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-856))))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799))))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799))))) (-12 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-479)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-479)))) (-12 (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#2| (QUOTE (-732))))) (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-373)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-479)))) (-12 (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#2| (QUOTE (-732)))) (-12 (|HasCategory| |#1| (QUOTE (-799))) (|HasCategory| |#2| (QUOTE (-799))))) (-12 (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#2| (QUOTE (-732)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-132))) (|HasCategory| |#2| (QUOTE (-132)))) (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-856))))) 
 (-972) 
 ((|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|) $) "\\spad{name(p)} returns the name of property \\spad{p}"))) 
 NIL 
 NIL 
 (-973 T$) 
-((|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|) $) "\\spad{isTerm f} returns a value \\spad{v} such that \\spad{v case T} holds if the formula \\spad{f} is a term."))) 
+((|constructor| (NIL "This domain implements propositional formula build over a term domain,{} that itself belongs to PropositionalLogic")) (|disjunction| (($ $ $) "\\spad{disjunction(p,q)} returns a formula denoting the disjunction of \\spad{p} and \\spad{q}.")) (|conjunction| (($ $ $) "\\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.")) (|isAtom| (((|Maybe| |#1|) $) "\\spad{isAtom f} returns a value \\spad{v} such that \\spad{v case T} holds if the formula \\spad{f} is a term."))) 
 NIL 
 NIL 
 (-974 T$) 
-((|constructor| (NIL "This package collects unary functions operating on propositional formulae.")) (|simplify| (((|PropositionalFormula| |#1|) (|PropositionalFormula| |#1|)) "\\spad{simplify f} returns a formula logically equivalent to \\spad{f} where obvious tautologies have been removed.")) (|terms| (((|Set| |#1|) (|PropositionalFormula| |#1|)) "\\spad{terms f} \\spad{++} returns the set of terms appearing in the formula \\spad{f}.")) (|dual| (((|PropositionalFormula| |#1|) (|PropositionalFormula| |#1|)) "\\spad{dual f} returns the dual of the proposition \\spad{f}."))) 
+((|constructor| (NIL "This package collects unary functions operating on propositional formulae.")) (|simplify| (((|PropositionalFormula| |#1|) (|PropositionalFormula| |#1|)) "\\spad{simplify f} returns a formula logically equivalent to \\spad{f} where obvious tautologies have been removed.")) (|atoms| (((|Set| |#1|) (|PropositionalFormula| |#1|)) "\\spad{atoms f} \\spad{++} returns the set of atoms appearing in the formula \\spad{f}.")) (|dual| (((|PropositionalFormula| |#1|) (|PropositionalFormula| |#1|)) "\\spad{dual f} returns the dual of the proposition \\spad{f}."))) 
 NIL 
 NIL 
 (-975 S T$) 
-((|constructor| (NIL "This package collects binary functions operating on propositional formulae.")) (|map| (((|PropositionalFormula| |#2|) (|Mapping| |#2| |#1|) (|PropositionalFormula| |#1|)) "\\spad{map(f,x)} returns a propositional formula where all terms in \\spad{x} have been replaced by the result of applying the function \\spad{f} to them."))) 
+((|constructor| (NIL "This package collects binary functions operating on propositional formulae.")) (|map| (((|PropositionalFormula| |#2|) (|Mapping| |#2| |#1|) (|PropositionalFormula| |#1|)) "\\spad{map(f,x)} returns a propositional formula where all atoms in \\spad{x} have been replaced by the result of applying the function \\spad{f} to them."))) 
 NIL 
 NIL 
 (-976) 
@@ -3838,7 +3838,7 @@ NIL
 NIL 
 (-977 S) 
 ((|constructor| (NIL "A priority queue is a bag of items from an ordered set where the item extracted is always the maximum element.")) (|merge!| (($ $ $) "\\spad{merge!(q,q1)} destructively changes priority queue \\spad{q} to include the values from priority queue \\spad{q1}.")) (|merge| (($ $ $) "\\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}."))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-978 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}}"))) 
@@ -3849,7 +3849,7 @@ NIL
 NIL 
 NIL 
 (-980) 
-((|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"))) 
+((|constructor| (NIL "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|) $) "\\spad{pdct(a1**n1 a2**n2 ...)} returns \\spad{n1! * a1**n1 * n2! * a2**n2 * ...}. This function is used in the package \\spadtype{CycleIndicators}.")) (|powers| (((|List| (|Pair| (|Integer|) (|PositiveInteger|))) (|List| (|Integer|))) "\\spad{powers(li)} returns a list of pairs. The second component of each pair is the multiplicity with which the first component occurs in \\spad{li}.")) (|partition| (($ (|List| (|Integer|))) "\\spad{partition(li)} converts a list of integers \\spad{li} to a partition"))) 
 NIL 
 NIL 
 (-981 S |Coef| |Expon| |Var|) 
@@ -3858,7 +3858,7 @@ NIL
 NIL 
 (-982 |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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-983) 
 ((|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}."))) 
@@ -3870,7 +3870,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-562)))) 
 (-985 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."))) 
-((-4449 . T)) 
+((-4448 . T)) 
 NIL 
 (-986 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."))) 
@@ -3886,7 +3886,7 @@ NIL
 NIL 
 (-989 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}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-990 R1 R2) 
 ((|constructor| (NIL "This package \\undocumented")) (|map| (((|Point| |#2|) (|Mapping| |#2| |#1|) (|Point| |#1|)) "\\spad{map(f,p)} \\undocumented"))) 
@@ -3904,7 +3904,7 @@ NIL
 ((|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 
-(-994 K R UP -1673) 
+(-994 K R UP -1674) 
 ((|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 
@@ -3934,7 +3934,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-1031))) (|HasCategory| |#2| (QUOTE (-826))) (|HasCategory| |#2| (QUOTE (-856))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-1161)))) 
 (-1001 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}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1002 |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}."))) 
@@ -3946,7 +3946,7 @@ NIL
 NIL 
 (-1004 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."))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-1005 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}."))) 
@@ -3954,7 +3954,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (QUOTE (-1069))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-294)))) 
 (-1006 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}."))) 
-((-4442 |has| |#1| (-294)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 |has| |#1| (-294)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1007 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}."))) 
@@ -3962,12 +3962,12 @@ NIL
 NIL 
 (-1008 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.}"))) 
-((-4442 |has| |#1| (-294)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-368))) (-2738 (|HasCategory| |#1| (QUOTE (-294))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (QUOTE (-294))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-1069))) (|HasCategory| |#1| (QUOTE (-551)))) 
+((-4441 |has| |#1| (-294)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-368))) (-2740 (|HasCategory| |#1| (QUOTE (-294))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (QUOTE (-294))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#1| (LIST (QUOTE -290) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-1069))) (|HasCategory| |#1| (QUOTE (-551)))) 
 (-1009 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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1010 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| (($ $) "\\spad{sqrt(x)} returns the square root of \\spad{x}."))) 
 NIL 
@@ -3976,14 +3976,14 @@ NIL
 ((|constructor| (NIL "The \\spad{RadicalCategory} is a model for the rational numbers.")) (** (($ $ (|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 
-(-1012 -1673 UP UPUP |radicnd| |n|) 
+(-1012 -1674 UP UPUP |radicnd| |n|) 
 ((|constructor| (NIL "Function field defined by y**n = \\spad{f}(\\spad{x})."))) 
-((-4442 |has| (-413 |#2|) (-368)) (-4447 |has| (-413 |#2|) (-368)) (-4441 |has| (-413 |#2|) (-368)) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-413 |#2|) (QUOTE (-146))) (|HasCategory| (-413 |#2|) (QUOTE (-148))) (|HasCategory| (-413 |#2|) (QUOTE (-354))) (-2738 (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-373))) (-2738 (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (-2738 (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-354))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -645) (QUOTE (-570)))) (-2738 (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368))))) 
+((-4441 |has| (-413 |#2|) (-368)) (-4446 |has| (-413 |#2|) (-368)) (-4440 |has| (-413 |#2|) (-368)) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-413 |#2|) (QUOTE (-146))) (|HasCategory| (-413 |#2|) (QUOTE (-148))) (|HasCategory| (-413 |#2|) (QUOTE (-354))) (-2740 (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (|HasCategory| (-413 |#2|) (QUOTE (-368))) (|HasCategory| (-413 |#2|) (QUOTE (-373))) (-2740 (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (QUOTE (-354)))) (-2740 (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-354))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -645) (QUOTE (-570)))) (-2740 (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 |#2|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-12 (|HasCategory| (-413 |#2|) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-413 |#2|) (QUOTE (-368)))) (-12 (|HasCategory| (-413 |#2|) (QUOTE (-235))) (|HasCategory| (-413 |#2|) (QUOTE (-368))))) 
 (-1013 |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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2738 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-570) (QUOTE (-916))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-1186)))) (|HasCategory| (-570) (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-148))) (|HasCategory| (-570) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-1031))) (|HasCategory| (-570) (QUOTE (-826))) (-2740 (|HasCategory| (-570) (QUOTE (-826))) (|HasCategory| (-570) (QUOTE (-856)))) (|HasCategory| (-570) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-1161))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| (-570) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| (-570) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| (-570) (QUOTE (-235))) (|HasCategory| (-570) (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| (-570) (LIST (QUOTE -520) (QUOTE (-1186)) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -313) (QUOTE (-570)))) (|HasCategory| (-570) (LIST (QUOTE -290) (QUOTE (-570)) (QUOTE (-570)))) (|HasCategory| (-570) (QUOTE (-311))) (|HasCategory| (-570) (QUOTE (-551))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-570) (LIST (QUOTE -645) (QUOTE (-570)))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| (-570) (QUOTE (-916)))) (|HasCategory| (-570) (QUOTE (-146))))) 
 (-1014) 
 ((|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 
@@ -4003,7 +4003,7 @@ NIL
 (-1018 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 -4450)) (|HasCategory| |#2| (QUOTE (-1109)))) 
+((|HasAttribute| |#1| (QUOTE -4449)) (|HasCategory| |#2| (QUOTE (-1109)))) 
 (-1019 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 
@@ -4014,21 +4014,21 @@ NIL
 NIL 
 (-1021) 
 ((|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}}"))) 
-((-4442 . T) (-4447 . T) (-4441 . T) (-4444 . T) (-4443 . T) ((-4451 "*") . T) (-4446 . T)) 
+((-4441 . T) (-4446 . T) (-4440 . T) (-4443 . T) (-4442 . T) ((-4450 "*") . T) (-4445 . T)) 
 NIL 
-(-1022 R -1673) 
+(-1022 R -1674) 
 ((|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 
-(-1023 R -1673) 
+(-1023 R -1674) 
 ((|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 
-(-1024 -1673 UP) 
+(-1024 -1674 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 
-(-1025 -1673 UP) 
+(-1025 -1674 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 
@@ -4062,9 +4062,9 @@ NIL
 NIL 
 (-1033 |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"))) 
-((-4442 . T) (-4447 . T) (-4441 . T) (-4444 . T) (-4443 . T) ((-4451 "*") . T) (-4446 . T)) 
-((-2738 (|HasCategory| (-413 (-570)) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-413 (-570)) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 (-570)) (LIST (QUOTE -1047) (QUOTE (-570))))) 
-(-1034 -1673 L) 
+((-4441 . T) (-4446 . T) (-4440 . T) (-4443 . T) (-4442 . T) ((-4450 "*") . T) (-4445 . T)) 
+((-2740 (|HasCategory| (-413 (-570)) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-413 (-570)) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-413 (-570)) (LIST (QUOTE -1047) (QUOTE (-570))))) 
+(-1034 -1674 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 
@@ -4074,12 +4074,12 @@ NIL
 ((|HasCategory| |#1| (QUOTE (-1109)))) 
 (-1036 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."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#4| (LIST (QUOTE -313) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1037 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 (-4451 "*")))) 
+((|HasAttribute| |#1| (QUOTE (-4450 "*")))) 
 (-1038 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 
@@ -4100,14 +4100,14 @@ NIL
 ((|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 
-(-1043 -1673 |Expon| |VarSet| |FPol| |LFPol|) 
+(-1043 -1674 |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"))) 
-(((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1044) 
 ((|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.}"))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1186))) (LIST (QUOTE |:|) (QUOTE -2223) (QUOTE (-52))))))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-52) (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -313) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-1186) (QUOTE (-856))) (|HasCategory| (-52) (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1186))) (LIST (QUOTE |:|) (QUOTE -2223) (QUOTE (-52))))))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-52) (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -313) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-1186) (QUOTE (-856))) (|HasCategory| (-52) (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1045) 
 ((|constructor| (NIL "This domain represents `return' expressions.")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression returned by `e'."))) 
 NIL 
@@ -4150,7 +4150,7 @@ NIL
 NIL 
 (-1055 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}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| (-786 |#1| (-870 |#2|)) (QUOTE (-1109))) (|HasCategory| (-786 |#1| (-870 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -786) (|devaluate| |#1|) (LIST (QUOTE -870) (|devaluate| |#2|)))))) (|HasCategory| (-786 |#1| (-870 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-786 |#1| (-870 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| (-870 |#2|) (QUOTE (-373))) (|HasCategory| (-786 |#1| (-870 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1056) 
 ((|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"))) 
@@ -4162,9 +4162,9 @@ NIL
 NIL 
 (-1058) 
 ((|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."))) 
-((-4446 . T)) 
+((-4445 . T)) 
 NIL 
-(-1059 |xx| -1673) 
+(-1059 |xx| -1674) 
 ((|constructor| (NIL "This package exports rational interpolation algorithms"))) 
 NIL 
 NIL 
@@ -4178,12 +4178,12 @@ NIL
 ((|HasCategory| |#4| (QUOTE (-311))) (|HasCategory| |#4| (QUOTE (-368))) (|HasCategory| |#4| (QUOTE (-562))) (|HasCategory| |#4| (QUOTE (-174)))) 
 (-1062 |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"))) 
-((-4449 . T) (-4444 . T) (-4443 . T)) 
+((-4448 . T) (-4443 . T) (-4442 . T)) 
 NIL 
 (-1063 |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}."))) 
-((-4449 . T) (-4444 . T) (-4443 . T)) 
-((|HasCategory| |#3| (QUOTE (-174))) (-2738 (-12 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-368))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|))))) (|HasCategory| |#3| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-368)))) (|HasCategory| |#3| (QUOTE (-368))) (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (QUOTE (-311))) (|HasCategory| |#3| (QUOTE (-562))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (|HasCategory| |#3| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4443 . T) (-4442 . T)) 
+((|HasCategory| |#3| (QUOTE (-174))) (-2740 (-12 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-368))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|))))) (|HasCategory| |#3| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (QUOTE (-368)))) (|HasCategory| |#3| (QUOTE (-368))) (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (QUOTE (-311))) (|HasCategory| |#3| (QUOTE (-562))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (|HasCategory| |#3| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1064 |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 
@@ -4206,7 +4206,7 @@ NIL
 NIL 
 (-1069) 
 ((|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."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1070 |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| $) "\\axiom{left(rootChar)} is the left bound of the isolating interval"))) 
@@ -4214,19 +4214,19 @@ NIL
 NIL 
 (-1071) 
 ((|constructor| (NIL "\\spadtype{RomanNumeral} provides functions for converting \\indented{1}{integers to roman numerals.}")) (|roman| (($ (|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."))) 
-((-4437 . T) (-4441 . T) (-4436 . T) (-4447 . T) (-4448 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4436 . T) (-4440 . T) (-4435 . T) (-4446 . T) (-4447 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1072) 
 ((|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}"))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1186))) (LIST (QUOTE |:|) (QUOTE -2223) (QUOTE (-52))))))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-52) (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -313) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-1186) (QUOTE (-856))) (|HasCategory| (-52) (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1186))) (LIST (QUOTE |:|) (QUOTE -2223) (QUOTE (-52))))))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-52) (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| (-52) (QUOTE (-1109))) (|HasCategory| (-52) (LIST (QUOTE -313) (QUOTE (-52))))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (QUOTE (-1109))) (|HasCategory| (-1186) (QUOTE (-856))) (|HasCategory| (-52) (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-52) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1186)) (|:| -2223 (-52))) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1073 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| $) "\\axiom{mvar(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in \\axiom{\\spad{V}}."))) 
 NIL 
 ((|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-551))) (|HasCategory| |#2| (LIST (QUOTE -38) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -1001) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-1186))))) 
 (-1074 R E V) 
 ((|constructor| (NIL "A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \\axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.")) (|mainSquareFreePart| (($ $) "\\axiom{mainSquareFreePart(\\spad{p})} returns the square free part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainPrimitivePart| (($ $) "\\axiom{mainPrimitivePart(\\spad{p})} returns the primitive part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainContent| (($ $) "\\axiom{mainContent(\\spad{p})} returns the content of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|primitivePart!| (($ $) "\\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}}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
 (-1075) 
 ((|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'."))) 
@@ -4250,7 +4250,7 @@ NIL
 NIL 
 (-1080 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}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-1081 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."))) 
@@ -4268,11 +4268,11 @@ NIL
 ((|constructor| (NIL "This domain implements named rules")) (|name| (((|Symbol|) $) "\\spad{name(x)} returns the symbol"))) 
 NIL 
 NIL 
-(-1085 |Base| R -1673) 
+(-1085 |Base| R -1674) 
 ((|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 
-(-1086 |Base| R -1673) 
+(-1086 |Base| R -1674) 
 ((|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 
@@ -4286,8 +4286,8 @@ NIL
 NIL 
 (-1089 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."))) 
-((-4442 |has| |#1| (-368)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-354))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (QUOTE (-354)))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-12 (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (QUOTE (-368))))) 
+((-4441 |has| |#1| (-368)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-354))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-354)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-373))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (QUOTE (-354)))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-12 (|HasCategory| |#1| (QUOTE (-354))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368)))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186))))) (-12 (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (QUOTE (-368))))) 
 (-1090 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 
@@ -4314,8 +4314,8 @@ NIL
 NIL 
 (-1096 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"))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-916))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4447)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1097 (-1186)) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-235))) (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4446)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
 (-1097 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 
@@ -4358,7 +4358,7 @@ NIL
 NIL 
 (-1107 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}."))) 
-((-4439 . T)) 
+((-4438 . T)) 
 NIL 
 (-1108 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}."))) 
@@ -4374,8 +4374,8 @@ NIL
 NIL 
 (-1111 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)}}"))) 
-((-4449 . T) (-4439 . T) (-4450 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4448 . T) (-4438 . T) (-4449 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#1| (QUOTE (-373))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-1112 |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 
@@ -4402,7 +4402,7 @@ NIL
 NIL 
 (-1118 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.}"))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-1119) 
 ((|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."))) 
@@ -4418,8 +4418,8 @@ NIL
 NIL 
 (-1122 |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}."))) 
-((-4443 |has| |#3| (-1058)) (-4444 |has| |#3| (-1058)) (-4446 |has| |#3| (-6 -4446)) ((-4451 "*") |has| |#3| (-174)) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-132))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-174))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-235))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-368))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-732))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-799))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-854))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1058))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|))) (|HasCategory| |#3| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|))) (|HasCategory| |#3| (LIST (QUOTE -907) (QUOTE (-1186)))))) (-2738 (-12 (|HasCategory| |#3| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#3| (QUOTE (-1109)))) (-12 (|HasCategory| |#3| (QUOTE (-235))) (|HasCategory| |#3| (QUOTE (-1058)))) (-12 (|HasCategory| |#3| (QUOTE (-1058))) (|HasCategory| |#3| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#3| (QUOTE (-1058))) (|HasCategory| |#3| (LIST (QUOTE -907) (QUOTE (-1186))))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -313) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1109))) (|HasCategory| |#3| (LIST (QUOTE -1047) (QUOTE (-570))))) (|HasCategory| 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 (-1123 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 
@@ -4428,7 +4428,7 @@ NIL
 ((|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 
-(-1125 R -1673) 
+(-1125 R -1674) 
 ((|constructor| (NIL "This package provides functions to determine the sign of an elementary function around a point or infinity.")) (|sign| (((|Union| (|Integer|) "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|) "failed") |#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|) "failed") |#2|) "\\spad{sign(f)} returns the sign of \\spad{f} if it is constant everywhere."))) 
 NIL 
 NIL 
@@ -4446,19 +4446,19 @@ NIL
 NIL 
 (-1129) 
 ((|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}.")) (|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."))) 
-((-4437 . T) (-4441 . T) (-4436 . T) (-4447 . T) (-4448 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4436 . T) (-4440 . T) (-4435 . T) (-4446 . T) (-4447 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1130 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| $) "\\spad{push!(x,s)} pushes \\spad{x} onto stack \\spad{s},{} \\spadignore{i.e.} destructively changing \\spad{s} so as to have a new first (top) element \\spad{x}. Afterwards,{} pop!(\\spad{s}) produces \\spad{x} and pop!(\\spad{s}) produces the original \\spad{s}."))) 
-((-4449 . T) (-4450 . T)) 
+((-4448 . T) (-4449 . T)) 
 NIL 
 (-1131 S |ndim| R |Row| |Col|) 
 ((|constructor| (NIL "\\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.")) (** (($ $ (|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 (-368))) (|HasAttribute| |#3| (QUOTE (-4451 "*"))) (|HasCategory| |#3| (QUOTE (-174)))) 
+((|HasCategory| |#3| (QUOTE (-368))) (|HasAttribute| |#3| (QUOTE (-4450 "*"))) (|HasCategory| |#3| (QUOTE (-174)))) 
 (-1132 |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."))) 
