$SPAD/src/algebra fspace.spad

Manuel Bronstein October 2, 2018

Abstract

Contents

1 category ES ExpressionSpace 3 2 ES.lsp BOOTSTRAP 10 3 ES-.lsp BOOTSTRAP 13 4 package ES1 ExpressionSpaceFunctions1 30 5 package ES2 ExpressionSpaceFunctions2 31 6 category FS FunctionSpace 32 7 package FS2 FunctionSpaceFunctions2 46 8 License 47

1 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)

2 ES.lsp BOOTSTRAP

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 ES category which we can write into the MID directory. We compile the lisp code and copy the ES.o file to the 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)

3 ES-.lsp BOOTSTRAP

ES- depends on ES. We need to break this cycle to build the algebra. So we keep a cached copy of the translated ES- category which we can write into the MID directory. We compile the lisp code and copy the ES-.o file to the 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| |$|)
(PROG (#1=#:G82377 |k| #2=#:G82378)
(RETURN
(SEQ
(COND
((SPADCALL |s|
(PROGN
(LETT #1# NIL |ES-;freeOf?;SSB;9|)
(SEQ
(LETT |k| NIL |ES-;freeOf?;SSB;9|)
(LETT #2# (|ES-;listk| |x| |$|) |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 |$| 67)))
(DEFUN |ES-;eval;SSMS;21| (|x| |s| |f| |$|)
(SPADCALL |x|
(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
(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| |$|)
(PROG (#1=#:G82412 |s| #2=#:G82413)
(RETURN
(SEQ
(SPADCALL |x|
(PROGN
(LETT #1# NIL |ES-;eval;SLLS;26|)
(SEQ
(LETT |s| NIL |ES-;eval;SLLS;26|)
(LETT #2# |ls| |ES-;eval;SLLS;26|)
G190
(COND
 ((OR (ATOM #2#)
      (PROGN (LETT |s| (CAR #2#) |ES-;eval;SLLS;26|) NIL)
 ) (GO G191)))
(SEQ
 (EXIT
  (LETT #1#
   (CONS (SPADCALL |s| (QREFELT |$| 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|
(SPADCALL
(LETT |k| (|SPADfirst| |l|) |ES-;mainKernel;SU;29|)
(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| |$|)
(PROG (|l| |k| #1=#:G82458 |u| |s0| |n| |arg| |t| |s|)
(RETURN
(SEQ
(LETT |s|
(SPADCALL
(LETT |l| (SPADCALL |f| (QREFELT |$| 38)) |ES-;allKernels|)
(QREFELT |$| 29))
|ES-;allKernels|)
(SEQ
(LETT |k| NIL |ES-;allKernels|)
(LETT #1# |l| |ES-;allKernels|)
G190
(COND
((OR (ATOM #1#)
  (PROGN (LETT |k| (CAR #1#) |ES-;allKernels|) NIL))
(GO G191)))
(SEQ
(LETT |t|
(SEQ
(LETT |u|
(SPADCALL
 (SPADCALL |k| (QREFELT |$| 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
(SPADCALL
(SPADCALL |op| |l|
(|+| 1
(SPADCALL
(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 |$| 107)) |ES-;retract;SK;34|)
(EXIT
 (COND
  ((OR (QEQCAR |k| 1)
       (NULL
        (SPADCALL
         (SPADCALL (QCDR |k|) (QREFELT |$| 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 |$| 107)) |ES-;retractIfCan;SU;35|)
(EXIT
 (COND
  ((OR (QEQCAR |k| 1)
       (NULL
        (SPADCALL
         (SPADCALL (QCDR |k|) (QREFELT |$| 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 |k|
          (SPADCALL "%paren" (QREFELT |$| 9))
          (QREFELT |$| 112))
         (SPADCALL |ky|
          (|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
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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
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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|)))

4 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"

5 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
      ++ \spad{k = op(a1,...,an)}.
  == add
    map(f, k) ==
      (operator(operator k)$F) [f x for x in argument k]$List(F)

