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## Re: [Axiom-developer] RE: [fspace.spad] dvi file display

 From: root Subject: Re: [Axiom-developer] RE: [fspace.spad] dvi file display Date: Mon, 7 Nov 2005 12:22:31 -0500

\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra fspace.spad} \author{Manuel Bronstein} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{category ES ExpressionSpace} <<category ES ExpressionSpace>>= )abbrev category ES ExpressionSpace ++ Category for domains on which operators can be applied ++ Author: Manuel Bronstein ++ Date Created: 22 March 1988 ++ Date Last Updated: 27 May 1994 ++ Description: ++ An expression space is a set which is closed under certain operators; ++ Keywords: operator, kernel, expression, space. ExpressionSpace(): Category == Defn where N ==> NonNegativeInteger K ==> Kernel % OP ==> BasicOperator SY ==> Symbol PAREN ==> "%paren"::SY BOX ==> "%box"::SY DUMMYVAR ==> "%dummyVar" Defn ==> Join(OrderedSet, RetractableTo K, InnerEvalable(K, %), Evalable %) with elt : (OP, %) -> % ++ elt(op,x) or op(x) applies the unary operator op to x. elt : (OP, %, %) -> % ++ elt(op,x,y) or op(x, y) applies the binary operator op to x and y. elt : (OP, %, %, %) -> % ++ elt(op,x,y,z) or op(x, y, z) applies the ternary operator op to x, y and z. elt : (OP, %, %, %, %) -> % ++ elt(op,x,y,z,t) or op(x, y, z, t) applies the 4-ary operator op to x, y, z and t. elt : (OP, List %) -> % ++ elt(op,[x1,...,xn]) or op([x1,...,xn]) applies the n-ary operator op to x1,...,xn. subst : (%, Equation %) -> % ++ subst(f, k = g) replaces the kernel k by g formally in f. subst : (%, List Equation %) -> % ++ subst(f, [k1 = g1,...,kn = gn]) replaces the kernels k1,...,kn ++ by g1,...,gn formally in f. subst : (%, List K, List %) -> % ++ subst(f, [k1...,kn], [g1,...,gn]) replaces the kernels k1,...,kn ++ by g1,...,gn formally in f. box : % -> % ++ box(f) returns f with a 'box' around it that prevents f from ++ being evaluated when operators are applied to it. For example, ++ \spad{log(1)} returns 0, but \spad{log(box 1)} ++ returns the formal kernel log(1). box : List % -> % ++ box([f1,...,fn]) returns \spad{(f1,...,fn)} with a 'box' ++ around them that ++ prevents the fi from being evaluated when operators are applied to ++ them, and makes them applicable to a unary operator. For example, ++ \spad{atan(box [x, 2])} returns the formal kernel \spad{atan(x, 2)}. paren : % -> % ++ paren(f) returns (f). This prevents f from ++ being evaluated when operators are applied to it. For example, ++ \spad{log(1)} returns 0, but \spad{log(paren 1)} returns the ++ formal kernel log((1)). paren : List % -> % ++ paren([f1,...,fn]) returns \spad{(f1,...,fn)}. This ++ prevents the fi from being evaluated when operators are applied to ++ them, and makes them applicable to a unary operator. For example, ++ \spad{atan(paren [x, 2])} returns the formal ++ kernel \spad{atan((x, 2))}. distribute : % -> % ++ distribute(f) expands all the kernels in f that are ++ formally enclosed by a \spadfunFrom{box}{ExpressionSpace} ++ or \spadfunFrom{paren}{ExpressionSpace} expression. distribute : (%, %) -> % ++ distribute(f, g) expands all the kernels in f that contain g in their ++ arguments and that are formally ++ enclosed by a \spadfunFrom{box}{ExpressionSpace} ++ or a \spadfunFrom{paren}{ExpressionSpace} expression. height : % -> N ++ height(f) returns the highest nesting level appearing in f. ++ Constants have height 0. Symbols have height 1. For any ++ operator op and expressions f1,...,fn, \spad{op(f1,...,fn)} has ++ height equal to \spad{1 + max(height(f1),...,height(fn))}. mainKernel : % -> Union(K, "failed") ++ mainKernel(f) returns a kernel of f with maximum nesting level, or ++ if f has no kernels (i.e. f is a constant). kernels : % -> List K ++ kernels(f) returns the list of all the top-level kernels ++ appearing in f, but not the ones appearing in the arguments ++ of the top-level kernels. tower : % -> List K ++ tower(f) returns all the kernels appearing in f, no matter ++ what their levels are. operators : % -> List OP ++ operators(f) returns all the basic operators appearing in f, ++ no matter what their levels are. operator : OP -> OP ++ operator(op) returns a copy of op with the domain-dependent ++ properties appropriate for %. belong? : OP -> Boolean ++ belong?(op) tests if % accepts op as applicable to its ++ elements. is? : (%, OP) -> Boolean ++ is?(x, op) tests if x is a kernel and is its operator is op. is? : (%, SY) -> Boolean ++ is?(x, s) tests if x is a kernel and is the name of its ++ operator is s. kernel : (OP, %) -> % ++ kernel(op, x) constructs op(x) without evaluating it. kernel : (OP, List %) -> % ++ kernel(op, [f1,...,fn]) constructs \spad{op(f1,...,fn)} without ++ evaluating it. map : (% -> %, K) -> % ++ map(f, k) returns \spad{op(f(x1),...,f(xn))} where ++ \spad{k = op(x1,...,xn)}. freeOf? : (%, %) -> Boolean ++ freeOf?(x, y) tests if x does not contain any occurrence of y, ++ where y is a single kernel. freeOf? : (%, SY) -> Boolean ++ freeOf?(x, s) tests if x does not contain any operator ++ whose name is s. eval : (%, List SY, List(% -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a)} in x by \spad{fi(a)} for any \spad{a}. eval : (%, List SY, List(List % -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a1,...,an)} in x by ++ \spad{fi(a1,...,an)} for any \spad{a1},...,\spad{an}. eval : (%, SY, List % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a1,..,am)} in x ++ by \spad{f(a1,..,am)} for any \spad{a1},...,\spad{am}. eval : (%, SY, % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a)} in x by \spad{f(a)} ++ for any \spad{a}. eval : (%, List OP, List(% -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a)} in x by \spad{fi(a)} for any \spad{a}. eval : (%, List OP, List(List % -> %)) -> % ++ eval(x, [s1,...,sm], [f1,...,fm]) replaces ++ every \spad{si(a1,...,an)} in x by ++ \spad{fi(a1,...,an)} for any \spad{a1},...,\spad{an}. eval : (%, OP, List % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a1,..,am)} in x ++ by \spad{f(a1,..,am)} for any \spad{a1},...,\spad{am}. eval : (%, OP, % -> %) -> % ++ eval(x, s, f) replaces every \spad{s(a)} in x by \spad{f(a)} ++ for any \spad{a}. if % has Ring then minPoly: K -> SparseUnivariatePolynomial % ++ minPoly(k) returns p such that \spad{p(k) = 0}. definingPolynomial: % -> % ++ definingPolynomial(x) returns an expression p such that ++ \spad{p(x) = 0}. if % has RetractableTo Integer then even?: % -> Boolean ++ even? x is true if x is an even integer. odd? : % -> Boolean ++ odd? x is true if x is an odd integer. add -- the 7 functions not provided are: -- kernels minPoly definingPolynomial -- coerce:K -> % eval:(%, List K, List %) -> % -- subst:(%, List K, List %) -> % -- eval:(%, List Symbol, List(List % -> %)) -> % allKernels: % -> Set K listk : % -> List K allk : List % -> Set K unwrap : (List K, %) -> % okkernel : (OP, List %) -> % mkKerLists: List Equation % -> Record(lstk: List K, lstv:List %) oppren := operator(PAREN)$CommonOperators()
opbox  := operator(BOX)$CommonOperators() box(x:%) == box [x] paren(x:%) == paren [x] belong? op == op = oppren or op = opbox listk f == parts allKernels f tower f == sort_! listk f allk l == reduce("union", [allKernels f for f in l], {}) operators f == [operator k for k in listk f] height f == reduce("max", [height k for k in kernels f], 0) freeOf?(x:%, s:SY) == not member?(s, [name k for k in listk x]) distribute x == unwrap([k for k in listk x | is?(k, oppren)], x) box(l:List %) == opbox l paren(l:List %) == oppren l freeOf?(x:%, k:%) == not member?(retract k, listk x) kernel(op:OP, arg:%) == kernel(op, [arg]) elt(op:OP, x:%) == op [x] elt(op:OP, x:%, y:%) == op [x, y] elt(op:OP, x:%, y:%, z:%) == op [x, y, z] elt(op:OP, x:%, y:%, z:%, t:%) == op [x, y, z, t] eval(x:%, s:SY, f:List % -> %) == eval(x, [s], [f]) eval(x:%, s:OP, f:List % -> %) == eval(x, [name s], [f]) eval(x:%, s:SY, f:% -> %) == eval(x, [s], [f first #1]) eval(x:%, s:OP, f:% -> %) == eval(x, [s], [f first #1]) subst(x:%, e:Equation %) == subst(x, [e]) eval(x:%, ls:List OP, lf:List(% -> %)) == eval(x, ls, [f first #1 for f in lf]$List(List % -> %))

eval(x:%, ls:List SY, lf:List(% -> %)) ==
eval(x, ls, [f first #1 for f in lf]$List(List % -> %)) eval(x:%, ls:List OP, lf:List(List % -> %)) == eval(x, [name s for s in ls]$List(SY), lf)

