\documentclass{article} \usepackage{axiom} \begin{document} \title{src/algebra list.spad} \author{Michael Monagon, Manuel Bronstein} \maketitle \begin{abstract} \end{abstract} \tableofcontents \eject \section{domain ILIST IndexedList} <>= import Type import ListAggregate )abbrev domain ILIST IndexedList ++ Author: Michael Monagan ++ Date Created: Sep 1987 ++ Change History: ++ Basic Operations: ++ \#, concat, concat!, construct, copy, elt, elt, empty, ++ empty?, eq?, first, member?, merge!, mergeSort, minIndex, ++ parts, removeDuplicates!, rest, rest, reverse, reverse!, ++ setelt, setfirst!, setrest!, sort!, split! ++ Related Constructors: List ++ Also See: ++ AMS Classification: ++ Keywords: list, aggregate, index ++ Description: ++ \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. IndexedList(S:Type, mn:Integer): Exports == Implementation where cycleMax ==> 1000 -- value used in checking for cycles -- The following seems to be a bit out of date, but is kept in case -- a knowledgeable person wants to update it: -- The following LISP dependencies are divided into two groups -- Those that are required -- CONS, EQ, NIL, NULL, QCAR, QCDR, RPLACA, RPLACD -- Those that are included for efficiency only -- NEQ, LIST, CAR, CDR, NCONC2, NREVERSE, LENGTH -- Also REVERSE, since it's called in Polynomial Ring Qfirst ==> QCAR$Lisp Qrest ==> QCDR$Lisp Qnull ==> NULL$Lisp Qeq ==> EQ$Lisp Qneq ==> NEQ$Lisp Qcons ==> CONS$Lisp Qpush ==> PUSH$Lisp Exports ==> ListAggregate S Implementation ==> add #x == LENGTH(x)$Lisp concat(s:S,x:%) == CONS(s,x)$Lisp eq?(x,y) == EQ(x,y)$Lisp first x == SPADfirst(x)$Lisp elt(x,"first") == SPADfirst(x)$Lisp empty() == NIL$Lisp empty? x == NULL(x)$Lisp rest x == CDR(x)$Lisp elt(x,"rest") == CDR(x)$Lisp setfirst!(x,s) == empty? x => error "Cannot update an empty list" Qfirst RPLACA(x,s)$Lisp setelt(x,"first",s) == empty? x => error "Cannot update an empty list" Qfirst RPLACA(x,s)$Lisp setrest!(x,y) == empty? x => error "Cannot update an empty list" Qrest RPLACD(x,y)$Lisp setelt(x,"rest",y) == empty? x => error "Cannot update an empty list" Qrest RPLACD(x,y)$Lisp construct l == l pretend % parts s == s pretend List S reverse! x == NREVERSE(x)$Lisp reverse x == REVERSE(x)$Lisp minIndex x == mn rest(x, n) == for i in 1..n repeat if Qnull x then error "index out of range" x := Qrest x x copy x == y := empty() for i in 0.. while not Qnull x repeat if Qeq(i,cycleMax) and cyclic? x then error "cyclic list" y := Qcons(Qfirst x,y) x := Qrest x (NREVERSE(y)$Lisp)@% if S has CoercibleTo(OutputForm) then coerce(x):OutputForm == -- displays cycle with overbar over the cycle y := empty()$List(OutputForm) s := cycleEntry x while Qneq(x, s) repeat y := concat((first x)::OutputForm, y) x := rest x y := reverse! y empty? s => bracket y -- cyclic case: z is cylic part z := list((first x)::OutputForm) while Qneq(s, rest x) repeat x := rest x z := concat((first x)::OutputForm, z) bracket concat!(y, overbar commaSeparate reverse! z) if S has SetCategory then x = y == Qeq(x,y) => true while not Qnull x and not Qnull y repeat Qfirst x ~=$S Qfirst y => return false x := Qrest x y := Qrest y Qnull x and Qnull y latex(x : %): String == s : String := "\left[" while not Qnull x repeat s := concat(s, latex(Qfirst x)$S)$String x := Qrest x if not Qnull x then s := concat(s, ", ")$String concat(s, " \right]")$String member?