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## Unevaluated mathematical expressions

Author: Bill Page

Date Created: 3 February 2016

## Description:

Symbolic is a domain that represents expressions involving symbolic operations. These expressions remain unevaluated and are displayed as entered unless otherwise specified. Rewrite rules implementing various common axioms can be specified that will be automatically applied when expressions are evaluated by the interpreter. Full simplification of expressions if/when desired is performed by means of interpretation in the Expression domain.

SymbolicExpression is a Symbolic domain with automatic rewrite rules that approximate Expression itself.

Keywords: expression, evaluation, operator, function.

## From:

spad
-- Type safe representation
rep(x) ==> x @ % pretend Rep
per(x) ==> x @ Rep pretend %
-- Applicative-style macros
Maybe(S) ==> Union(S,"failed")
applyIf(f,x) ==> if (%r:=x) case "failed" then "failed" else f(%r)
returnIf(x) ==> if not((%r:=x) case "failed") then return %r
orReturn(x) ==> if x case "failed" then return "failed"
SIN ==> InputForm
Pair SIN ==> Record(left:SIN,right:SIN)
)abbrev package SINF SINFunctions
-- This should be built-in to InputForm, i.e. InputForm should satisfy Comparable
SINFunctions(): with
smaller? : (SIN,SIN) -> Boolean
rank: (Symbol,Integer) -> SIN
rank: Symbol -> SIN
== add
rank(x:Symbol,n:Integer):SIN == PUT(x,'rank,convert(n)@SIN)$Lisp rank(x:Symbol):SIN == GET(x,'rank)$Lisp
--null < list < integer < float < symbol rank < symbol rank
smaller1?(x:SIN,y:SIN):Boolean ==
--output("smaller1? ",paren [x::OutputForm,y::OutputForm])$OutputPackage null? y => return false list? y and null? x => return true -- depth first list? y and list? x => --output("cdr x, cdr y", paren [(cdr x)::OutputForm,(cdr y)::OutputForm])$OutputPackage
(# cdr x > # cdr y or (
# cdr x = # cdr y and (smaller?(cdr x,cdr y) or
not smaller?(cdr y, cdr x) and smaller?(car x,car y))))
integer? y => null? x or list? x
or integer? x and integer x < integer y
float? y => null? x or list? x or integer? x
or float? x and float x < float y
symbol? y => null? x or list? x or integer? x or float? x
or symbol? y and smaller?(rank symbol x, rank symbol y)
or not smaller?(rank symbol y, rank symbol x) and symbol x < symbol y
false
smaller?(x:SIN,y:SIN):Boolean ==
if smaller1?(x,y) and smaller1?(y,x) then
output("smaller? ",paren [x::OutputForm,y::OutputForm])$OutputPackage error("smaller? total order failed") else return smaller1?(x,y) )abbrev category SYMCAT SymbolicCategory SymbolicCategory(R : Comparable) : Category == Exports where F ==> Expression R Exports ==> Join(FunctionSpace R, CoercibleTo OutputForm, ConvertibleTo SIN) with axioms: () -> List Equation SIN -- Do we want these even if R not a Ring, e.g. Symbol ? _+: (%,%) -> % _-: (%,%) -> % _-: % -> % _*: (%,%) -> % _*:(PositiveInteger,%) -> % _*:(NonNegativeInteger,%) -> % _/: (%,%) -> % _^: (%,%) -> % _^: (%,Integer) -> % nthRoot: (%, %) -> % nthRoot: (%, Integer) -> % if R has IntegralDomain then Ring AlgebraicallyClosedFunctionSpace R TranscendentalFunctionCategory CombinatorialOpsCategory LiouvillianFunctionCategory SpecialFunctionCategory coerce: Polynomial R -> % -- transformations expand: % -> % if % has FunctionSpace(Integer) then factor: % -> % if % has TranscendentalFunctionCategory and R has GcdDomain then simplify : % -> % -- coercions coerce: Pi -> % coerce: % -> F coerce: F -> % convert: Equation % -> Equation SIN convert: Equation SIN -> Equation % coerce: SIN -> % -- equal?: Equation % -> Boolean )abbrev category AXIOM Axiom ++ Properties and axioms for symbolic rewrite rules Axiom(): Category == Type with axioms: () -> List Equation SIN )abbrev category COMAX CommutativeAxiom ++ Commutative CommutativeAxiom(): Category == Axiom with rewriteCommutative: ((SIN,SIN)->SIN, SIN, SIN) -> Maybe(SIN) add rewriteCommutative(f:(SIN,SIN)->SIN, x:SIN, y:SIN): Maybe(SIN) == if smaller?(y,x)$SINFunctions then
--output("rewriteCommutative: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage return f(y,x) else return "failed" )abbrev category COMAX+ CommutativeAxiom+ CommutativeAxiom_+():Category == CommutativeAxiom )abbrev category COMAX1 CommutativeAxiom* CommutativeAxiom_*():Category == CommutativeAxiom )abbrev category COMAX2 CommutativeAxiom** CommutativeAxiom_*_*():Category == CommutativeAxiom )abbrev category COMAX3 CommutativeAxiom/\ CommutativeAxiom_/_\():Category == CommutativeAxiom )abbrev category ALTAX AntiCommutativeAxiom ++ AntiCommutative AntiCommutativeAxiom(): Category == Axiom with rewriteAntiCommutative: (SIN->SIN,(SIN,SIN)->SIN, SIN, SIN) -> Maybe(SIN) add rewriteAntiCommutative(m:SIN->SIN,f:(SIN,SIN)->SIN, x:SIN, y:SIN): Maybe(SIN) == if symbol? x and symbol? y and smaller?(0,rank symbol x)$SINFunctions
and smaller?(0,rank symbol y)$SINFunctions then -- non-commutative symbols anti-commute if smaller?(y,x)$SINFunctions then return m(f(y,x))
if not smaller?(x,y) then return 0
-- everythning else commutes