map(fn, k) ==
(l := [fn x for x in argument k]$List(%)) = argument k => k::% (operator k) l operator op == is?(op, PAREN) => oppren is?(op, BOX) => opbox error "Unknown operator" mainKernel x == empty?(l := kernels x) => "failed" n := height(k := first l) for kk in rest l repeat if height(kk) > n then n := height kk k := kk k -- takes all the kernels except for the dummy variables, which are second -- arguments of rootOf's, integrals, sums and products which appear only in -- their first arguments allKernels f == s := brace(l := kernels f) for k in l repeat t := (u := property(operator k, DUMMYVAR)) case None => arg := argument k s0 := remove_!(retract(second arg)@K, allKernels first arg) arg := rest rest arg n := (u::None) pretend N if n > 1 then arg := rest arg union(s0, allk arg) allk argument k s := union(s, t) s kernel(op:OP, args:List %) == not belong? op => error "Unknown operator" okkernel(op, args) okkernel(op, l) == kernel(op, l, 1 + reduce("max", [height f for f in l], 0))$K :: %

elt(op:OP, args:List %) ==
not belong? op => error "Unknown operator"
((u := arity op) case N) and (#args ^= u::N)
=> error "Wrong number of arguments"
(v := evaluate(op,args)$BasicOperatorFunctions1(%)) case % => v::% okkernel(op, args) retract f == (k := mainKernel f) case "failed" => error "not a kernel" k::K::% ^= f => error "not a kernel" k::K retractIfCan f == (k := mainKernel f) case "failed" => "failed" k::K::% ^= f => "failed" k is?(f:%, s:SY) == (k := retractIfCan f) case "failed" => false is?(k::K, s) is?(f:%, op:OP) == (k := retractIfCan f) case "failed" => false is?(k::K, op) unwrap(l, x) == for k in reverse_! l repeat x := eval(x, k, first argument k) x distribute(x, y) == ky := retract y unwrap([k for k in listk x | is?(k, "%paren"::SY) and member?(ky, listk(k::%))], x) -- in case of conflicting substitutions e.g. [x = a, x = b], -- the first one prevails. -- this is not part of the semantics of the function, but just -- a feature of this implementation. eval(f:%, leq:List Equation %) == rec := mkKerLists leq eval(f, rec.lstk, rec.lstv) subst(f:%, leq:List Equation %) == rec := mkKerLists leq subst(f, rec.lstk, rec.lstv) mkKerLists leq == lk := empty()$List(K)
lv := empty()$List(%) for eq in leq repeat (k := retractIfCan(lhs eq)@Union(K, "failed")) case "failed" => error "left hand side must be a single kernel" if not member?(k::K, lk) then lk := concat(k::K, lk) lv := concat(rhs eq, lv) [lk, lv] if % has RetractableTo Integer then intpred?: (%, Integer -> Boolean) -> Boolean even? x == intpred?(x, even?) odd? x == intpred?(x, odd?) intpred?(x, pred?) == (u := retractIfCan(x)@Union(Integer, "failed")) case Integer and pred?(u::Integer) @ \section{ES.lsp BOOTSTRAP} {\bf ES} depends on a chain of files. We need to break this cycle to build the algebra. So we keep a cached copy of the translated {\bf ES} category which we can write into the {\bf MID} directory. We compile the lisp code and copy the {\bf ES.o} file to the {\bf OUT} directory. This is eventually forcibly replaced by a recompiled version. Note that this code is not included in the generated catdef.spad file. <<ES.lsp BOOTSTRAP>>= (|/VERSIONCHECK| 2) (SETQ |ExpressionSpace;AL| (QUOTE NIL)) (DEFUN |ExpressionSpace| NIL (LET (#:G82344) (COND (|ExpressionSpace;AL|) (T (SETQ |ExpressionSpace;AL| (|ExpressionSpace;|)))))) (DEFUN |ExpressionSpace;| NIL (PROG (#1=#:G82342) (RETURN (PROG1 (LETT #1# (|sublisV| (PAIR (QUOTE (#2=#:G82340 #3=#:G82341)) (LIST (QUOTE (|Kernel| |$|)) (QUOTE (|Kernel| |$|)))) (|Join| (|OrderedSet|) (|RetractableTo| (QUOTE #2#)) (|InnerEvalable| (QUOTE #3#) (QUOTE |$|))
(|Evalable| (QUOTE |$|)) (|mkCategory| (QUOTE |domain|) (QUOTE ( ((|elt| (|$| (|BasicOperator|) |$|)) T) ((|elt| (|$| (|BasicOperator|) |$| |$|)) T)
((|elt| (|$| (|BasicOperator|) |$| |$| |$|)) T)
((|elt| (|$| (|BasicOperator|) |$| |$| |$| |$|)) T) ((|elt| (|$| (|BasicOperator|) (|List| |$|))) T) ((|subst| (|$| |$| (|Equation| |$|))) T)
((|subst| (|$| |$| (|List| (|Equation| |$|)))) T) ((|subst| (|$| |$| (|List| (|Kernel| |$|)) (|List| |$|))) T) ((|box| (|$| |$|)) T) ((|box| (|$| (|List| |$|))) T) ((|paren| (|$| |$|)) T) ((|paren| (|$| (|List| |$|))) T) ((|distribute| (|$| |$|)) T) ((|distribute| (|$| |$| |$|)) T)
((|height| ((|NonNegativeInteger|) |$|)) T) ((|mainKernel| ((|Union| (|Kernel| |$|) "failed") |$|)) T) ((|kernels| ((|List| (|Kernel| |$|)) |$|)) T) ((|tower| ((|List| (|Kernel| |$|)) |$|)) T) ((|operators| ((|List| (|BasicOperator|)) |$|)) T)
((|operator| ((|BasicOperator|) (|BasicOperator|))) T)
((|belong?| ((|Boolean|) (|BasicOperator|))) T)
((|is?| ((|Boolean|) |$| (|BasicOperator|))) T) ((|is?| ((|Boolean|) |$| (|Symbol|))) T)
((|kernel| (|$| (|BasicOperator|) |$|)) T)
((|kernel| (|$| (|BasicOperator|) (|List| |$|))) T)
((|map| (|$| (|Mapping| |$| |$|) (|Kernel| |$|))) T)
((|freeOf?| ((|Boolean|) |$| |$|)) T)
((|freeOf?| ((|Boolean|) |$| (|Symbol|))) T) ((|eval| (|$| |$| (|List| (|Symbol|)) (|List| (|Mapping| |$| |$|)))) T) ((|eval| (|$| |$| (|List| (|Symbol|)) (|List| (|Mapping| |$| (|List| |$|))))) T) ((|eval| (|$| |$| (|Symbol|) (|Mapping| |$| (|List| |$|)))) T) ((|eval| (|$| |$| (|Symbol|) (|Mapping| |$| |$|))) T) ((|eval| (|$| |$| (|List| (|BasicOperator|)) (|List| (|Mapping| |$| |$|)))) T) ((|eval| (|$| |$| (|List| (|BasicOperator|)) (|List| (|Mapping| |$| (|List| |$|))))) T) ((|eval| (|$| |$| (|BasicOperator|) (|Mapping| |$| (|List| |$|)))) T) ((|eval| (|$| |$| (|BasicOperator|) (|Mapping| |$| |$|))) T) ((|minPoly| ((|SparseUnivariatePolynomial| |$|) (|Kernel| |$|))) (|has| |$| (|Ring|)))
((|definingPolynomial| (|$| |$|)) (|has| |$| (|Ring|))) ((|even?| ((|Boolean|) |$|))
(|has| |$| (|RetractableTo| (|Integer|)))) ((|odd?| ((|Boolean|) |$|))
(|has| |$| (|RetractableTo| (|Integer|)))))) NIL (QUOTE ( (|Boolean|) (|SparseUnivariatePolynomial| |$|)
(|Kernel| |$|) (|BasicOperator|) (|List| (|BasicOperator|)) (|List| (|Mapping| |$| (|List| |$|))) (|List| (|Mapping| |$| |$|)) (|Symbol|) (|List| (|Symbol|)) (|List| |$|)
(|List| (|Kernel| |$|)) (|NonNegativeInteger|) (|List| (|Equation| |$|))
(|Equation| |$|))) NIL))) |ExpressionSpace|) (SETELT #1# 0 (QUOTE (|ExpressionSpace|))))))) (MAKEPROP (QUOTE |ExpressionSpace|) (QUOTE NILADIC) T) @ \section{ES-.lsp BOOTSTRAP} {\bf ES-} depends on {\bf ES}. We need to break this cycle to build the algebra. So we keep a cached copy of the translated {\bf ES-} category which we can write into the {\bf MID} directory. We compile the lisp code and copy the {\bf ES-.o} file to the {\bf OUT} directory. This is eventually forcibly replaced by a recompiled version. Note that this code is not included in the generated catdef.spad file. <<ES-.lsp BOOTSTRAP>>= (|/VERSIONCHECK| 2) (DEFUN |ES-;box;2S;1| (|x| |$|) (SPADCALL (LIST |x|) (QREFELT |$| 16))) (DEFUN |ES-;paren;2S;2| (|x| |$|) (SPADCALL (LIST |x|) (QREFELT |$| 18))) (DEFUN |ES-;belong?;BoB;3| (|op| |$|)
(COND
((SPADCALL |op| (QREFELT |$| 13) (QREFELT |$| 21)) (QUOTE T))
((QUOTE T) (SPADCALL |op| (QREFELT |$| 14) (QREFELT |$| 21)))))

(DEFUN |ES-;listk| (|f| |$|) (SPADCALL (|ES-;allKernels| |f| |$|) (QREFELT |$| 25))) (DEFUN |ES-;tower;SL;5| (|f| |$|)
(SPADCALL (|ES-;listk| |f| |$|) (QREFELT |$| 26)))