(s,x) == while not Qnull x repeat if s = Qfirst x then return true else x := Qrest x false -- Lots of code from parts of AGGCAT, repeated here to -- get faster compilation concat!(x:%,y:%) == Qnull x => Qnull y => x Qpush(first y,x) QRPLACD(x,rest y)$Lisp x z:=x while not Qnull Qrest z repeat z:=Qrest z QRPLACD(z,y)$Lisp x -- Then a quicky: if S has SetCategory then removeDuplicates! l == p := l while not Qnull p repeat -- p := setrest!(p, remove!(#1 = Qfirst p, Qrest p)) -- far too expensive - builds closures etc. pp:=p f:S:=Qfirst p p:=Qrest p while not Qnull (pr:=Qrest pp) repeat if (Qfirst pr)@S = f then QRPLACD(pp,Qrest pr)$Lisp else pp:=pr l -- then sorting mergeSort: ((S, S) -> Boolean, %, Integer) -> % sort!(f, l) == mergeSort(f, l, #l) merge!(f, p, q) == Qnull p => q Qnull q => p Qeq(p, q) => error "cannot merge a list into itself" if f(Qfirst p, Qfirst q) then (r := t := p; p := Qrest p) else (r := t := q; q := Qrest q) while not Qnull p and not Qnull q repeat if f(Qfirst p, Qfirst q) then (QRPLACD(t, p)$Lisp; t := p; p := Qrest p) else (QRPLACD(t, q)$Lisp; t := q; q := Qrest q) QRPLACD(t, if Qnull p then q else p)$Lisp r split!(p, n) == n < 1 => error "index out of range" p := rest(p, (n - 1)::NonNegativeInteger) q := Qrest p QRPLACD(p, NIL$Lisp)$Lisp q mergeSort(f, p, n) == if n = 2 and f(first rest p, first p) then p := reverse! p n < 3 => p l := (n quo 2)::NonNegativeInteger q := split!(p, l) p := mergeSort(f, p, l) q := mergeSort(f, q, n - l) merge!(f, p, q) @ \section{domain LIST List} <>= import Type import ListAggregate )abbrev domain LIST List ++ Author: Michael Monagan ++ Date Created: Sep 1987 ++ Change History: ++ Basic Operations: ++ \#, append, concat, concat!, cons, construct, copy, elt, elt, ++ empty, empty?, eq?, first, member?, merge!, mergeSort, minIndex, ++ nil, null, parts, removeDuplicates!, rest, rest, reverse, ++ reverse!, setDifference, setIntersection, setUnion, setelt, ++ setfirst!, setrest!, sort!, split! ++ Related Constructors: ListFunctions2, ListFunctions3, ListToMap ++ Also See: IndexList, ListAggregate ++ AMS Classification: ++ Keywords: list, index, aggregate, lisp ++ Description: ++ \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}. List(S:Type): Exports == Implementation where LISTMININDEX ==> 1 -- this is the minimum list index Exports ==> ListAggregate S with nil : % ++ \spad{nil} is the empty list. null : % -> Boolean ++ null(u) tests if list \spad{u} is the ++ empty list. cons : (S, %) -> % ++ 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. append : (%, %) -> % ++ 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. if S has SetCategory then setUnion : (%, %) -> % ++ setUnion(u1,u2) appends the two lists u1 and u2, then ++ removes all duplicates. The order of elements in the ++ resulting list is unspecified. setIntersection : (%, %) -> % ++ 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. setDifference : (%, %) -> % ++ 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. if S has OpenMath then OpenMath Implementation ==> IndexedList(S, LISTMININDEX) add nil == NIL$Lisp null l == NULL(l)$Lisp cons(s, l) == CONS(s, l)$Lisp append(l:%, t:%) == APPEND(l, t)$Lisp if S has OpenMath then writeOMList(dev: OpenMathDevice, x: %): Void == OMputApp(dev) OMputSymbol(dev, "list1", "list") -- The following didn't compile because the compiler isn't -- convinced that `xval' is a S. Duhhh! MCD. --for xval in x