-((-4449 . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4448 . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1133 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}."))) 
@@ -4466,17 +4466,17 @@ NIL
 NIL 
 (-1134 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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-916))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4447)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2738 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-916))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (|HasCategory| |#1| (QUOTE (-458))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#1| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4446)) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-916)))) (|HasCategory| |#1| (QUOTE (-146))))) 
 (-1135 |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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-368)))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-1136 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}"))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
-(-1137 UP -1673) 
+(-1137 UP -1674) 
 ((|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 
@@ -4530,19 +4530,19 @@ NIL
 NIL 
 (-1150 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") $) "\\axiom{extractSplittingLeaf(a)} returns the left most leaf (as a tree) whose status is \\spad{false} if any,{} else \"failed\" is returned."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -313) (LIST (QUOTE -1149) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1149 |#1| |#2|) (QUOTE (-1109)))) (|HasCategory| (-1149 |#1| |#2|) (QUOTE (-1109))) (-2738 (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -313) (LIST (QUOTE -1149) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1149 |#1| |#2|) (QUOTE (-1109))))) (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -313) (LIST (QUOTE -1149) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1149 |#1| |#2|) (QUOTE (-1109)))) (|HasCategory| (-1149 |#1| |#2|) (QUOTE (-1109))) (-2740 (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -313) (LIST (QUOTE -1149) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1149 |#1| |#2|) (QUOTE (-1109))))) (|HasCategory| (-1149 |#1| |#2|) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1151 |ndim| R) 
 ((|constructor| (NIL "\\spadtype{SquareMatrix} is a matrix domain of square matrices,{} where the number of rows (= number of columns) is a parameter of the type.")) (|unitsKnown| ((|attribute|) "the invertible matrices are simply the matrices whose determinants are units in the Ring \\spad{R}.")) (|central| ((|attribute|) "the elements of the Ring \\spad{R},{} viewed as diagonal matrices,{} commute with all matrices and,{} indeed,{} are the only matrices which commute with all matrices.")) (|squareMatrix| (($ (|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}."))) 
-((-4446 . T) (-4438 |has| |#2| (-6 (-4451 "*"))) (-4449 . T) (-4443 . T) (-4444 . T)) 
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+((-4445 . T) (-4437 |has| |#2| (-6 (-4450 "*"))) (-4448 . T) (-4442 . T) (-4443 . T)) 
+((|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235))) (|HasAttribute| |#2| (QUOTE (-4450 "*"))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (-12 (|HasCategory| |#2| (QUOTE (-235))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570))))) (-12 (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-368))) (-2740 (|HasAttribute| |#2| (QUOTE (-4450 "*"))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-174)))) 
 (-1152 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|) $ $) "\\spad{prefix?(s,t)} tests if the string \\spad{s} is the initial substring of \\spad{t}. Note: \\axiom{prefix?(\\spad{s},{}\\spad{t}) \\spad{==} reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.\\spad{i} for \\spad{i} in 0..maxIndex \\spad{s}])}.")) (|upperCase!| (($ $) "\\spad{upperCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by upper case characters.")) (|upperCase| (($ $) "\\spad{upperCase(s)} returns the string with all characters in upper case.")) (|lowerCase!| (($ $) "\\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by lower case.")) (|lowerCase| (($ $) "\\spad{lowerCase(s)} returns the string with all characters in lower case."))) 
 NIL 
 NIL 
 (-1153) 
 ((|constructor| (NIL "A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.")) (|elt| (($ $ $) "\\spad{elt(s,t)} returns the concatenation of \\spad{s} and \\spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \\axiom{\"smoo\" \"shed\"} returns \\axiom{\"smooshed\"}.")) (|rightTrim| (($ $ (|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."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-1154 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.}"))) 
@@ -4550,12 +4550,12 @@ NIL
 NIL 
 (-1155 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."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#4| (LIST (QUOTE -313) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1156 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}."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1157 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 
@@ -4566,8 +4566,8 @@ NIL
 NIL 
 (-1159 |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."))) 
-((-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-856))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109)))) 
+((-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-856))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109)))) 
 (-1160) 
 ((|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|) $) "\\spad{iterationVar(i)} returns the name of the iterating variable of the arithmetic progression iterator \\spad{i}."))) 
 NIL 
@@ -4594,20 +4594,20 @@ NIL
 NIL 
 (-1166 S) 
 ((|constructor| (NIL "A stream is an implementation of an infinite sequence using a list of terms that have been computed and a function closure to compute additional terms when needed.")) (|filterUntil| (($ (|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."))) 
-((-4450 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4449 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1167) 
 ((|constructor| (NIL "A category for string-like objects")) (|string| (($ (|Integer|)) "\\spad{string(i)} returns the decimal representation of \\spad{i} in a string"))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-1168) 
 NIL 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145))))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) (|HasCategory| (-145) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| (-145) (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| (-145) (QUOTE (-1109))) (|HasCategory| (-145) (LIST (QUOTE -313) (QUOTE (-145)))))) 
 (-1169 |Entry|) 
 ((|constructor| (NIL "This domain provides tables where the keys are strings. A specialized hash function for strings is used."))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1168))) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#1|)))))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| (-1168) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (QUOTE (-1168))) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#1|)))))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (QUOTE (-1109))) (|HasCategory| (-1168) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 (-1168)) (|:| -2223 |#1|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1170 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 
@@ -4638,9 +4638,9 @@ NIL
 NIL 
 (-1177 |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."))) 
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+(-1178 R -1674) 
 ((|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 
@@ -4658,16 +4658,16 @@ NIL
 NIL 
 (-1182 R) 
 ((|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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 (-1183 |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}."))) 
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 (-1184 |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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|)))) (|HasCategory| (-777) (QUOTE (-1121))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasCategory| |#1| (QUOTE (-368))) (-2738 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3665) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|)))) (|HasCategory| (-777) (QUOTE (-1121))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasCategory| |#1| (QUOTE (-368))) (-2740 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3555) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
 (-1185) 
 ((|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 
@@ -4682,8 +4682,8 @@ NIL
 NIL 
 (-1188 R) 
 ((|constructor| (NIL "This domain implements symmetric polynomial"))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-6 -4447)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| (-980) (QUOTE (-132))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasAttribute| |#1| (QUOTE -4447))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-6 -4446)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-2740 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-458))) (-12 (|HasCategory| (-980) (QUOTE (-132))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasAttribute| |#1| (QUOTE -4446))) 
 (-1189) 
 ((|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| "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| "void"))) "\\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| "void"))) "\\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| "void")) $) "\\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 
@@ -4726,8 +4726,8 @@ NIL
 NIL 
 (-1199 |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}"))) 
-((-4449 . T) (-4450 . T)) 
-((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2738 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4448 . T) (-4449 . T)) 
+((-12 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -313) (LIST (QUOTE -2) (LIST (QUOTE |:|) (QUOTE -2013) (|devaluate| |#1|)) (LIST (QUOTE |:|) (QUOTE -2223) (|devaluate| |#2|)))))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#2| (QUOTE (-1109)))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -620) (QUOTE (-542)))) (-12 (|HasCategory| |#2| (QUOTE (-1109))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#2| (QUOTE (-1109))) (-2740 (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#2| (LIST (QUOTE -619) (QUOTE (-868)))) (|HasCategory| (-2 (|:| -2013 |#1|) (|:| -2223 |#2|)) (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1200 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 
@@ -4738,7 +4738,7 @@ NIL
 NIL 
 (-1202 |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})}."))) 
-((-4450 . T)) 
+((-4449 . T)) 
 NIL 
 (-1203 |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."))) 
@@ -4778,8 +4778,8 @@ NIL
 NIL 
 (-1212 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| $)) "\\spad{tree(nd,ls)} creates a tree with value \\spad{nd},{} and children \\spad{ls}."))) 
-((-4450 . T) (-4449 . T)) 
-((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
+((-4449 . T) (-4448 . T)) 
+((-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1109))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1213 S) 
 ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) 
 NIL 
@@ -4788,7 +4788,7 @@ NIL
 ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) 
 NIL 
 NIL 
-(-1215 R -1673) 
+(-1215 R -1674) 
 ((|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 
@@ -4796,7 +4796,7 @@ NIL
 ((|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 
-(-1217 R -1673) 
+(-1217 R -1674) 
 ((|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 -620) (LIST (QUOTE -899) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -893) (|devaluate| |#1|))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (|devaluate| |#1|)))) (|HasCategory| |#2| (LIST (QUOTE -893) (|devaluate| |#1|))))) 
@@ -4806,12 +4806,12 @@ NIL
 ((|HasCategory| |#4| (QUOTE (-373)))) 
 (-1219 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|) $ $) "\\axiom{infRittWu?(\\spad{ts1},{}\\spad{ts2})} returns \\spad{true} iff \\axiom{\\spad{ts2}} has higher rank than \\axiom{\\spad{ts1}} in Wu Wen Tsun sense."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-1220 |Coef|) 
 ((|constructor| (NIL "\\spadtype{TaylorSeries} is a general multivariate Taylor series domain over the ring Coef and with variables of type Symbol.")) (|fintegrate| (($ (|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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-368)))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-148))) (|HasCategory| |#1| (QUOTE (-146))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-1221 |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 
@@ -4824,7 +4824,7 @@ NIL
 ((|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 (-1109))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) 
-(-1224 -1673) 
+(-1224 -1674) 
 ((|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 
@@ -4850,7 +4850,7 @@ NIL
 NIL 
 (-1230) 
 ((|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|) $) "\\spad{prime?(x)} tests if \\spad{x} can never be written as the product of two non-units of the ring,{} \\spadignore{i.e.} \\spad{x} is an irreducible element."))) 
-((-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1231) 
 ((|constructor| (NIL "This domain is a datatype for (unsigned) integer values of precision 16 bits."))) 
@@ -4874,7 +4874,7 @@ NIL
 NIL 
 (-1236 |Coef|) 
 ((|constructor| (NIL "\\spadtype{UnivariateLaurentSeriesCategory} is the category of Laurent series in one variable.")) (|integrate| (($ $ (|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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1237 S |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| |#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)}."))) 
@@ -4882,16 +4882,16 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-368)))) 
 (-1238 |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)}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1239 |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)}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((-2738 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -290) (|devaluate| |#2|) (|devaluate| |#2|)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -520) (QUOTE (-1186)) (|devaluate| |#2|)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-826)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-856)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-916)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-1031)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-1161)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -313) (|devaluate| |#2|)))) (-12 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#2| (LIST (QUOTE -1047) 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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 
@@ -4926,8 +4926,8 @@ NIL
 NIL 
 (-1249 |x| R) 
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+((|HasCategory| |#2| (QUOTE (-916))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-562)))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-384)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-384))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -893) (QUOTE (-570)))) (|HasCategory| |#2| (LIST (QUOTE -893) (QUOTE (-570))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-384)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -620) (LIST (QUOTE -899) (QUOTE (-570)))))) (-12 (|HasCategory| (-1091) (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#2| (LIST (QUOTE -620) (QUOTE (-542))))) (|HasCategory| |#2| (LIST (QUOTE -645) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-148))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (QUOTE (-570)))) (-2740 (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| |#2| (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (-2740 (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-1161))) (|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasCategory| |#2| (QUOTE (-235))) (|HasAttribute| |#2| (QUOTE -4446)) (|HasCategory| |#2| (QUOTE (-458))) (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (-2740 (-12 (|HasCategory| $ (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-916)))) (|HasCategory| |#2| (QUOTE (-146))))) 
 (-1250 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 
@@ -4938,7 +4938,7 @@ NIL
 ((|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368))) (|HasCategory| |#2| (QUOTE (-458))) (|HasCategory| |#2| (QUOTE (-562))) (|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (QUOTE (-1161)))) 
 (-1252 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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4445 |has| |#1| (-368)) (-4447 |has| |#1| (-6 -4447)) (-4444 . T) (-4443 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4444 |has| |#1| (-368)) (-4446 |has| |#1| (-6 -4446)) (-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
 (-1253 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."))) 
@@ -4946,7 +4946,7 @@ NIL
 ((|HasCategory| |#2| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#2| (LIST (QUOTE *) (LIST (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#2|)))) (|HasCategory| |#3| (QUOTE (-1121))) (|HasSignature| |#2| (LIST (QUOTE **) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (|devaluate| |#3|)))) (|HasSignature| |#2| (LIST (QUOTE -3735) (LIST (|devaluate| |#2|) (QUOTE (-1186)))))) 
 (-1254 |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|))) $) "\\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1255 RC P) 
 ((|constructor| (NIL "This package provides for square-free decomposition of univariate polynomials over arbitrary rings,{} \\spadignore{i.e.} a partial factorization such that each factor is a product of irreducibles with multiplicity one and the factors are pairwise relatively prime. If the ring has characteristic zero,{} the result is guaranteed to satisfy this condition. If the ring is an infinite ring of finite characteristic,{} then it may not be possible to decide when polynomials contain factors which are \\spad{p}th powers. In this case,{} the flag associated with that polynomial is set to \"nil\" (meaning that that polynomials are not guaranteed to be square-free).")) (|BumInSepFFE| (((|Record| (|:| |flg| (|Union| "nil" "sqfr" "irred" "prime")) (|:| |fctr| |#2|) (|:| |xpnt| (|Integer|))) (|Record| (|:| |flg| (|Union| "nil" "sqfr" "irred" "prime")) (|:| |fctr| |#2|) (|:| |xpnt| (|Integer|)))) "\\spad{BumInSepFFE(f)} is a local function,{} exported only because it has multiple conditional definitions.")) (|squareFreePart| ((|#2| |#2|) "\\spad{squareFreePart(p)} returns a polynomial which has the same irreducible factors as the univariate polynomial \\spad{p},{} but each factor has multiplicity one.")) (|squareFree| (((|Factored| |#2|) |#2|) "\\spad{squareFree(p)} computes the square-free factorization of the univariate polynomial \\spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime.")) (|gcd| (($ $ $) "\\spad{gcd(p,q)} computes the greatest-common-divisor of \\spad{p} and \\spad{q}."))) 
@@ -4958,7 +4958,7 @@ NIL
 NIL 
 (-1257 |Coef|) 
 ((|constructor| (NIL "\\spadtype{UnivariatePuiseuxSeriesCategory} is the category of Puiseux series in one variable.")) (|integrate| (($ $ (|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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1258 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)}."))) 
@@ -4966,24 +4966,24 @@ NIL
 NIL 
 (-1259 |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)}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1260 |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)}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2738 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3665) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|)))))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2740 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3555) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|)))))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) 
 (-1261 |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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4447 |has| |#1| (-368)) (-4441 |has| |#1| (-368)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2738 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2738 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3665) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4446 |has| |#1| (-368)) (-4440 |has| |#1| (-368)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#1| (QUOTE (-174))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570))) (|devaluate| |#1|)))) (|HasCategory| (-413 (-570)) (QUOTE (-1121))) (|HasCategory| |#1| (QUOTE (-368))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-2740 (|HasCategory| |#1| (QUOTE (-368))) (|HasCategory| |#1| (QUOTE (-562)))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (LIST (QUOTE -413) (QUOTE (-570)))))) (-2740 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3555) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
 (-1262 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))}."))) 
-(((-4451 "*") |has| (-1261 |#2| |#3| |#4|) (-174)) (-4442 |has| (-1261 |#2| |#3| |#4|) (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-146))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-148))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-174))) (-2738 (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-368))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-458))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-562)))) 
+(((-4450 "*") |has| (-1261 |#2| |#3| |#4|) (-174)) (-4441 |has| (-1261 |#2| |#3| |#4|) (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-146))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-148))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-174))) (-2740 (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570)))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -1047) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| (-1261 |#2| |#3| |#4|) (LIST (QUOTE -1047) (QUOTE (-570)))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-368))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-458))) (|HasCategory| (-1261 |#2| |#3| |#4|) (QUOTE (-562)))) 
 (-1263 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| $) "\\spad{concat(x,u)} returns aggregate consisting of \\spad{x} followed by the elements of \\spad{u}. Note: if \\axiom{\\spad{v} = concat(\\spad{x},{}\\spad{u})} then \\axiom{\\spad{x} = first \\spad{v}} and \\axiom{\\spad{u} = rest \\spad{v}}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate \\spad{w} consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: \\axiom{\\spad{v} = rest(\\spad{w},{}\\#a)}."))) 
 NIL 
-((|HasAttribute| |#1| (QUOTE -4450))) 
+((|HasAttribute| |#1| (QUOTE -4449))) 
 (-1264 S) 
 ((|constructor| (NIL "A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \\spadfun{value} of the list designating the head,{} or \\spadfun{first},{} of the list,{} and the child designating the tail,{} or \\spadfun{rest},{} of the list. A node with no child then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.")) (|split!| (($ $ (|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| $) "\\spad{concat(x,u)} returns aggregate consisting of \\spad{x} followed by the elements of \\spad{u}. Note: if \\axiom{\\spad{v} = concat(\\spad{x},{}\\spad{u})} then \\axiom{\\spad{x} = first \\spad{v}} and \\axiom{\\spad{u} = rest \\spad{v}}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate \\spad{w} consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: \\axiom{\\spad{v} = rest(\\spad{w},{}\\#a)}."))) 
 NIL 
@@ -4995,20 +4995,20 @@ NIL
 (-1266 S |Coef|) 
 ((|constructor| (NIL "\\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.")) (|integrate| (($ $ (|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 (-570)))) (|HasCategory| |#2| (QUOTE (-966))) (|HasCategory| |#2| (QUOTE (-1211))) (|HasSignature| |#2| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#2|)))) (|HasSignature| |#2| (LIST (QUOTE -3665) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1186))))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368)))) 
+((|HasCategory| |#2| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#2| (QUOTE (-966))) (|HasCategory| |#2| (QUOTE (-1211))) (|HasSignature| |#2| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#2|)))) (|HasSignature| |#2| (LIST (QUOTE -3555) (LIST (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1186))))) (|HasCategory| |#2| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#2| (QUOTE (-368)))) 
 (-1267 |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."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1268 |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}."))) 
-(((-4451 "*") |has| |#1| (-174)) (-4442 |has| |#1| (-562)) (-4443 . T) (-4444 . T) (-4446 . T)) 
-((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2738 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|)))) (|HasCategory| (-777) (QUOTE (-1121))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasCategory| |#1| (QUOTE (-368))) (-2738 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3665) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
+(((-4450 "*") |has| |#1| (-174)) (-4441 |has| |#1| (-562)) (-4442 . T) (-4443 . T) (-4445 . T)) 
+((|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasCategory| |#1| (QUOTE (-562))) (-2740 (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-562)))) (|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-148))) (-12 (|HasCategory| |#1| (LIST (QUOTE -907) (QUOTE (-1186)))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|))))) (|HasSignature| |#1| (LIST (QUOTE *) (LIST (|devaluate| |#1|) (QUOTE (-777)) (|devaluate| |#1|)))) (|HasCategory| (-777) (QUOTE (-1121))) (-12 (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasSignature| |#1| (LIST (QUOTE -3735) (LIST (|devaluate| |#1|) (QUOTE (-1186)))))) (|HasSignature| |#1| (LIST (QUOTE **) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-777))))) (|HasCategory| |#1| (QUOTE (-368))) (-2740 (-12 (|HasCategory| |#1| (LIST (QUOTE -29) (QUOTE (-570)))) (|HasCategory| |#1| (QUOTE (-966))) (|HasCategory| |#1| (QUOTE (-1211))) (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570)))))) (-12 (|HasCategory| |#1| (LIST (QUOTE -38) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasSignature| |#1| (LIST (QUOTE -3555) (LIST (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1186))))) (|HasSignature| |#1| (LIST (QUOTE -1716) (LIST (LIST (QUOTE -650) (QUOTE (-1186))) (|devaluate| |#1|))))))) 
 (-1269 |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 
-(-1270 -1673 UP L UTS) 
+(-1270 -1674 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 (-562)))) 
@@ -5026,7 +5026,7 @@ NIL
 ((|HasCategory| |#2| (QUOTE (-1011))) (|HasCategory| |#2| (QUOTE (-1058))) (|HasCategory| |#2| (QUOTE (-732))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-25)))) 
 (-1274 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}.")) (+ (($ $ $) "\\spad{x + y} returns the component-wise sum of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 NIL 
 (-1275 A B) 
 ((|constructor| (NIL "\\indented{2}{This package provides operations which all take as arguments} vectors of elements of some type \\spad{A} and functions from \\spad{A} to another of type \\spad{B}. The operations all iterate over their vector argument and either return a value of type \\spad{B} or a vector over \\spad{B}.")) (|map| (((|Union| (|Vector| |#2|) "failed") (|Mapping| (|Union| |#2| "failed") |#1|) (|Vector| |#1|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values or \\spad{\"failed\"}.") (((|Vector| |#2|) (|Mapping| |#2| |#1|) (|Vector| |#1|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|Vector| |#1|) |#2|) "\\spad{reduce(func,vec,ident)} combines the elements in \\spad{vec} using the binary function \\spad{func}. Argument \\spad{ident} is returned if \\spad{vec} is empty.")) (|scan| (((|Vector| |#2|) (|Mapping| |#2| |#1| |#2|) (|Vector| |#1|) |#2|) "\\spad{scan(func,vec,ident)} creates a new vector whose elements are the result of applying reduce to the binary function \\spad{func},{} increasing initial subsequences of the vector \\spad{vec},{} and the element \\spad{ident}."))) 