(DEFUN |ES-;allk| (|l| |$|) (PROG (#1=#:G82361 |f| #2=#:G82362) (RETURN (SEQ (SPADCALL (ELT |$| 30)
(PROGN
(LETT #1# NIL |ES-;allk|)
(SEQ
(LETT |f| NIL |ES-;allk|)
(LETT #2# |l| |ES-;allk|)
G190
(COND
((OR (ATOM #2#)
(PROGN (LETT |f| (CAR #2#) |ES-;allk|) NIL))
(GO G191)))
(SEQ
(EXIT
(LETT #1# (CONS (|ES-;allKernels| |f| |$|) #1#) |ES-;allk|))) (LETT #2# (CDR #2#) |ES-;allk|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) (SPADCALL NIL (QREFELT |$| 29))
(QREFELT |$| 33)))))) (DEFUN |ES-;operators;SL;7| (|f| |$|)
(PROG (#1=#:G82365 |k| #2=#:G82366)
(RETURN
(SEQ
(PROGN
(LETT #1# NIL |ES-;operators;SL;7|)
(SEQ
(LETT |k| NIL |ES-;operators;SL;7|)
(LETT #2# (|ES-;listk| |f| |$|) |ES-;operators;SL;7|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |k| (CAR #2#) |ES-;operators;SL;7|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (SPADCALL |k| (QREFELT |$| 35)) #1#)
|ES-;operators;SL;7|)))
(LETT #2# (CDR #2#) |ES-;operators;SL;7|)
(GO G190)
G191
(EXIT (NREVERSE0 #1#))))))))

(DEFUN |ES-;height;SNni;8| (|f| |$|) (PROG (#1=#:G82371 |k| #2=#:G82372) (RETURN (SEQ (SPADCALL (ELT |$| 41)
(PROGN
(LETT #1# NIL |ES-;height;SNni;8|)
(SEQ
(LETT |k| NIL |ES-;height;SNni;8|)
(LETT #2# (SPADCALL |f| (QREFELT |$| 38)) |ES-;height;SNni;8|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |k| (CAR #2#) |ES-;height;SNni;8|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (SPADCALL |k| (QREFELT |$| 40)) #1#)
|ES-;height;SNni;8|)))
(LETT #2# (CDR #2#) |ES-;height;SNni;8|)
(GO G190)
G191
(EXIT (NREVERSE0 #1#))))
0
(QREFELT |$| 44)))))) (DEFUN |ES-;freeOf?;SSB;9| (|x| |s| |$|)
(PROG (#1=#:G82377 |k| #2=#:G82378)
(RETURN
(SEQ
(COND
(PROGN
(LETT #1# NIL |ES-;freeOf?;SSB;9|)
(SEQ
(LETT |k| NIL |ES-;freeOf?;SSB;9|)
(LETT #2# (|ES-;listk| |x| |$|) |ES-;freeOf?;SSB;9|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |k| (CAR #2#) |ES-;freeOf?;SSB;9|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (SPADCALL |k| (QREFELT |$| 46)) #1#)
|ES-;freeOf?;SSB;9|)))
(LETT #2# (CDR #2#) |ES-;freeOf?;SSB;9|)
(GO G190)
G191
(EXIT (NREVERSE0 #1#))))
(QREFELT |$| 48)) (QUOTE NIL)) ((QUOTE T) (QUOTE T))))))) (DEFUN |ES-;distribute;2S;10| (|x| |$|)
(PROG (#1=#:G82381 |k| #2=#:G82382)
(RETURN
(SEQ
(|ES-;unwrap|
(PROGN
(LETT #1# NIL |ES-;distribute;2S;10|)
(SEQ
(LETT |k| NIL |ES-;distribute;2S;10|)
(LETT #2# (|ES-;listk| |x| |$|) |ES-;distribute;2S;10|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |k| (CAR #2#) |ES-;distribute;2S;10|) NIL)) (GO G191))) (SEQ (EXIT (COND ((SPADCALL |k| (QREFELT |$| 13) (QREFELT |$| 50)) (LETT #1# (CONS |k| #1#) |ES-;distribute;2S;10|))))) (LETT #2# (CDR #2#) |ES-;distribute;2S;10|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) |x| |$|)))))

(DEFUN |ES-;box;LS;11| (|l| |$|) (SPADCALL (QREFELT |$| 14) |l| (QREFELT |$| 52))) (DEFUN |ES-;paren;LS;12| (|l| |$|)
(SPADCALL (QREFELT |$| 13) |l| (QREFELT |$| 52)))

(DEFUN |ES-;freeOf?;2SB;13| (|x| |k| |$|) (COND ((SPADCALL (SPADCALL |k| (QREFELT |$| 56))
(|ES-;listk| |x| |$|) (QREFELT |$| 57))
(QUOTE NIL))
((QUOTE T) (QUOTE T))))

(DEFUN |ES-;kernel;Bo2S;14| (|op| |arg| |$|) (SPADCALL |op| (LIST |arg|) (QREFELT |$| 59)))

(DEFUN |ES-;elt;Bo2S;15| (|op| |x| |$|) (SPADCALL |op| (LIST |x|) (QREFELT |$| 52)))

(DEFUN |ES-;elt;Bo3S;16| (|op| |x| |y| |$|) (SPADCALL |op| (LIST |x| |y|) (QREFELT |$| 52)))

(DEFUN |ES-;elt;Bo4S;17| (|op| |x| |y| |z| |$|) (SPADCALL |op| (LIST |x| |y| |z|) (QREFELT |$| 52)))

(DEFUN |ES-;elt;Bo5S;18| (|op| |x| |y| |z| |t| |$|) (SPADCALL |op| (LIST |x| |y| |z| |t|) (QREFELT |$| 52)))

(DEFUN |ES-;eval;SSMS;19| (|x| |s| |f| |$|) (SPADCALL |x| (LIST |s|) (LIST |f|) (QREFELT |$| 67)))

(DEFUN |ES-;eval;SBoMS;20| (|x| |s| |f| |$|) (SPADCALL |x| (LIST (SPADCALL |s| (QREFELT |$| 69)))
(LIST |f|)
(QREFELT |$| 67))) (DEFUN |ES-;eval;SSMS;21| (|x| |s| |f| |$|)
(LIST |s|)
(LIST (CONS (FUNCTION |ES-;eval;SSMS;21!0|) (VECTOR |f| |$|))) (QREFELT |$| 67)))

(DEFUN |ES-;eval;SSMS;21!0| (|#1| |$$|) (SPADCALL (SPADCALL |#1| (QREFELT (QREFELT |$$| 1) 72))
(QREFELT |$$| 0))) (DEFUN |ES-;eval;SBoMS;22| (|x| |s| |f| ||) (SPADCALL |x| (LIST |s|) (LIST (CONS (FUNCTION |ES-;eval;SBoMS;22!0|) (VECTOR |f| ||))) (QREFELT || 75))) (DEFUN |ES-;eval;SBoMS;22!0| (|#1| |$$|)
(SPADCALL |#1| (QREFELT (QREFELT |$$| 1) 72)) (QREFELT |$$| 0)))

(DEFUN |ES-;subst;SES;23| (|x| |e| |$|) (SPADCALL |x| (LIST |e|) (QREFELT |$| 78)))

(DEFUN |ES-;eval;SLLS;24| (|x| |ls| |lf| |$|) (PROG (#1=#:G82403 |f| #2=#:G82404) (RETURN (SEQ (SPADCALL |x| |ls| (PROGN (LETT #1# NIL |ES-;eval;SLLS;24|) (SEQ (LETT |f| NIL |ES-;eval;SLLS;24|) (LETT #2# |lf| |ES-;eval;SLLS;24|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |f| (CAR #2#) |ES-;eval;SLLS;24|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (CONS (FUNCTION |ES-;eval;SLLS;24!0|) (VECTOR |f| |$|)) #1#)
|ES-;eval;SLLS;24|)))
(LETT #2# (CDR #2#) |ES-;eval;SLLS;24|)
(GO G190)
G191
(EXIT (NREVERSE0 #1#))))
(QREFELT |$| 75)))))) (DEFUN |ES-;eval;SLLS;24!0| (|#1| |$$|) (SPADCALL (SPADCALL |#1| (QREFELT (QREFELT |$$| 1) 72)) (QREFELT |$$| 0))) (DEFUN |ES-;eval;SLLS;25| (|x| |ls| |lf| ||) (PROG (#1=#:G82407 |f| #2=#:G82408) (RETURN (SEQ (SPADCALL |x| |ls| (PROGN (LETT #1# NIL |ES-;eval;SLLS;25|) (SEQ (LETT |f| NIL |ES-;eval;SLLS;25|) (LETT #2# |lf| |ES-;eval;SLLS;25|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |f| (CAR #2#) |ES-;eval;SLLS;25|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (CONS (FUNCTION |ES-;eval;SLLS;25!0|) (VECTOR |f| ||)) #1#) |ES-;eval;SLLS;25|))) (LETT #2# (CDR #2#) |ES-;eval;SLLS;25|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) (QREFELT || 67)))))) (DEFUN |ES-;eval;SLLS;25!0| (|#1| |$$|) (SPADCALL (SPADCALL |#1| (QREFELT (QREFELT |$$| 1) 72)) (QREFELT |$$| 0))) (DEFUN |ES-;eval;SLLS;26| (|x| |ls| |lf| |$|)
(PROG (#1=#:G82412 |s| #2=#:G82413)
(RETURN
(SEQ
(PROGN
(LETT #1# NIL |ES-;eval;SLLS;26|)
(SEQ
(LETT |s| NIL |ES-;eval;SLLS;26|)
(LETT #2# |ls| |ES-;eval;SLLS;26|)
G190
(COND
((OR (ATOM #2#)
(PROGN (LETT |s| (CAR #2#) |ES-;eval;SLLS;26|) NIL)
) (GO G191)))
(SEQ
(EXIT
(LETT #1#
(CONS (SPADCALL |s| (QREFELT |$| 69)) #1#) |ES-;eval;SLLS;26|))) (LETT #2# (CDR #2#) |ES-;eval;SLLS;26|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) |lf| (QREFELT |$| 67))))))