if smaller?(y,x)$SINFunctions then return f(y,x) "failed" )abbrev category DISAX DistributiveAxiom ++ Distributive DistributiveAxiom(): Category == Axiom with rewriteDistributive: (SIN->Maybe Pair SIN, (SIN,SIN)->SIN, (SIN,SIN)->SIN, SIN, SIN) -> Maybe SIN add rewriteDistributive(terms:SIN->Maybe Pair SIN, f:(SIN,SIN)->SIN, g:(SIN,SIN)->SIN, x:SIN, y:SIN): Maybe(SIN) == if (t:=terms(x)) case Pair SIN then return f(g(t.left,y),g(t.right,y)) if (t:=terms(y)) case Pair SIN then return f(g(x,t.left),g(x,t.right)) "failed" )abbrev category ASSAX AssociativeAxiom ++ Associative AssociativeAxiom(): Category == Axiom with rewriteAssociative: (SIN->Maybe Pair SIN, (SIN,SIN)->SIN, SIN, SIN) -> Maybe SIN add rewriteAssociative(terms:SIN->Maybe Pair SIN, f:(SIN,SIN)->SIN, x:SIN, y:SIN):Maybe(SIN) == --output("rewriteAssociative: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage
if (t:=terms(y)) case Pair SIN then
--output("terms: ",paren [coerce(t.left)@OutputForm,coerce(t.right)@OutputForm])$OutputPackage return f(f(x,t.left),t.right) "failed" )abbrev category ASSAX+ AssociativeAxiom+ AssociativeAxiom_+():Category == AssociativeAxiom )abbrev category ASSAXS AssociativeAxiom* AssociativeAxiom_*():Category == AssociativeAxiom )abbrev category IDAXL IdentityLeftAxiom ++ Left Identities IdentityLeftAxiom(): Category == Axiom with rewriteIdentityLeft: (SIN,SIN,SIN)-> Maybe SIN add rewriteIdentityLeft(u:SIN,x:SIN,y:SIN): Maybe SIN == --output("rewriteIdentityLeft: ",paren [u::OutputForm,x::OutputForm,y::OutputForm])$OutputPackage
if x=u then y else "failed"
)abbrev category IDAXR IdentityRightAxiom
++ Right Identities
IdentityRightAxiom(): Category == Axiom with
rewriteIdentityRight: (SIN,SIN,SIN)-> Maybe SIN
add
rewriteIdentityRight(u:SIN,x:SIN,y:SIN): Maybe SIN ==
if y=u then x else "failed"
)abbrev category IDAX IdentityAxiom
++ Identities
IdentityAxiom(): Category == Join(IdentityLeftAxiom,IdentityRightAxiom)
)abbrev category INVAX InverseAxiom
++ Inverse
InverseAxiom(): Category == Axiom with
rewriteInverse: (SIN->SIN,SIN,SIN,SIN)-> Maybe SIN
rewriteInverseInverse: (SIN->SIN,SIN) -> Maybe SIN
rewriteInverseBinary: (SIN->SIN,(SIN,SIN)->SIN,SIN,SIN)-> Maybe SIN
add
rewriteInverse(inv:SIN->SIN,u:SIN,x:SIN,y:SIN): Maybe SIN ==
if inv(x) = y then return u
if inv(y) = x then return u
"failed"
rewriteInverseInverse(inv:SIN->SIN,x:SIN) ==
--output("rewriteInverseInverse: ",coerce(x)@OutputForm)$OutputPackage if list? x and inv(car cdr x)=x then return car cdr x "failed" rewriteInverseBinary(inv:SIN->SIN,f:(SIN,SIN)->SIN,x:SIN,y:SIN): Maybe SIN == if list? y and y = inv(car cdr y) then return f(x,inv y) "failed" )abbrev category DBLAX DoublesAxiom ++ Doubling DoublesAxiom(): Category == Axiom with rewriteDoubles: ((SIN,SIN)->SIN,SIN,SIN) -> Maybe SIN add rewriteDoubles(mul:(SIN,SIN)->SIN,x:SIN,y:SIN):Maybe SIN == if integer? x and integer? y then return convert(integer(x)+integer(y)) if float? x and float? y then return convert(float(x)+float(y)) if x = y then return mul(convert(2),x) "failed" )abbrev category CANAX CancelsAxiom ++ Cancelling CancelsAxiom(): Category == Axiom with rewriteCancels: (SIN,SIN) -> Maybe SIN add rewriteCancels(x:SIN,y:SIN):Maybe SIN == if integer? x and integer? y then return convert(integer(x)-integer(y)) if float? x and float? y then return convert(float(x)-float(y)) if x = y then return 0 "failed" )abbrev category SQAX SquaresAxiom ++ Squares SquaresAxiom(): Category == Axiom with rewriteSquares: ((SIN,SIN)->SIN,SIN, SIN) -> Maybe SIN add rewriteSquares(pow:(SIN,SIN)->SIN,x:SIN, y:SIN): Maybe SIN == --if eq(x,0) then return 0 if eq(y,0) then return 0 if integer? x and integer? y then return convert(integer(x)*integer(y)) if float? x and float? y then return convert(float(x)*float(y)) if x = y then return pow(x,convert(2)) "failed" )abbrev category DIVAX DividesAxiom ++ Divides DividesAxiom(): Category == Axiom with rewriteDivides: (SIN, SIN) -> Maybe SIN add rewriteDivides(x:SIN, y:SIN): Maybe SIN == if eq(y,0) then return "failed" if eq(x,0) then return 0 if integer? x and integer? y then orReturn(r:=integer(x) exquo integer(y)) return convert r if float? x and float? y then return convert(float(x)/float(y)) if x = y then return 1 "failed" )abbrev domain SYMB Symbolic ++ Unevaluated mathematical expressions ++ Author: Bill Page ++ Date Created: 4 December 2016 ++ Description: ++ Expressions involving symbolic functions that remain unevaluated and are ++ displayed as entered. Evaluation is delayed until an operation is performed ++ which requires a canonical form such as testing for equality or ++ simplification. The domain depends directly internally on the Expression ++ domain for evaluation of expressions. ++ Keywords: expression, evaluation, operator, function. Symbolic(R:Comparable, AxiomList:Type):SymbolicCategory(R) == Implementation where F ==> Expression R K ==> Kernel % KF ==> Kernel F SMP ==> SparseMultivariatePolynomial(R, K) LIFT ==> PolynomialCategoryLifting(IndexedExponents(KF),KF,R,SparseMultivariatePolynomial(R,KF),SMP) --AF ==> AlgebraicFunction(R, F) --EF ==> ElementaryFunction(R, F) CF ==> CombinatorialFunction(R, F) --LF ==> LiouvillianFunction(R, F) --AN ==> AlgebraicNumber --KAN ==> Kernel AN --ESD ==> ExpressionSpace_&(F) --FSD ==> FunctionSpace_&(F, R) Implementation ==> add Rep := SIN --rep(x:%):Rep == x pretend Rep --per(x:Rep):% == x pretend % smaller?(x:%,y:%):Boolean == smaller?(rep x, rep y)$SINFunctions
if AxiomList has Axiom then
-- It might be nice if this was displayed like Symbolic(Integer,None)
-- but that would be recursive and require bootstrap mode.