@@ -5034,8 +5034,8 @@ NIL
 NIL 
 (-1276 R) 
 ((|constructor| (NIL "This type represents vector like objects with varying lengths and indexed by a finite segment of integers starting at 1.")) (|vector| (($ (|List| |#1|)) "\\spad{vector(l)} converts the list \\spad{l} to a vector."))) 
-((-4450 . T) (-4449 . T)) 
-((-2738 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2738 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2738 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
+((-4449 . T) (-4448 . T)) 
+((-2740 (-12 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) (-2740 (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868))))) (|HasCategory| |#1| (LIST (QUOTE -620) (QUOTE (-542)))) (-2740 (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109)))) (|HasCategory| |#1| (QUOTE (-856))) (|HasCategory| (-570) (QUOTE (-856))) (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-732))) (|HasCategory| |#1| (QUOTE (-1058))) (-12 (|HasCategory| |#1| (QUOTE (-1011))) (|HasCategory| |#1| (QUOTE (-1058)))) (|HasCategory| |#1| (LIST (QUOTE -619) (QUOTE (-868)))) (-12 (|HasCategory| |#1| (QUOTE (-1109))) (|HasCategory| |#1| (LIST (QUOTE -313) (|devaluate| |#1|))))) 
 (-1277) 
 ((|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|))) $) "\\spad{getPickedPoints(x)} returns a list of small floats for the points the user interactively picked on the viewport for full integration into the system,{} some design issues need to be addressed: \\spadignore{e.g.} how to go through the GraphImage interface,{} how to default to graphs,{} etc."))) 
 NIL 
@@ -5062,13 +5062,13 @@ NIL
 NIL 
 (-1283 S) 
 ((|constructor| (NIL "Vector Spaces (not necessarily finite dimensional) over a field.")) (|dimension| (((|CardinalNumber|)) "\\spad{dimension()} returns the dimensionality of the vector space.")) (/ (($ $ |#1|) "\\spad{x/y} divides the vector \\spad{x} by the scalar \\spad{y}."))) 
-((-4444 . T) (-4443 . T)) 
+((-4443 . T) (-4442 . T)) 
 NIL 
 (-1284 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 
-(-1285 K R UP -1673) 
+(-1285 K R UP -1674) 
 ((|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 
@@ -5082,56 +5082,56 @@ NIL
 NIL 
 (-1288 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)"))) 
-((-4444 |has| |#1| (-174)) (-4443 |has| |#1| (-174)) (-4446 . T)) 
+((-4443 |has| |#1| (-174)) (-4442 |has| |#1| (-174)) (-4445 . T)) 
 ((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368)))) 
 (-1289 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}}."))) 
-((-4450 . T) (-4449 . T)) 
+((-4449 . T) (-4448 . T)) 
 ((-12 (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#4| (LIST (QUOTE -313) (|devaluate| |#4|)))) (|HasCategory| |#4| (LIST (QUOTE -620) (QUOTE (-542)))) (|HasCategory| |#4| (QUOTE (-1109))) (|HasCategory| |#1| (QUOTE (-562))) (|HasCategory| |#3| (QUOTE (-373))) (|HasCategory| |#4| (LIST (QUOTE -619) (QUOTE (-868))))) 
 (-1290 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})"))) 
-((-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1291 |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."))) 
-((-4446 . T) (-4442 |has| |#2| (-6 -4442)) (-4444 . T) (-4443 . T)) 
-((|HasCategory| |#2| (QUOTE (-174))) (|HasAttribute| |#2| (QUOTE -4442))) 
+((-4445 . T) (-4441 |has| |#2| (-6 -4441)) (-4443 . T) (-4442 . T)) 
+((|HasCategory| |#2| (QUOTE (-174))) (|HasAttribute| |#2| (QUOTE -4441))) 
 (-1292 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 
 (-1293 |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| $) "\\spad{v * x} returns the product of a variable \\spad{x} by \\spad{x}."))) 
-((-4442 |has| |#2| (-6 -4442)) (-4444 . T) (-4443 . T) (-4446 . T)) 
+((-4441 |has| |#2| (-6 -4441)) (-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
-(-1294 S -1673) 
+(-1294 S -1674) 
 ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|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|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) 
 NIL 
 ((|HasCategory| |#2| (QUOTE (-373))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-148)))) 
-(-1295 -1673) 
+(-1295 -1674) 
 ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|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|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) 
-((-4441 . T) (-4447 . T) (-4442 . T) ((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+((-4440 . T) (-4446 . T) (-4441 . T) ((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 (-1296 |VarSet| R) 
 ((|constructor| (NIL "This domain constructor implements polynomials in non-commutative variables written in the Poincare-Birkhoff-Witt basis from the Lyndon basis. These polynomials can be used to compute Baker-Campbell-Hausdorff relations. \\newline Author: Michel Petitot (petitot@lifl.\\spad{fr}).")) (|log| (($ $ (|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}}."))) 
-((-4442 |has| |#2| (-6 -4442)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (LIST (QUOTE -723) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasAttribute| |#2| (QUOTE -4442))) 
+((-4441 |has| |#2| (-6 -4441)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#2| (QUOTE (-174))) (|HasCategory| |#2| (LIST (QUOTE -723) (LIST (QUOTE -413) (QUOTE (-570))))) (|HasAttribute| |#2| (QUOTE -4441))) 
 (-1297 |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|) $) "\\spad{maxdeg(p)} returns the greatest leading word in the support of \\spad{p}."))) 
-((-4442 |has| |#2| (-6 -4442)) (-4444 . T) (-4443 . T) (-4446 . T)) 
+((-4441 |has| |#2| (-6 -4441)) (-4443 . T) (-4442 . T) (-4445 . T)) 
 NIL 
 (-1298 R) 
 ((|constructor| (NIL "\\indented{2}{This type supports multivariate polynomials} whose set of variables is \\spadtype{Symbol}. The representation is recursive. The coefficient ring may be non-commutative and the variables do not commute. However,{} coefficients and variables commute."))) 
-((-4442 |has| |#1| (-6 -4442)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#1| (QUOTE (-174))) (|HasAttribute| |#1| (QUOTE -4442))) 
+((-4441 |has| |#1| (-6 -4441)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#1| (QUOTE (-174))) (|HasAttribute| |#1| (QUOTE -4441))) 
 (-1299 R E) 
 ((|constructor| (NIL "This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and words belonging to an arbitrary \\spadtype{OrderedMonoid}. This type is used,{} for instance,{} by the \\spadtype{XDistributedPolynomial} domain constructor where the Monoid is free.")) (|canonicalUnitNormal| ((|attribute|) "canonicalUnitNormal guarantees that the function unitCanonical returns the same representative for all associates of any particular element.")) (/ (($ $ |#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}."))) 
-((-4446 . T) (-4447 |has| |#1| (-6 -4447)) (-4442 |has| |#1| (-6 -4442)) (-4444 . T) (-4443 . T)) 
-((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4446)) (|HasAttribute| |#1| (QUOTE -4447)) (|HasAttribute| |#1| (QUOTE -4442))) 
+((-4445 . T) (-4446 |has| |#1| (-6 -4446)) (-4441 |has| |#1| (-6 -4441)) (-4443 . T) (-4442 . T)) 
+((|HasCategory| |#1| (QUOTE (-174))) (|HasCategory| |#1| (QUOTE (-368))) (|HasAttribute| |#1| (QUOTE -4445)) (|HasAttribute| |#1| (QUOTE -4446)) (|HasAttribute| |#1| (QUOTE -4441))) 
 (-1300 |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."))) 
-((-4442 |has| |#2| (-6 -4442)) (-4444 . T) (-4443 . T) (-4446 . T)) 
-((|HasCategory| |#2| (QUOTE (-174))) (|HasAttribute| |#2| (QUOTE -4442))) 
+((-4441 |has| |#2| (-6 -4441)) (-4443 . T) (-4442 . T) (-4445 . T)) 
+((|HasCategory| |#2| (QUOTE (-174))) (|HasAttribute| |#2| (QUOTE -4441))) 
 (-1301 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 
@@ -5146,7 +5146,7 @@ NIL
 NIL 
 (-1304 |p|) 
 ((|constructor| (NIL "IntegerMod(\\spad{n}) creates the ring of integers reduced modulo the integer \\spad{n}."))) 
-(((-4451 "*") . T) (-4443 . T) (-4444 . T) (-4446 . T)) 
+(((-4450 "*") . T) (-4442 . T) (-4443 . T) (-4445 . T)) 
 NIL 
 NIL 
 NIL 
@@ -5164,4 +5164,4 @@ NIL
 NIL 
 NIL 
 NIL 
-((-3 NIL 2267909 2267914 2267919 2267924) (-2 NIL 2267889 2267894 2267899 2267904) (-1 NIL 2267869 2267874 2267879 2267884) (0 NIL 2267849 2267854 2267859 2267864) (-1304 "ZMOD.spad" 2267658 2267671 2267787 2267844) (-1303 "ZLINDEP.spad" 2266724 2266735 2267648 2267653) (-1302 "ZDSOLVE.spad" 2256669 2256691 2266714 2266719) (-1301 "YSTREAM.spad" 2256164 2256175 2256659 2256664) (-1300 "XRPOLY.spad" 2255384 2255404 2256020 2256089) (-1299 "XPR.spad" 2253179 2253192 2255102 2255201) (-1298 "XPOLY.spad" 2252734 2252745 2253035 2253104) (-1297 "XPOLYC.spad" 2252053 2252069 2252660 2252729) (-1296 "XPBWPOLY.spad" 2250490 2250510 2251833 2251902) (-1295 "XF.spad" 2248953 2248968 2250392 2250485) (-1294 "XF.spad" 2247396 2247413 2248837 2248842) (-1293 "XFALG.spad" 2244444 2244460 2247322 2247391) (-1292 "XEXPPKG.spad" 2243695 2243721 2244434 2244439) (-1291 "XDPOLY.spad" 2243309 2243325 2243551 2243620) (-1290 "XALG.spad" 2242969 2242980 2243265 2243304) (-1289 "WUTSET.spad" 2238808 2238825 2242615 2242642) (-1288 "WP.spad" 2238007 2238051 2238666 2238733) (-1287 "WHILEAST.spad" 2237805 2237814 2237997 2238002) (-1286 "WHEREAST.spad" 2237476 2237485 2237795 2237800) (-1285 "WFFINTBS.spad" 2235139 2235161 2237466 2237471) (-1284 "WEIER.spad" 2233361 2233372 2235129 2235134) (-1283 "VSPACE.spad" 2233034 2233045 2233329 2233356) (-1282 "VSPACE.spad" 2232727 2232740 2233024 2233029) (-1281 "VOID.spad" 2232404 2232413 2232717 2232722) (-1280 "VIEW.spad" 2230084 2230093 2232394 2232399) (-1279 "VIEWDEF.spad" 2225285 2225294 2230074 2230079) (-1278 "VIEW3D.spad" 2209246 2209255 2225275 2225280) (-1277 "VIEW2D.spad" 2197137 2197146 2209236 2209241) (-1276 "VECTOR.spad" 2195811 2195822 2196062 2196089) (-1275 "VECTOR2.spad" 2194450 2194463 2195801 2195806) (-1274 "VECTCAT.spad" 2192354 2192365 2194418 2194445) (-1273 "VECTCAT.spad" 2190065 2190078 2192131 2192136) (-1272 "VARIABLE.spad" 2189845 2189860 2190055 2190060) (-1271 "UTYPE.spad" 2189489 2189498 2189835 2189840) (-1270 "UTSODETL.spad" 2188784 2188808 2189445 2189450) (-1269 "UTSODE.spad" 2187000 2187020 2188774 2188779) (-1268 "UTS.spad" 2181804 2181832 2185467 2185564) (-1267 "UTSCAT.spad" 2179283 2179299 2181702 2181799) (-1266 "UTSCAT.spad" 2176406 2176424 2178827 2178832) (-1265 "UTS2.spad" 2176001 2176036 2176396 2176401) (-1264 "URAGG.spad" 2170674 2170685 2175991 2175996) (-1263 "URAGG.spad" 2165311 2165324 2170630 2170635) (-1262 "UPXSSING.spad" 2162956 2162982 2164392 2164525) (-1261 "UPXS.spad" 2160110 2160138 2161088 2161237) (-1260 "UPXSCONS.spad" 2157869 2157889 2158242 2158391) (-1259 "UPXSCCA.spad" 2156440 2156460 2157715 2157864) (-1258 "UPXSCCA.spad" 2155153 2155175 2156430 2156435) (-1257 "UPXSCAT.spad" 2153742 2153758 2154999 2155148) (-1256 "UPXS2.spad" 2153285 2153338 2153732 2153737) (-1255 "UPSQFREE.spad" 2151699 2151713 2153275 2153280) (-1254 "UPSCAT.spad" 2149310 2149334 2151597 2151694) (-1253 "UPSCAT.spad" 2146627 2146653 2148916 2148921) (-1252 "UPOLYC.spad" 2141667 2141678 2146469 2146622) (-1251 "UPOLYC.spad" 2136599 2136612 2141403 2141408) (-1250 "UPOLYC2.spad" 2136070 2136089 2136589 2136594) (-1249 "UP.spad" 2133269 2133284 2133656 2133809) (-1248 "UPMP.spad" 2132169 2132182 2133259 2133264) (-1247 "UPDIVP.spad" 2131734 2131748 2132159 2132164) (-1246 "UPDECOMP.spad" 2129979 2129993 2131724 2131729) (-1245 "UPCDEN.spad" 2129188 2129204 2129969 2129974) (-1244 "UP2.spad" 2128552 2128573 2129178 2129183) (-1243 "UNISEG.spad" 2127905 2127916 2128471 2128476) (-1242 "UNISEG2.spad" 2127402 2127415 2127861 2127866) (-1241 "UNIFACT.spad" 2126505 2126517 2127392 2127397) (-1240 "ULS.spad" 2117063 2117091 2118150 2118579) (-1239 "ULSCONS.spad" 2109459 2109479 2109829 2109978) (-1238 "ULSCCAT.spad" 2107196 2107216 2109305 2109454) (-1237 "ULSCCAT.spad" 2105041 2105063 2107152 2107157) (-1236 "ULSCAT.spad" 2103273 2103289 2104887 2105036) (-1235 "ULS2.spad" 2102787 2102840 2103263 2103268) (-1234 "UINT8.spad" 2102664 2102673 2102777 2102782) (-1233 "UINT64.spad" 2102540 2102549 2102654 2102659) (-1232 "UINT32.spad" 2102416 2102425 2102530 2102535) (-1231 "UINT16.spad" 2102292 2102301 2102406 2102411) (-1230 "UFD.spad" 2101357 2101366 2102218 2102287) (-1229 "UFD.spad" 2100484 2100495 2101347 2101352) (-1228 "UDVO.spad" 2099365 2099374 2100474 2100479) (-1227 "UDPO.spad" 2096858 2096869 2099321 2099326) (-1226 "TYPE.spad" 2096790 2096799 2096848 2096853) (-1225 "TYPEAST.spad" 2096709 2096718 2096780 2096785) (-1224 "TWOFACT.spad" 2095361 2095376 2096699 2096704) (-1223 "TUPLE.spad" 2094847 2094858 2095260 2095265) (-1222 "TUBETOOL.spad" 2091714 2091723 2094837 2094842) (-1221 "TUBE.spad" 2090361 2090378 2091704 2091709) (-1220 "TS.spad" 2088960 2088976 2089926 2090023) (-1219 "TSETCAT.spad" 2076087 2076104 2088928 2088955) (-1218 "TSETCAT.spad" 2063200 2063219 2076043 2076048) (-1217 "TRMANIP.spad" 2057566 2057583 2062906 2062911) (-1216 "TRIMAT.spad" 2056529 2056554 2057556 2057561) (-1215 "TRIGMNIP.spad" 2055056 2055073 2056519 2056524) (-1214 "TRIGCAT.spad" 2054568 2054577 2055046 2055051) (-1213 "TRIGCAT.spad" 2054078 2054089 2054558 2054563) (-1212 "TREE.spad" 2052653 2052664 2053685 2053712) (-1211 "TRANFUN.spad" 2052492 2052501 2052643 2052648) (-1210 "TRANFUN.spad" 2052329 2052340 2052482 2052487) (-1209 "TOPSP.spad" 2052003 2052012 2052319 2052324) (-1208 "TOOLSIGN.spad" 2051666 2051677 2051993 2051998) (-1207 "TEXTFILE.spad" 2050227 2050236 2051656 2051661) (-1206 "TEX.spad" 2047373 2047382 2050217 2050222) (-1205 "TEX1.spad" 2046929 2046940 2047363 2047368) (-1204 "TEMUTL.spad" 2046484 2046493 2046919 2046924) (-1203 "TBCMPPK.spad" 2044577 2044600 2046474 2046479) (-1202 "TBAGG.spad" 2043627 2043650 2044557 2044572) (-1201 "TBAGG.spad" 2042685 2042710 2043617 2043622) (-1200 "TANEXP.spad" 2042093 2042104 2042675 2042680) (-1199 "TABLE.spad" 2040504 2040527 2040774 2040801) (-1198 "TABLEAU.spad" 2039985 2039996 2040494 2040499) (-1197 "TABLBUMP.spad" 2036788 2036799 2039975 2039980) (-1196 "SYSTEM.spad" 2036016 2036025 2036778 2036783) (-1195 "SYSSOLP.spad" 2033499 2033510 2036006 2036011) (-1194 "SYSPTR.spad" 2033398 2033407 2033489 2033494) (-1193 "SYSNNI.spad" 2032580 2032591 2033388 2033393) (-1192 "SYSINT.spad" 2031984 2031995 2032570 2032575) (-1191 "SYNTAX.spad" 2028190 2028199 2031974 2031979) (-1190 "SYMTAB.spad" 2026258 2026267 2028180 2028185) (-1189 "SYMS.spad" 2022281 2022290 2026248 2026253) (-1188 "SYMPOLY.spad" 2021288 2021299 2021370 2021497) (-1187 "SYMFUNC.spad" 2020789 2020800 2021278 2021283) (-1186 "SYMBOL.spad" 2018292 2018301 2020779 2020784) (-1185 "SWITCH.spad" 2015063 2015072 2018282 2018287) (-1184 "SUTS.spad" 2011968 2011996 2013530 2013627) (-1183 "SUPXS.spad" 2009109 2009137 2010100 2010249) (-1182 "SUP.spad" 2005922 2005933 2006695 2006848) (-1181 "SUPFRACF.spad" 2005027 2005045 2005912 2005917) (-1180 "SUP2.spad" 2004419 2004432 2005017 2005022) (-1179 "SUMRF.spad" 2003393 2003404 2004409 2004414) (-1178 "SUMFS.spad" 2003030 2003047 2003383 2003388) (-1177 "SULS.spad" 1993575 1993603 1994675 1995104) (-1176 "SUCHTAST.spad" 1993344 1993353 1993565 1993570) (-1175 "SUCH.spad" 1993026 1993041 1993334 1993339) (-1174 "SUBSPACE.spad" 1985141 1985156 1993016 1993021) (-1173 "SUBRESP.spad" 1984311 1984325 1985097 1985102) (-1172 "STTF.spad" 1980410 1980426 1984301 1984306) (-1171 "STTFNC.spad" 1976878 1976894 1980400 1980405) (-1170 "STTAYLOR.spad" 1969513 1969524 1976759 1976764) (-1169 "STRTBL.spad" 1968018 1968035 1968167 1968194) (-1168 "STRING.spad" 1967427 1967436 1967441 1967468) (-1167 "STRICAT.spad" 1967215 1967224 1967395 1967422) (-1166 "STREAM.spad" 1964133 1964144 1966740 1966755) (-1165 "STREAM3.spad" 1963706 1963721 1964123 1964128) (-1164 "STREAM2.spad" 1962834 1962847 1963696 1963701) (-1163 "STREAM1.spad" 1962540 1962551 1962824 1962829) (-1162 "STINPROD.spad" 1961476 1961492 1962530 1962535) (-1161 "STEP.spad" 1960677 1960686 1961466 1961471) (-1160 "STEPAST.spad" 1959911 1959920 1960667 1960672) (-1159 "STBL.spad" 1958437 1958465 1958604 1958619) (-1158 "STAGG.spad" 1957512 1957523 1958427 1958432) (-1157 "STAGG.spad" 1956585 1956598 1957502 1957507) (-1156 "STACK.spad" 1955942 1955953 1956192 1956219) (-1155 "SREGSET.spad" 1953646 1953663 1955588 1955615) (-1154 "SRDCMPK.spad" 1952207 1952227 1953636 1953641) (-1153 "SRAGG.spad" 1947350 1947359 1952175 1952202) (-1152 "SRAGG.spad" 1942513 1942524 1947340 1947345) (-1151 "SQMATRIX.spad" 1940129 1940147 1941045 1941132) (-1150 "SPLTREE.spad" 1934681 1934694 1939565 1939592) (-1149 "SPLNODE.spad" 1931269 1931282 1934671 1934676) (-1148 "SPFCAT.spad" 1930078 1930087 1931259 1931264) (-1147 "SPECOUT.spad" 1928630 1928639 1930068 1930073) (-1146 "SPADXPT.spad" 1920225 1920234 1928620 1928625) (-1145 "spad-parser.spad" 1919690 1919699 1920215 1920220) (-1144 "SPADAST.spad" 1919391 1919400 1919680 1919685) (-1143 "SPACEC.spad" 1903590 1903601 1919381 1919386) (-1142 "SPACE3.spad" 1903366 1903377 1903580 1903585) (-1141 "SORTPAK.spad" 1902915 1902928 1903322 1903327) (-1140 "SOLVETRA.spad" 1900678 1900689 1902905 1902910) (-1139 "SOLVESER.spad" 1899206 1899217 1900668 1900673) (-1138 "SOLVERAD.spad" 1895232 1895243 1899196 1899201) (-1137 "SOLVEFOR.spad" 1893694 1893712 1895222 1895227) (-1136 "SNTSCAT.spad" 1893294 1893311 1893662 1893689) (-1135 "SMTS.spad" 1891566 1891592 1892859 1892956) (-1134 "SMP.spad" 1889041 1889061 1889431 1889558) (-1133 "SMITH.spad" 1887886 1887911 1889031 1889036) (-1132 "SMATCAT.spad" 1885996 1886026 1887830 1887881) (-1131 "SMATCAT.spad" 1884038 1884070 1885874 1885879) (-1130 "SKAGG.spad" 1883001 1883012 1884006 1884033) (-1129 "SINT.spad" 1881941 1881950 1882867 1882996) (-1128 "SIMPAN.spad" 1881669 1881678 1881931 1881936) (-1127 "SIG.spad" 1880999 1881008 1881659 1881664) (-1126 "SIGNRF.spad" 1880117 1880128 1880989 1880994) (-1125 "SIGNEF.spad" 1879396 1879413 1880107 1880112) (-1124 "SIGAST.spad" 1878781 1878790 1879386 1879391) (-1123 "SHP.spad" 1876709 1876724 1878737 1878742) (-1122 "SHDP.spad" 1866420 1866447 1866929 1867060) (-1121 "SGROUP.spad" 1866028 1866037 1866410 1866415) (-1120 "SGROUP.spad" 1865634 1865645 1866018 1866023) (-1119 "SGCF.spad" 1858797 1858806 1865624 1865629) (-1118 "SFRTCAT.spad" 1857727 1857744 1858765 1858792) (-1117 "SFRGCD.spad" 1856790 1856810 1857717 1857722) (-1116 "SFQCMPK.spad" 1851427 1851447 1856780 1856785) (-1115 "SFORT.spad" 1850866 1850880 1851417 1851422) (-1114 "SEXOF.spad" 1850709 1850749 1850856 1850861) (-1113 "SEX.spad" 1850601 1850610 1850699 1850704) (-1112 "SEXCAT.spad" 1848202 1848242 1850591 1850596) (-1111 "SET.spad" 1846526 1846537 1847623 1847662) (-1110 "SETMN.spad" 1844976 1844993 1846516 1846521) (-1109 "SETCAT.spad" 1844298 1844307 1844966 1844971) (-1108 "SETCAT.spad" 1843618 1843629 1844288 1844293) (-1107 "SETAGG.spad" 1840167 1840178 1843598 1843613) (-1106 "SETAGG.spad" 1836724 1836737 1840157 1840162) (-1105 "SEQAST.spad" 1836427 1836436 1836714 1836719) (-1104 "SEGXCAT.spad" 1835583 1835596 1836417 1836422) (-1103 "SEG.spad" 1835396 1835407 1835502 1835507) (-1102 "SEGCAT.spad" 1834321 1834332 1835386 1835391) (-1101 "SEGBIND.spad" 1834079 1834090 1834268 1834273) (-1100 "SEGBIND2.spad" 1833777 1833790 1834069 1834074) (-1099 "SEGAST.spad" 1833491 1833500 1833767 1833772) (-1098 "SEG2.spad" 1832926 1832939 1833447 1833452) (-1097 "SDVAR.spad" 1832202 1832213 1832916 1832921) (-1096 "SDPOL.spad" 1829628 1829639 1829919 1830046) (-1095 "SCPKG.spad" 1827717 1827728 1829618 1829623) (-1094 "SCOPE.spad" 1826870 1826879 1827707 1827712) (-1093 "SCACHE.spad" 1825566 1825577 1826860 1826865) (-1092 "SASTCAT.spad" 1825475 1825484 1825556 1825561) (-1091 "SAOS.spad" 1825347 1825356 1825465 1825470) (-1090 "SAERFFC.spad" 1825060 1825080 1825337 1825342) (-1089 "SAE.spad" 1823235 1823251 1823846 1823981) (-1088 "SAEFACT.spad" 1822936 1822956 1823225 1823230) (-1087 "RURPK.spad" 1820595 1820611 1822926 1822931) (-1086 "RULESET.spad" 1820048 1820072 1820585 1820590) (-1085 "RULE.spad" 1818288 1818312 1820038 1820043) (-1084 "RULECOLD.spad" 1818140 1818153 1818278 1818283) (-1083 "RTVALUE.spad" 1817875 1817884 1818130 1818135) (-1082 "RSTRCAST.spad" 1817592 1817601 1817865 1817870) (-1081 "RSETGCD.spad" 1813970 1813990 1817582 1817587) (-1080 "RSETCAT.spad" 1803906 1803923 1813938 1813965) (-1079 "RSETCAT.spad" 1793862 1793881 1803896 1803901) (-1078 "RSDCMPK.spad" 1792314 1792334 1793852 1793857) (-1077 "RRCC.spad" 1790698 1790728 1792304 1792309) (-1076 "RRCC.spad" 1789080 1789112 1790688 1790693) (-1075 "RPTAST.spad" 1788782 1788791 1789070 1789075) (-1074 "RPOLCAT.spad" 1768142 1768157 1788650 1788777) (-1073 "RPOLCAT.spad" 1747215 1747232 1767725 1767730) (-1072 "ROUTINE.spad" 1743098 1743107 1745862 1745889) (-1071 "ROMAN.spad" 1742426 1742435 1742964 1743093) (-1070 "ROIRC.spad" 1741506 1741538 1742416 1742421) (-1069 "RNS.spad" 1740409 1740418 1741408 1741501) (-1068 "RNS.spad" 1739398 1739409 1740399 1740404) (-1067 "RNG.spad" 1739133 1739142 1739388 1739393) (-1066 "RNGBIND.spad" 1738293 1738307 1739088 1739093) (-1065 "RMODULE.spad" 1738058 1738069 1738283 1738288) (-1064 "RMCAT2.spad" 1737478 1737535 1738048 1738053) (-1063 "RMATRIX.spad" 1736302 1736321 1736645 1736684) (-1062 "RMATCAT.spad" 1731881 1731912 1736258 1736297) (-1061 "RMATCAT.spad" 1727350 1727383 1731729 1731734) (-1060 "RLINSET.spad" 1726744 1726755 1727340 1727345) (-1059 "RINTERP.spad" 1726632 1726652 1726734 1726739) (-1058 "RING.spad" 1726102 1726111 1726612 1726627) (-1057 "RING.spad" 1725580 1725591 1726092 1726097) (-1056 "RIDIST.spad" 1724972 1724981 1725570 1725575) (-1055 "RGCHAIN.spad" 1723555 1723571 1724457 1724484) (-1054 "RGBCSPC.spad" 1723336 1723348 1723545 1723550) (-1053 "RGBCMDL.spad" 1722866 1722878 1723326 1723331) (-1052 "RF.spad" 1720508 1720519 1722856 1722861) (-1051 "RFFACTOR.spad" 1719970 1719981 1720498 1720503) (-1050 "RFFACT.spad" 1719705 1719717 1719960 1719965) (-1049 "RFDIST.spad" 1718701 1718710 1719695 1719700) (-1048 "RETSOL.spad" 1718120 1718133 1718691 1718696) (-1047 "RETRACT.spad" 1717548 1717559 1718110 1718115) (-1046 "RETRACT.spad" 1716974 1716987 1717538 1717543) (-1045 "RETAST.spad" 1716786 1716795 1716964 1716969) (-1044 "RESULT.spad" 1714846 1714855 1715433 1715460) (-1043 "RESRING.spad" 1714193 1714240 1714784 1714841) (-1042 "RESLATC.spad" 1713517 1713528 1714183 1714188) (-1041 "REPSQ.spad" 1713248 1713259 1713507 1713512) (-1040 "REP.spad" 1710802 1710811 1713238 1713243) (-1039 "REPDB.spad" 1710509 1710520 1710792 1710797) (-1038 "REP2.spad" 1700167 1700178 1710351 1710356) (-1037 "REP1.spad" 1694363 1694374 1700117 1700122) (-1036 "REGSET.spad" 1692160 1692177 1694009 1694036) (-1035 "REF.spad" 1691495 1691506 1692115 1692120) (-1034 "REDORDER.spad" 1690701 1690718 1691485 1691490) (-1033 "RECLOS.spad" 1689484 1689504 1690188 1690281) (-1032 "REALSOLV.spad" 1688624 1688633 1689474 1689479) (-1031 "REAL.spad" 1688496 1688505 1688614 1688619) (-1030 "REAL0Q.spad" 1685794 1685809 1688486 1688491) (-1029 "REAL0.spad" 1682638 1682653 1685784 1685789) (-1028 "RDUCEAST.spad" 1682359 1682368 1682628 1682633) (-1027 "RDIV.spad" 1682014 1682039 1682349 1682354) (-1026 "RDIST.spad" 1681581 1681592 1682004 1682009) (-1025 "RDETRS.spad" 1680445 1680463 1681571 1681576) (-1024 "RDETR.spad" 1678584 1678602 1680435 1680440) (-1023 "RDEEFS.spad" 1677683 1677700 1678574 1678579) (-1022 "RDEEF.spad" 1676693 1676710 1677673 1677678) (-1021 "RCFIELD.spad" 1673879 1673888 1676595 1676688) (-1020 "RCFIELD.spad" 1671151 1671162 1673869 1673874) (-1019 "RCAGG.spad" 1669079 1669090 1671141 1671146) (-1018 "RCAGG.spad" 1666934 1666947 1668998 1669003) (-1017 "RATRET.spad" 1666294 1666305 1666924 1666929) (-1016 "RATFACT.spad" 1665986 1665998 1666284 1666289) (-1015 "RANDSRC.spad" 1665305 1665314 1665976 1665981) (-1014 "RADUTIL.spad" 1665061 1665070 1665295 1665300) (-1013 "RADIX.spad" 1661982 1661996 1663528 1663621) (-1012 "RADFF.spad" 1660395 1660432 1660514 1660670) (-1011 "RADCAT.spad" 1659990 1659999 1660385 1660390) (-1010 "RADCAT.spad" 1659583 1659594 1659980 1659985) (-1009 "QUEUE.spad" 1658931 1658942 1659190 1659217) (-1008 "QUAT.spad" 1657512 1657523 1657855 1657920) (-1007 "QUATCT2.spad" 1657132 1657151 1657502 1657507) (-1006 "QUATCAT.spad" 1655302 1655313 1657062 1657127) (-1005 "QUATCAT.spad" 1653223 1653236 1654985 1654990) (-1004 "QUAGG.spad" 1652050 1652061 1653191 1653218) (-1003 "QQUTAST.spad" 1651818 1651827 1652040 1652045) (-1002 "QFORM.spad" 1651282 1651297 1651808 1651813) (-1001 "QFCAT.spad" 1649984 1649995 1651184 1651277) (-1000 "QFCAT.spad" 1648277 1648290 1649479 1649484) (-999 "QFCAT2.spad" 1647970 1647986 1648267 1648272) (-998 "QEQUAT.spad" 1647529 1647537 1647960 1647965) (-997 "QCMPACK.spad" 1642276 1642295 1647519 1647524) (-996 "QALGSET.spad" 1638355 1638387 1642190 1642195) (-995 "QALGSET2.spad" 1636351 1636369 1638345 1638350) (-994 "PWFFINTB.spad" 1633767 1633788 1636341 1636346) (-993 "PUSHVAR.spad" 1633106 1633125 1633757 1633762) (-992 "PTRANFN.spad" 1629234 1629244 1633096 1633101) (-991 "PTPACK.spad" 1626322 1626332 1629224 1629229) (-990 "PTFUNC2.spad" 1626145 1626159 1626312 1626317) (-989 "PTCAT.spad" 1625400 1625410 1626113 1626140) (-988 "PSQFR.spad" 1624707 1624731 1625390 1625395) (-987 "PSEUDLIN.spad" 1623593 1623603 1624697 1624702) (-986 "PSETPK.spad" 1609026 1609042 1623471 1623476) (-985 "PSETCAT.spad" 1602946 1602969 1609006 1609021) (-984 "PSETCAT.spad" 1596840 1596865 1602902 1602907) (-983 "PSCURVE.spad" 1595823 1595831 1596830 1596835) (-982 "PSCAT.spad" 1594606 1594635 1595721 1595818) (-981 "PSCAT.spad" 1593479 1593510 1594596 1594601) (-980 "PRTITION.spad" 1592440 1592448 1593469 1593474) (-979 "PRTDAST.spad" 1592159 1592167 1592430 1592435) (-978 "PRS.spad" 1581721 1581738 1592115 1592120) (-977 "PRQAGG.spad" 1581156 1581166 1581689 1581716) (-976 "PROPLOG.spad" 1580728 1580736 1581146 1581151) (-975 "PROPFUN2.spad" 1580351 1580364 1580718 1580723) (-974 "PROPFUN1.spad" 1579749 1579760 1580341 1580346) (-973 "PROPFRML.spad" 1578317 1578328 1579739 1579744) (-972 "PROPERTY.spad" 1577805 1577813 1578307 1578312) (-971 "PRODUCT.spad" 1575487 1575499 1575771 1575826) (-970 "PR.spad" 1573879 1573891 1574578 1574705) (-969 "PRINT.spad" 1573631 1573639 1573869 1573874) (-968 "PRIMES.spad" 1571884 1571894 1573621 1573626) (-967 "PRIMELT.spad" 1569965 1569979 1571874 1571879) (-966 "PRIMCAT.spad" 1569592 1569600 1569955 1569960) (-965 "PRIMARR.spad" 1568597 1568607 1568775 1568802) (-964 "PRIMARR2.spad" 1567364 1567376 1568587 1568592) (-963 "PREASSOC.spad" 1566746 1566758 1567354 1567359) (-962 "PPCURVE.spad" 1565883 1565891 1566736 1566741) (-961 "PORTNUM.spad" 1565658 1565666 1565873 1565878) (-960 "POLYROOT.spad" 1564507 1564529 1565614 1565619) (-959 "POLY.spad" 1561842 1561852 1562357 1562484) (-958 "POLYLIFT.spad" 1561107 1561130 1561832 1561837) (-957 "POLYCATQ.spad" 1559225 1559247 1561097 1561102) (-956 "POLYCAT.spad" 1552695 1552716 1559093 1559220) (-955 "POLYCAT.spad" 1545503 1545526 1551903 1551908) (-954 "POLY2UP.spad" 1544955 1544969 1545493 1545498) (-953 "POLY2.spad" 1544552 1544564 1544945 1544950) (-952 "POLUTIL.spad" 1543493 1543522 1544508 1544513) (-951 "POLTOPOL.spad" 1542241 1542256 1543483 1543488) (-950 "POINT.spad" 1541079 1541089 1541166 1541193) (-949 "PNTHEORY.spad" 1537781 1537789 1541069 1541074) (-948 "PMTOOLS.spad" 1536556 1536570 1537771 1537776) (-947 "PMSYM.spad" 1536105 1536115 1536546 1536551) (-946 "PMQFCAT.spad" 1535696 1535710 1536095 1536100) (-945 "PMPRED.spad" 1535175 1535189 1535686 1535691) (-944 "PMPREDFS.spad" 1534629 1534651 1535165 1535170) (-943 "PMPLCAT.spad" 1533709 1533727 1534561 1534566) (-942 "PMLSAGG.spad" 1533294 1533308 1533699 1533704) (-941 "PMKERNEL.spad" 1532873 1532885 1533284 1533289) (-940 "PMINS.spad" 1532453 1532463 1532863 1532868) (-939 "PMFS.spad" 1532030 1532048 1532443 1532448) (-938 "PMDOWN.spad" 1531320 1531334 1532020 1532025) (-937 "PMASS.spad" 1530330 1530338 1531310 1531315) (-936 "PMASSFS.spad" 1529297 1529313 1530320 1530325) (-935 "PLOTTOOL.spad" 1529077 1529085 1529287 1529292) (-934 "PLOT.spad" 1524000 1524008 1529067 1529072) (-933 "PLOT3D.spad" 1520464 1520472 1523990 1523995) (-932 "PLOT1.spad" 1519621 1519631 1520454 1520459) (-931 "PLEQN.spad" 1506911 1506938 1519611 1519616) (-930 "PINTERP.spad" 1506533 1506552 1506901 1506906) (-929 "PINTERPA.spad" 1506317 1506333 1506523 1506528) (-928 "PI.spad" 1505926 1505934 1506291 1506312) (-927 "PID.spad" 1504896 1504904 1505852 1505921) (-926 "PICOERCE.spad" 1504553 1504563 1504886 1504891) (-925 "PGROEB.spad" 1503154 1503168 1504543 1504548) (-924 "PGE.spad" 1494771 1494779 1503144 1503149) (-923 "PGCD.spad" 1493661 1493678 1494761 1494766) (-922 "PFRPAC.spad" 1492810 1492820 1493651 1493656) (-921 "PFR.spad" 1489473 1489483 1492712 1492805) (-920 "PFOTOOLS.spad" 1488731 1488747 1489463 1489468) (-919 "PFOQ.spad" 1488101 1488119 1488721 1488726) (-918 "PFO.spad" 1487520 1487547 1488091 1488096) (-917 "PF.spad" 1487094 1487106 1487325 1487418) (-916 "PFECAT.spad" 1484776 1484784 1487020 1487089) (-915 "PFECAT.spad" 1482486 1482496 1484732 1484737) (-914 "PFBRU.spad" 1480374 1480386 1482476 1482481) (-913 "PFBR.spad" 1477934 1477957 1480364 1480369) (-912 "PERM.spad" 1473619 1473629 1477764 1477779) (-911 "PERMGRP.spad" 1468381 1468391 1473609 1473614) (-910 "PERMCAT.spad" 1466939 1466949 1468361 1468376) (-909 "PERMAN.spad" 1465471 1465485 1466929 1466934) (-908 "PENDTREE.spad" 1464812 1464822 1465100 1465105) (-907 "PDRING.spad" 1463363 1463373 1464792 1464807) (-906 "PDRING.spad" 1461922 1461934 1463353 1463358) (-905 "PDEPROB.spad" 1460937 1460945 1461912 1461917) (-904 "PDEPACK.spad" 1454977 1454985 1460927 1460932) (-903 "PDECOMP.spad" 1454447 1454464 1454967 1454972) (-902 "PDECAT.spad" 1452803 1452811 1454437 1454442) (-901 "PCOMP.spad" 1452656 1452669 1452793 1452798) (-900 "PBWLB.spad" 1451244 1451261 1452646 1452651) (-899 "PATTERN.spad" 1445783 1445793 1451234 1451239) (-898 "PATTERN2.spad" 1445521 1445533 1445773 1445778) (-897 "PATTERN1.spad" 1443857 1443873 1445511 1445516) (-896 "PATRES.spad" 1441432 1441444 1443847 1443852) (-895 "PATRES2.spad" 1441104 1441118 1441422 1441427) (-894 "PATMATCH.spad" 1439301 1439332 1440812 1440817) (-893 "PATMAB.spad" 1438730 1438740 1439291 1439296) (-892 "PATLRES.spad" 1437816 1437830 1438720 1438725) (-891 "PATAB.spad" 1437580 1437590 1437806 1437811) (-890 "PARTPERM.spad" 1434980 1434988 1437570 1437575) (-889 "PARSURF.spad" 1434414 1434442 1434970 1434975) (-888 "PARSU2.spad" 1434211 1434227 1434404 1434409) (-887 "script-parser.spad" 1433731 1433739 1434201 1434206) (-886 "PARSCURV.spad" 1433165 1433193 1433721 1433726) (-885 "PARSC2.spad" 1432956 1432972 1433155 1433160) (-884 "PARPCURV.spad" 1432418 1432446 1432946 1432951) (-883 "PARPC2.spad" 1432209 1432225 1432408 1432413) (-882 "PARAMAST.spad" 1431337 1431345 1432199 1432204) (-881 "PAN2EXPR.spad" 1430749 1430757 1431327 1431332) (-880 "PALETTE.spad" 1429719 1429727 1430739 1430744) (-879 "PAIR.spad" 1428706 1428719 1429307 1429312) (-878 "PADICRC.spad" 1426040 1426058 1427211 1427304) (-877 "PADICRAT.spad" 1424055 1424067 1424276 