(DEFUN |ES-;map;MKS;27| (|fn| |k| |$|) (PROG (#1=#:G82428 |x| #2=#:G82429 |l|) (RETURN (SEQ (COND ((SPADCALL (LETT |l| (PROGN (LETT #1# NIL |ES-;map;MKS;27|) (SEQ (LETT |x| NIL |ES-;map;MKS;27|) (LETT #2# (SPADCALL |k| (QREFELT |$| 85)) |ES-;map;MKS;27|)
G190
(COND
((OR (ATOM #2#)
(PROGN (LETT |x| (CAR #2#) |ES-;map;MKS;27|) NIL))
(GO G191)))
(SEQ
(EXIT
(LETT #1# (CONS (SPADCALL |x| |fn|) #1#) |ES-;map;MKS;27|)))
(LETT #2# (CDR #2#) |ES-;map;MKS;27|)
(GO G190)
G191
(EXIT (NREVERSE0 #1#))))
|ES-;map;MKS;27|)
(SPADCALL |k| (QREFELT |$| 85)) (QREFELT |$| 86))
(SPADCALL |k| (QREFELT |$| 87))) ((QUOTE T) (SPADCALL (SPADCALL |k| (QREFELT |$| 35)) |l| (QREFELT |$| 52)))))))) (DEFUN |ES-;operator;2Bo;28| (|op| |$|)
(COND
((SPADCALL |op| (SPADCALL "%paren" (QREFELT |$| 9)) (QREFELT |$| 89))
(QREFELT |$| 13)) ((SPADCALL |op| (SPADCALL "%box" (QREFELT |$| 9)) (QREFELT |$| 89)) (QREFELT |$| 14))
((QUOTE T) (|error| "Unknown operator"))))

(DEFUN |ES-;mainKernel;SU;29| (|x| |$|) (PROG (|l| |kk| #1=#:G82445 |n| |k|) (RETURN (SEQ (COND ((NULL (LETT |l| (SPADCALL |x| (QREFELT |$| 38)) |ES-;mainKernel;SU;29|))
(CONS 1 "failed"))
((QUOTE T)
(SEQ
(LETT |n|
(QREFELT |$| 40)) |ES-;mainKernel;SU;29|) (SEQ (LETT |kk| NIL |ES-;mainKernel;SU;29|) (LETT #1# (CDR |l|) |ES-;mainKernel;SU;29|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |kk| (CAR #1#) |ES-;mainKernel;SU;29|) NIL)) (GO G191))) (SEQ (EXIT (COND ((|<| |n| (SPADCALL |kk| (QREFELT |$| 40)))
(SEQ
(LETT |n|
(SPADCALL |kk| (QREFELT |$| 40)) |ES-;mainKernel;SU;29|) (EXIT (LETT |k| |kk| |ES-;mainKernel;SU;29|))))))) (LETT #1# (CDR #1#) |ES-;mainKernel;SU;29|) (GO G190) G191 (EXIT NIL)) (EXIT (CONS 0 |k|))))))))) (DEFUN |ES-;allKernels| (|f| |$|)
(PROG (|l| |k| #1=#:G82458 |u| |s0| |n| |arg| |t| |s|)
(RETURN
(SEQ
(LETT |s|
(LETT |l| (SPADCALL |f| (QREFELT |$| 38)) |ES-;allKernels|) (QREFELT |$| 29))
|ES-;allKernels|)
(SEQ
(LETT |k| NIL |ES-;allKernels|)
(LETT #1# |l| |ES-;allKernels|)
G190
(COND
((OR (ATOM #1#)
(PROGN (LETT |k| (CAR #1#) |ES-;allKernels|) NIL))
(GO G191)))
(SEQ
(LETT |t|
(SEQ
(LETT |u|
(SPADCALL |k| (QREFELT |$| 35)) "%dummyVar" (QREFELT |$| 94))
|ES-;allKernels|)
(EXIT
(COND
((QEQCAR |u| 0)
(SEQ
(LETT |arg| (SPADCALL |k| (QREFELT |$| 85)) |ES-;allKernels|) (LETT |s0| (SPADCALL (SPADCALL (SPADCALL |arg| (QREFELT |$| 95)) (QREFELT |$| 56)) (|ES-;allKernels| (|SPADfirst| |arg|) |$|)
(QREFELT |$| 96)) |ES-;allKernels|) (LETT |arg| (CDR (CDR |arg|)) |ES-;allKernels|) (LETT |n| (QCDR |u|) |ES-;allKernels|) (COND ((|<| 1 |n|) (LETT |arg| (CDR |arg|) |ES-;allKernels|))) (EXIT (SPADCALL |s0| (|ES-;allk| |arg| |$|) (QREFELT |$| 30))))) ((QUOTE T) (|ES-;allk| (SPADCALL |k| (QREFELT |$| 85)) |$|))))) |ES-;allKernels|) (EXIT (LETT |s| (SPADCALL |s| |t| (QREFELT |$| 30)) |ES-;allKernels|)))
(LETT #1# (CDR #1#) |ES-;allKernels|)
(GO G190)
G191
(EXIT NIL))
(EXIT |s|)))))

(DEFUN |ES-;kernel;BoLS;31| (|op| |args| |$|) (COND ((NULL (SPADCALL |op| (QREFELT |$| 97))) (|error| "Unknown operator"))
((QUOTE T) (|ES-;okkernel| |op| |args| |$|)))) (DEFUN |ES-;okkernel| (|op| |l| |$|)
(PROG (#1=#:G82465 |f| #2=#:G82466)
(RETURN
(SEQ
(|+| 1
(ELT |$| 41) (PROGN (LETT #1# NIL |ES-;okkernel|) (SEQ (LETT |f| NIL |ES-;okkernel|) (LETT #2# |l| |ES-;okkernel|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |f| (CAR #2#) |ES-;okkernel|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (SPADCALL |f| (QREFELT |$| 99)) #1#) |ES-;okkernel|)))
(LETT #2# (CDR #2#) |ES-;okkernel|)
(GO G190)
G191
(EXIT (NREVERSE0 #1#))))
0
(QREFELT |$| 44))) (QREFELT |$| 100))
(QREFELT |$| 87)))))) (DEFUN |ES-;elt;BoLS;33| (|op| |args| |$|)
(PROG (|u| #1=#:G82482 |v|)
(RETURN
(SEQ
(EXIT
(COND
((NULL (SPADCALL |op| (QREFELT |$| 97))) (|error| "Unknown operator")) ((QUOTE T) (SEQ (SEQ (LETT |u| (SPADCALL |op| (QREFELT |$| 102)) |ES-;elt;BoLS;33|)
(EXIT
(COND
((QEQCAR |u| 0)
(COND
((NULL (EQL (LENGTH |args|) (QCDR |u|)))
(PROGN
(LETT #1#
(|error| "Wrong number of arguments")
|ES-;elt;BoLS;33|)
(GO #1#))))))))
(LETT |v| (SPADCALL |op| |args| (QREFELT |$| 105)) |ES-;elt;BoLS;33|) (EXIT (COND ((QEQCAR |v| 0) (QCDR |v|)) ((QUOTE T) (|ES-;okkernel| |op| |args| |$|))))))))
#1#
(EXIT #1#)))))

(DEFUN |ES-;retract;SK;34| (|f| |$|) (PROG (|k|) (RETURN (SEQ (LETT |k| (SPADCALL |f| (QREFELT |$| 107)) |ES-;retract;SK;34|)
(EXIT
(COND
((OR (QEQCAR |k| 1)
(NULL
(SPADCALL (QCDR |k|) (QREFELT |$| 87)) |f| (QREFELT |$| 108))))
(|error| "not a kernel"))
((QUOTE T) (QCDR |k|))))))))

(DEFUN |ES-;retractIfCan;SU;35| (|f| |$|) (PROG (|k|) (RETURN (SEQ (LETT |k| (SPADCALL |f| (QREFELT |$| 107)) |ES-;retractIfCan;SU;35|)
(EXIT
(COND
((OR (QEQCAR |k| 1)
(NULL
(SPADCALL (QCDR |k|) (QREFELT |$| 87)) |f| (QREFELT |$| 108))))
(CONS 1 "failed"))
((QUOTE T) |k|)))))))

(DEFUN |ES-;is?;SSB;36| (|f| |s| |$|) (PROG (|k|) (RETURN (SEQ (LETT |k| (SPADCALL |f| (QREFELT |$| 111)) |ES-;is?;SSB;36|)
(EXIT
(COND
((QEQCAR |k| 1) (QUOTE NIL))
((QUOTE T) (SPADCALL (QCDR |k|) |s| (QREFELT |$| 112))))))))) (DEFUN |ES-;is?;SBoB;37| (|f| |op| |$|)
(PROG (|k|)
(RETURN
(SEQ
(LETT |k| (SPADCALL |f| (QREFELT |$| 111)) |ES-;is?;SBoB;37|) (EXIT (COND ((QEQCAR |k| 1) (QUOTE NIL)) ((QUOTE T) (SPADCALL (QCDR |k|) |op| (QREFELT |$| 50)))))))))

(DEFUN |ES-;unwrap| (|l| |x| |$|) (PROG (|k| #1=#:G82507) (RETURN (SEQ (SEQ (LETT |k| NIL |ES-;unwrap|) (LETT #1# (NREVERSE |l|) |ES-;unwrap|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |k| (CAR #1#) |ES-;unwrap|) NIL)) (GO G191))) (SEQ (EXIT (LETT |x| (SPADCALL |x| |k| (|SPADfirst| (SPADCALL |k| (QREFELT |$| 85)))
(QREFELT |$| 115)) |ES-;unwrap|))) (LETT #1# (CDR #1#) |ES-;unwrap|) (GO G190) G191 (EXIT NIL)) (EXIT |x|))))) (DEFUN |ES-;distribute;3S;39| (|x| |y| |$|)
(PROG (|ky| #1=#:G82512 |k| #2=#:G82513)
(RETURN
(SEQ
(LETT |ky| (SPADCALL |y| (QREFELT |$| 56)) |ES-;distribute;3S;39|) (EXIT (|ES-;unwrap| (PROGN (LETT #1# NIL |ES-;distribute;3S;39|) (SEQ (LETT |k| NIL |ES-;distribute;3S;39|) (LETT #2# (|ES-;listk| |x| |$|) |ES-;distribute;3S;39|)
G190
(COND
((OR (ATOM #2#)
(PROGN (LETT |k| (CAR #2#) |ES-;distribute;3S;39|) NIL))
(GO G191)))
(SEQ
(EXIT
(COND
((COND
(SPADCALL "%paren" (QREFELT |$| 9)) (QREFELT |$| 112))
(|ES-;listk| (SPADCALL |k| (QREFELT |$| 87)) |$|)
(QREFELT |$| 57))) ((QUOTE T) (QUOTE NIL))) (LETT #1# (CONS |k| #1#) |ES-;distribute;3S;39|))))) (LETT #2# (CDR #2#) |ES-;distribute;3S;39|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) |x| |$|))))))