repeat -- OMwrite(dev, xval, false) while not null x repeat OMwrite(dev,first x,false) x := rest x OMputEndApp(dev) OMwrite(x: %): String == s: String := "" sp := OM_-STRINGTOSTRINGPTR(s)$Lisp dev: OpenMathDevice := OMopenString(sp pretend String, OMencodingXML) OMputObject(dev) writeOMList(dev, x) OMputEndObject(dev) OMclose(dev) s := OM_-STRINGPTRTOSTRING(sp)$Lisp pretend String s OMwrite(x: %, wholeObj: Boolean): String == s: String := "" sp := OM_-STRINGTOSTRINGPTR(s)$Lisp dev: OpenMathDevice := OMopenString(sp pretend String, OMencodingXML) if wholeObj then OMputObject(dev) writeOMList(dev, x) if wholeObj then OMputEndObject(dev) OMclose(dev) s := OM_-STRINGPTRTOSTRING(sp)$Lisp pretend String s OMwrite(dev: OpenMathDevice, x: %): Void == OMputObject(dev) writeOMList(dev, x) OMputEndObject(dev) OMwrite(dev: OpenMathDevice, x: %, wholeObj: Boolean): Void == if wholeObj then OMputObject(dev) writeOMList(dev, x) if wholeObj then OMputEndObject(dev) if S has SetCategory then setUnion(l1:%,l2:%) == removeDuplicates concat(l1,l2) setIntersection(l1:%,l2:%) == u :% := empty() l1 := removeDuplicates l1 while not empty? l1 repeat if member?(first l1,l2) then u := cons(first l1,u) l1 := rest l1 u setDifference(l1:%,l2:%) == l1 := removeDuplicates l1 lu:% := empty() while not empty? l1 repeat l11:=l1.1 if not member?(l11,l2) then lu := concat(l11,lu) l1 := rest l1 lu if S has ConvertibleTo InputForm then convert(x:%):InputForm == convert concat(convert('construct)@InputForm, [convert a for a in (x pretend List S)]$List(InputForm)) @ \section{package LIST2 ListFunctions2} <>= import Type import FiniteLinearAggregateFunctions2 )abbrev package LIST2 ListFunctions2 ++ Author: ++ Date Created: ++ Change History: ++ Basic Operations: map, reduce, scan ++ Related Constructors: List ++ Also See: ListFunctions3 ++ AMS Classification: ++ Keywords: list, aggregate, map, reduce ++ Description: ++ \spadtype{ListFunctions2} implements utility functions that ++ operate on two kinds of lists, each with a possibly different ++ type of element. ListFunctions2(A:Type, B:Type): public == private where LA ==> List A LB ==> List B O2 ==> FiniteLinearAggregateFunctions2(A, LA, B, LB) public ==> with scan: ((A, B) -> B, LA, B) -> LB ++ scan(fn,u,ident) successively uses the binary function ++ \spad{fn} to reduce more and more of list \spad{u}. ++ \spad{ident} is returned if the \spad{u} is empty. ++ The result is a list of the reductions at each step. See ++ \spadfun{reduce} for more information. Examples: ++ \spad{scan(fn,[1,2],0) = [fn(2,fn(1,0)),fn(1,0)]} and ++ \spad{scan(*,[2,3],1) = [2 * 1, 3 * (2 * 1)]}. reduce: ((A, B) -> B, LA, B) -> B ++ reduce(fn,u,ident) successively uses the binary function ++ \spad{fn} on the elements of list \spad{u} and the result ++ of previous applications. \spad{ident} is returned if the ++ \spad{u} is empty. Note the order of application in ++ the following examples: ++ \spad{reduce(fn,[1,2,3],0) = fn(3,fn(2,fn(1,0)))} and ++ \spad{reduce(*,[2,3],1) = 3 * (2 * 1)}. map: (A -> B, LA) -> LB ++ map(fn,u) applies \spad{fn} to each element of ++ list \spad{u} and returns a new list with the results. ++ For example \spad{map(square,[1,2,3]) = [1,4,9]}. private ==> add map(f, l) == map(f, l)$O2 scan(f, l, b) == scan(f, l, b)$O2 reduce(f, l, b) == reduce(f, l, b)$O2 @ \section{package LIST3 ListFunctions3} <>= import Type import Type )abbrev