1424369) (-876 "PADIC.spad" 1423750 1423762 1423981 1424050) (-875 "PADICCT.spad" 1422299 1422311 1423676 1423745) (-874 "PADEPAC.spad" 1420988 1421007 1422289 1422294) (-873 "PADE.spad" 1419740 1419756 1420978 1420983) (-872 "OWP.spad" 1418980 1419010 1419598 1419665) (-871 "OVERSET.spad" 1418553 1418561 1418970 1418975) (-870 "OVAR.spad" 1418334 1418357 1418543 1418548) (-869 "OUT.spad" 1417420 1417428 1418324 1418329) (-868 "OUTFORM.spad" 1406812 1406820 1417410 1417415) (-867 "OUTBFILE.spad" 1406230 1406238 1406802 1406807) (-866 "OUTBCON.spad" 1405236 1405244 1406220 1406225) (-865 "OUTBCON.spad" 1404240 1404250 1405226 1405231) (-864 "OSI.spad" 1403715 1403723 1404230 1404235) (-863 "OSGROUP.spad" 1403633 1403641 1403705 1403710) (-862 "ORTHPOL.spad" 1402118 1402128 1403550 1403555) (-861 "OREUP.spad" 1401571 1401599 1401798 1401837) (-860 "ORESUP.spad" 1400872 1400896 1401251 1401290) (-859 "OREPCTO.spad" 1398729 1398741 1400792 1400797) (-858 "OREPCAT.spad" 1392876 1392886 1398685 1398724) (-857 "OREPCAT.spad" 1386913 1386925 1392724 1392729) (-856 "ORDSET.spad" 1386085 1386093 1386903 1386908) (-855 "ORDSET.spad" 1385255 1385265 1386075 1386080) (-854 "ORDRING.spad" 1384645 1384653 1385235 1385250) (-853 "ORDRING.spad" 1384043 1384053 1384635 1384640) (-852 "ORDMON.spad" 1383898 1383906 1384033 1384038) (-851 "ORDFUNS.spad" 1383030 1383046 1383888 1383893) (-850 "ORDFIN.spad" 1382850 1382858 1383020 1383025) (-849 "ORDCOMP.spad" 1381315 1381325 1382397 1382426) (-848 "ORDCOMP2.spad" 1380608 1380620 1381305 1381310) (-847 "OPTPROB.spad" 1379246 1379254 1380598 1380603) (-846 "OPTPACK.spad" 1371655 1371663 1379236 1379241) (-845 "OPTCAT.spad" 1369334 1369342 1371645 1371650) (-844 "OPSIG.spad" 1368988 1368996 1369324 1369329) (-843 "OPQUERY.spad" 1368537 1368545 1368978 1368983) (-842 "OP.spad" 1368279 1368289 1368359 1368426) (-841 "OPERCAT.spad" 1367745 1367755 1368269 1368274) (-840 "OPERCAT.spad" 1367209 1367221 1367735 1367740) (-839 "ONECOMP.spad" 1365954 1365964 1366756 1366785) (-838 "ONECOMP2.spad" 1365378 1365390 1365944 1365949) (-837 "OMSERVER.spad" 1364384 1364392 1365368 1365373) (-836 "OMSAGG.spad" 1364172 1364182 1364340 1364379) (-835 "OMPKG.spad" 1362788 1362796 1364162 1364167) (-834 "OM.spad" 1361761 1361769 1362778 1362783) (-833 "OMLO.spad" 1361186 1361198 1361647 1361686) (-832 "OMEXPR.spad" 1361020 1361030 1361176 1361181) (-831 "OMERR.spad" 1360565 1360573 1361010 1361015) (-830 "OMERRK.spad" 1359599 1359607 1360555 1360560) (-829 "OMENC.spad" 1358943 1358951 1359589 1359594) (-828 "OMDEV.spad" 1353252 1353260 1358933 1358938) (-827 "OMCONN.spad" 1352661 1352669 1353242 1353247) (-826 "OINTDOM.spad" 1352424 1352432 1352587 1352656) (-825 "OFMONOID.spad" 1350547 1350557 1352380 1352385) (-824 "ODVAR.spad" 1349808 1349818 1350537 1350542) (-823 "ODR.spad" 1349452 1349478 1349620 1349769) (-822 "ODPOL.spad" 1346834 1346844 1347174 1347301) (-821 "ODP.spad" 1336681 1336701 1337054 1337185) (-820 "ODETOOLS.spad" 1335330 1335349 1336671 1336676) (-819 "ODESYS.spad" 1333024 1333041 1335320 1335325) (-818 "ODERTRIC.spad" 1329033 1329050 1332981 1332986) (-817 "ODERED.spad" 1328432 1328456 1329023 1329028) (-816 "ODERAT.spad" 1326047 1326064 1328422 1328427) (-815 "ODEPRRIC.spad" 1323084 1323106 1326037 1326042) (-814 "ODEPROB.spad" 1322341 1322349 1323074 1323079) (-813 "ODEPRIM.spad" 1319675 1319697 1322331 1322336) (-812 "ODEPAL.spad" 1319061 1319085 1319665 1319670) (-811 "ODEPACK.spad" 1305727 1305735 1319051 1319056) (-810 "ODEINT.spad" 1305162 1305178 1305717 1305722) (-809 "ODEIFTBL.spad" 1302557 1302565 1305152 1305157) (-808 "ODEEF.spad" 1298048 1298064 1302547 1302552) (-807 "ODECONST.spad" 1297585 1297603 1298038 1298043) (-806 "ODECAT.spad" 1296183 1296191 1297575 1297580) (-805 "OCT.spad" 1294319 1294329 1295033 1295072) (-804 "OCTCT2.spad" 1293965 1293986 1294309 1294314) (-803 "OC.spad" 1291761 1291771 1293921 1293960) (-802 "OC.spad" 1289282 1289294 1291444 1291449) (-801 "OCAMON.spad" 1289130 1289138 1289272 1289277) (-800 "OASGP.spad" 1288945 1288953 1289120 1289125) (-799 "OAMONS.spad" 1288467 1288475 1288935 1288940) (-798 "OAMON.spad" 1288328 1288336 1288457 1288462) (-797 "OAGROUP.spad" 1288190 1288198 1288318 1288323) (-796 "NUMTUBE.spad" 1287781 1287797 1288180 1288185) (-795 "NUMQUAD.spad" 1275757 1275765 1287771 1287776) (-794 "NUMODE.spad" 1267111 1267119 1275747 1275752) (-793 "NUMINT.spad" 1264677 1264685 1267101 1267106) (-792 "NUMFMT.spad" 1263517 1263525 1264667 1264672) (-791 "NUMERIC.spad" 1255631 1255641 1263322 1263327) (-790 "NTSCAT.spad" 1254139 1254155 1255599 1255626) (-789 "NTPOLFN.spad" 1253690 1253700 1254056 1254061) (-788 "NSUP.spad" 1246736 1246746 1251276 1251429) (-787 "NSUP2.spad" 1246128 1246140 1246726 1246731) (-786 "NSMP.spad" 1242358 1242377 1242666 1242793) (-785 "NREP.spad" 1240736 1240750 1242348 1242353) (-784 "NPCOEF.spad" 1239982 1240002 1240726 1240731) (-783 "NORMRETR.spad" 1239580 1239619 1239972 1239977) (-782 "NORMPK.spad" 1237482 1237501 1239570 1239575) (-781 "NORMMA.spad" 1237170 1237196 1237472 1237477) (-780 "NONE.spad" 1236911 1236919 1237160 1237165) (-779 "NONE1.spad" 1236587 1236597 1236901 1236906) (-778 "NODE1.spad" 1236074 1236090 1236577 1236582) (-777 "NNI.spad" 1234969 1234977 1236048 1236069) (-776 "NLINSOL.spad" 1233595 1233605 1234959 1234964) (-775 "NIPROB.spad" 1232136 1232144 1233585 1233590) (-774 "NFINTBAS.spad" 1229696 1229713 1232126 1232131) (-773 "NETCLT.spad" 1229670 1229681 1229686 1229691) (-772 "NCODIV.spad" 1227886 1227902 1229660 1229665) (-771 "NCNTFRAC.spad" 1227528 1227542 1227876 1227881) (-770 "NCEP.spad" 1225694 1225708 1227518 1227523) (-769 "NASRING.spad" 1225290 1225298 1225684 1225689) (-768 "NASRING.spad" 1224884 1224894 1225280 1225285) (-767 "NARNG.spad" 1224236 1224244 1224874 1224879) (-766 "NARNG.spad" 1223586 1223596 1224226 1224231) (-765 "NAGSP.spad" 1222663 1222671 1223576 1223581) (-764 "NAGS.spad" 1212324 1212332 1222653 1222658) (-763 "NAGF07.spad" 1210755 1210763 1212314 1212319) (-762 "NAGF04.spad" 1205157 1205165 1210745 1210750) (-761 "NAGF02.spad" 1199226 1199234 1205147 1205152) (-760 "NAGF01.spad" 1194987 1194995 1199216 1199221) (-759 "NAGE04.spad" 1188687 1188695 1194977 1194982) (-758 "NAGE02.spad" 1179347 1179355 1188677 1188682) (-757 "NAGE01.spad" 1175349 1175357 1179337 1179342) (-756 "NAGD03.spad" 1173353 1173361 1175339 1175344) (-755 "NAGD02.spad" 1166100 1166108 1173343 1173348) (-754 "NAGD01.spad" 1160393 1160401 1166090 1166095) (-753 "NAGC06.spad" 1156268 1156276 1160383 1160388) (-752 "NAGC05.spad" 1154769 1154777 1156258 1156263) (-751 "NAGC02.spad" 1154036 1154044 1154759 1154764) (-750 "NAALG.spad" 1153577 1153587 1154004 1154031) (-749 "NAALG.spad" 1153138 1153150 1153567 1153572) (-748 "MULTSQFR.spad" 1150096 1150113 1153128 1153133) (-747 "MULTFACT.spad" 1149479 1149496 1150086 1150091) (-746 "MTSCAT.spad" 1147573 1147594 1149377 1149474) (-745 "MTHING.spad" 1147232 1147242 1147563 1147568) (-744 "MSYSCMD.spad" 1146666 1146674 1147222 1147227) (-743 "MSET.spad" 1144624 1144634 1146372 1146411) (-742 "MSETAGG.spad" 1144469 1144479 1144592 1144619) (-741 "MRING.spad" 1141446 1141458 1144177 1144244) (-740 "MRF2.spad" 1141016 1141030 1141436 1141441) (-739 "MRATFAC.spad" 1140562 1140579 1141006 1141011) (-738 "MPRFF.spad" 1138602 1138621 1140552 1140557) (-737 "MPOLY.spad" 1136073 1136088 1136432 1136559) (-736 "MPCPF.spad" 1135337 1135356 1136063 1136068) (-735 "MPC3.spad" 1135154 1135194 1135327 1135332) (-734 "MPC2.spad" 1134800 1134833 1135144 1135149) (-733 "MONOTOOL.spad" 1133151 1133168 1134790 1134795) (-732 "MONOID.spad" 1132470 1132478 1133141 1133146) (-731 "MONOID.spad" 1131787 1131797 1132460 1132465) (-730 "MONOGEN.spad" 1130535 1130548 1131647 1131782) (-729 "MONOGEN.spad" 1129305 1129320 1130419 1130424) (-728 "MONADWU.spad" 1127335 1127343 1129295 1129300) (-727 "MONADWU.spad" 1125363 1125373 1127325 1127330) (-726 "MONAD.spad" 1124523 1124531 1125353 1125358) (-725 "MONAD.spad" 1123681 1123691 1124513 1124518) (-724 "MOEBIUS.spad" 1122417 1122431 1123661 1123676) (-723 "MODULE.spad" 1122287 1122297 1122385 1122412) (-722 "MODULE.spad" 1122177 1122189 1122277 1122282) (-721 "MODRING.spad" 1121512 1121551 1122157 1122172) (-720 "MODOP.spad" 1120177 1120189 1121334 1121401) (-719 "MODMONOM.spad" 1119908 1119926 1120167 1120172) (-718 "MODMON.spad" 1116703 1116719 1117422 1117575) (-717 "MODFIELD.spad" 1116065 1116104 1116605 1116698) (-716 "MMLFORM.spad" 1114925 1114933 1116055 1116060) (-715 "MMAP.spad" 1114667 1114701 1114915 1114920) (-714 "MLO.spad" 1113126 1113136 1114623 1114662) (-713 "MLIFT.spad" 1111738 1111755 1113116 1113121) (-712 "MKUCFUNC.spad" 1111273 1111291 1111728 1111733) (-711 "MKRECORD.spad" 1110877 1110890 1111263 1111268) (-710 "MKFUNC.spad" 1110284 1110294 1110867 1110872) (-709 "MKFLCFN.spad" 1109252 1109262 1110274 1110279) (-708 "MKBCFUNC.spad" 1108747 1108765 1109242 1109247) (-707 "MINT.spad" 1108186 1108194 1108649 1108742) (-706 "MHROWRED.spad" 1106697 1106707 1108176 1108181) (-705 "MFLOAT.spad" 1105217 1105225 1106587 1106692) (-704 "MFINFACT.spad" 1104617 1104639 1105207 1105212) (-703 "MESH.spad" 1102399 1102407 1104607 1104612) (-702 "MDDFACT.spad" 1100610 1100620 1102389 1102394) (-701 "MDAGG.spad" 1099901 1099911 1100590 1100605) (-700 "MCMPLX.spad" 1095912 1095920 1096526 1096727) (-699 "MCDEN.spad" 1095122 1095134 1095902 1095907) (-698 "MCALCFN.spad" 1092244 1092270 1095112 1095117) (-697 "MAYBE.spad" 1091528 1091539 1092234 1092239) (-696 "MATSTOR.spad" 1088836 1088846 1091518 1091523) (-695 "MATRIX.spad" 1087540 1087550 1088024 1088051) (-694 "MATLIN.spad" 1084884 1084908 1087424 1087429) (-693 "MATCAT.spad" 1076613 1076635 1084852 1084879) (-692 "MATCAT.spad" 1068214 1068238 1076455 1076460) (-691 "MATCAT2.spad" 1067496 1067544 1068204 1068209) (-690 "MAPPKG3.spad" 1066411 1066425 1067486 1067491) (-689 "MAPPKG2.spad" 1065749 1065761 1066401 1066406) (-688 "MAPPKG1.spad" 1064577 1064587 1065739 1065744) (-687 "MAPPAST.spad" 1063892 1063900 1064567 1064572) (-686 "MAPHACK3.spad" 1063704 1063718 1063882 1063887) (-685 "MAPHACK2.spad" 1063473 1063485 1063694 1063699) (-684 "MAPHACK1.spad" 1063117 1063127 1063463 1063468) (-683 "MAGMA.spad" 1060907 1060924 1063107 1063112) (-682 "MACROAST.spad" 1060486 1060494 1060897 1060902) (-681 "M3D.spad" 1058206 1058216 1059864 1059869) (-680 "LZSTAGG.spad" 1055444 1055454 1058196 1058201) (-679 "LZSTAGG.spad" 1052680 1052692 1055434 1055439) (-678 "LWORD.spad" 1049385 1049402 1052670 1052675) (-677 "LSTAST.spad" 1049169 1049177 1049375 1049380) (-676 "LSQM.spad" 1047399 1047413 1047793 1047844) (-675 "LSPP.spad" 1046934 1046951 1047389 1047394) (-674 "LSMP.spad" 1045784 1045812 1046924 1046929) (-673 "LSMP1.spad" 1043602 1043616 1045774 1045779) (-672 "LSAGG.spad" 1043271 1043281 1043570 1043597) (-671 "LSAGG.spad" 1042960 1042972 1043261 1043266) (-670 "LPOLY.spad" 1041914 1041933 1042816 1042885) (-669 "LPEFRAC.spad" 1041185 1041195 1041904 1041909) (-668 "LO.spad" 1040586 1040600 1041119 1041146) (-667 "LOGIC.spad" 1040188 1040196 1040576 1040581) (-666 "LOGIC.spad" 1039788 1039798 1040178 1040183) (-665 "LODOOPS.spad" 1038718 1038730 1039778 1039783) (-664 "LODO.spad" 1038102 1038118 1038398 1038437) (-663 "LODOF.spad" 1037148 1037165 1038059 1038064) (-662 "LODOCAT.spad" 1035814 1035824 1037104 1037143) (-661 "LODOCAT.spad" 1034478 1034490 1035770 1035775) (-660 "LODO2.spad" 1033751 1033763 1034158 1034197) (-659 "LODO1.spad" 1033151 1033161 1033431 1033470) (-658 "LODEEF.spad" 1031953 1031971 1033141 1033146) (-657 "LNAGG.spad" 1027785 1027795 1031943 1031948) (-656 "LNAGG.spad" 1023581 1023593 1027741 1027746) (-655 "LMOPS.spad" 1020349 1020366 1023571 1023576) (-654 "LMODULE.spad" 1020117 1020127 1020339 1020344) (-653 "LMDICT.spad" 1019404 1019414 1019668 1019695) (-652 "LLINSET.spad" 1018801 1018811 1019394 1019399) (-651 "LITERAL.spad" 1018707 1018718 1018791 1018796) (-650 "LIST.spad" 1016442 1016452 1017854 1017881) (-649 "LIST3.spad" 1015753 1015767 1016432 1016437) (-648 "LIST2.spad" 1014455 1014467 1015743 1015748) (-647 "LIST2MAP.spad" 1011358 1011370 1014445 1014450) (-646 "LINSET.spad" 1010980 1010990 1011348 1011353) (-645 "LINEXP.spad" 1010414 1010424 1010960 1010975) (-644 "LINDEP.spad" 1009223 1009235 1010326 1010331) (-643 "LIMITRF.spad" 1007151 1007161 1009213 1009218) (-642 "LIMITPS.spad" 1006054 1006067 1007141 1007146) (-641 "LIE.spad" 1004070 1004082 1005344 1005489) (-640 "LIECAT.spad" 1003546 1003556 1003996 1004065) (-639 "LIECAT.spad" 1003050 1003062 1003502 1003507) (-638 "LIB.spad" 1001100 1001108 1001709 1001724) (-637 "LGROBP.spad" 998453 998472 1001090 1001095) (-636 "LF.spad" 997408 997424 998443 998448) (-635 "LFCAT.spad" 996467 996475 997398 997403) (-634 "LEXTRIPK.spad" 991970 991985 996457 996462) (-633 "LEXP.spad" 989973 990000 991950 991965) (-632 "LETAST.spad" 989672 989680 989963 989968) (-631 "LEADCDET.spad" 988070 988087 989662 989667) (-630 "LAZM3PK.spad" 986774 986796 988060 988065) (-629 "LAUPOL.spad" 985467 985480 986367 986436) (-628 "LAPLACE.spad" 985050 985066 985457 985462) (-627 "LA.spad" 984490 984504 984972 985011) (-626 "LALG.spad" 984266 984276 984470 984485) (-625 "LALG.spad" 984050 984062 984256 984261) (-624 "KVTFROM.spad" 983785 983795 984040 984045) (-623 "KTVLOGIC.spad" 983297 983305 983775 983780) (-622 "KRCFROM.spad" 983035 983045 983287 983292) (-621 "KOVACIC.spad" 981758 981775 983025 983030) (-620 "KONVERT.spad" 981480 981490 981748 981753) (-619 "KOERCE.spad" 981217 981227 981470 981475) (-618 "KERNEL.spad" 979872 979882 981001 981006) (-617 "KERNEL2.spad" 979575 979587 979862 979867) (-616 "KDAGG.spad" 978684 978706 979555 979570) (-615 "KDAGG.spad" 977801 977825 978674 978679) (-614 "KAFILE.spad" 976764 976780 976999 977026) (-613 "JORDAN.spad" 974593 974605 976054 976199) (-612 "JOINAST.spad" 974287 974295 974583 974588) (-611 "JAVACODE.spad" 974153 974161 974277 974282) (-610 "IXAGG.spad" 972286 972310 974143 974148) (-609 "IXAGG.spad" 970274 970300 972133 972138) (-608 "IVECTOR.spad" 969044 969059 969199 969226) (-607 "ITUPLE.spad" 968205 968215 969034 969039) (-606 "ITRIGMNP.spad" 967044 967063 968195 968200) (-605 "ITFUN3.spad" 966550 966564 967034 967039) (-604 "ITFUN2.spad" 966294 966306 966540 966545) (-603 "ITFORM.spad" 965649 965657 966284 966289) (-602 "ITAYLOR.spad" 963643 963658 965513 965610) (-601 "ISUPS.spad" 956080 956095 962617 962714) (-600 "ISUMP.spad" 955581 955597 956070 956075) (-599 "ISTRING.spad" 954669 954682 954750 954777) (-598 "ISAST.spad" 954388 954396 954659 954664) (-597 "IRURPK.spad" 953105 953124 954378 954383) (-596 "IRSN.spad" 951109 951117 953095 953100) (-595 "IRRF2F.spad" 949594 949604 951065 951070) (-594 "IRREDFFX.spad" 949195 949206 949584 949589) (-593 "IROOT.spad" 947534 947544 949185 949190) (-592 "IR.spad" 945335 945349 947389 947416) (-591 "IRFORM.spad" 944659 944667 945325 945330) (-590 "IR2.spad" 943687 943703 944649 944654) (-589 "IR2F.spad" 942893 942909 943677 943682) (-588 "IPRNTPK.spad" 942653 942661 942883 942888) (-587 "IPF.spad" 942218 942230 942458 942551) (-586 "IPADIC.spad" 941979 942005 942144 942213) (-585 "IP4ADDR.spad" 941536 941544 941969 941974) (-584 "IOMODE.spad" 941058 941066 941526 941531) (-583 "IOBFILE.spad" 940419 940427 941048 941053) (-582 "IOBCON.spad" 940284 940292 940409 940414) (-581 "INVLAPLA.spad" 939933 939949 940274 940279) (-580 "INTTR.spad" 933315 933332 939923 939928) (-579 "INTTOOLS.spad" 931070 931086 932889 932894) (-578 "INTSLPE.spad" 930390 930398 931060 931065) (-577 "INTRVL.spad" 929956 929966 930304 930385) (-576 "INTRF.spad" 928380 928394 929946 929951) (-575 "INTRET.spad" 927812 927822 928370 928375) (-574 "INTRAT.spad" 926539 926556 927802 927807) (-573 "INTPM.spad" 924924 924940 926182 926187) (-572 "INTPAF.spad" 922788 922806 924856 924861) (-571 "INTPACK.spad" 913162 913170 922778 922783) (-570 "INT.spad" 912610 912618 913016 913157) (-569 "INTHERTR.spad" 911884 911901 912600 912605) (-568 "INTHERAL.spad" 911554 911578 911874 911879) (-567 "INTHEORY.spad" 907993 908001 911544 911549) (-566 "INTG0.spad" 901726 901744 907925 907930) (-565 "INTFTBL.spad" 895755 895763 901716 901721) (-564 "INTFACT.spad" 894814 894824 895745 895750) (-563 "INTEF.spad" 893199 893215 894804 894809) (-562 "INTDOM.spad" 891822 891830 893125 893194) (-561 "INTDOM.spad" 890507 890517 891812 891817) (-560 "INTCAT.spad" 888766 888776 890421 890502) (-559 "INTBIT.spad" 888273 888281 888756 888761) (-558 "INTALG.spad" 887461 887488 888263 888268) (-557 "INTAF.spad" 886961 886977 887451 887456) (-556 "INTABL.spad" 885479 885510 885642 885669) (-555 "INT8.spad" 885359 885367 885469 885474) (-554 "INT64.spad" 885238 885246 885349 885354) (-553 "INT32.spad" 885117 885125 885228 885233) (-552 "INT16.spad" 884996 885004 885107 885112) (-551 "INS.spad" 882499 882507 884898 884991) (-550 "INS.spad" 880088 880098 882489 882494) (-549 "INPSIGN.spad" 879536 879549 880078 880083) (-548 "INPRODPF.spad" 878632 878651 879526 879531) (-547 "INPRODFF.spad" 877720 877744 878622 878627) (-546 "INNMFACT.spad" 876695 876712 877710 877715) (-545 "INMODGCD.spad" 876183 876213 876685 876690) (-544 "INFSP.spad" 874480 874502 876173 876178) (-543 "INFPROD0.spad" 873560 873579 874470 874475) (-542 "INFORM.spad" 870759 870767 873550 873555) (-541 "INFORM1.spad" 870384 870394 870749 870754) (-540 "INFINITY.spad" 869936 869944 870374 870379) (-539 "INETCLTS.spad" 869913 869921 869926 869931) (-538 "INEP.spad" 868451 868473 869903 869908) (-537 "INDE.spad" 868180 868197 868441 868446) (-536 "INCRMAPS.spad" 867601 867611 868170 868175) (-535 "INBFILE.spad" 866673 866681 867591 867596) (-534 "INBFF.spad" 862467 862478 866663 866668) (-533 "INBCON.spad" 860757 860765 862457 862462) (-532 "INBCON.spad" 859045 859055 860747 860752) (-531 "INAST.spad" 858706 858714 859035 859040) (-530 "IMPTAST.spad" 858414 858422 858696 858701) (-529 "IMATRIX.spad" 857359 857385 857871 857898) (-528 "IMATQF.spad" 856453 856497 857315 857320) (-527 "IMATLIN.spad" 855058 855082 856409 856414) (-526 "ILIST.spad" 853716 853731 854241 854268) (-525 "IIARRAY2.spad" 853104 853142 853323 853350) (-524 "IFF.spad" 852514 852530 852785 852878) (-523 "IFAST.spad" 852128 852136 852504 852509) (-522 "IFARRAY.spad" 849621 849636 851311 851338) (-521 "IFAMON.spad" 849483 849500 849577 849582) (-520 "IEVALAB.spad" 848888 848900 849473 849478) (-519 "IEVALAB.spad" 848291 848305 848878 848883) (-518 "IDPO.spad" 848089 848101 848281 848286) (-517 "IDPOAMS.spad" 847845 847857 848079 848084) (-516 "IDPOAM.spad" 847565 847577 847835 847840) (-515 "IDPC.spad" 846503 846515 847555 847560) (-514 "IDPAM.spad" 846248 846260 846493 846498) (-513 "IDPAG.spad" 845995 846007 846238 846243) (-512 "IDENT.spad" 845645 845653 845985 845990) (-511 "IDECOMP.spad" 842884 842902 845635 845640) (-510 "IDEAL.spad" 837833 837872 842819 842824) (-509 "ICDEN.spad" 837022 837038 837823 837828) (-508 "ICARD.spad" 836213 836221 837012 837017) (-507 "IBPTOOLS.spad" 834820 834837 836203 836208) (-506 "IBITS.spad" 834023 834036 834456 834483) (-505 "IBATOOL.spad" 831000 831019 834013 834018) (-504 "IBACHIN.spad" 829507 829522 830990 830995) (-503 "IARRAY2.spad" 828495 828521 829114 829141) (-502 "IARRAY1.spad" 827540 827555 827678 827705) (-501 "IAN.spad" 825763 825771 827356 827449) (-500 "IALGFACT.spad" 825366 825399 825753 825758) (-499 "HYPCAT.spad" 824790 824798 825356 825361) (-498 "HYPCAT.spad" 824212 824222 824780 824785) (-497 "HOSTNAME.spad" 824020 824028 824202 824207) (-496 "HOMOTOP.spad" 823763 823773 824010 824015) (-495 "HOAGG.spad" 821045 821055 823753 823758) (-494 "HOAGG.spad" 818102 818114 820812 820817) (-493 "HEXADEC.spad" 816204 816212 816569 816662) (-492 "HEUGCD.spad" 815239 815250 816194 816199) (-491 "HELLFDIV.spad" 814829 814853 815229 815234) (-490 "HEAP.spad" 814221 814231 814436 814463) (-489 "HEADAST.spad" 813754 813762 814211 814216) (-488 "HDP.spad" 803597 803613 803974 804105) (-487 "HDMP.spad" 800811 800826 801427 801554) (-486 "HB.spad" 799062 799070 800801 800806) (-485 "HASHTBL.spad" 797532 797563 797743 797770) (-484 "HASAST.spad" 797248 797256 797522 797527) (-483 "HACKPI.spad" 796739 796747 797150 797243) (-482 "GTSET.spad" 795678 795694 796385 796412) (-481 "GSTBL.spad" 794197 794232 794371 794386) (-480 "GSERIES.spad" 791368 791395 792329 792478) (-479 "GROUP.spad" 790641 790649 791348 791363) (-478 "GROUP.spad" 789922 789932 790631 790636) (-477 "GROEBSOL.spad" 788416 788437 789912 789917) (-476 "GRMOD.spad" 786987 786999 788406 788411) (-475 "GRMOD.spad" 785556 785570 786977 786982) (-474 "GRIMAGE.spad" 778445 778453 785546 785551) (-473 "GRDEF.spad" 776824 776832 778435 778440) (-472 "GRAY.spad" 775287 775295 776814 776819) (-471 "GRALG.spad" 774364 774376 775277 775282) (-470 "GRALG.spad" 773439 773453 774354 774359) (-469 "GPOLSET.spad" 772893 772916 773121 773148) (-468 "GOSPER.spad" 772162 772180 772883 772888) (-467 "GMODPOL.spad" 771310 771337 772130 772157) (-466 "GHENSEL.spad" 770393 770407 771300 771305) (-465 "GENUPS.spad" 766686 766699 770383 770388) (-464 "GENUFACT.spad" 766263 766273 766676 766681) (-463 "GENPGCD.spad" 765849 765866 766253 766258) (-462 "GENMFACT.spad" 765301 765320 765839 765844) (-461 "GENEEZ.spad" 763252 763265 765291 765296) (-460 "GDMP.spad" 760308 760325 761082 761209) (-459 "GCNAALG.spad" 754231 754258 760102 760169) (-458 "GCDDOM.spad" 753407 753415 754157 754226) (-457 "GCDDOM.spad" 752645 752655 753397 753402) (-456 "GB.spad" 750171 750209 752601 752606) (-455 "GBINTERN.spad" 746191 746229 750161 750166) (-454 "GBF.spad" 741958 741996 746181 746186) (-453 "GBEUCLID.spad" 739840 739878 741948 741953) (-452 "GAUSSFAC.spad" 739153 739161 739830 739835) (-451 "GALUTIL.spad" 737479 737489 739109 739114) (-450 "GALPOLYU.spad" 735933 735946 737469 737474) (-449 "GALFACTU.spad" 734106 734125 735923 735928) (-448 "GALFACT.spad" 724295 724306 734096 734101) (-447 "FVFUN.spad" 721318 721326 724285 724290) (-446 "FVC.spad" 720370 720378 721308 721313) (-445 "FUNDESC.spad" 720048 720056 720360 720365) (-444 "FUNCTION.spad" 719897 719909 720038 720043) (-443 "FT.spad" 718194 718202 719887 719892) (-442 "FTEM.spad" 717359 717367 718184 718189) (-441 "FSUPFACT.spad" 716259 716278 717295 717300) (-440 "FST.spad" 714345 714353 716249 716254) (-439 "FSRED.spad" 713825 713841 714335 714340) (-438 "FSPRMELT.spad" 712707 712723 713782 713787) (-437 "FSPECF.spad" 710798 710814 712697 712702) (-436 "FS.spad" 705066 705076 710573 710793) (-435 "FS.spad" 699112 699124 704621 704626) (-434 "FSINT.spad" 698772 698788 699102 699107) (-433 "FSERIES.spad" 697963 697975 698592 698691) (-432 "FSCINT.spad" 697280 697296 697953 697958) (-431 "FSAGG.spad" 696397 696407 697236 697275) (-430 "FSAGG.spad" 695476 695488 696317 696322) (-429 "FSAGG2.spad" 694219 694235 695466 695471) (-428 "FS2UPS.spad" 688710 688744 694209 694214) (-427 "FS2.spad" 688357 688373 688700 688705) (-426 "FS2EXPXP.spad" 687482 687505 688347 688352) (-425 "FRUTIL.spad" 686436 686446 687472 687477) (-424 "FR.spad" 680152 680162 685460 685529) (-423 "FRNAALG.spad" 675271 675281 680094 680147) (-422 "FRNAALG.spad" 670402 670414 675227 675232) (-421 "FRNAAF2.spad" 669858 669876 670392 670397) (-420 "FRMOD.spad" 669268 669298 669789 669794) (-419 "FRIDEAL.spad" 668493 668514 669248 669263) (-418 "FRIDEAL2.spad" 668097 668129 668483 668488) (-417 "FRETRCT.spad" 667608 667618 668087 668092) (-416 "FRETRCT.spad" 666985 666997 667466 667471) (-415 "FRAMALG.spad" 665333 665346 666941 666980) (-414 "FRAMALG.spad" 663713 663728 665323 665328) (-413 "FRAC.spad" 660812 660822 661215 661388) (-412 "FRAC2.spad" 660417 660429 660802 660807) (-411 "FR2.spad" 659753 659765 660407 660412) (-410 "FPS.spad" 656568 656576 659643 659748) (-409 "FPS.spad" 653411 653421 656488 656493) (-408 "FPC.spad" 652457 652465 653313 653406) (-407 "FPC.spad" 651589 651599 652447 652452) (-406 "FPATMAB.spad" 651351 651361 651579 651584) (-405 "FPARFRAC.spad" 649838 649855 651341 651346) (-404 "FORTRAN.spad" 648344 648387 649828 649833) (-403 "FORT.spad" 647293 647301 648334 648339) (-402 "FORTFN.spad" 644463 644471 647283 647288) (-401 "FORTCAT.spad" 644147 644155 644453 644458) (-400 "FORMULA.spad" 641621 641629 644137 644142) (-399 "FORMULA1.spad" 641100 641110 641611 641616) (-398 "FORDER.spad" 640791 640815 641090 641095) (-397 "FOP.spad" 639992 640000 640781 640786) (-396 "FNLA.spad" 639416 639438 639960 639987) (-395 "FNCAT.spad" 638011 638019 639406 639411) (-394 "FNAME.spad" 637903 637911 638001 638006) (-393 "FMTC.spad" 637701 637709 637829 637898) (-392 "FMONOID.spad" 637366 637376 637657 637662) (-391 "FMONCAT.spad" 634519 634529 637356 637361) (-390 "FM.spad" 634214 634226 634453 634480) (-389 "FMFUN.spad" 631244 631252 634204 634209) (-388 "FMC.spad" 630296 630304 631234 631239) (-387 "FMCAT.spad" 627964 627982 630264 630291) (-386 "FM1.spad" 627321 627333 627898 627925) (-385 "FLOATRP.spad" 625056 625070 627311 627316) (-384 "FLOAT.spad" 618370 618378 624922 625051) (-383 "FLOATCP.spad" 615801 615815 618360 618365) (-382 "FLINEXP.spad" 615513 615523 615781 615796) (-381 "FLINEXP.spad" 615179 615191 615449 615454) (-380 "FLASORT.spad" 614505 614517 615169 615174) (-379 "FLALG.spad" 612151 612170 614431 614500) (-378 "FLAGG.spad" 609193 609203 612131 612146) (-377 "FLAGG.spad" 606136 606148 609076 609081) (-376 "FLAGG2.spad" 604861 604877 606126 606131) (-375 "FINRALG.spad" 602922 602935 604817 604856) (-374 "FINRALG.spad" 600909 600924 602806 602811) (-373 "FINITE.spad" 600061 600069 600899 600904) (-372 "FINAALG.spad" 589182 589192 600003 600056) (-371 "FINAALG.spad" 578315 578327 589138 589143) (-370 "FILE.spad" 577898 577908 578305 578310) (-369 "FILECAT.spad" 576424 576441 577888 577893) (-368 "FIELD.spad" 575830 575838 576326 576419) (-367 "FIELD.spad" 575322 575332 575820 575825) (-366 "FGROUP.spad" 573969 573979 575302 575317) (-365 "FGLMICPK.spad" 572756 572771 573959 573964) (-364 "FFX.spad" 572131 572146 572472 572565) (-363 "FFSLPE.spad" 571634 571655 572121 572126) (-362 "FFPOLY.spad" 562896 562907 571624 571629) (-361 "FFPOLY2.spad" 561956 561973 562886 562891) (-360 "FFP.spad" 561353 561373 561672 561765) (-359 "FF.spad" 560801 560817 561034 561127) (-358 "FFNBX.spad" 559313 559333 560517 560610) (-357 "FFNBP.spad" 557826 557843 559029 559122) (-356 "FFNB.spad" 556291 556312 557507 557600) (-355 "FFINTBAS.spad" 553805 553824 556281 556286) (-354 "FFIELDC.spad" 551382 551390 553707 553800) (-353 "FFIELDC.spad" 549045 549055 551372 551377) (-352 "FFHOM.spad" 547793 547810 549035 549040) (-351 "FFF.spad" 545228 545239 547783 547788) (-350 "FFCGX.spad" 544075 544095 544944 545037) (-349 "FFCGP.spad" 542964 542984 543791 543884) (-348 "FFCG.spad" 541756 541777 542645 542738) (-347 "FFCAT.spad" 534929 534951 541595 541751) (-346 "FFCAT.spad" 528181 528205 534849 534854) (-345 "FFCAT2.spad" 527928 527968 528171 528176) (-344 "FEXPR.spad" 519645 519691 527684 527723) (-343 "FEVALAB.spad" 519353 519363 519635 519640) (-342 "FEVALAB.spad" 518846 518858 519130 519135) (-341 "FDIV.spad" 518288 518312 518836 518841) (-340 "FDIVCAT.spad" 516352 516376 518278 518283) (-339 "FDIVCAT.spad" 514414 514440 516342 516347) (-338 "FDIV2.spad" 514070 514110 514404 514409) (-337 "FCTRDATA.spad" 513078 513086 514060 514065) (-336 "FCPAK1.spad" 511645 511653 513068 513073) (-335 "FCOMP.spad" 511024 511034 511635 511640) (-334 "FC.spad" 501031 501039 511014 511019) (-333 "FAXF.spad" 494002 494016 500933 501026) (-332 "FAXF.spad" 487025 487041 493958 493963) (-331 "FARRAY.spad" 485175 485185 486208 486235) (-330 "FAMR.spad" 483311 483323 485073 485170) (-329 "FAMR.spad" 481431 481445 483195 483200) (-328 "FAMONOID.spad" 481099 481109 481385 481390) (-327 "FAMONC.spad" 479395 479407 481089 481094) (-326 "FAGROUP.spad" 479019 479029 479291 479318) (-325 "FACUTIL.spad" 477223 477240 479009 479014) (-324 "FACTFUNC.spad" 476417 476427 477213 477218) (-323 "EXPUPXS.spad" 473250 473273 474549 474698) (-322 "EXPRTUBE.spad" 470538 470546 473240 473245) (-321 "EXPRODE.spad" 467698 467714 470528 470533) (-320 "EXPR.spad" 462973 462983 463687 464094) (-319 "EXPR2UPS.spad" 459095 459108 462963 462968) (-318 "EXPR2.spad" 458800 458812 459085 459090) (-317 "EXPEXPAN.spad" 455740 455765 456372 456465) (-316 "EXIT.spad" 455411 455419 455730 455735) (-315 "EXITAST.spad" 455147 455155 455401 455406) (-314 "EVALCYC.spad" 454607 454621 455137 455142) (-313 "EVALAB.spad" 454179 454189 454597 454602) (-312 "EVALAB.spad" 453749 453761 454169 454174) (-311 "EUCDOM.spad" 451323 451331 453675 453744) (-310 "EUCDOM.spad" 448959 448969 451313 451318) (-309 "ESTOOLS.spad" 440805 440813 448949 448954) (-308 "ESTOOLS2.spad" 440408 440422 440795 440800) (-307 "ESTOOLS1.spad" 440093 440104 440398 440403) (-306 "ES.spad" 432908 432916 440083 440088) (-305 "ES.spad" 425629 425639 432806 432811) (-304 "ESCONT.spad" 422422 422430 425619 425624) (-303 "ESCONT1.spad" 422171 422183 422412 422417) (-302 "ES2.spad" 421676 421692 422161 422166) (-301 "ES1.spad" 421246 421262 421666 421671) (-300 "ERROR.spad" 418573 418581 421236 421241) (-299 "EQTBL.spad" 417045 417067 417254 417281) (-298 "EQ.spad" 411850 411860 414637 414749) (-297 "EQ2.spad" 411568 411580 411840 411845) (-296 "EP.spad" 407894 407904 411558 411563) (-295 "ENV.spad" 406572 406580 407884 407889) (-294 "ENTIRER.spad" 406240 406248 406516 406567) (-293 "EMR.spad" 405447 405488 406166 406235) (-292 "ELTAGG.spad" 403701 403720 405437 405442) (-291 "ELTAGG.spad" 401919 401940 403657 403662) (-290 "ELTAB.spad" 401368 401386 401909 401914) (-289 "ELFUTS.spad" 400755 400774 401358 401363) (-288 "ELEMFUN.spad" 400444 400452 400745 400750) (-287 "ELEMFUN.spad" 400131 400141 400434 400439) (-286 "ELAGG.spad" 398102 398112 400111 400126) (-285 "ELAGG.spad" 396010 396022 398021 398026) (-284 "ELABOR.spad" 395356 395364 396000 396005) (-283 "ELABEXPR.spad" 394288 394296 395346 395351) (-282 "EFUPXS.spad" 391064 391094 394244 394249) (-281 "EFULS.spad" 387900 387923 391020 391025) (-280 "EFSTRUC.spad" 385915 385931 387890 387895) (-279 "EF.spad" 380691 380707 385905 385910) (-278 "EAB.spad" 378967 378975 380681 380686) (-277 "E04UCFA.spad" 378503 378511 378957 378962) (-276 "E04NAFA.spad" 378080 378088 378493 378498) (-275 "E04MBFA.spad" 377660 377668 378070 378075) (-274 "E04JAFA.spad" 377196 377204 377650 377655) (-273 "E04GCFA.spad" 376732 376740 377186 377191) (-272 "E04FDFA.spad" 376268 376276 376722 376727) (-271 "E04DGFA.spad" 375804 375812 376258 376263) (-270 "E04AGNT.spad" 371654 371662 375794 375799) (-269 "DVARCAT.spad" 368343 368353 371644 371649) (-268 "DVARCAT.spad" 365030 365042 368333 368338) (-267 "DSMP.spad" 362497 362511 362802 362929) (-266 "DROPT.spad" 356456 356464 362487 362492) (-265 "DROPT1.spad" 356121 356131 356446 356451) (-264 "DROPT0.spad" 350978 350986 356111 356116) (-263 "DRAWPT.spad" 349151 349159 350968 350973) (-262 "DRAW.spad" 342027 342040 349141 349146) (-261 "DRAWHACK.spad" 341335 341345 342017 342022) (-260 "DRAWCX.spad" 338805 338813 341325 341330) (-259 "DRAWCURV.spad" 