axioms() == axioms()$AxiomList else axioms() == [] belong? op == true if R has AbelianSemiGroup then 0:% == per 0$Rep
zero? x == x = 0
if R has SemiGroup then
1:% == per 1$Rep one? x == x = 1 hash(x:%):SingleInteger == hash(rep x)$SIN
equal?(eq:Equation %):Boolean == not smaller?(lhs eq,rhs eq) and not smaller?(rhs eq,lhs eq)
(x:% = y:%):Boolean == not smaller?(x,y) and not smaller?(y,x)
subst(x:%,e:Equation %):% == per SUBST(rep rhs e,rep lhs e,rep x)$Lisp subst(x:%,ks:List Kernel %, vs:List %) == if #ks > 1 then return subst(subst(x,equation(first(ks)::%,first(vs))),rest ks, rest vs) else return subst(x,equation(first(ks)::%,first(vs))) subst(x:%, es:List Equation %):% == if #es > 1 then return subst(subst(x, first es),rest es) else subst(x, first es) -- eval(x:%,k:Kernel(%),s:%):% == --output("eval 1:", paren [x::OutputForm, k::OutputForm, s::OutputForm])$OutputPackage
coerce rep subst(x,[k],[s])
eval(x:%,ks:List Kernel %, vs:List %) ==
--output("eval 2:", paren [x::OutputForm, ks::OutputForm, vs::OutputForm])$OutputPackage r:=x -- Use temporary symbols in case of parallel substitutions ts:List Kernel % := [kernel(new()$Symbol) for k in ks]
for k in ks for t in ts repeat
r:=eval(r,k,coerce t)
for t in ts for v in vs repeat
r:=eval(r,t,v)
return r
if R has ConvertibleTo SIN then
eval(x:%):% ==
--output("eval 0:", x::OutputForm)$OutputPackage coerce(convert(x)@SIN)@% eval(x:%,a:%,b:%):% == --output("eval 3:", paren [x::OutputForm, a::OutputForm, b::OutputForm])$OutputPackage
eval(x, retract a, b)
eval(x:%,e:Equation %):% == eval(x,lhs e,rhs e)
--
-- need to handle subscripted Symbol
retractIfCan(x:%):Maybe(Kernel %) ==
--output("retractIfCan Kernel: ",x::OutputForm)$OutputPackage if atom? rep x and symbol? rep x then return kernel symbol rep x if list? rep x and symbol? car rep x then if list? cdr rep x then s := symbol car rep x -- Why not '_- ?? if not member?(s,['_+,'_*,'_/])$List(Symbol) then
k:Kernel(%) := kernel(operator(s)$CommonOperators, destruct cdr rep x, #cdr(rep x)::NonNegativeInteger) return k else return "failed" else return "failed" else return "failed" retractIfCan(x:%):Maybe(R) == --output("retractIfCan R: ",x::OutputForm)$OutputPackage
if list? rep x and # cdr rep x = 0 then error "stop"
if R is Integer then
if integer? rep x then return (integer rep x) pretend R
if R is DoubleFloat then
if float? rep x then return (float rep x) pretend R
if R is Symbol then
if symbol? rep x then return (symbol rep x) pretend R
--
return "failed"
retractIfCan(x:%):Maybe(Symbol) ==
--output("retractIfCan Symbol: ",x::OutputForm)$OutputPackage if symbol? rep x then symbol rep x else "failed" retract(x:%):Symbol == --output("retract Symbol: ",x::OutputForm)$OutputPackage
s:=retractIfCan(x)@Maybe(Symbol)
if s case Symbol then return s
error "not a Symbol"
--
isPlus1(x:SIN):Maybe(Pair SIN) ==
list? x and symbol? car x and symbol car x = '_+ => [car cdr x, car cdr cdr x]
"failed"
isTimes1(x:SIN):Maybe(Pair SIN) ==
list? x and symbol? car x and symbol car x = '_* => [car cdr x, car cdr cdr x]
"failed"
isDivide1(x:SIN):Maybe(Pair SIN) ==
list? x and symbol? car x and symbol car x = '_/ => [car cdr x, car cdr cdr x]
"failed"
if R has IntegralDomain then
kk2(k:KF):SMP ==
ak:List % := [coerce x for x in argument k]
return coerce kernel(operator k,ak,position k)$K rr2(r:R):SMP == coerce r -- Symbolic is syntactical! numerator(x:%):% == r := isDivide1 rep x if r case Pair SIN then return per r.left else return x denominator(x:%):% == r := isDivide1 rep x if r case Pair SIN then return per r.right else return 1 makeKernel(x:%):Kernel % == --output("makeKernel: ",x::OutputForm)$OutputPackage
if symbol? rep x then return kernel symbol rep x
k:Maybe(Kernel %):=retractIfCan(x)
if k case Kernel % then
return k
else
-- We must wrap up this term in a paren box for SMP.
return coerce kernel(operator('%box)$CommonOperators,[x],1)$K
smpTerm(x:%):SparseMultivariatePolynomial(R,Kernel %) ==
--output("smpTerm: ",x::OutputForm)$OutputPackage t:=isTimes x if t case List % then return reduce("*",[smpTerm(i) for i in t]) t2:=isPower x if t2 case Record(val:%,exponent:Integer) then if t2.exponent > 0 then return smpTerm(t2.val)^(t2.exponent::NonNegativeInteger) -- this test is not quite adequate if R is Integer and integer? rep x then return coerce integer rep x --if atom?(rep x) and not symbol?(rep x) then -- return rr2 retract x else return coerce makeKernel(x) numer(x:%):SparseMultivariatePolynomial(R,Kernel %) == --output("numer: ",x::OutputForm)$OutputPackage
r:=numerator(x)
s:=isPlus r
if s case List % then
return reduce("+",[smpTerm(i) for i in s])
else
return smpTerm(r)
denom(x:%):SparseMultivariatePolynomial(R,Kernel %) ==
--output("denom: ",x::OutputForm)$OutputPackage r:=denominator(x) s:=isPlus r if s case List % then return reduce("+",[smpTerm(i) for i in s]) else return smpTerm(r) (x:SMP / y:SMP):% == coerce(x) / coerce(y) isPlus(x:%):Maybe(List %) == r := isPlus1 rep x if r case Pair SIN then r1:=isPlus per(r.left) r2:=isPlus per(r.right) if r1 case List % then if r2 case List % then concat(r1,r2) else concat(r1, [per(r.right)]) else if r2 case List % then concat([per(r.left)],r2) else [per(r.left),per(r.right)] else "failed" factorials(x:%):% == coerce factorials(coerce x)$CF
factorials(x:%, n:Symbol):% == coerce factorials(coerce x, n)$CF expand(x:%):% == n := isPlus numerator(coerce x)$F
if n case List F then
d:% := denominator(x)
if d=1 then return reduce("+",[coerce(s) for s in n])
return reduce("+",[coerce(s)/d for s in n])
else
return x
--
if R has ConvertibleTo SIN then
coerce(x:SMP):% == per convert(x)$SMP coerce(x:Polynomial R):% == per convert(x)$Polynomial(R)
coerce(x:Fraction R):% == per convert(x)$Fraction(R) if R has RetractableTo Integer or R has RetractableTo Fraction Integer then retractIfCan(x:%):Maybe(Fraction Integer) == r:Maybe(R):=retractIfCan(x) if R has RetractableTo Fraction Integer then return retractIfCan(r::R)@Maybe(Fraction Integer) if R has RetractableTo Integer then r2:Maybe(Integer):=retractIfCan(r::R) if r2 case Integer then return r2::Integer::Fraction Integer return "failed" if R has PatternMatchable Integer then patternMatch(x : %, p : Pattern Integer, l : PatternMatchResult(Integer, %)) == --output("patternMatch: ",paren [x::OutputForm,p::OutputForm,l::OutputForm])$OutputPackage
patternMatch(x, p, l)$PatternMatchFunctionSpace(Integer, R, %) if R has PatternMatchable Float then patternMatch(x : %, p : Pattern Float, l : PatternMatchResult(Float, %)) == patternMatch(x, p, l)$PatternMatchFunctionSpace(Float, R, %)
if R has SemiGroup then
isTimes(x:%):Maybe(List %) ==
r := isTimes1 rep x
if r case Pair SIN then
r1:=isTimes per(r.left)
r2:=isTimes per(r.right)
if r1 case List % then
if r2 case List % then
concat(r1,r2)
else
concat(r1, per(r.right))
else
if r2 case List % then
concat(per(r.left),r2)
else
[per(r.left),per(r.right)]
else
"failed"
if F has FunctionSpace(Integer) then
factor(x:%):% ==
(factor(numer(coerce x)$F)::OutputForm pretend SIN / factor(denom(coerce x)$F)::OutputForm pretend SIN)$SIN coerce(x:Pi):% == per convert(x)$Pi
coerce(x:Symbol):% == per convert(x)$Symbol coerce(k:Kernel %):% == coerce name(k) coerceOutputForm1:(Rep,Boolean)->OutputForm coerceOutputForm(x:Rep):OutputForm ==coerceOutputForm1(x,false) coerceOutputForm1(x:Rep,topLevel:Boolean):OutputForm == if null? x then return empty()$OutputForm
if atom? x then return (x pretend OutputForm)
if list? x and symbol? car x then
s := symbol car x
-- scripted?