6 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(foo, x) returns \spad{'foo(x)}.
     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 %) -> %
++ applyQuote(foo, [x1,...,xn]) returns \spad{'foo(x1,...,xn)}.
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)}
  ++ with \spad{y} replaced by \spad{a} for any \spad{a}.
eval     : (%, List OP, List %, SY) -> %
  ++ eval(x, [s1,...,sm], [f1,...,fm], y) replaces every
  ++ \spad{si(a)} in x by \spad{fi(y)}
  ++ with \spad{y} replaced by \spad{a} for any \spad{a}.
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")
  ++ isTimes(p) returns \spad{[a1,...,an]}
  ++ if \spad{p = a1*...*an} and \spad{n > 1}.
isExpt : % -> Union(Record(var:K,exponent:Z),"failed")
  ++ isExpt(p) returns \spad{[x, n]} if \spad{p = x**n}
  ++ and \spad{n <> 0}.
if R has Group then Group
if R has AbelianSemiGroup then
AbelianMonoid
isPlus: % -> Union(List %, "failed")
  ++ isPlus(p) returns \spad{[m1,...,mn]}
  ++ if \spad{p = m1 +...+ mn} and \spad{n > 1}.
isMult: % -> Union(Record(coef:Z, var:K),"failed")
  ++ isMult(p) returns \spad{[n, x]} if \spad{p = n * x}
  ++ and \spad{n <> 0}.
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(p,op) returns \spad{[x, n]} if \spad{p = x**n}
++ and \spad{n <> 0} and \spad{x = op(a)}.
isExpt:(%,SY) -> Union(Record(var:K,exponent:Z),"failed")
++ isExpt(p,f) returns \spad{[x, n]} if \spad{p = x**n}
++ and \spad{n <> 0} and \spad{x = f(a)}.
isPower   : % -> Union(Record(val:%,exponent:Z),"failed")
++ isPower(p) returns \spad{[x, n]} if \spad{p = x**n}
++ and \spad{n <> 0}.
eval: (%, List SY, List N, List(% -> %)) -> %
++ eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm]) replaces
++ every \spad{si(a)**ni} in x by \spad{fi(a)} for any \spad{a}.
eval: (%, List SY, List N, List(List % -> %)) -> %
++ eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm]) replaces
++ every \spad{si(a1,...,an)**ni} in x by \spad{fi(a1,...,an)}
++ 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
++ by \spad{f(a)} for any \spad{a}.
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
++ created using a \spadfunFrom{paren}{ExpressionSpace}.
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
add
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]$List(%))
operator op ==
  is?(op, "%diff"::SY) => opdiff
  is?(op, "%quote"::SY) => opquote
  error "Unknown operator"
if R has ConvertibleTo InputForm then
  INP==>InputForm
  import MakeUnaryCompiledFunction(%, %, %)
  indiff: List % -> INP
  pint  : List INP-> INP
  differentiand: List % -> %
  differentiand l    == eval(first l, retract(second l)@K, third l)
  pint l  == convert concat(convert("D"::SY)@INP, l)
  indiff l ==
     r2:= convert([convert("::"::SY)@INP,convert(third l)@INP,convert("Symbol"::SY)@INP]@List INP)@INP
     pint [convert(differentiand l)@INP, r2]
  eval(f:%, s:SY)            == eval(f, [s])
  eval(f:%, s:OP, g:%, x:SY) == eval(f, [s], [g], x)
  eval(f:%, ls:List OP, lg:List %, x:SY) ==
    eval(f, ls, [compiledFunction(g, x) for g in lg])
  setProperty(opdiff,SPECIALINPUT,indiff@(List % -> InputForm) pretend None)
variables x ==
  l := empty()$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)$PolynomialCategoryLifting(IndexedExponents K,K,R,MP,%)
-- this is called on k when k is not a member of lk
notfound(fn, lk, k) ==
empty? setIntersection(tower(f := k::%), lk) => f
fn k
if R has ConvertibleTo InputForm then
pushunq(l, arg) ==
   empty? l => [eval a for a in arg]
   [eval(a, l) for a in arg]
kunq(k, l, givenlist?) ==
  givenlist? and empty? l => k::%
  is?(k, opquote) and
    (member?(s:=retract(first argument k)@SY, l) or empty? l) =>
      interpret(convert(concat(convert(s)@InputForm,
        [convert a for a in pushunq(l, rest argument k)
           ]@List(InputForm)))@InputForm)$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)
      and zero?(leadingCoefficient reductum q)
        and ((r:=retractIfCan(leadingCoefficient q)@Union(R,"failed"))
           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)

7 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)

8 License

⟨license⟩≡

--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--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.

⟨* ⟩≡ ⟨license⟩

-- 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 ⟩

References^{[1] nothing}

49