/usr/share/axiom-20170501/src/algebra/ES.spad is in axiom-source 20170501-3.
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++ 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;
ExpressionSpace() : Category == SIG where
N ==> NonNegativeInteger
K ==> Kernel %
OP ==> BasicOperator
SY ==> Symbol
PAREN ==> "%paren"::SY
BOX ==> "%box"::SY
DUMMYVAR ==> "%dummyVar"
OS ==> OrderedSet
RT ==> RetractableTo(K)
IE ==> InnerEvalable(K, %)
EV ==> Evalable(%)
SIG ==> Join(OS,RT,IE,EV) 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 (that is, 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], [(y:List %):% +-> f(first y)])
eval(x:%, s:OP, f:% -> %) ==
eval(x, [s], [(y:List %):% +-> f(first y)])
subst(x:%, e:Equation %) == subst(x, [e])
eval(x:%, ls:List OP, lf:List(% -> %)) ==
eval(x, ls, [y +-> f(first y) for f in lf]$List(List % -> %))
eval(x:%, ls:List SY, lf:List(% -> %)) ==
eval(x, ls, [y +-> f(first y) for f in lf]$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, for example, [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)
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