338352 338367 338795 338800) (-258 "DRAWCFUN.spad" 327884 327892 338342 338347) (-257 "DQAGG.spad" 326062 326072 327852 327879) (-256 "DPOLCAT.spad" 321411 321427 325930 326057) (-255 "DPOLCAT.spad" 316846 316864 321367 321372) (-254 "DPMO.spad" 309072 309088 309210 309511) (-253 "DPMM.spad" 301311 301329 301436 301737) (-252 "DOMTMPLT.spad" 300971 300979 301301 301306) (-251 "DOMCTOR.spad" 300726 300734 300961 300966) (-250 "DOMAIN.spad" 299813 299821 300716 300721) (-249 "DMP.spad" 297073 297088 297643 297770) (-248 "DLP.spad" 296425 296435 297063 297068) (-247 "DLIST.spad" 295004 295014 295608 295635) (-246 "DLAGG.spad" 293421 293431 294994 294999) (-245 "DIVRING.spad" 292963 292971 293365 293416) (-244 "DIVRING.spad" 292549 292559 292953 292958) (-243 "DISPLAY.spad" 290739 290747 292539 292544) (-242 "DIRPROD.spad" 280319 280335 280959 281090) (-241 "DIRPROD2.spad" 279137 279155 280309 280314) (-240 "DIRPCAT.spad" 278081 278097 279001 279132) (-239 "DIRPCAT.spad" 276754 276772 277676 277681) (-238 "DIOSP.spad" 275579 275587 276744 276749) (-237 "DIOPS.spad" 274575 274585 275559 275574) (-236 "DIOPS.spad" 273545 273557 274531 274536) (-235 "DIFRING.spad" 272841 272849 273525 273540) (-234 "DIFRING.spad" 272145 272155 272831 272836) (-233 "DIFEXT.spad" 271316 271326 272125 272140) (-232 "DIFEXT.spad" 270404 270416 271215 271220) (-231 "DIAGG.spad" 270034 270044 270384 270399) (-230 "DIAGG.spad" 269672 269684 270024 270029) (-229 "DHMATRIX.spad" 267984 267994 269129 269156) (-228 "DFSFUN.spad" 261624 261632 267974 267979) (-227 "DFLOAT.spad" 258355 258363 261514 261619) (-226 "DFINTTLS.spad" 256586 256602 258345 258350) (-225 "DERHAM.spad" 254500 254532 256566 256581) (-224 "DEQUEUE.spad" 253824 253834 254107 254134) (-223 "DEGRED.spad" 253441 253455 253814 253819) (-222 "DEFINTRF.spad" 250978 250988 253431 253436) (-221 "DEFINTEF.spad" 249488 249504 250968 250973) (-220 "DEFAST.spad" 248856 248864 249478 249483) (-219 "DECIMAL.spad" 246962 246970 247323 247416) (-218 "DDFACT.spad" 244775 244792 246952 246957) (-217 "DBLRESP.spad" 244375 244399 244765 244770) (-216 "DBASE.spad" 243039 243049 244365 244370) (-215 "DATAARY.spad" 242501 242514 243029 243034) (-214 "D03FAFA.spad" 242329 242337 242491 242496) (-213 "D03EEFA.spad" 242149 242157 242319 242324) (-212 "D03AGNT.spad" 241235 241243 242139 242144) (-211 "D02EJFA.spad" 240697 240705 241225 241230) (-210 "D02CJFA.spad" 240175 240183 240687 240692) (-209 "D02BHFA.spad" 239665 239673 240165 240170) (-208 "D02BBFA.spad" 239155 239163 239655 239660) (-207 "D02AGNT.spad" 233969 233977 239145 239150) (-206 "D01WGTS.spad" 232288 232296 233959 233964) (-205 "D01TRNS.spad" 232265 232273 232278 232283) (-204 "D01GBFA.spad" 231787 231795 232255 232260) (-203 "D01FCFA.spad" 231309 231317 231777 231782) (-202 "D01ASFA.spad" 230777 230785 231299 231304) (-201 "D01AQFA.spad" 230223 230231 230767 230772) (-200 "D01APFA.spad" 229647 229655 230213 230218) (-199 "D01ANFA.spad" 229141 229149 229637 229642) (-198 "D01AMFA.spad" 228651 228659 229131 229136) (-197 "D01ALFA.spad" 228191 228199 228641 228646) (-196 "D01AKFA.spad" 227717 227725 228181 228186) (-195 "D01AJFA.spad" 227240 227248 227707 227712) (-194 "D01AGNT.spad" 223307 223315 227230 227235) (-193 "CYCLOTOM.spad" 222813 222821 223297 223302) (-192 "CYCLES.spad" 219669 219677 222803 222808) (-191 "CVMP.spad" 219086 219096 219659 219664) (-190 "CTRIGMNP.spad" 217586 217602 219076 219081) (-189 "CTOR.spad" 217277 217285 217576 217581) (-188 "CTORKIND.spad" 216880 216888 217267 217272) (-187 "CTORCAT.spad" 216129 216137 216870 216875) (-186 "CTORCAT.spad" 215376 215386 216119 216124) (-185 "CTORCALL.spad" 214965 214975 215366 215371) (-184 "CSTTOOLS.spad" 214210 214223 214955 214960) (-183 "CRFP.spad" 207934 207947 214200 214205) (-182 "CRCEAST.spad" 207654 207662 207924 207929) (-181 "CRAPACK.spad" 206705 206715 207644 207649) (-180 "CPMATCH.spad" 206209 206224 206630 206635) (-179 "CPIMA.spad" 205914 205933 206199 206204) (-178 "COORDSYS.spad" 200923 200933 205904 205909) (-177 "CONTOUR.spad" 200334 200342 200913 200918) (-176 "CONTFRAC.spad" 196084 196094 200236 200329) (-175 "CONDUIT.spad" 195842 195850 196074 196079) (-174 "COMRING.spad" 195516 195524 195780 195837) (-173 "COMPPROP.spad" 195034 195042 195506 195511) (-172 "COMPLPAT.spad" 194801 194816 195024 195029) (-171 "COMPLEX.spad" 188938 188948 189182 189443) (-170 "COMPLEX2.spad" 188653 188665 188928 188933) (-169 "COMPILER.spad" 188202 188210 188643 188648) (-168 "COMPFACT.spad" 187804 187818 188192 188197) (-167 "COMPCAT.spad" 185876 185886 187538 187799) (-166 "COMPCAT.spad" 183676 183688 185340 185345) (-165 "COMMUPC.spad" 183424 183442 183666 183671) (-164 "COMMONOP.spad" 182957 182965 183414 183419) (-163 "COMM.spad" 182768 182776 182947 182952) (-162 "COMMAAST.spad" 182531 182539 182758 182763) (-161 "COMBOPC.spad" 181446 181454 182521 182526) (-160 "COMBINAT.spad" 180213 180223 181436 181441) (-159 "COMBF.spad" 177595 177611 180203 180208) (-158 "COLOR.spad" 176432 176440 177585 177590) (-157 "COLONAST.spad" 176098 176106 176422 176427) (-156 "CMPLXRT.spad" 175809 175826 176088 176093) (-155 "CLLCTAST.spad" 175471 175479 175799 175804) (-154 "CLIP.spad" 171579 171587 175461 175466) (-153 "CLIF.spad" 170234 170250 171535 171574) (-152 "CLAGG.spad" 166739 166749 170224 170229) (-151 "CLAGG.spad" 163115 163127 166602 166607) (-150 "CINTSLPE.spad" 162446 162459 163105 163110) (-149 "CHVAR.spad" 160584 160606 162436 162441) (-148 "CHARZ.spad" 160499 160507 160564 160579) (-147 "CHARPOL.spad" 160009 160019 160489 160494) (-146 "CHARNZ.spad" 159762 159770 159989 160004) (-145 "CHAR.spad" 157636 157644 159752 159757) (-144 "CFCAT.spad" 156964 156972 157626 157631) (-143 "CDEN.spad" 156160 156174 156954 156959) (-142 "CCLASS.spad" 154309 154317 155571 155610) (-141 "CATEGORY.spad" 153351 153359 154299 154304) (-140 "CATCTOR.spad" 153242 153250 153341 153346) (-139 "CATAST.spad" 152860 152868 153232 153237) (-138 "CASEAST.spad" 152574 152582 152850 152855) (-137 "CARTEN.spad" 147861 147885 152564 152569) (-136 "CARTEN2.spad" 147251 147278 147851 147856) (-135 "CARD.spad" 144546 144554 147225 147246) (-134 "CAPSLAST.spad" 144320 144328 144536 144541) (-133 "CACHSET.spad" 143944 143952 144310 144315) (-132 "CABMON.spad" 143499 143507 143934 143939) (-131 "BYTEORD.spad" 143174 143182 143489 143494) (-130 "BYTE.spad" 142601 142609 143164 143169) (-129 "BYTEBUF.spad" 140460 140468 141770 141797) (-128 "BTREE.spad" 139533 139543 140067 140094) (-127 "BTOURN.spad" 138538 138548 139140 139167) (-126 "BTCAT.spad" 137930 137940 138506 138533) (-125 "BTCAT.spad" 137342 137354 137920 137925) (-124 "BTAGG.spad" 136808 136816 137310 137337) (-123 "BTAGG.spad" 136294 136304 136798 136803) (-122 "BSTREE.spad" 135035 135045 135901 135928) (-121 "BRILL.spad" 133232 133243 135025 135030) (-120 "BRAGG.spad" 132172 132182 133222 133227) (-119 "BRAGG.spad" 131076 131088 132128 132133) (-118 "BPADICRT.spad" 129057 129069 129312 129405) (-117 "BPADIC.spad" 128721 128733 128983 129052) (-116 "BOUNDZRO.spad" 128377 128394 128711 128716) (-115 "BOP.spad" 123559 123567 128367 128372) (-114 "BOP1.spad" 121025 121035 123549 123554) (-113 "BOOLE.spad" 120675 120683 121015 121020) (-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 "ASP8.spad" 96485 96498 97432 97437) (-87 "ASP80.spad" 95807 95820 96475 96480) (-86 "ASP7.spad" 94967 94980 95797 95802) (-85 "ASP78.spad" 94418 94431 94957 94962) (-84 "ASP77.spad" 93787 93800 94408 94413) (-83 "ASP74.spad" 92879 92892 93777 93782) (-82 "ASP73.spad" 92150 92163 92869 92874) (-81 "ASP6.spad" 91017 91030 92140 92145) (-80 "ASP55.spad" 89526 89539 91007 91012) (-79 "ASP50.spad" 87343 87356 89516 89521) (-78 "ASP4.spad" 86638 86651 87333 87338) (-77 "ASP49.spad" 85637 85650 86628 86633) (-76 "ASP42.spad" 84044 84083 85627 85632) (-75 "ASP41.spad" 82623 82662 84034 84039) (-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 "ASP1.spad" 66072 66085 66681 66686) (-63 "ASP19.spad" 60758 60771 66062 66067) (-62 "ASP12.spad" 60172 60185 60748 60753) (-61 "ASP10.spad" 59443 59456 60162 60167) (-60 "ARRAY2.spad" 58803 58812 59050 59077) (-59 "ARRAY1.spad" 57640 57649 57986 58013) (-58 "ARRAY12.spad" 56353 56364 57630 57635) (-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 "ANY.spad" 45031 45038 46162 46167) (-51 "ANY1.spad" 44102 44111 45021 45026) (-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 2267781 2267786 2267791 2267796) (-2 NIL 2267761 2267766 2267771 2267776) (-1 NIL 2267741 2267746 2267751 2267756) (0 NIL 2267721 2267726 2267731 2267736) (-1304 "ZMOD.spad" 2267530 2267543 2267659 2267716) (-1303 "ZLINDEP.spad" 2266596 2266607 2267520 2267525) (-1302 "ZDSOLVE.spad" 2256541 2256563 2266586 2266591) (-1301 "YSTREAM.spad" 2256036 2256047 2256531 2256536) (-1300 "XRPOLY.spad" 2255256 2255276 2255892 2255961) (-1299 "XPR.spad" 2253051 2253064 2254974 2255073) (-1298 "XPOLY.spad" 2252606 2252617 2252907 2252976) (-1297 "XPOLYC.spad" 2251925 2251941 2252532 2252601) (-1296 "XPBWPOLY.spad" 2250362 2250382 2251705 2251774) (-1295 "XF.spad" 2248825 2248840 2250264 2250357) (-1294 "XF.spad" 2247268 2247285 2248709 2248714) (-1293 "XFALG.spad" 2244316 2244332 2247194 2247263) (-1292 "XEXPPKG.spad" 2243567 2243593 2244306 2244311) (-1291 "XDPOLY.spad" 2243181 2243197 2243423 2243492) (-1290 "XALG.spad" 2242841 2242852 2243137 2243176) (-1289 "WUTSET.spad" 2238680 2238697 2242487 2242514) (-1288 "WP.spad" 2237879 2237923 2238538 2238605) (-1287 "WHILEAST.spad" 2237677 2237686 2237869 2237874) (-1286 "WHEREAST.spad" 2237348 2237357 2237667 2237672) (-1285 "WFFINTBS.spad" 2235011 2235033 2237338 2237343) (-1284 "WEIER.spad" 2233233 2233244 2235001 2235006) (-1283 "VSPACE.spad" 2232906 2232917 2233201 2233228) (-1282 "VSPACE.spad" 2232599 2232612 2232896 2232901) (-1281 "VOID.spad" 2232276 2232285 2232589 2232594) (-1280 "VIEW.spad" 2229956 2229965 2232266 2232271) (-1279 "VIEWDEF.spad" 2225157 2225166 2229946 2229951) (-1278 "VIEW3D.spad" 2209118 2209127 2225147 2225152) (-1277 "VIEW2D.spad" 2197009 2197018 2209108 2209113) (-1276 "VECTOR.spad" 2195683 2195694 2195934 2195961) (-1275 "VECTOR2.spad" 2194322 2194335 2195673 2195678) (-1274 "VECTCAT.spad" 2192226 2192237 2194290 2194317) (-1273 "VECTCAT.spad" 2189937 2189950 2192003 2192008) (-1272 "VARIABLE.spad" 2189717 2189732 2189927 2189932) (-1271 "UTYPE.spad" 2189361 2189370 2189707 2189712) (-1270 "UTSODETL.spad" 2188656 2188680 2189317 2189322) (-1269 "UTSODE.spad" 2186872 2186892 2188646 2188651) (-1268 "UTS.spad" 2181676 2181704 2185339 2185436) (-1267 "UTSCAT.spad" 2179155 2179171 2181574 2181671) (-1266 "UTSCAT.spad" 2176278 2176296 2178699 2178704) (-1265 "UTS2.spad" 2175873 2175908 2176268 2176273) (-1264 "URAGG.spad" 2170546 2170557 2175863 2175868) (-1263 "URAGG.spad" 2165183 2165196 2170502 2170507) (-1262 "UPXSSING.spad" 2162828 2162854 2164264 2164397) (-1261 "UPXS.spad" 2159982 2160010 2160960 2161109) (-1260 "UPXSCONS.spad" 2157741 2157761 2158114 2158263) (-1259 "UPXSCCA.spad" 2156312 2156332 2157587 2157736) (-1258 "UPXSCCA.spad" 2155025 2155047 2156302 2156307) (-1257 "UPXSCAT.spad" 2153614 2153630 2154871 2155020) (-1256 "UPXS2.spad" 2153157 2153210 2153604 2153609) (-1255 "UPSQFREE.spad" 2151571 2151585 2153147 2153152) (-1254 "UPSCAT.spad" 2149182 2149206 2151469 2151566) (-1253 "UPSCAT.spad" 2146499 2146525 2148788 2148793) (-1252 "UPOLYC.spad" 2141539 2141550 2146341 2146494) (-1251 "UPOLYC.spad" 2136471 2136484 2141275 2141280) (-1250 "UPOLYC2.spad" 2135942 2135961 2136461 2136466) (-1249 "UP.spad" 2133141 2133156 2133528 2133681) (-1248 "UPMP.spad" 2132041 2132054 2133131 2133136) (-1247 "UPDIVP.spad" 2131606 2131620 2132031 2132036) (-1246 "UPDECOMP.spad" 2129851 2129865 2131596 2131601) (-1245 "UPCDEN.spad" 2129060 2129076 2129841 2129846) (-1244 "UP2.spad" 2128424 2128445 2129050 2129055) (-1243 "UNISEG.spad" 2127777 2127788 2128343 2128348) (-1242 "UNISEG2.spad" 2127274 2127287 2127733 2127738) (-1241 "UNIFACT.spad" 2126377 2126389 2127264 2127269) (-1240 "ULS.spad" 2116935 2116963 2118022 2118451) (-1239 "ULSCONS.spad" 2109331 2109351 2109701 2109850) (-1238 "ULSCCAT.spad" 2107068 2107088 2109177 2109326) (-1237 "ULSCCAT.spad" 2104913 2104935 2107024 2107029) (-1236 "ULSCAT.spad" 2103145 2103161 2104759 2104908) (-1235 "ULS2.spad" 2102659 2102712 2103135 2103140) (-1234 "UINT8.spad" 2102536 2102545 2102649 2102654) (-1233 "UINT64.spad" 2102412 2102421 2102526 2102531) (-1232 "UINT32.spad" 2102288 2102297 2102402 2102407) (-1231 "UINT16.spad" 2102164 2102173 2102278 2102283) (-1230 "UFD.spad" 2101229 2101238 2102090 2102159) (-1229 "UFD.spad" 2100356 2100367 2101219 2101224) (-1228 "UDVO.spad" 2099237 2099246 2100346 2100351) (-1227 "UDPO.spad" 2096730 2096741 2099193 2099198) (-1226 "TYPE.spad" 2096662 2096671 2096720 2096725) (-1225 "TYPEAST.spad" 2096581 2096590 2096652 2096657) (-1224 "TWOFACT.spad" 2095233 2095248 2096571 2096576) (-1223 "TUPLE.spad" 2094719 2094730 2095132 2095137) (-1222 "TUBETOOL.spad" 2091586 2091595 2094709 2094714) (-1221 "TUBE.spad" 2090233 2090250 2091576 2091581) (-1220 "TS.spad" 2088832 2088848 2089798 2089895) (-1219 "TSETCAT.spad" 2075959 2075976 2088800 2088827) (-1218 "TSETCAT.spad" 2063072 2063091 2075915 2075920) (-1217 "TRMANIP.spad" 2057438 2057455 2062778 2062783) (-1216 "TRIMAT.spad" 2056401 2056426 2057428 2057433) (-1215 "TRIGMNIP.spad" 2054928 2054945 2056391 2056396) (-1214 "TRIGCAT.spad" 2054440 2054449 2054918 2054923) (-1213 "TRIGCAT.spad" 2053950 2053961 2054430 2054435) (-1212 "TREE.spad" 2052525 2052536 2053557 2053584) (-1211 "TRANFUN.spad" 2052364 2052373 2052515 2052520) (-1210 "TRANFUN.spad" 2052201 2052212 2052354 2052359) (-1209 "TOPSP.spad" 2051875 2051884 2052191 2052196) (-1208 "TOOLSIGN.spad" 2051538 2051549 2051865 2051870) (-1207 "TEXTFILE.spad" 2050099 2050108 2051528 2051533) (-1206 "TEX.spad" 2047245 2047254 2050089 2050094) (-1205 "TEX1.spad" 2046801 2046812 2047235 2047240) (-1204 "TEMUTL.spad" 2046356 2046365 2046791 2046796) (-1203 "TBCMPPK.spad" 2044449 2044472 2046346 2046351) (-1202 "TBAGG.spad" 2043499 2043522 2044429 2044444) (-1201 "TBAGG.spad" 2042557 2042582 2043489 2043494) (-1200 "TANEXP.spad" 2041965 2041976 2042547 2042552) (-1199 "TABLE.spad" 2040376 2040399 2040646 2040673) (-1198 "TABLEAU.spad" 2039857 2039868 2040366 2040371) (-1197 "TABLBUMP.spad" 2036660 2036671 2039847 2039852) (-1196 "SYSTEM.spad" 2035888 2035897 2036650 2036655) (-1195 "SYSSOLP.spad" 2033371 2033382 2035878 2035883) (-1194 "SYSPTR.spad" 2033270 2033279 2033361 2033366) (-1193 "SYSNNI.spad" 2032452 2032463 2033260 2033265) (-1192 "SYSINT.spad" 2031856 2031867 2032442 2032447) (-1191 "SYNTAX.spad" 2028062 2028071 2031846 2031851) (-1190 "SYMTAB.spad" 2026130 2026139 2028052 2028057) (-1189 "SYMS.spad" 2022153 2022162 2026120 2026125) (-1188 "SYMPOLY.spad" 2021160 2021171 2021242 2021369) (-1187 "SYMFUNC.spad" 2020661 2020672 2021150 2021155) (-1186 "SYMBOL.spad" 2018164 2018173 2020651 2020656) (-1185 "SWITCH.spad" 2014935 2014944 2018154 2018159) (-1184 "SUTS.spad" 2011840 2011868 2013402 2013499) (-1183 "SUPXS.spad" 2008981 2009009 2009972 2010121) (-1182 "SUP.spad" 2005794 2005805 2006567 2006720) (-1181 "SUPFRACF.spad" 2004899 2004917 2005784 2005789) (-1180 "SUP2.spad" 2004291 2004304 2004889 2004894) (-1179 "SUMRF.spad" 2003265 2003276 2004281 2004286) (-1178 "SUMFS.spad" 2002902 2002919 2003255 2003260) (-1177 "SULS.spad" 1993447 1993475 1994547 1994976) (-1176 "SUCHTAST.spad" 1993216 1993225 1993437 1993442) (-1175 "SUCH.spad" 1992898 