(DEFUN |ES-;eval;SLS;40| (|f| |leq| |$|) (PROG (|rec|) (RETURN (SEQ (LETT |rec| (|ES-;mkKerLists| |leq| |$|) |ES-;eval;SLS;40|)
(EXIT (SPADCALL |f| (QCAR |rec|) (QCDR |rec|) (QREFELT |$| 117))))))) (DEFUN |ES-;subst;SLS;41| (|f| |leq| |$|)
(PROG (|rec|)
(RETURN
(SEQ
(LETT |rec| (|ES-;mkKerLists| |leq| |$|) |ES-;subst;SLS;41|) (EXIT (SPADCALL |f| (QCAR |rec|) (QCDR |rec|) (QREFELT |$| 119)))))))

(DEFUN |ES-;mkKerLists| (|leq| |$|) (PROG (|eq| #1=#:G82530 |k| |lk| |lv|) (RETURN (SEQ (LETT |lk| NIL |ES-;mkKerLists|) (LETT |lv| NIL |ES-;mkKerLists|) (SEQ (LETT |eq| NIL |ES-;mkKerLists|) (LETT #1# |leq| |ES-;mkKerLists|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |eq| (CAR #1#) |ES-;mkKerLists|) NIL)) (GO G191))) (SEQ (LETT |k| (SPADCALL (SPADCALL |eq| (QREFELT |$| 122)) (QREFELT |$| 111)) |ES-;mkKerLists|) (EXIT (COND ((QEQCAR |k| 1) (|error| "left hand side must be a single kernel")) ((NULL (SPADCALL (QCDR |k|) |lk| (QREFELT |$| 57)))
(SEQ
(LETT |lk| (CONS (QCDR |k|) |lk|) |ES-;mkKerLists|)
(EXIT
(LETT |lv|
(CONS (SPADCALL |eq| (QREFELT |$| 123)) |lv|) |ES-;mkKerLists|))))))) (LETT #1# (CDR #1#) |ES-;mkKerLists|) (GO G190) G191 (EXIT NIL)) (EXIT (CONS |lk| |lv|)))))) (DEFUN |ES-;even?;SB;43| (|x| |$|) (|ES-;intpred?| |x| (ELT |$| 125) |$|))

(DEFUN |ES-;odd?;SB;44| (|x| |$|) (|ES-;intpred?| |x| (ELT |$| 127) |$|)) (DEFUN |ES-;intpred?| (|x| |pred?| |$|)
(PROG (|u|)
(RETURN
(SEQ
(LETT |u| (SPADCALL |x| (QREFELT |$| 130)) |ES-;intpred?|) (EXIT (COND ((QEQCAR |u| 0) (SPADCALL (QCDR |u|) |pred?|)) ((QUOTE T) (QUOTE NIL)))))))) (DEFUN |ExpressionSpace&| (|#1|) (PROG (|DV$1| |dv$| |$| |pv$|) (RETURN (PROGN (LETT |DV$1| (|devaluate| |#1|) . #1=(|ExpressionSpace&|))
(LETT |dv$| (LIST (QUOTE |ExpressionSpace&|) |DV$1|) . #1#)
(LETT |$| (GETREFV 131) . #1#) (QSETREFV |$| 0 |dv$|) (QSETREFV |$| 3
(LETT |pv$| (|buildPredVector| 0 0 (LIST (|HasCategory| |#1| (QUOTE (|RetractableTo| (|Integer|)))) (|HasCategory| |#1| (QUOTE (|Ring|))))) . #1#)) (|stuffDomainSlots| |$|)
(QSETREFV |$| 6 |#1|) (QSETREFV |$| 13
(SPADCALL (SPADCALL "%paren" (QREFELT |$| 9)) (QREFELT |$| 12)))
(QSETREFV |$| 14 (SPADCALL (SPADCALL "%box" (QREFELT |$| 9)) (QREFELT |$| 12))) (COND ((|testBitVector| |pv$| 1)
(PROGN
(QSETREFV |$| 126 (CONS (|dispatchFunction| |ES-;even?;SB;43|) |$|))
(QSETREFV |$| 128 (CONS (|dispatchFunction| |ES-;odd?;SB;44|) |$|)))))
|$|)))) (MAKEPROP (QUOTE |ExpressionSpace&|) (QUOTE |infovec|) (LIST (QUOTE #( NIL NIL NIL NIL NIL NIL (|local| |#1|) (|String|) (|Symbol|) (0 . |coerce|) (|BasicOperator|) (|CommonOperators|) (5 . |operator|) (QUOTE |oppren|) (QUOTE |opbox|) (|List| |$|) (10 . |box|) |ES-;box;2S;1| (15 . |paren|)
|ES-;paren;2S;2| (|Boolean|) (20 . |=|) |ES-;belong?;BoB;3| (|List| 34)
(|Set| 34) (26 . |parts|) (31 . |sort!|) (|List| 55) |ES-;tower;SL;5|
(36 . |brace|) (41 . |union|) (|Mapping| 24 24 24) (|List| 24)
(47 . |reduce|) (|Kernel| 6) (54 . |operator|) (|List| 10)
|ES-;operators;SL;7| (59 . |kernels|) (|NonNegativeInteger|)
(64 . |height|) (69 . |max|) (|Mapping| 39 39 39) (|List| 39)
(75 . |reduce|) |ES-;height;SNni;8| (82 . |name|) (|List| 8)
(87 . |member?|) |ES-;freeOf?;SSB;9| (93 . |is?|) |ES-;distribute;2S;10|
(99 . |elt|) |ES-;box;LS;11| |ES-;paren;LS;12| (|Kernel| |$|) (105 . |retract|) (110 . |member?|) |ES-;freeOf?;2SB;13| (116 . |kernel|) |ES-;kernel;Bo2S;14| |ES-;elt;Bo2S;15| |ES-;elt;Bo3S;16| |ES-;elt;Bo4S;17| |ES-;elt;Bo5S;18| (|Mapping| |$| 15) (|List| 65) (122 . |eval|)
|ES-;eval;SSMS;19| (129 . |name|) |ES-;eval;SBoMS;20| (|List| 6)
(134 . |first|) (|Mapping| |$| |$|) |ES-;eval;SSMS;21| (139 . |eval|)
|ES-;eval;SBoMS;22| (|List| 79) (146 . |subst|) (|Equation| |$|) |ES-;subst;SES;23| (|List| 73) |ES-;eval;SLLS;24| |ES-;eval;SLLS;25| |ES-;eval;SLLS;26| (152 . |argument|) (157 . |=|) (163 . |coerce|) |ES-;map;MKS;27| (168 . |is?|) |ES-;operator;2Bo;28| (|Union| 55 (QUOTE "failed")) |ES-;mainKernel;SU;29| (|Union| (|None|) (QUOTE "failed")) (174 . |property|) (180 . |second|) (185 . |remove!|) (191 . |belong?|) |ES-;kernel;BoLS;31| (196 . |height|) (201 . |kernel|) (|Union| 39 (QUOTE "failed")) (208 . |arity|) (|Union| 6 (QUOTE "failed")) (|BasicOperatorFunctions1| 6) (213 . |evaluate|) |ES-;elt;BoLS;33| (219 . |mainKernel|) (224 . |=|) |ES-;retract;SK;34| |ES-;retractIfCan;SU;35| (230 . |retractIfCan|) (235 . |is?|) |ES-;is?;SSB;36| |ES-;is?;SBoB;37| (241 . |eval|) |ES-;distribute;3S;39| (248 . |eval|) |ES-;eval;SLS;40| (255 . |subst|) |ES-;subst;SLS;41| (|Equation| 6) (262 . |lhs|) (267 . |rhs|) (|Integer|) (272 . |even?|) (277 . |even?|) (282 . |odd?|) (287 . |odd?|) (|Union| 124 (QUOTE "failed")) (292 . |retractIfCan|))) (QUOTE #( |tower| 297 |subst| 302 |retractIfCan| 314 |retract| 319 |paren| 324 |operators| 334 |operator| 339 |odd?| 344 |map| 349 |mainKernel| 355 |kernel| 360 |is?| 372 |height| 384 |freeOf?| 389 |even?| 401 |eval| 406 |elt| 461 |distribute| 497 |box| 508 |belong?| 518)) (QUOTE NIL) (CONS (|makeByteWordVec2| 1 (QUOTE NIL)) (CONS (QUOTE #()) (CONS (QUOTE #()) (|makeByteWordVec2| 130 (QUOTE (1 8 0 7 9 1 11 10 8 12 1 6 0 15 16 1 6 0 15 18 2 10 20 0 0 21 1 24 23 0 25 1 23 0 0 26 1 24 0 23 29 2 24 0 0 0 30 3 32 24 31 0 24 33 1 34 10 0 35 1 6 27 0 38 1 34 39 0 40 2 39 0 0 0 41 3 43 39 42 0 39 44 1 34 8 0 46 2 47 20 8 0 48 2 34 20 0 10 50 2 6 0 10 15 52 1 6 55 0 56 2 23 20 34 0 57 2 6 0 10 15 59 3 6 0 0 47 66 67 1 10 8 0 69 1 71 6 0 72 3 6 0 0 36 66 75 2 6 0 0 77 78 1 34 71 0 85 2 71 20 0 0 86 1 6 0 55 87 2 10 20 0 8 89 2 10 93 0 7 94 1 71 6 0 95 2 24 0 34 0 96 1 6 20 10 97 1 6 39 0 99 3 34 0 10 71 39 100 1 10 101 0 102 2 104 103 10 71 105 1 6 91 0 107 2 6 20 0 0 108 1 6 91 0 111 2 34 20 0 8 112 3 6 0 0 55 0 115 3 6 0 0 27 15 117 3 6 0 0 27 15 119 1 121 6 0 122 1 121 6 0 123 1 124 20 0 125 1 0 20 0 126 1 124 20 0 127 1 0 20 0 128 1 6 129 0 130 1 0 27 0 28 2 0 0 0 77 120 2 0 0 0 79 80 1 0 91 0 110 1 0 55 0 109 1 0 0 0 19 1 0 0 15 54 1 0 36 0 37 1 0 10 10 90 1 0 20 0 128 2 0 0 73 55 88 1 0 91 0 92 2 0 0 10 15 98 2 0 0 10 0 60 2 0 20 0 8 113 2 0 20 0 10 114 1 0 39 0 45 2 0 20 0 8 49 2 0 20 0 0 58 1 0 20 0 126 3 0 0 0 10 73 76 3 0 0 0 36 66 84 3 0 0 0 10 65 70 3 0 0 0 36 81 82 3 0 0 0 8 65 68 3 0 0 0 8 73 74 3 0 0 0 47 81 83 2 0 0 0 77 118 2 0 0 10 15 106 5 0 0 10 0 0 0 0 64 3 0 0 10 0 0 62 4 0 0 10 0 0 0 63 2 0 0 10 0 61 2 0 0 0 0 116 1 0 0 0 51 1 0 0 15 53 1 0 0 0 17 1 0 20 10 22)))))) (QUOTE |lookupComplete|))) @ \section{package ES1 ExpressionSpaceFunctions1} <<package ES1 ExpressionSpaceFunctions1>>= )abbrev package ES1 ExpressionSpaceFunctions1 ++ Lifting of maps from expression spaces to kernels over them ++ Author: Manuel Bronstein ++ Date Created: 23 March 1988 ++ Date Last Updated: 19 April 1991 ++ Description: ++ 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. -- should not be exposed ExpressionSpaceFunctions1(F:ExpressionSpace, S:Type): with map: (F -> S, String, Kernel F) -> S ++ map(f, p, k) uses the property p of the operator ++ of k, in order to lift f and apply it to k. == add -- prop contains an evaluation function List S -> S map(F2S, prop, k) == args := [F2S x for x in argument k]$List(S)
(p := property(operator k, prop)) case None =>
((p::None) pretend (List S -> S)) args
error "Operator does not have required property"