if #string(s)>2 and ord string(s).1 = 42 and digit? string(s).2 then
return coerce(x pretend Symbol)
if s='%paren then
return paren coerceOutputForm car cdr x
if s='%box then
-- make box visible (for now)
return bracket coerceOutputForm car cdr x
if s='float then
return interpret(x)$InputFormFunctions1(Float)::OutputForm if s='factorial then if list? car(cdr x) then return postfix(outputForm '_!, paren coerceOutputForm car cdr x) else return postfix(outputForm '_!, coerceOutputForm car cdr x) if s='sum or s='summation then return sum(coerceOutputForm car cdr x, coerceOutputForm car cdr cdr x) if s='product then return prod(coerceOutputForm car cdr x, coerceOutputForm car cdr cdr x) if s='binomial then return binomial(coerceOutputForm car cdr x, coerceOutputForm car cdr cdr x) if s='sqrt then return root coerceOutputForm car cdr x if s='nthRoot then return root(coerceOutputForm car cdr x, coerceOutputForm car cdr cdr x) if s='_^ and car cdr cdr x = convert(1/2) then return root(coerceOutputForm car cdr x) if s='exp and car(cdr x)=1 then return outputForm '%e if member?(s,['_+,'_-,'_*,'_/,'_^])$List(Symbol) and #cdr(x)>1 then
if s='_/ then -- assume )set output fraction vertical (should check flag?)
return infix(outputForm s, [coerceOutputForm1(i,true) for i in destruct cdr x])
else if AxiomList has AssociativeAxiom or topLevel then
return infix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x])
else -- ( ) required if non-associative
return paren infix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x])
else
if list? car(cdr x) or s ~= '_- then
return prefix(outputForm s,
[coerceOutputForm(i) for i in destruct cdr x])
else
return hconcat([outputForm s, hspace(1), coerceOutputForm car cdr x])
else
return hconcat(message("what is:")$OutputForm,x pretend OutputForm) coerce(x:%):OutputForm == coerceOutputForm1(rep x,true) -- +0 is a hack to avoid premature conversion (do we need it?) hack(x:%):Rep == binary(convert('_+),[rep x,0$Rep])
coerce(x:%):F ==
--output("coerce: ",x::OutputForm)$OutputPackage v:List Symbol := variables x if #v = 0 then return interpret(hack x)$InputFormFunctions1(F)
-- we need to substitute variables with unknown modes
s:List Symbol := [new()$Symbol for vi in v] ks:List Equation % := [equation(coerce(vi)$%, coerce(si)$%) for vi in v for si in s] sk:List Equation F := [equation(coerce(si)$F, coerce(vi)$F) for vi in v for si in s] return subst(interpret(hack subst(x,ks))$InputFormFunctions1(F),sk)
if R has ConvertibleTo SIN then
coerce(x:F):% == per convert(x)$F coerce(x:R):% == per convert(x)$R
convert(x:%):SIN == rep x
coerce(x:SIN):% == interpret(hack x)$InputFormFunctions1(%) -- factorial(x:%):% == per convert([convert('factorial)@SIN,rep x]) binomial(n:%, m:%):% == binary(convert('binomial),[n::Rep,m::Rep])$SIN
permutation(n:%, m:%):%   == binary(convert('permutation),[n::Rep,m::Rep])$SIN -- sum(x : %, n : Symbol):% == per binary(convert('sum),[rep x,convert(n)@SIN])$SIN
sum(x : %, s : SegmentBinding %):% == per binary(convert('sum),[rep x,convert(s)@SIN])$SIN summation(x : %, n : Symbol):% == per binary(convert('summation),[rep x,convert(n)@SIN])$SIN
summation(x : %, s : SegmentBinding %):% == per binary(convert('summation),[rep x,convert(s)@SIN])$SIN product(x : %, n : Symbol):% == per binary(convert('product),[rep x,convert(n)@SIN])$SIN
product(x : %, s : SegmentBinding %):%   == per binary(convert('product),[rep x,convert(s)@SIN])$SIN -- We don't want the InputForm auto-simplifications because we may or -- may not be doing our own simplifications. power: (Rep,Rep) -> Rep power(x:Rep,y:Rep):Rep == binary(convert('_^),[x,y])$SIN
(x:% ^ y:%):% == per power(rep x,rep y)
(x:% ^ y:PositiveInteger):% == expt(x, y)$RepeatedSquaring(%) (x:% ^ y:NonNegativeInteger):% == if y = 0 then x^0 else x ^ y::PositiveInteger (x:% ^ y:Integer):% == if y < 0 then 1/(x ^ (-y)::PositiveInteger) else x ^ y::NonNegativeInteger (x:% ^ y:Fraction Integer):% == nthRoot(x,denom(y))^numer(y) isPower1(x:SIN):Maybe(Pair SIN) == list? x and symbol? car x and symbol car x = '_^ => [car cdr x, car cdr cdr x] "failed" isPower(x:%):Maybe(Record(val:%,exponent:Integer)) == r:=isPower1(rep x) if r case Pair SIN then if integer?(r.right) then return [per r.left,integer(r.right)] "failed" -- plus: (Rep,Rep) -> Rep minus: (Rep,Rep) -> Rep uminus: Rep -> Rep times(x:Rep,y:Rep):Rep == --output("times: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage
if AxiomList has IdentityAxiom then
returnIf rewriteIdentityLeft(1,x,y)
returnIf rewriteIdentityRight(1,x,y)
if AxiomList has AssociativeAxiom_* then
returnIf rewriteAssociative(isTimes1,times,x,y)
if AxiomList has CommutativeAxiom_* then
returnIf rewriteCommutative(times,x,y)
if AxiomList has AntiCommutativeAxiom then
returnIf rewriteAntiCommutative(uminus,times,x,y)
if AxiomList has DistributiveAxiom then
returnIf rewriteDistributive(isPlus1,plus,times,x,y)
if AxiomList has SquaresAxiom then
returnIf rewriteSquares(power,x,y)
--if AxiomList has AssociativeAxiom then
--  if (t:=isTimes1(x)) case Pair SIN then
--    return times(t.left,times(t.right,y))
binary(convert('_*),[x,y])$SIN (x:% * y:%):% == per times(rep x,rep y) -- (x:PositiveInteger * y:%):% == double(x,y)$RepeatedDoubling(%)
(x:NonNegativeInteger * y:%):% ==
if x=0 then 0*y
else x::PositiveInteger * y
(x:Integer * y:%):% ==
if x < 0 then
(-x)::PositiveInteger * y
else
x::NonNegativeInteger * y
--
plus(x:Rep,y:Rep):Rep ==
--output("plus: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentityLeft(0,x,y) returnIf rewriteIdentityRight(0,x,y) if AxiomList has AssociativeAxiom_+ then returnIf rewriteAssociative(isPlus1,plus,x,y) if AxiomList has CommutativeAxiom_+ then returnIf rewriteCommutative(plus,x,y) if AxiomList has InverseAxiom then returnIf rewriteInverse(uminus,0,x,y) returnIf rewriteInverseBinary(uminus,minus,x,y) if AxiomList has DoublesAxiom then returnIf rewriteDoubles(times,x,y) binary(convert('_+),[x,y])$SIN
(x:% + y:%):% == per plus(rep x,rep y)
minus(x:Rep,y:Rep):Rep ==
--output("minus: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage if AxiomList has IdentityRightAxiom then returnIf rewriteIdentityRight(0,x,y) if AxiomList has CancelsAxiom then returnIf rewriteCancels(x,y) if AxiomList has InverseAxiom then returnIf rewriteInverseBinary(uminus,plus,x,y) binary(convert('_-),[x,y])$SIN
(x:% - y:%):% == per minus(rep x,rep y)
uminus(x:Rep):Rep ==
if AxiomList has InverseAxiom then
returnIf rewriteInverseInverse(uminus,x)
convert([convert('_-)@SIN,x])
_-(x:%):% == per uminus rep x
div(x:Rep,y:Rep):Rep ==
--output("divide: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage if AxiomList has IdentityRightAxiom then returnIf rewriteIdentityRight(1,x,y) if AxiomList has DividesAxiom then returnIf rewriteDivides(x,y) binary(convert('_/),[x,y])$SIN
(x:% / y:%):% == per div(rep x,rep y)
elt(f:BasicOperator,x:List %):% ==
--output("elt: ", paren [f::OutputForm,x::OutputForm])$OutputPackage --output("elt: properties: ",properties(f)::OutputForm)$OutputPackage
if not property(f,'%symbol) case "failed" then
return per convert name f
else
-- Should we actually evaluate the operator??