package LIST3 ListFunctions3 ++ Author: ++ Date Created: ++ Change History: ++ Basic Operations: map ++ Related Constructors: List ++ Also See: ListFunctions2 ++ AMS Classification: ++ Keywords: list, aggregate, map ++ Description: ++ \spadtype{ListFunctions3} implements utility functions that ++ operate on three kinds of lists, each with a possibly different ++ type of element. ListFunctions3(A:Type, B:Type, C:Type): public == private where LA ==> List A LB ==> List B LC ==> List C public ==> with map: ( (A,B)->C, LA, LB) -> LC ++ map(fn,list1, u2) applies the binary function \spad{fn} ++ to corresponding elements of lists \spad{u1} and \spad{u2} ++ and returns a list of the results (in the same order). Thus ++ \spad{map(/,[1,2,3],[4,5,6]) = [1/4,2/4,1/2]}. The computation ++ terminates when the end of either list is reached. That is, ++ the length of the result list is equal to the minimum of the ++ lengths of \spad{u1} and \spad{u2}. private ==> add map(fn : (A,B) -> C, la : LA, lb : LB): LC == empty?(la) or empty?(lb) => empty()$LC concat(fn(first la, first lb), map(fn, rest la, rest lb)) @ \section{package LIST2MAP ListToMap} <>= import Type import SetCategory import List )abbrev package LIST2MAP ListToMap ++ Author: Manuel Bronstein ++ Date Created: 22 Mar 1988 ++ Change History: ++ 11 Oct 1989 MB ? ++ Basic Operations: match ++ Related Constructors: List ++ Also See: ++ AMS Classification: ++ Keywords: mapping, list ++ Description: ++ \spadtype{ListToMap} allows mappings to be described by a pair of ++ lists of equal lengths. The image of an element \spad{x}, ++ which appears in position \spad{n} in the first list, is then ++ the \spad{n}th element of the second list. A default value or ++ default function can be specified to be used when \spad{x} ++ does not appear in the first list. In the absence of defaults, ++ an error will occur in that case. ListToMap(A:SetCategory, B:Type): Exports == Implementation where LA ==> List A LB ==> List B AB ==> (A -> B) Exports ==> with match: (LA, LB ) -> AB ++ match(la, lb) creates a map with no default source or target values ++ defined by lists la and lb of equal length. ++ The target of a source value \spad{x} in la is the ++ value y with the same index lb. ++ Error: if la and lb are not of equal length. ++ Note: when this map is applied, an error occurs when ++ applied to a value missing from la. match: (LA, LB, A) -> B ++ match(la, lb, a) creates a map ++ defined by lists la and lb of equal length, where \spad{a} is used ++ as the default source value if the given one is not in \spad{la}. ++ The target of a source value \spad{x} in la is the ++ value y with the same index lb. ++ Error: if la and lb are not of equal length. match: (LA, LB, B) -> AB ++ match(la, lb, b) creates a map ++ defined by lists la and lb of equal length, where \spad{b} is used ++ as the default target value if the given function argument is ++ not in \spad{la}. ++ The target of a source value \spad{x} in la is the ++ value y with the same index lb. ++ Error: if la and lb are not of equal length. match: (LA, LB, A, B) -> B ++ match(la, lb, a, b) creates a map ++ defined by lists la and lb of equal length. ++ and applies this map to a. ++ The target of a source value \spad{x} in la is the ++ value y with the same index lb. ++ Argument b is the default target value if a is not in la. ++ Error: if la and lb are not of