1992913 1993206 1993211) (-1174 "SUBSPACE.spad" 1985013 1985028 1992888 1992893) (-1173 "SUBRESP.spad" 1984183 1984197 1984969 1984974) (-1172 "STTF.spad" 1980282 1980298 1984173 1984178) (-1171 "STTFNC.spad" 1976750 1976766 1980272 1980277) (-1170 "STTAYLOR.spad" 1969385 1969396 1976631 1976636) (-1169 "STRTBL.spad" 1967890 1967907 1968039 1968066) (-1168 "STRING.spad" 1967299 1967308 1967313 1967340) (-1167 "STRICAT.spad" 1967087 1967096 1967267 1967294) (-1166 "STREAM.spad" 1964005 1964016 1966612 1966627) (-1165 "STREAM3.spad" 1963578 1963593 1963995 1964000) (-1164 "STREAM2.spad" 1962706 1962719 1963568 1963573) (-1163 "STREAM1.spad" 1962412 1962423 1962696 1962701) (-1162 "STINPROD.spad" 1961348 1961364 1962402 1962407) (-1161 "STEP.spad" 1960549 1960558 1961338 1961343) (-1160 "STEPAST.spad" 1959783 1959792 1960539 1960544) (-1159 "STBL.spad" 1958309 1958337 1958476 1958491) (-1158 "STAGG.spad" 1957384 1957395 1958299 1958304) (-1157 "STAGG.spad" 1956457 1956470 1957374 1957379) (-1156 "STACK.spad" 1955814 1955825 1956064 1956091) (-1155 "SREGSET.spad" 1953518 1953535 1955460 1955487) (-1154 "SRDCMPK.spad" 1952079 1952099 1953508 1953513) (-1153 "SRAGG.spad" 1947222 1947231 1952047 1952074) (-1152 "SRAGG.spad" 1942385 1942396 1947212 1947217) (-1151 "SQMATRIX.spad" 1940001 1940019 1940917 1941004) (-1150 "SPLTREE.spad" 1934553 1934566 1939437 1939464) (-1149 "SPLNODE.spad" 1931141 1931154 1934543 1934548) (-1148 "SPFCAT.spad" 1929950 1929959 1931131 1931136) (-1147 "SPECOUT.spad" 1928502 1928511 1929940 1929945) (-1146 "SPADXPT.spad" 1920097 1920106 1928492 1928497) (-1145 "spad-parser.spad" 1919562 1919571 1920087 1920092) (-1144 "SPADAST.spad" 1919263 1919272 1919552 1919557) (-1143 "SPACEC.spad" 1903462 1903473 1919253 1919258) (-1142 "SPACE3.spad" 1903238 1903249 1903452 1903457) (-1141 "SORTPAK.spad" 1902787 1902800 1903194 1903199) (-1140 "SOLVETRA.spad" 1900550 1900561 1902777 1902782) (-1139 "SOLVESER.spad" 1899078 1899089 1900540 1900545) (-1138 "SOLVERAD.spad" 1895104 1895115 1899068 1899073) (-1137 "SOLVEFOR.spad" 1893566 1893584 1895094 1895099) (-1136 "SNTSCAT.spad" 1893166 1893183 1893534 1893561) (-1135 "SMTS.spad" 1891438 1891464 1892731 1892828) (-1134 "SMP.spad" 1888913 1888933 1889303 1889430) (-1133 "SMITH.spad" 1887758 1887783 1888903 1888908) (-1132 "SMATCAT.spad" 1885868 1885898 1887702 1887753) (-1131 "SMATCAT.spad" 1883910 1883942 1885746 1885751) (-1130 "SKAGG.spad" 1882873 1882884 1883878 1883905) (-1129 "SINT.spad" 1881813 1881822 1882739 1882868) (-1128 "SIMPAN.spad" 1881541 1881550 1881803 1881808) (-1127 "SIG.spad" 1880871 1880880 1881531 1881536) (-1126 "SIGNRF.spad" 1879989 1880000 1880861 1880866) (-1125 "SIGNEF.spad" 1879268 1879285 1879979 1879984) (-1124 "SIGAST.spad" 1878653 1878662 1879258 1879263) (-1123 "SHP.spad" 1876581 1876596 1878609 1878614) (-1122 "SHDP.spad" 1866292 1866319 1866801 1866932) (-1121 "SGROUP.spad" 1865900 1865909 1866282 1866287) (-1120 "SGROUP.spad" 1865506 1865517 1865890 1865895) (-1119 "SGCF.spad" 1858669 1858678 1865496 1865501) (-1118 "SFRTCAT.spad" 1857599 1857616 1858637 1858664) (-1117 "SFRGCD.spad" 1856662 1856682 1857589 1857594) (-1116 "SFQCMPK.spad" 1851299 1851319 1856652 1856657) (-1115 "SFORT.spad" 1850738 1850752 1851289 1851294) (-1114 "SEXOF.spad" 1850581 1850621 1850728 1850733) (-1113 "SEX.spad" 1850473 1850482 1850571 1850576) (-1112 "SEXCAT.spad" 1848074 1848114 1850463 1850468) (-1111 "SET.spad" 1846398 1846409 1847495 1847534) (-1110 "SETMN.spad" 1844848 1844865 1846388 1846393) (-1109 "SETCAT.spad" 1844170 1844179 1844838 1844843) (-1108 "SETCAT.spad" 1843490 1843501 1844160 1844165) (-1107 "SETAGG.spad" 1840039 1840050 1843470 1843485) (-1106 "SETAGG.spad" 1836596 1836609 1840029 1840034) (-1105 "SEQAST.spad" 1836299 1836308 1836586 1836591) (-1104 "SEGXCAT.spad" 1835455 1835468 1836289 1836294) (-1103 "SEG.spad" 1835268 1835279 1835374 1835379) (-1102 "SEGCAT.spad" 1834193 1834204 1835258 1835263) (-1101 "SEGBIND.spad" 1833951 1833962 1834140 1834145) (-1100 "SEGBIND2.spad" 1833649 1833662 1833941 1833946) (-1099 "SEGAST.spad" 1833363 1833372 1833639 1833644) (-1098 "SEG2.spad" 1832798 1832811 1833319 1833324) (-1097 "SDVAR.spad" 1832074 1832085 1832788 1832793) (-1096 "SDPOL.spad" 1829500 1829511 1829791 1829918) (-1095 "SCPKG.spad" 1827589 1827600 1829490 1829495) (-1094 "SCOPE.spad" 1826742 1826751 1827579 1827584) (-1093 "SCACHE.spad" 1825438 1825449 1826732 1826737) (-1092 "SASTCAT.spad" 1825347 1825356 1825428 1825433) (-1091 "SAOS.spad" 1825219 1825228 1825337 1825342) (-1090 "SAERFFC.spad" 1824932 1824952 1825209 1825214) (-1089 "SAE.spad" 1823107 1823123 1823718 1823853) (-1088 "SAEFACT.spad" 1822808 1822828 1823097 1823102) (-1087 "RURPK.spad" 1820467 1820483 1822798 1822803) (-1086 "RULESET.spad" 1819920 1819944 1820457 1820462) (-1085 "RULE.spad" 1818160 1818184 1819910 1819915) (-1084 "RULECOLD.spad" 1818012 1818025 1818150 1818155) (-1083 "RTVALUE.spad" 1817747 1817756 1818002 1818007) (-1082 "RSTRCAST.spad" 1817464 1817473 1817737 1817742) (-1081 "RSETGCD.spad" 1813842 1813862 1817454 1817459) (-1080 "RSETCAT.spad" 1803778 1803795 1813810 1813837) (-1079 "RSETCAT.spad" 1793734 1793753 1803768 1803773) (-1078 "RSDCMPK.spad" 1792186 1792206 1793724 1793729) (-1077 "RRCC.spad" 1790570 1790600 1792176 1792181) (-1076 "RRCC.spad" 1788952 1788984 1790560 1790565) (-1075 "RPTAST.spad" 1788654 1788663 1788942 1788947) (-1074 "RPOLCAT.spad" 1768014 1768029 1788522 1788649) (-1073 "RPOLCAT.spad" 1747087 1747104 1767597 1767602) (-1072 "ROUTINE.spad" 1742970 1742979 1745734 1745761) (-1071 "ROMAN.spad" 1742298 1742307 1742836 1742965) (-1070 "ROIRC.spad" 1741378 1741410 1742288 1742293) (-1069 "RNS.spad" 1740281 1740290 1741280 1741373) (-1068 "RNS.spad" 1739270 1739281 1740271 1740276) (-1067 "RNG.spad" 1739005 1739014 1739260 1739265) (-1066 "RNGBIND.spad" 1738165 1738179 1738960 1738965) (-1065 "RMODULE.spad" 1737930 1737941 1738155 1738160) (-1064 "RMCAT2.spad" 1737350 1737407 1737920 1737925) (-1063 "RMATRIX.spad" 1736174 1736193 1736517 1736556) (-1062 "RMATCAT.spad" 1731753 1731784 1736130 1736169) (-1061 "RMATCAT.spad" 1727222 1727255 1731601 1731606) (-1060 "RLINSET.spad" 1726616 1726627 1727212 1727217) (-1059 "RINTERP.spad" 1726504 1726524 1726606 1726611) (-1058 "RING.spad" 1725974 1725983 1726484 1726499) (-1057 "RING.spad" 1725452 1725463 1725964 1725969) (-1056 "RIDIST.spad" 1724844 1724853 1725442 1725447) (-1055 "RGCHAIN.spad" 1723427 1723443 1724329 1724356) (-1054 "RGBCSPC.spad" 1723208 1723220 1723417 1723422) (-1053 "RGBCMDL.spad" 1722738 1722750 1723198 1723203) (-1052 "RF.spad" 1720380 1720391 1722728 1722733) (-1051 "RFFACTOR.spad" 1719842 1719853 1720370 1720375) (-1050 "RFFACT.spad" 1719577 1719589 1719832 1719837) (-1049 "RFDIST.spad" 1718573 1718582 1719567 1719572) (-1048 "RETSOL.spad" 1717992 1718005 1718563 1718568) (-1047 "RETRACT.spad" 1717420 1717431 1717982 1717987) (-1046 "RETRACT.spad" 1716846 1716859 1717410 1717415) (-1045 "RETAST.spad" 1716658 1716667 1716836 1716841) (-1044 "RESULT.spad" 1714718 1714727 1715305 1715332) (-1043 "RESRING.spad" 1714065 1714112 1714656 1714713) (-1042 "RESLATC.spad" 1713389 1713400 1714055 1714060) (-1041 "REPSQ.spad" 1713120 1713131 1713379 1713384) (-1040 "REP.spad" 1710674 1710683 1713110 1713115) (-1039 "REPDB.spad" 1710381 1710392 1710664 1710669) (-1038 "REP2.spad" 1700039 1700050 1710223 1710228) (-1037 "REP1.spad" 1694235 1694246 1699989 1699994) (-1036 "REGSET.spad" 1692032 1692049 1693881 1693908) (-1035 "REF.spad" 1691367 1691378 1691987 1691992) (-1034 "REDORDER.spad" 1690573 1690590 1691357 1691362) (-1033 "RECLOS.spad" 1689356 1689376 1690060 1690153) (-1032 "REALSOLV.spad" 1688496 1688505 1689346 1689351) (-1031 "REAL.spad" 1688368 1688377 1688486 1688491) (-1030 "REAL0Q.spad" 1685666 1685681 1688358 1688363) (-1029 "REAL0.spad" 1682510 1682525 1685656 1685661) (-1028 "RDUCEAST.spad" 1682231 1682240 1682500 1682505) (-1027 "RDIV.spad" 1681886 1681911 1682221 1682226) (-1026 "RDIST.spad" 1681453 1681464 1681876 1681881) (-1025 "RDETRS.spad" 1680317 1680335 1681443 1681448) (-1024 "RDETR.spad" 1678456 1678474 1680307 1680312) (-1023 "RDEEFS.spad" 1677555 1677572 1678446 1678451) (-1022 "RDEEF.spad" 1676565 1676582 1677545 1677550) (-1021 "RCFIELD.spad" 1673751 1673760 1676467 1676560) (-1020 "RCFIELD.spad" 1671023 1671034 1673741 1673746) (-1019 "RCAGG.spad" 1668951 1668962 1671013 1671018) (-1018 "RCAGG.spad" 1666806 1666819 1668870 1668875) (-1017 "RATRET.spad" 1666166 1666177 1666796 1666801) (-1016 "RATFACT.spad" 1665858 1665870 1666156 1666161) (-1015 "RANDSRC.spad" 1665177 1665186 1665848 1665853) (-1014 "RADUTIL.spad" 1664933 1664942 1665167 1665172) (-1013 "RADIX.spad" 1661854 1661868 1663400 1663493) (-1012 "RADFF.spad" 1660267 1660304 1660386 1660542) (-1011 "RADCAT.spad" 1659862 1659871 1660257 1660262) (-1010 "RADCAT.spad" 1659455 1659466 1659852 1659857) (-1009 "QUEUE.spad" 1658803 1658814 1659062 1659089) (-1008 "QUAT.spad" 1657384 1657395 1657727 1657792) (-1007 "QUATCT2.spad" 1657004 1657023 1657374 1657379) (-1006 "QUATCAT.spad" 1655174 1655185 1656934 1656999) (-1005 "QUATCAT.spad" 1653095 1653108 1654857 1654862) (-1004 "QUAGG.spad" 1651922 1651933 1653063 1653090) (-1003 "QQUTAST.spad" 1651690 1651699 1651912 1651917) (-1002 "QFORM.spad" 1651154 1651169 1651680 1651685) (-1001 "QFCAT.spad" 1649856 1649867 1651056 1651149) (-1000 "QFCAT.spad" 1648149 1648162 1649351 1649356) (-999 "QFCAT2.spad" 1647842 1647858 1648139 1648144) (-998 "QEQUAT.spad" 1647401 1647409 1647832 1647837) (-997 "QCMPACK.spad" 1642148 1642167 1647391 1647396) (-996 "QALGSET.spad" 1638227 1638259 1642062 1642067) (-995 "QALGSET2.spad" 1636223 1636241 1638217 1638222) (-994 "PWFFINTB.spad" 1633639 1633660 1636213 1636218) (-993 "PUSHVAR.spad" 1632978 1632997 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(-349 "FFCGP.spad" 542964 542984 543791 543884) (-348 "FFCG.spad" 541756 541777 542645 542738) (-347 "FFCAT.spad" 534929 534951 541595 541751) (-346 "FFCAT.spad" 528181 528205 534849 534854) (-345 "FFCAT2.spad" 527928 527968 528171 528176) (-344 "FEXPR.spad" 519645 519691 527684 527723) (-343 "FEVALAB.spad" 519353 519363 519635 519640) (-342 "FEVALAB.spad" 518846 518858 519130 519135) (-341 "FDIV.spad" 518288 518312 518836 518841) (-340 "FDIVCAT.spad" 516352 516376 518278 518283) (-339 "FDIVCAT.spad" 514414 514440 516342 516347) (-338 "FDIV2.spad" 514070 514110 514404 514409) (-337 "FCTRDATA.spad" 513078 513086 514060 514065) (-336 "FCPAK1.spad" 511645 511653 513068 513073) (-335 "FCOMP.spad" 511024 511034 511635 511640) (-334 "FC.spad" 501031 501039 511014 511019) (-333 "FAXF.spad" 494002 494016 500933 501026) (-332 "FAXF.spad" 487025 487041 493958 493963) (-331 "FARRAY.spad" 485175 485185 486208 486235) (-330 "FAMR.spad" 483311 483323 485073 485170) (-329 "FAMR.spad" 481431 481445 483195 483200) (-328 "FAMONOID.spad" 481099 481109 481385 481390) (-327 "FAMONC.spad" 479395 479407 481089 481094) (-326 "FAGROUP.spad" 479019 479029 479291 479318) (-325 "FACUTIL.spad" 477223 477240 479009 479014) (-324 "FACTFUNC.spad" 476417 476427 477213 477218) (-323 "EXPUPXS.spad" 473250 473273 474549 474698) (-322 "EXPRTUBE.spad" 470538 470546 473240 473245) (-321 "EXPRODE.spad" 467698 467714 470528 470533) (-320 "EXPR.spad" 462973 462983 463687 464094) (-319 "EXPR2UPS.spad" 459095 459108 462963 462968) (-318 "EXPR2.spad" 458800 458812 459085 459090) (-317 "EXPEXPAN.spad" 455740 455765 456372 456465) (-316 "EXIT.spad" 455411 455419 455730 455735) (-315 "EXITAST.spad" 455147 455155 455401 455406) (-314 "EVALCYC.spad" 454607 454621 455137 455142) (-313 "EVALAB.spad" 454179 454189 454597 454602) (-312 "EVALAB.spad" 453749 453761 454169 454174) (-311 "EUCDOM.spad" 451323 451331 453675 453744) (-310 "EUCDOM.spad" 448959 448969 451313 451318) (-309 "ESTOOLS.spad" 440805 440813 448949 448954) (-308 "ESTOOLS2.spad" 440408 440422 440795 440800) (-307 "ESTOOLS1.spad" 440093 440104 440398 440403) (-306 "ES.spad" 432908 432916 440083 440088) (-305 "ES.spad" 425629 425639 432806 432811) (-304 "ESCONT.spad" 422422 422430 425619 425624) (-303 "ESCONT1.spad" 422171 422183 422412 422417) (-302 "ES2.spad" 421676 421692 422161 422166) (-301 "ES1.spad" 421246 421262 421666 421671) (-300 "ERROR.spad" 418573 418581 421236 421241) (-299 "EQTBL.spad" 417045 417067 417254 417281) (-298 "EQ.spad" 411850 411860 414637 414749) (-297 "EQ2.spad" 411568 411580 411840 411845) (-296 "EP.spad" 407894 407904 411558 411563) (-295 "ENV.spad" 406572 406580 407884 407889) (-294 "ENTIRER.spad" 406240 406248 406516 406567) (-293 "EMR.spad" 405447 405488 406166 406235) (-292 "ELTAGG.spad" 403701 403720 405437 405442) (-291 "ELTAGG.spad" 401919 401940 403657 403662) (-290 "ELTAB.spad" 401368 401386 401909 401914) (-289 "ELFUTS.spad" 400755 400774 401358 401363) (-288 "ELEMFUN.spad" 400444 400452 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(-267 "DSMP.spad" 362497 362511 362802 362929) (-266 "DROPT.spad" 356456 356464 362487 362492) (-265 "DROPT1.spad" 356121 356131 356446 356451) (-264 "DROPT0.spad" 350978 350986 356111 356116) (-263 "DRAWPT.spad" 349151 349159 350968 350973) (-262 "DRAW.spad" 342027 342040 349141 349146) (-261 "DRAWHACK.spad" 341335 341345 342017 342022) (-260 "DRAWCX.spad" 338805 338813 341325 341330) (-259 "DRAWCURV.spad" 338352 338367 338795 338800) (-258 "DRAWCFUN.spad" 327884 327892 338342 338347) (-257 "DQAGG.spad" 326062 326072 327852 327879) (-256 "DPOLCAT.spad" 321411 321427 325930 326057) (-255 "DPOLCAT.spad" 316846 316864 321367 321372) (-254 "DPMO.spad" 309072 309088 309210 309511) (-253 "DPMM.spad" 301311 301329 301436 301737) (-252 "DOMTMPLT.spad" 300971 300979 301301 301306) (-251 "DOMCTOR.spad" 300726 300734 300961 300966) (-250 "DOMAIN.spad" 299813 299821 300716 300721) (-249 "DMP.spad" 297073 297088 297643 297770) (-248 "DLP.spad" 296425 296435 297063 297068) (-247 "DLIST.spad" 295004 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(-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 "ASP8.spad" 96485 96498 97432 97437) (-87 "ASP80.spad" 95807 95820 96475 96480) (-86 "ASP7.spad" 94967 94980 95797 95802) (-85 "ASP78.spad" 94418 94431 94957 94962) (-84 "ASP77.spad" 93787 93800 94408 94413) (-83 "ASP74.spad" 92879 92892 93777 93782) (-82 "ASP73.spad" 92150 92163 92869 92874) (-81 "ASP6.spad" 91017 91030 92140 92145) (-80 "ASP55.spad" 89526 89539 91007 91012) (-79 "ASP50.spad" 87343 87356 89516 89521) (-78 "ASP4.spad" 86638 86651 87333 87338) (-77 "ASP49.spad" 85637 85650 86628 86633) (-76 "ASP42.spad" 84044 84083 85627 85632) (-75 "ASP41.spad" 82623 82662 84034 84039) (-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 "ASP1.spad" 66072 66085 66681 66686) (-63 "ASP19.spad" 60758 60771 66062 66067) (-62 "ASP12.spad" 60172 60185 60748 60753) (-61 "ASP10.spad" 59443 59456 60162 60167) (-60 "ARRAY2.spad" 58803 58812 59050 59077) (-59 "ARRAY1.spad" 57640 57649 57986 58013) (-58 "ARRAY12.spad" 56353 56364 57630 57635) (-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 "ANY.spad" 45031 45038 46162 46167) (-51 "ANY1.spad" 44102 44111 45021 45026) (-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
author	dos-reis <gdr@axiomatics.org>	2010-04-17 23:03:55 +0000
committer	dos-reis <gdr@axiomatics.org>	2010-04-17 23:03:55 +0000
commit	5b24e39b978891b7a8dc22b9ab6ffa5eec341e6a (patch)
tree	450e549494784c805ce24f0b319598a7e25cb0e8 /src/share/algebra/browse.daase
parent	423125f1638b432ccfb944f954fd4304ddfb0183 (diff)
download	open-axiom-5b24e39b978891b7a8dc22b9ab6ffa5eec341e6a.tar.gz