return per convert(concat(convert(name f)@SIN,[rep i for i in x]))
-- Elementary Functions
acos(x:%):% == per convert([convert('acos)@SIN,rep x])
acosh(x:%):% == per convert([convert('acosh)@SIN,rep x])
acot(x:%):% == per convert([convert('acot)@SIN,rep x])
acoth(x:%):% == per convert([convert('acoth)@SIN,rep x])
acsc(x:%):% == per convert([convert('acsc)@SIN,rep x])
acsch(x:%):% == per convert([convert('acsch)@SIN,rep x])
asec(x:%):% == per convert([convert('asec)@SIN,rep x])
asech(x:%):% == per convert([convert('asech)@SIN,rep x])
asin(x:%):% == per convert([convert('asin)@SIN,rep x])
asinh(x:%):% == per convert([convert('asinh)@SIN,rep x])
atan(x:%):% == per convert([convert('atan)@SIN,rep x])
atanh(x:%):% == per convert([convert('atanh)@SIN,rep x])
cos(x:%):% == per convert([convert('cos)@SIN,rep x])
cosh(x:%):% == per convert([convert('cosh)@SIN,rep x])
cot(x:%):% == per convert([convert('cot)@SIN,rep x])
coth(x:%):% == per convert([convert('coth)@SIN,rep x])
csc(x:%):% == per convert([convert('csc)@SIN,rep x])
csch(x:%):% == per convert([convert('csch)@SIN,rep x])
exp(x:%):% == per convert([convert('exp)@SIN,rep x])
log(x:%):% == per convert([convert('log)@SIN,rep x])
nthRoot(x:%, n:Integer) == binary(convert('nthRoot),[rep x,n::Rep])$SIN nthRoot(x:%, n:%) == binary(convert('nthRoot),[rep x,n::Rep])$SIN
--pi():% == per convert([convert('pi)@SIN])
sec(x:%):% == per convert([convert('sec)@SIN,rep x])
sech(x:%):% == per convert([convert('sech)@SIN,rep x])
sin(x:%):% == per convert([convert('sin)@SIN,rep x])
sinh(x:%):% == per convert([convert('sinh)@SIN,rep x])
sqrt(x:%):% == per convert([convert('sqrt)@SIN,rep x])
tan(x:%):% == per convert([convert('tan)@SIN,rep x])
tanh(x:%):% == per convert([convert('tanh)@SIN,rep x])
--
retract(x:%):R == retract(coerce x)$F variables1(x:Rep):Set Symbol == if null? x then return empty() if atom? x and symbol? x then return set [symbol x] if list? x and symbol? car x then if list? cdr x then return reduce("union", [variables1(i) for i in destruct cdr x],empty()) else return variables1 car cdr x return empty() variables(x:%):List Symbol == members variables1(rep x) -- still need to handle non symbols and scripted symbols kernels1(x:Rep):Set Kernel % == if null? x then return empty() if atom? x and symbol? x then return set [kernel symbol x] if list? x and symbol? car x then if list? cdr x then s := symbol car x if member?(s,['_+,'_-,'_*,'_/])$List(Symbol) then
r:Set Kernel % := reduce("union",
[kernels1(i) for i in destruct cdr x])
return r
else
k:Kernel(%) := kernel(operator(s)$CommonOperators, destruct cdr x, #cdr(x)::NonNegativeInteger) return set [k] else return kernels1 car cdr x return empty() kernels(x:%):List Kernel % == members kernels1(rep x) if F has TranscendentalFunctionCategory and R has GcdDomain then simplify(x:%):% == coerce(simplify(coerce x)$TranscendentalManipulations(R,F))
convert(eq:Equation %): Equation SIN == equation(convert lhs eq, convert rhs eq)
convert(eq:Equation SIN): Equation % == equation(convert lhs eq, convert rhs eq)
)abbrev domain COM+ Commutative+
Commutative_+():CommutativeAxiom_+ == add
import Symbolic(Integer,None)
axioms() ==
x:Symbolic(Integer,None) := coerce 'x
y:Symbolic(Integer,None) := coerce 'y
[ convert equation (x+y, y+x) ]
)abbrev domain COMS Commutative*
Commutative_*():CommutativeAxiom_* == add
import Symbolic(Integer,None)
axioms() ==
x:Symbolic(Integer,None) := coerce 'x
y:Symbolic(Integer,None) := coerce 'y
[ convert equation (x*y, y*x) ]
)abbrev domain ALT AntiCommutative
AntiCommutative():AntiCommutativeAxiom == add
import Symbolic(Integer,None)
axioms() ==
x := coerce 'x
y := coerce 'y
[ convert equation (x+y, y+x),
convert equation (x*y, y*x) ]
)abbrev domain DIS Distributive
Distributive():DistributiveAxiom == add
import Symbolic(Integer,None)
axioms() ==
x := coerce 'x
y := coerce 'y
z := coerce 'z
[ convert equation (x*(y+z), (x*y)+(x*z)) ]
)abbrev domain ASS+ Associative+
Associative_+():AssociativeAxiom_+ == add
import Symbolic(Integer,None)
x := coerce 'x
y := coerce 'y
z := coerce 'z
axioms() ==
[ convert equation ((x+y)+z, x+(y+z)) ]
)abbrev domain ASSS Associative*
Associative_*():AssociativeAxiom_* == add
import Symbolic(Integer,None)
x := coerce 'x
y := coerce 'y
z := coerce 'z
axioms() ==
[ convert equation ((x*y)*z, (x*(y*z))) ]
)abbrev domain LID LeftIdentities
LeftIdentities():IdentityLeftAxiom == add
import Symbolic(Integer,None)
x := coerce 'x
axioms() ==
[ convert equation (0+x, x),
convert equation (1*x, x) ]
)abbrev domain RID RightIdentities
RightIdentities():IdentityRightAxiom == add
import Symbolic(Integer,None)
x := coerce 'x
axioms() ==
[ convert     equation (x+0, x),
convert     equation (x-0, x),
convert   equation (x*1, x),
convert equation (x/1, x) ]
)abbrev domain INV Inverse
Inverse():InverseAxiom == add
import Symbolic(Integer,None)
x := coerce 'x
axioms() ==
[ convert equation (x + (-x), 0),
convert equation (x * (1/x), 1) ]
)abbrev domain DBL Doubles
Doubles():DoublesAxiom == add
import Symbolic(Integer,None)
x := coerce 'x
axioms() == [ convert equation (x+x, 2*x) ]
)abbrev domain CAN Cancels
Cancels():CancelsAxiom == add
import Symbolic(Integer,None)
x := coerce 'x
axioms() == [ convert equation (x-x, 0) ]
)abbrev domain SQ Squares
Squares():SquaresAxiom == add
import Symbolic(Integer,None)
x := coerce 'x
axioms() == [ convert equation (x*x, x^2) ]
)abbrev domain DIV Divides
Divides():DividesAxiom == add
import Symbolic(Integer,None)
x := coerce 'x
axioms() == [ convert equation (x/x, 1) ]
)abbrev domain AND /\
++ Conjunction of rules
_/_\(A:Axiom,B:Axiom):Axiom with
if A has CommutativeAxiom_+ then CommutativeAxiom_+
if B has CommutativeAxiom_+ then CommutativeAxiom_+
if A has CommutativeAxiom_* then CommutativeAxiom_*
if B has CommutativeAxiom_* then CommutativeAxiom_*
if A has AntiCommutativeAxiom then AntiCommutativeAxiom
if B has AntiCommutativeAxiom then AntiCommutativeAxiom
if A has DistributiveAxiom then DistributiveAxiom
if B has DistributiveAxiom then DistributiveAxiom
if A has AssociativeAxiom_+ then AssociativeAxiom_+
if B has AssociativeAxiom_+ then AssociativeAxiom_+
if A has AssociativeAxiom_* then AssociativeAxiom_*
if B has AssociativeAxiom_* then AssociativeAxiom_*
if A has IdentityAxiom then IdentityAxiom
if B has IdentityAxiom then IdentityAxiom
if A has IdentityRightAxiom then IdentityRightAxiom
if B has IdentityRightAxiom then IdentityRightAxiom
if A has IdentityLeftAxiom then IdentityLeftAxiom
if B has IdentityLeftAxiom then IdentityLeftAxiom
if A has InverseAxiom then InverseAxiom
if B has InverseAxiom then InverseAxiom