@
\section{package ES2 ExpressionSpaceFunctions2}
<<package ES2 ExpressionSpaceFunctions2>>=
)abbrev package ES2 ExpressionSpaceFunctions2
++ Lifting of maps from expression spaces to kernels over them
++ Author: Manuel Bronstein
++ Date Created: 23 March 1988
++ Date Last Updated: 19 April 1991
++ Description:
++ This package allows a mapping E -> F to be lifted to a kernel over E;
++ This lifting can fail if the operator of the kernel cannot be applied
++ in F; Do not use this package with E = F, since this may
++ drop some properties of the operators.
ExpressionSpaceFunctions2(E:ExpressionSpace, F:ExpressionSpace): with
map: (E -> F, Kernel E) -> F
++ map(f, k) returns \spad{g = op(f(a1),...,f(an))} where
map(f, k) ==
(operator(operator k)$F) [f x for x in argument k]$List(F)

@
\section{category FS FunctionSpace}
<<category FS FunctionSpace>>=
)abbrev category FS FunctionSpace
++ Category for formal functions
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Date Last Updated: 14 February 1994
++ Description:
++   A space of formal functions with arguments in an arbitrary
++   ordered set.
++ Keywords: operator, kernel, function.
FunctionSpace(R:OrderedSet): Category == Definition where
OP ==> BasicOperator
O  ==> OutputForm
SY ==> Symbol
N  ==> NonNegativeInteger
Z  ==> Integer
K  ==> Kernel %
Q  ==> Fraction R
PR ==> Polynomial R
MP ==> SparseMultivariatePolynomial(R, K)
QF==> PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,MP,%)

ODD  ==> "odd"
EVEN ==> "even"

SPECIALDIFF  ==> "%specialDiff"
SPECIALDISP  ==> "%specialDisp"
SPECIALEQUAL ==> "%specialEqual"
SPECIALINPUT ==> "%specialInput"

Definition ==> Join(ExpressionSpace, RetractableTo SY, Patternable R,
FullyPatternMatchable R, FullyRetractableTo R) with
ground?   : % -> Boolean
++ ground?(f) tests if f is an element of R.
ground    : % -> R
++ ground(f) returns f as an element of R.
++ An error occurs if f is not an element of R.
variables : %  -> List SY
++ variables(f) returns the list of all the variables of f.
applyQuote: (SY, %) -> %
applyQuote: (SY, %, %) -> %
++ applyQuote(foo, x, y) returns \spad{'foo(x,y)}.
applyQuote: (SY, %, %, %) -> %
++ applyQuote(foo, x, y, z) returns \spad{'foo(x,y,z)}.
applyQuote: (SY, %, %, %, %) -> %
++ applyQuote(foo, x, y, z, t) returns \spad{'foo(x,y,z,t)}.
applyQuote: (SY, List %) -> %
if R has ConvertibleTo InputForm then
ConvertibleTo InputForm
eval     : (%, SY) -> %
++ eval(f, foo) unquotes all the foo's in f.
eval     : (%, List SY) -> %
++ eval(f, [foo1,...,foon]) unquotes all the \spad{fooi}'s in f.
eval     : % -> %
++ eval(f) unquotes all the quoted operators in f.
eval     : (%, OP, %, SY) -> %
++ eval(x, s, f, y) replaces every \spad{s(a)} in x by \spad{f(y)}
eval     : (%, List OP, List %, SY) -> %
++ eval(x, [s1,...,sm], [f1,...,fm], y) replaces every
if R has SemiGroup then
Monoid
-- the following line is necessary because of a compiler bug
"**"   : (%, N) -> %
++ x**n returns x * x * x * ... * x (n times).
isTimes: % -> Union(List %, "failed")
isExpt : % -> Union(Record(var:K,exponent:Z),"failed")
if R has Group then Group
if R has AbelianSemiGroup then
AbelianMonoid
isPlus: % -> Union(List %, "failed")
isMult: % -> Union(Record(coef:Z, var:K),"failed")
if R has AbelianGroup then AbelianGroup
if R has Ring then
Ring
RetractableTo PR
PartialDifferentialRing SY
FullyLinearlyExplicitRingOver R
coerce    : MP -> %
++ coerce(p) returns p as an element of %.
numer     : %  -> MP
++ numer(f) returns the
++ numerator of f viewed as a polynomial in the kernels over R
++ if R is an integral domain. If not, then numer(f) = f viewed
++ as a polynomial in the kernels over R.
-- DO NOT change this meaning of numer!  MB 1/90
numerator : % -> %
++ numerator(f) returns the numerator of \spad{f} converted to %.
isExpt:(%,OP) -> Union(Record(var:K,exponent:Z),"failed")
isExpt:(%,SY) -> Union(Record(var:K,exponent:Z),"failed")
isPower   : % -> Union(Record(val:%,exponent:Z),"failed")
eval: (%, List SY, List N, List(% -> %)) -> %
++ eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm]) replaces
eval: (%, List SY, List N, List(List % -> %)) -> %
++ eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm]) replaces
++ for any a1,...,am.
eval: (%, SY, N, List % -> %) -> %
++ eval(x, s, n, f) replaces every \spad{s(a1,...,am)**n} in x
++ by \spad{f(a1,...,am)} for any a1,...,am.
eval: (%, SY, N, % -> %) -> %
++ eval(x, s, n, f) replaces every \spad{s(a)**n} in x
if R has CharacteristicZero then CharacteristicZero
if R has CharacteristicNonZero then CharacteristicNonZero
if R has CommutativeRing then
Algebra R
if R has IntegralDomain then
Field
RetractableTo Fraction PR
convert   : Factored % -> %
++ convert(f1\^e1 ... fm\^em) returns \spad{(f1)\^e1 ... (fm)\^em}
++ as an element of %, using formal kernels
denom     : %  -> MP
++ denom(f) returns the denominator of f viewed as a
++ polynomial in the kernels over R.
denominator : % -> %
++ denominator(f) returns the denominator of \spad{f} converted to %.
"/"       : (MP, MP) -> %
++ p1/p2 returns the quotient of p1 and p2 as an element of %.
coerce    : Q  -> %
++ coerce(q) returns q as an element of %.
coerce    : Polynomial Q -> %
++ coerce(p) returns p as an element of %.
coerce    : Fraction Polynomial Q -> %
++ coerce(f) returns f as an element of %.
univariate: (%, K) -> Fraction SparseUnivariatePolynomial %
++ univariate(f, k) returns f viewed as a univariate fraction in k.
if R has RetractableTo Z then RetractableTo Fraction Z
import BasicOperatorFunctions1(%)