equal length. match: (LA, LB, AB) -> AB ++ match(la, lb, f) creates a map ++ defined by lists la and lb of equal length. ++ The target of a source value \spad{x} in la is the ++ value y with the same index lb. ++ Argument \spad{f} is used as the ++ function to call when the given function argument is not in ++ \spad{la}. ++ The value returned is f applied to that argument. match: (LA, LB, A, AB) -> B ++ match(la, lb, a, f) creates a map ++ defined by lists la and lb of equal length. ++ and applies this map to a. ++ The target of a source value \spad{x} in la is the ++ value y with the same index lb. ++ Argument \spad{f} is a default function to call if a is not in la. ++ The value returned is then obtained by applying f to argument a. Implementation ==> add match(la, lb) == match(la, lb, #1) match(la:LA, lb:LB, a:A) == lb.position(a, la) match(la:LA, lb:LB, b:B) == match(la, lb, #1, b) match(la:LA, lb:LB, f:AB) == match(la, lb, #1, f) match(la:LA, lb:LB, a:A, b:B) == (p := position(a, la)) < minIndex(la) => b lb.p match(la:LA, lb:LB, a:A, f:AB) == (p := position(a, la)) < minIndex(la) => f a lb.p @ \section{domain ALIST AssociationList} <>= import SetCategory import List import Reference )abbrev domain ALIST AssociationList ++ Author: ++ Date Created: ++ Change History: ++ Basic Operations: empty, empty?, keys, \#, concat, first, rest, ++ setrest!, search, setelt, remove! ++ Related Constructors: ++ Also See: List ++ AMS Classification: ++ Keywords: list, association list ++ Description: ++ \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. AssociationList(Key:SetCategory, Entry:SetCategory): AssociationListAggregate(Key, Entry) == add Pair ==> Record(key:Key, entry:Entry) Rep := Reference List Pair dictionary() == ref empty() empty() == dictionary() empty? t == empty? deref t entries(t:%):List(Pair) == deref t parts(t:%):List(Pair) == deref t keys t == [k.key for k in deref t] # t == # deref t first(t:%):Pair == first deref t rest t == ref rest deref t concat(p:Pair, t:%) == ref concat(p, deref t) setrest!(a:%, b:%) == ref setrest!(deref a, deref b) setfirst!(a:%, p:Pair) == setfirst!(deref a,p) minIndex(a:%):Integer == minIndex(deref a) maxIndex(a:%):Integer == maxIndex(deref a) search(k, t) == for r in deref t repeat k = r.key => return(r.entry) "failed" latex(a : %) : String == l : List Pair := entries a s : String := "\left[" while not empty?(l) repeat r : Pair := first l l := rest l s := concat(s, concat(latex r.key, concat(" = ", latex r.entry)$String)$String)$String if not empty?(l) then s := concat(s, ", ")$String concat(s, " \right]")$String -- assoc(k, l) == -- (r := find(#1.key=k, l)) case "failed" => "failed" -- r assoc(k, t) == for r in deref t repeat k = r.key => return r "failed" setelt(t:%, k:Key, e:Entry) == (r := assoc(k, t)) case Pair => (r::Pair).entry := e setref(t, concat([k, e], deref t)) e remove!(k:Key, t:%) == empty?(l := deref t) => "failed" k = first(l).key => setref(t, rest l) first(l).entry prev := l curr := rest l while not empty? curr and first(curr).key ~= k repeat prev := curr curr := rest curr empty? curr => "failed" setrest!(prev, rest curr) first(curr).entry @ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. --Copyright (C) 2007-2009, Gabriel Dos Reis. --All rights reserved. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. @ <<*>>= <> <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}