if A has DoublesAxiom then DoublesAxiom
if B has DoublesAxiom then DoublesAxiom
if A has CancelsAxiom then CancelsAxiom
if B has CancelsAxiom then CancelsAxiom
if A has SquaresAxiom then SquaresAxiom
if B has SquaresAxiom then SquaresAxiom
if A has DividesAxiom then DividesAxiom
if B has DividesAxiom then DividesAxiom
== add
axioms() == concat(axioms()$A,axioms()$B)
)abbrev domain ID Identities
Identities():IdentityAxiom == LeftIdentities /\ RightIdentities
)abbrev domain SEXPR SymbolicExpression
SymbolicExpression(R:Comparable) : SymbolicCategory(R) == Implementation where
F ==> Expression R
Implementation ==> Symbolic(R, Commutative_+/\Commutative_*/\Distributive/\Associative_+/\Associative_*/\Identities/\Inverse/\Doubles/\Cancels/\Squares/\Divides) add
-- use equality from Expression(R)
(x:% = y:%):Boolean ==
--output("=",paren [x::OutputForm,y::OutputForm])$OutputPackage coerce(x)@F = coerce(y)@F )abbrev package SYMPKG1 SymbolicFunctions1 SymbolicFunctions1(R: Join(Comparable,ConvertibleTo SIN), A:SymbolicCategory R, S: Join(Comparable,ConvertibleTo SIN), B:SymbolicCategory S): with convert: B -> A == add -- +0 is a hack to avoid premature conversion hack(x:B):SIN == binary(convert('_+),[convert x,0::SIN]) convert(x:B):A == v:List Symbol := variables(x) if #v = 0 then return interpret(hack x)$InputFormFunctions1(A)
-- we need to substitute variables with unknown modes
s:List Symbol := [new()$Symbol for vi in v] ks:List Equation B := [equation(coerce(vi)$B, coerce(si)$B) for vi in v for si in s] sk:List Equation A := [equation(coerce(si)$A, coerce(vi)$A) for vi in v for si in s] return subst(interpret(hack subst(x,ks))$InputFormFunctions1(A),sk)
spad
   Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/828338597200291887-25px001.spad
using old system compiler.
SINF abbreviates package SINFunctions
------------------------------------------------------------------------
initializing NRLIB SINF for SINFunctions
compiling into NRLIB SINF
compiling exported rank : (Symbol,Integer) -> InputForm
Time: 0 SEC.
compiling exported rank : Symbol -> InputForm
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compiling local smaller1? : (InputForm,InputForm) -> Boolean
Time: 0.01 SEC.
compiling exported smaller? : (InputForm,InputForm) -> Boolean
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(time taken in buildFunctor:  0)
;;;     ***       |SINFunctions| REDEFINED
;;;     ***       |SINFunctions| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor SINFunctions
Time: 0.01 seconds
finalizing NRLIB SINF
Processing SINFunctions for Browser database:
--->-->SINFunctions(constructor): Not documented!!!!
--->-->SINFunctions((smaller? ((Boolean) (InputForm) (InputForm)))): Not documented!!!!
--->-->SINFunctions((rank ((InputForm) (Symbol) (Integer)))): Not documented!!!!
--->-->SINFunctions((rank ((InputForm) (Symbol)))): Not documented!!!!
--->-->SINFunctions(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SINF.NRLIB/SINF.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/SINF.NRLIB/SINF.fasl written
; compilation finished in 0:00:00.048
------------------------------------------------------------------------
SINFunctions is now explicitly exposed in frame initial
SINFunctions will be automatically loaded when needed from
/var/aw/var/LatexWiki/SINF.NRLIB/SINF
SYMCAT abbreviates category SymbolicCategory
------------------------------------------------------------------------
initializing NRLIB SYMCAT for SymbolicCategory
compiling into NRLIB SYMCAT
;;;     ***       |SymbolicCategory| REDEFINED
Time: 0 SEC.
finalizing NRLIB SYMCAT
Processing SymbolicCategory for Browser database:
--->-->SymbolicCategory(constructor): Not documented!!!!
--->-->SymbolicCategory((axioms ((List (Equation (InputForm)))))): Not documented!!!!
--->-->SymbolicCategory((+ (% % %))): Not documented!!!!
--->-->SymbolicCategory((- (% % %))): Not documented!!!!
--->-->SymbolicCategory((- (% %))): Not documented!!!!
--->-->SymbolicCategory((* (% % %))): Not documented!!!!
--->-->SymbolicCategory((* (% (PositiveInteger) %))): Not documented!!!!
--->-->SymbolicCategory((* (% (NonNegativeInteger) %))): Not documented!!!!
--->-->SymbolicCategory((/ (% % %))): Not documented!!!!
--->-->SymbolicCategory((^ (% % %))): Not documented!!!!
--->-->SymbolicCategory((^ (% % (Integer)))): Not documented!!!!
--->-->SymbolicCategory((nthRoot (% % %))): Not documented!!!!
--->-->SymbolicCategory((nthRoot (% % (Integer)))): Not documented!!!!
--->-->SymbolicCategory((coerce (% (Polynomial R)))): Not documented!!!!
--->-->SymbolicCategory((expand (% %))): Not documented!!!!
--->-->SymbolicCategory((factor (% %))): Not documented!!!!
--->-->SymbolicCategory((simplify (% %))): Not documented!!!!
--->-->SymbolicCategory((coerce (% (Pi)))): Not documented!!!!
--->-->SymbolicCategory((coerce ((Expression R) %))): Not documented!!!!
--->-->SymbolicCategory((coerce (% (Expression R)))): Not documented!!!!
--->-->SymbolicCategory((convert ((Equation (InputForm)) (Equation %)))): Not documented!!!!
--->-->SymbolicCategory((convert ((Equation %) (Equation (InputForm))))): Not documented!!!!
--->-->SymbolicCategory((coerce (% (InputForm)))): Not documented!!!!
--->-->SymbolicCategory((equal? ((Boolean) (Equation %)))): Not documented!!!!
--->-->SymbolicCategory(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SYMCAT.NRLIB/SYMCAT.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/SYMCAT.NRLIB/SYMCAT.fasl written
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
SymbolicCategory is now explicitly exposed in frame initial
SymbolicCategory will be automatically loaded when needed from
/var/aw/var/LatexWiki/SYMCAT.NRLIB/SYMCAT
AXIOM abbreviates category Axiom
------------------------------------------------------------------------
initializing NRLIB AXIOM for Axiom
compiling into NRLIB AXIOM
;;;     ***       |Axiom| REDEFINED
Time: 0 SEC.
finalizing NRLIB AXIOM
Processing Axiom for Browser database:
--------constructor---------
--->-->Axiom((axioms ((List (Equation (InputForm)))))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/AXIOM.NRLIB/AXIOM.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/AXIOM.NRLIB/AXIOM.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
Axiom is now explicitly exposed in frame initial
Axiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/AXIOM.NRLIB/AXIOM
COMAX abbreviates category CommutativeAxiom
------------------------------------------------------------------------
initializing NRLIB COMAX for CommutativeAxiom
compiling into NRLIB COMAX
;;;     ***       |CommutativeAxiom| REDEFINED
Time: 0 SEC.