-- these are needed in Ring only, but need to be declared here
-- because of compiler bug: if they are declared inside the Ring
-- case, then they are not visible inside the IntegralDomain case.
smpIsMult : MP -> Union(Record(coef:Z, var:K),"failed")
smpret    : MP -> Union(PR, "failed")
smpeval   : (MP, List K, List %) -> %
smpsubst  : (MP, List K, List %) -> %
smpderiv  : (MP, SY) -> %
smpunq    : (MP, List SY, Boolean) -> %
kerderiv  : (K, SY)  -> %
kderiv    : K -> List %
opderiv   : (OP, N) -> List(List % -> %)
smp2O     : MP -> O
bestKernel: List K -> K
worse?    : (K, K) -> Boolean
diffArg   : (List %, OP, N) -> List %
substArg  : (OP, List %, Z, %) -> %
dispdiff  : List % -> Record(name:O, sub:O, arg:List O, level:N)
ddiff     : List % -> O
diffEval  : List % -> %
dfeval    : (List %, K) -> %
smprep    : (List SY, List N, List(List % -> %), MP) -> %
diffdiff  : (List %, SY) -> %
diffdiff0 : (List %, SY, %, K, List %) -> %
subs      : (% -> %, K) -> %
symsub    : (SY, Z) -> SY
kunq      : (K, List SY, Boolean) -> %
pushunq   : (List SY, List %) -> List %
notfound  : (K -> %, List K, K) -> %

equaldiff : (K,K)->Boolean
debugA: (List % ,List %,Boolean) -> Boolean
opdiff := operator("%diff"::SY)$CommonOperators() opquote := operator("applyQuote"::SY)$CommonOperators

ground? x                == retractIfCan(x)@Union(R,"failed") case R
ground  x                == retract x
coerce(x:SY):%             == kernel(x)@K :: %
retract(x:%):SY            == symbolIfCan(retract(x)@K)::SY
applyQuote(s:SY, x:%)      == applyQuote(s, [x])
applyQuote(s, x, y)        == applyQuote(s, [x, y])
applyQuote(s, x, y, z)     == applyQuote(s, [x, y, z])
applyQuote(s, x, y, z, t)  == applyQuote(s, [x, y, z, t])
applyQuote(s:SY, l:List %) == opquote concat(s::%, l)
belong? op                 == op = opdiff or op = opquote
subs(fn, k) == kernel(operator k,[fn x for x in argument k]$List(%)) operator op == is?(op, "%diff"::SY) => opdiff is?(op, "%quote"::SY) => opquote error "Unknown operator" if R has ConvertibleTo InputForm then INP==>InputForm import MakeUnaryCompiledFunction(%, %, %) indiff: List % -> INP pint : List INP-> INP differentiand: List % -> % differentiand l == eval(first l, retract(second l)@K, third l) pint l == convert concat(convert("D"::SY)@INP, l) indiff l == r2:= convert([convert("::"::SY)@INP,convert(third l)@INP,convert("Symbol"::SY)@address@hidden INP)@INP pint [convert(differentiand l)@INP, r2] eval(f:%, s:SY) == eval(f, [s]) eval(f:%, s:OP, g:%, x:SY) == eval(f, [s], [g], x) eval(f:%, ls:List OP, lg:List %, x:SY) == eval(f, ls, [compiledFunction(g, x) for g in lg]) setProperty(opdiff,SPECIALINPUT,indiff@(List % -> InputForm) pretend None) variables x == l := empty()$List(SY)
for k in tower x repeat
if ((s := symbolIfCan k) case SY) then l := concat(s::SY, l)
reverse_! l

retractIfCan(x:%):Union(SY, "failed") ==
(k := retractIfCan(x)@Union(K,"failed")) case "failed" => "failed"
symbolIfCan(k::K)

if R has Ring then
import UserDefinedPartialOrdering(SY)

-- cannot use new()$Symbol because of possible re-instantiation gendiff := "%%0"::SY characteristic() == characteristic()$R
coerce(k:K):%       == k::MP::%
symsub(sy, i)       == concat(string sy, convert(i)@String)::SY
numerator x         == numer(x)::%
eval(x:%, s:SY, n:N, f:% -> %)     == eval(x,[s],[n],[f first #1])
eval(x:%, s:SY, n:N, f:List % -> %) == eval(x, [s], [n], [f])
eval(x:%, l:List SY, f:List(List % -> %)) == eval(x, l, new(#l, 1), f)

elt(op:OP, args:List %) ==
unary? op and ((od? := has?(op, ODD)) or has?(op, EVEN)) and
leadingCoefficient(numer first args) < 0 =>
x := op(- first args)
od? => -x
x
elt(op, args)$ExpressionSpace_&(%) eval(x:%, s:List SY, n:List N, l:List(% -> %)) == eval(x, s, n, [f first #1 for f in l]$List(List % -> %))

-- op(arg)**m ==> func(arg)**(m quo n) * op(arg)**(m rem n)
smprep(lop, lexp, lfunc, p) ==
(v := mainVariable p) case "failed" => p::%
symbolIfCan(k := v::K) case SY => p::%
g := (op := operator k)
(arg := [eval(a,lop,lexp,lfunc) for a in argument k]$List(%)) q := map(eval(#1::%, lop, lexp, lfunc), univariate(p, k))$SparseUnivariatePolynomialFunctions2(MP, %)
(n := position(name op, lop)) < minIndex lop => q g
a:%  := 0
f    := eval((lfunc.n) arg, lop, lexp, lfunc)
e    := lexp.n
while q ^= 0 repeat
m  := degree q
qr := divide(m, e)
t1 := f ** (qr.quotient)::N
t2 := g ** (qr.remainder)::N
a  := a + leadingCoefficient(q) * t1 * t2
q  := reductum q
a

dispdiff l ==
s := second(l)::O
t := third(l)::O
a := argument(k := retract(first l)@K)
is?(k, opdiff) =>
rec := dispdiff a
i   := position(s, rec.arg)
rec.arg.i := t
[rec.name,
hconcat(rec.sub, hconcat(","::SY::O, (i+1-minIndex a)::O)),
rec.arg, (zero?(rec.level) => 0; rec.level + 1)]
i   := position(second l, a)
m   := [x::O for x in a]$List(O) m.i := t [name(operator k)::O, hconcat(","::SY::O, (i+1-minIndex a)::O), m, (empty? rest a => 1; 0)] ddiff l == rec := dispdiff l opname := zero?(rec.level) => sub(rec.name, rec.sub) differentiate(rec.name, rec.level) prefix(opname, rec.arg) substArg(op, l, i, g) == z := copy l z.i := g kernel(op, z) diffdiff(l, x) == f := kernel(opdiff, l) diffdiff0(l, x, f, retract(f)@K, empty()) diffdiff0(l, x, expr, kd, done) == op := operator(k := retract(first l)@K) gg := second l u := third l arg := argument k ans:% := 0 if (not member?(u,done)) and (ans := differentiate(u,x))^=0 then ans := ans * kernel(opdiff, [subst(expr, [kd], [kernel(opdiff, [first l, gg, gg])]), gg, u]) done := concat(gg, done) is?(k, opdiff) => ans + diffdiff0(arg, x, expr, k, done) for i in minIndex arg .. maxIndex arg for b in arg repeat if (not member?(b,done)) and (bp:=differentiate(b,x))^=0 then g := symsub(gendiff, i)::% ans := ans + bp * kernel(opdiff, [subst(expr, [kd], [kernel(opdiff, [substArg(op, arg, i, g), gg, u])]), g, b]) ans dfeval(l, g) == eval(differentiate(first l, symbolIfCan(g)::SY), g, third l) diffEval l == k:K g := retract(second l)@K ((u := retractIfCan(first l)@Union(K, "failed")) case "failed") or (u case K and symbolIfCan(k := u::K) case SY) => dfeval(l, g) op := operator k (ud := derivative op) case "failed" => -- possible trouble -- make sure it is a dummy var dumm:%:=symsub(gendiff,1)::% ss:=subst(l.1,l.2=dumm) -- output(nl::OutputForm)$OutputPackage
-- output("fixed"::OutputForm)$OutputPackage nl:=[ss,dumm,l.3] kernel(opdiff, nl) (n := position(second l,argument k)) < minIndex l => dfeval(l,g) d := ud::List(List % -> %) eval((d.n)(argument k), g, third l) diffArg(l, op, i) == n := i - 1 + minIndex l z := copy l z.n := g := symsub(gendiff, n)::% [kernel(op, z), g, l.n] opderiv(op, n) == -- one? n => (n = 1) => g := symsub(gendiff, n)::% [kernel(opdiff,[kernel(op, g), g, first #1])] [kernel(opdiff, diffArg(#1, op, i)) for i in 1..n] kderiv k == zero?(n := #(args := argument k)) => empty() op := operator k grad := (u := derivative op) case "failed" => opderiv(op, n) u::List(List % -> %) if #grad ^= n then grad := opderiv(op, n) [g args for g in grad] -- SPECIALDIFF contains a map (List %, Symbol) -> % -- it is used when the usual chain rule does not apply, -- for instance with implicit algebraics. kerderiv(k, x) == (v := symbolIfCan(k)) case SY => v::SY = x => 1 0 (fn := property(operator k, SPECIALDIFF)) case None => ((fn::None) pretend ((List %, SY) -> %)) (argument k, x) +/[g * differentiate(y,x) for g in kderiv k for y in argument k] smpderiv(p, x) == map(retract differentiate(#1::PR, x), p)::% + +/[differentiate(p,k)::% * kerderiv(k, x) for k in variables p] coerce(p:PR):% == map(#1::%, #1::%, p)$PolynomialCategoryLifting(
IndexedExponents SY, SY, R, PR, %)

worse?(k1, k2) ==
(u := less?(name operator k1,name operator k2)) case "failed" =>
k1 < k2
u::Boolean

bestKernel l ==
empty? rest l => first l
a := bestKernel rest l
worse?(first l, a) => a
first l

smp2O p ==
(r:=retractIfCan(p)@Union(R,"failed")) case R =>r::R::OutputForm
a :=
userOrdered?() => bestKernel variables p
mainVariable(p)::K
outputForm(map(#1::%, univariate(p,
a))$SparseUnivariatePolynomialFunctions2(MP, %), a::OutputForm) smpsubst(p, lk, lv) == map(match(lk, lv, #1, notfound(subs(subst(#1, lk, lv), #1), lk, #1))$ListToMap(K,%),
#1::%,p)$PolynomialCategoryLifting(IndexedExponents K,K,R,MP,%) smpeval(p, lk, lv) == map(match(lk, lv, #1, notfound(map(eval(#1, lk, lv), #1), lk, #1))$ListToMap(K,%),
#1::%,p)$PolynomialCategoryLifting(IndexedExponents K,K,R,MP,%) -- this is called on k when k is not a member of lk notfound(fn, lk, k) == empty? setIntersection(tower(f := k::%), lk) => f fn k if R has ConvertibleTo InputForm then pushunq(l, arg) == empty? l => [eval a for a in arg] [eval(a, l) for a in arg] kunq(k, l, givenlist?) == givenlist? and empty? l => k::% is?(k, opquote) and (member?(s:=retract(first argument k)@SY, l) or empty? l) => interpret(convert(concat(convert(s)@InputForm, [convert a for a in pushunq(l, rest argument k) address@hidden(InputForm)))@InputForm)$InputFormFunctions1(%)
(operator k) pushunq(l, argument k)