COMAX- abbreviates domain CommutativeAxiom&
------------------------------------------------------------------------
initializing NRLIB COMAX- for CommutativeAxiom&
compiling into NRLIB COMAX-
compiling exported rewriteCommutative : ((InputForm,InputForm) -> InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |CommutativeAxiom&| REDEFINED
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Cumulative Statistics for Constructor CommutativeAxiom&
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finalizing NRLIB COMAX-
Processing CommutativeAxiom& for Browser database:
--------constructor---------
--->-->CommutativeAxiom&((rewriteCommutative ((Union (InputForm) failed) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/COMAX-.NRLIB/COMAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/COMAX-.NRLIB/COMAX-.fasl written
; compilation finished in 0:00:00.007
------------------------------------------------------------------------
CommutativeAxiom& is now explicitly exposed in frame initial
CommutativeAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/COMAX-.NRLIB/COMAX-
finalizing NRLIB COMAX
Processing CommutativeAxiom for Browser database:
--------constructor---------
--->-->CommutativeAxiom((rewriteCommutative ((Union (InputForm) failed) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/COMAX.NRLIB/COMAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/COMAX.NRLIB/COMAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
CommutativeAxiom is now explicitly exposed in frame initial
CommutativeAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/COMAX.NRLIB/COMAX
COMAX+ abbreviates category CommutativeAxiom+
------------------------------------------------------------------------
initializing NRLIB COMAX+ for CommutativeAxiom+
compiling into NRLIB COMAX+
;;;     ***       |CommutativeAxiom+| REDEFINED
Time: 0.01 SEC.
finalizing NRLIB COMAX+
Processing CommutativeAxiom+ for Browser database:
--->-->CommutativeAxiom+(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX+.NRLIB/COMAX+.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/COMAX+.NRLIB/COMAX+.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
CommutativeAxiom+ is now explicitly exposed in frame initial
CommutativeAxiom+ will be automatically loaded when needed from
/var/aw/var/LatexWiki/COMAX+.NRLIB/COMAX+
COMAX1 abbreviates category CommutativeAxiom*
------------------------------------------------------------------------
initializing NRLIB COMAX1 for CommutativeAxiom*
compiling into NRLIB COMAX1
;;;     ***       |CommutativeAxiom*| REDEFINED
Time: 0.01 SEC.
finalizing NRLIB COMAX1
Processing CommutativeAxiom* for Browser database:
--->-->CommutativeAxiom*(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX1.NRLIB/COMAX1.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/COMAX1.NRLIB/COMAX1.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
CommutativeAxiom* is now explicitly exposed in frame initial
CommutativeAxiom* will be automatically loaded when needed from
/var/aw/var/LatexWiki/COMAX1.NRLIB/COMAX1
COMAX2 abbreviates category CommutativeAxiom**
------------------------------------------------------------------------
initializing NRLIB COMAX2 for CommutativeAxiom**
compiling into NRLIB COMAX2
;;;     ***       |CommutativeAxiom**| REDEFINED
Time: 0 SEC.
finalizing NRLIB COMAX2
Processing CommutativeAxiom** for Browser database:
--->-->CommutativeAxiom**(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX2.NRLIB/COMAX2.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/COMAX2.NRLIB/COMAX2.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
CommutativeAxiom** is now explicitly exposed in frame initial
CommutativeAxiom** will be automatically loaded when needed from
/var/aw/var/LatexWiki/COMAX2.NRLIB/COMAX2
COMAX3 abbreviates category CommutativeAxiom/\
------------------------------------------------------------------------
initializing NRLIB COMAX3 for CommutativeAxiom/\
compiling into NRLIB COMAX3
;;;     ***       |CommutativeAxiom/\\| REDEFINED
Time: 0 SEC.
finalizing NRLIB COMAX3
Processing CommutativeAxiom/\ for Browser database:
--->-->CommutativeAxiom/\(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX3.NRLIB/COMAX3.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/COMAX3.NRLIB/COMAX3.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
CommutativeAxiom/\ is now explicitly exposed in frame initial
CommutativeAxiom/\ will be automatically loaded when needed from
/var/aw/var/LatexWiki/COMAX3.NRLIB/COMAX3
ALTAX abbreviates category AntiCommutativeAxiom
------------------------------------------------------------------------
initializing NRLIB ALTAX for AntiCommutativeAxiom
compiling into NRLIB ALTAX
;;;     ***       |AntiCommutativeAxiom| REDEFINED
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ALTAX- abbreviates domain AntiCommutativeAxiom&
------------------------------------------------------------------------
initializing NRLIB ALTAX- for AntiCommutativeAxiom&
compiling into NRLIB ALTAX-
compiling exported rewriteAntiCommutative : (InputForm -> InputForm,(InputForm,InputForm) -> InputForm,InputForm,InputForm) -> Union(InputForm,failed)
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(time taken in buildFunctor:  0)
;;;     ***       |AntiCommutativeAxiom&| REDEFINED
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Cumulative Statistics for Constructor AntiCommutativeAxiom&
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finalizing NRLIB ALTAX-
Processing AntiCommutativeAxiom& for Browser database:
--------constructor---------
--->-->AntiCommutativeAxiom&((rewriteAntiCommutative ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/ALTAX-.NRLIB/ALTAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/ALTAX-.NRLIB/ALTAX-.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
AntiCommutativeAxiom& is now explicitly exposed in frame initial
AntiCommutativeAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/ALTAX-.NRLIB/ALTAX-
finalizing NRLIB ALTAX
Processing AntiCommutativeAxiom for Browser database:
--------constructor---------
--->-->AntiCommutativeAxiom((rewriteAntiCommutative ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/ALTAX.NRLIB/ALTAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/ALTAX.NRLIB/ALTAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
AntiCommutativeAxiom is now explicitly exposed in frame initial
AntiCommutativeAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/ALTAX.NRLIB/ALTAX
DISAX abbreviates category DistributiveAxiom
------------------------------------------------------------------------
initializing NRLIB DISAX for DistributiveAxiom
compiling into NRLIB DISAX
;;;     ***       |DistributiveAxiom| REDEFINED
Time: 0 SEC.
DISAX- abbreviates domain DistributiveAxiom&
------------------------------------------------------------------------
initializing NRLIB DISAX- for DistributiveAxiom&
compiling into NRLIB DISAX-
compiling exported rewriteDistributive : (InputForm -> Union(Record(left: InputForm,right: InputForm),failed),(InputForm,InputForm) -> InputForm,(InputForm,InputForm) -> InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |DistributiveAxiom&| REDEFINED
Time: 0 SEC.