smpunq(p, l, givenlist?) ==
givenlist? and empty? l => p::%
map(kunq(#1, l, givenlist?), #1::%,
p)$PolynomialCategoryLifting(IndexedExponents K,K,R,MP,%) smpret p == "or"/[symbolIfCan(k) case "failed" for k in variables p] => "failed" map(symbolIfCan(#1)::SY::PR, #1::PR, p)$PolynomialCategoryLifting(IndexedExponents K, K, R, MP, PR)

isExpt(x:%, op:OP) ==
(u := isExpt x) case "failed" => "failed"
is?((u::Record(var:K, exponent:Z)).var, op) => u
"failed"

isExpt(x:%, sy:SY) ==
(u := isExpt x) case "failed" => "failed"
is?((u::Record(var:K, exponent:Z)).var, sy) => u
"failed"

if R has RetractableTo Z then
smpIsMult p ==
--            (u := mainVariable p) case K and one?
degree(q:=univariate(p,u::K))
(u := mainVariable p) case K and (degree(q:=univariate(p,u::K))=1)
case R)
and (n := retractIfCan(r::R)@Union(Z, "failed")) case Z =>
[n::Z, u::K]
"failed"

evaluate(opdiff, diffEval)

debugA(a1,a2,t) ==
-- uncomment for debugging
-- output(hconcat
[a1::OutputForm,a2::OutputForm,t::OutputForm])$OutputPackage t equaldiff(k1,k2) == a1:=argument k1 a2:=argument k2 -- check the operator res:=operator k1 = operator k2 not res => debugA(a1,a2,res) -- check the evaluation point res:= (a1.3 = a2.3) not res => debugA(a1,a2,res) -- check all the arguments res:= (a1.1 = a2.1) and (a1.2 = a2.2) res => debugA(a1,a2,res) -- check the substituted arguments (subst(a1.1,[retract(a1.2)@K],[a2.2]) = a2.1) => debugA(a1,a2,true) debugA(a1,a2,false) setProperty(opdiff,SPECIALEQUAL, equaldiff@((K,K) -> Boolean) pretend None) setProperty(opdiff, SPECIALDIFF, diffdiff@((List %, SY) -> %) pretend None) setProperty(opdiff, SPECIALDISP, ddiff@(List % -> OutputForm) pretend None) if not(R has IntegralDomain) then mainKernel x == mainVariable numer x kernels x == variables numer x retract(x:%):R == retract numer x retract(x:%):PR == smpret(numer x)::PR retractIfCan(x:%):Union(R, "failed") == retract numer x retractIfCan(x:%):Union(PR, "failed") == smpret numer x eval(x:%, lk:List K, lv:List %) == smpeval(numer x, lk, lv) subst(x:%, lk:List K, lv:List %) == smpsubst(numer x, lk, lv) differentiate(x:%, s:SY) == smpderiv(numer x, s) coerce(x:%):OutputForm == smp2O numer x if R has ConvertibleTo InputForm then eval(f:%, l:List SY) == smpunq(numer f, l, true) eval f == smpunq(numer f, empty(), false) eval(x:%, s:List SY, n:List N, f:List(List % -> %)) == smprep(s, n, f, numer x) isPlus x == (u := isPlus numer x) case "failed" => "failed" [p::% for p in u::List(MP)] isTimes x == (u := isTimes numer x) case "failed" => "failed" [p::% for p in u::List(MP)] isExpt x == (u := isExpt numer x) case "failed" => "failed" r := u::Record(var:K, exponent:NonNegativeInteger) [r.var, r.exponent::Z] isPower x == (u := isExpt numer x) case "failed" => "failed" r := u::Record(var:K, exponent:NonNegativeInteger) [r.var::%, r.exponent::Z] if R has ConvertibleTo Pattern Z then convert(x:%):Pattern(Z) == convert numer x if R has ConvertibleTo Pattern Float then convert(x:%):Pattern(Float) == convert numer x if R has RetractableTo Z then isMult x == smpIsMult numer x if R has CommutativeRing then r:R * x:% == r::MP::% * x if R has IntegralDomain then par : % -> % mainKernel x == mainVariable(x)$QF
kernels x                       == variables(x)$QF univariate(x:%, k:K) == univariate(x, k)$QF
isPlus x                        == isPlus(x)$QF isTimes x == isTimes(x)$QF
isExpt x                        == isExpt(x)$QF isPower x == isPower(x)$QF
denominator x                   == denom(x)::%
coerce(q:Q):%                   == (numer q)::MP / (denom q)::MP
coerce(q:Fraction PR):%         == (numer q)::% / (denom q)::%
coerce(q:Fraction Polynomial Q) == (numer q)::% / (denom q)::%
retract(x:%):PR                == retract(retract(x)@Fraction(PR))
retract(x:%):Fraction(PR) == smpret(numer x)::PR / smpret(denom x)::PR
retract(x:%):R == (retract(numer x)@R exquo retract(denom x)@R)::R

coerce(x:%):OutputForm ==
--        one?(denom x) => smp2O numer x
((denom x) = 1) => smp2O numer x
smp2O(numer x) / smp2O(denom x)

retractIfCan(x:%):Union(R, "failed") ==
(n := retractIfCan(numer x)@Union(R, "failed")) case "failed" or
(d := retractIfCan(denom x)@Union(R, "failed")) case "failed"
or (r := n::R exquo d::R) case "failed" => "failed"
r::R

eval(f:%, l:List SY) ==
smpunq(numer f, l, true) / smpunq(denom f, l, true)

if R has ConvertibleTo InputForm then
eval f ==
smpunq(numer f, empty(), false) / smpunq(denom f, empty(), false)

eval(x:%, s:List SY, n:List N, f:List(List % -> %)) ==
smprep(s, n, f, numer x) / smprep(s, n, f, denom x)

differentiate(f:%, x:SY) ==
(smpderiv(numer f, x) * denom(f)::% -
numer(f)::% * smpderiv(denom f, x))
/ (denom(f)::% ** 2)

eval(x:%, lk:List K, lv:List %) ==
smpeval(numer x, lk, lv) / smpeval(denom x, lk, lv)

subst(x:%, lk:List K, lv:List %) ==
smpsubst(numer x, lk, lv) / smpsubst(denom x, lk, lv)

par x ==
(r := retractIfCan(x)@Union(R, "failed")) case R => x
paren x

convert(x:Factored %):% ==
par(unit x) * */[par(f.factor) ** f.exponent for f in factors x]

retractIfCan(x:%):Union(PR, "failed") ==
(u := retractIfCan(x)@Union(Fraction PR,"failed")) case "failed"
=> "failed"
retractIfCan(u::Fraction(PR))

retractIfCan(x:%):Union(Fraction PR, "failed") ==
(n := smpret numer x) case "failed" => "failed"
(d := smpret denom x) case "failed" => "failed"
n::PR / d::PR

coerce(p:Polynomial Q):% ==
map(#1::%, #1::%,
p)$PolynomialCategoryLifting(IndexedExponents SY, SY, Q, Polynomial Q, %) if R has RetractableTo Z then coerce(x:Fraction Z):% == numer(x)::MP / denom(x)::MP isMult x == (u := smpIsMult numer x) case "failed" or (v := retractIfCan(denom x)@Union(R, "failed")) case "failed" or (w := retractIfCan(v::R)@Union(Z, "failed")) case "failed" => "failed" r := u::Record(coef:Z, var:K) (q := r.coef exquo w::Z) case "failed" => "failed" [q::Z, r.var] if R has ConvertibleTo Pattern Z then convert(x:%):Pattern(Z) == convert(numer x) / convert(denom x) if R has ConvertibleTo Pattern Float then convert(x:%):Pattern(Float) == convert(numer x) / convert(denom x) @ \section{package FS2 FunctionSpaceFunctions2} <<package FS2 FunctionSpaceFunctions2>>= )abbrev package FS2 FunctionSpaceFunctions2 ++ Lifting of maps to function spaces ++ Author: Manuel Bronstein ++ Date Created: 22 March 1988 ++ Date Last Updated: 3 May 1994 ++ Description: ++ This package allows a mapping R -> S to be lifted to a mapping ++ from a function space over R to a function space over S; FunctionSpaceFunctions2(R, A, S, B): Exports == Implementation where R, S: Join(Ring, OrderedSet) A : FunctionSpace R B : FunctionSpace S K ==> Kernel A P ==> SparseMultivariatePolynomial(R, K) Exports ==> with map: (R -> S, A) -> B ++ map(f, a) applies f to all the constants in R appearing in \spad{a}. Implementation ==> add smpmap: (R -> S, P) -> B smpmap(fn, p) == map(map(map(fn, #1), #1)$ExpressionSpaceFunctions2(A,B),fn(#1)::B,
p)\$PolynomialCategoryLifting(IndexedExponents K, K, R, P, B)

if R has IntegralDomain then
if S has IntegralDomain then
map(f, x) == smpmap(f, numer x) / smpmap(f, denom x)
else
map(f, x) == smpmap(f, numer x) * (recip(smpmap(f, denom x))::B)
else
map(f, x) == smpmap(f, numer x)

@
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--
--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.
@
<<*>>=

-- SPAD files for the functional world should be compiled in the
-- following order:
--
--   op  kl  FSPACE  expr funcpkgs

<<category ES ExpressionSpace>>
<<package ES1 ExpressionSpaceFunctions1>>
<<package ES2 ExpressionSpaceFunctions2>>
<<category FS FunctionSpace>>
<<package FS2 FunctionSpaceFunctions2>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}