Warnings:
[1] rewriteDistributive:  left has no value
[2] rewriteDistributive:  right has no value
Cumulative Statistics for Constructor DistributiveAxiom&
Time: 0 seconds
finalizing NRLIB DISAX-
Processing DistributiveAxiom& for Browser database:
--------constructor---------
--->-->DistributiveAxiom&((rewriteDistributive ((Union (InputForm) failed) (Mapping (Union (Record (: left (InputForm)) (: right (InputForm))) failed) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/DISAX-.NRLIB/DISAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/DISAX-.NRLIB/DISAX-.fasl written
; compilation finished in 0:00:00.009
------------------------------------------------------------------------
DistributiveAxiom& is now explicitly exposed in frame initial
DistributiveAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/DISAX-.NRLIB/DISAX-
finalizing NRLIB DISAX
Processing DistributiveAxiom for Browser database:
--------constructor---------
--->-->DistributiveAxiom((rewriteDistributive ((Union (InputForm) failed) (Mapping (Union (Record (: left (InputForm)) (: right (InputForm))) failed) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/DISAX.NRLIB/DISAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/DISAX.NRLIB/DISAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
DistributiveAxiom is now explicitly exposed in frame initial
DistributiveAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/DISAX.NRLIB/DISAX
ASSAX abbreviates category AssociativeAxiom
------------------------------------------------------------------------
initializing NRLIB ASSAX for AssociativeAxiom
compiling into NRLIB ASSAX
;;;     ***       |AssociativeAxiom| REDEFINED
Time: 0 SEC.
ASSAX- abbreviates domain AssociativeAxiom&
------------------------------------------------------------------------
initializing NRLIB ASSAX- for AssociativeAxiom&
compiling into NRLIB ASSAX-
compiling exported rewriteAssociative : (InputForm -> Union(Record(left: InputForm,right: InputForm),failed),(InputForm,InputForm) -> InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |AssociativeAxiom&| REDEFINED
Time: 0 SEC.
Warnings:
[1] rewriteAssociative:  left has no value
[2] rewriteAssociative:  right has no value
Cumulative Statistics for Constructor AssociativeAxiom&
Time: 0 seconds
finalizing NRLIB ASSAX-
Processing AssociativeAxiom& for Browser database:
--------constructor---------
--->-->AssociativeAxiom&((rewriteAssociative ((Union (InputForm) failed) (Mapping (Union (Record (: left (InputForm)) (: right (InputForm))) failed) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/ASSAX-.NRLIB/ASSAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/ASSAX-.NRLIB/ASSAX-.fasl written
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
AssociativeAxiom& is now explicitly exposed in frame initial
AssociativeAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/ASSAX-.NRLIB/ASSAX-
finalizing NRLIB ASSAX
Processing AssociativeAxiom for Browser database:
--------constructor---------
--->-->AssociativeAxiom((rewriteAssociative ((Union (InputForm) failed) (Mapping (Union (Record (: left (InputForm)) (: right (InputForm))) failed) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/ASSAX.NRLIB/ASSAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/ASSAX.NRLIB/ASSAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
AssociativeAxiom is now explicitly exposed in frame initial
AssociativeAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/ASSAX.NRLIB/ASSAX
ASSAX+ abbreviates category AssociativeAxiom+
------------------------------------------------------------------------
initializing NRLIB ASSAX+ for AssociativeAxiom+
compiling into NRLIB ASSAX+
;;;     ***       |AssociativeAxiom+| REDEFINED
Time: 0 SEC.
finalizing NRLIB ASSAX+
Processing AssociativeAxiom+ for Browser database:
--->-->AssociativeAxiom+(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ASSAX+.NRLIB/ASSAX+.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/ASSAX+.NRLIB/ASSAX+.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
AssociativeAxiom+ is now explicitly exposed in frame initial
AssociativeAxiom+ will be automatically loaded when needed from
/var/aw/var/LatexWiki/ASSAX+.NRLIB/ASSAX+
ASSAXS abbreviates category AssociativeAxiom*
------------------------------------------------------------------------
initializing NRLIB ASSAXS for AssociativeAxiom*
compiling into NRLIB ASSAXS
;;;     ***       |AssociativeAxiom*| REDEFINED
Time: 0 SEC.
finalizing NRLIB ASSAXS
Processing AssociativeAxiom* for Browser database:
--->-->AssociativeAxiom*(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ASSAXS.NRLIB/ASSAXS.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/ASSAXS.NRLIB/ASSAXS.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
AssociativeAxiom* is now explicitly exposed in frame initial
AssociativeAxiom* will be automatically loaded when needed from
/var/aw/var/LatexWiki/ASSAXS.NRLIB/ASSAXS
IDAXL abbreviates category IdentityLeftAxiom
------------------------------------------------------------------------
initializing NRLIB IDAXL for IdentityLeftAxiom
compiling into NRLIB IDAXL
;;;     ***       |IdentityLeftAxiom| REDEFINED
Time: 0 SEC.
IDAXL- abbreviates domain IdentityLeftAxiom&
------------------------------------------------------------------------
initializing NRLIB IDAXL- for IdentityLeftAxiom&
compiling into NRLIB IDAXL-
compiling exported rewriteIdentityLeft : (InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |IdentityLeftAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor IdentityLeftAxiom&
Time: 0 seconds
finalizing NRLIB IDAXL-
Processing IdentityLeftAxiom& for Browser database:
--------constructor---------
--->-->IdentityLeftAxiom&((rewriteIdentityLeft ((Union (InputForm) failed) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/IDAXL-.NRLIB/IDAXL-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/IDAXL-.NRLIB/IDAXL-.fasl written
; compilation finished in 0:00:00.006
------------------------------------------------------------------------
IdentityLeftAxiom& is now explicitly exposed in frame initial
IdentityLeftAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/IDAXL-.NRLIB/IDAXL-
finalizing NRLIB IDAXL
Processing IdentityLeftAxiom for Browser database:
--------constructor---------
--->-->IdentityLeftAxiom((rewriteIdentityLeft ((Union (InputForm) failed) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/IDAXL.NRLIB/IDAXL.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/IDAXL.NRLIB/IDAXL.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
IdentityLeftAxiom is now explicitly exposed in frame initial
IdentityLeftAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/IDAXL.NRLIB/IDAXL
IDAXR abbreviates category IdentityRightAxiom
------------------------------------------------------------------------
initializing NRLIB IDAXR for IdentityRightAxiom
compiling into NRLIB IDAXR
;;;     ***       |IdentityRightAxiom| REDEFINED
Time: 0 SEC.
IDAXR- abbreviates domain IdentityRightAxiom&
------------------------------------------------------------------------
initializing NRLIB IDAXR- for IdentityRightAxiom&
compiling into NRLIB IDAXR-
compiling exported rewriteIdentityRight : (InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |IdentityRightAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor IdentityRightAxiom&
Time: 0 seconds
finalizing NRLIB IDAXR-
Processing IdentityRightAxiom& for Browser database:
--------constructor---------
--->-->IdentityRightAxiom&((rewriteIdentityRight ((Union (InputForm) failed) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/IDAXR-.NRLIB/IDAXR-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/IDAXR-.NRLIB/IDAXR-.fasl written
; compilation finished in 0:00:00.006
------------------------------------------------------------------------
IdentityRightAxiom& is now explicitly exposed in frame initial
IdentityRightAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/IDAXR-.NRLIB/IDAXR-
finalizing NRLIB IDAXR
Processing IdentityRightAxiom for Browser database:
--------constructor---------
--->-->IdentityRightAxiom((rewriteIdentityRight ((Union (InputForm) failed) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/IDAXR.NRLIB/IDAXR.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/IDAXR.NRLIB/IDAXR.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
IdentityRightAxiom is now explicitly exposed in frame initial
IdentityRightAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/IDAXR.NRLIB/IDAXR
IDAX abbreviates category IdentityAxiom
------------------------------------------------------------------------
initializing NRLIB IDAX for IdentityAxiom
compiling into NRLIB IDAX
;;;     ***       |IdentityAxiom| REDEFINED
Time: 0 SEC.
finalizing NRLIB IDAX
Processing IdentityAxiom for Browser database:
--------constructor---------
; compiling file "/var/aw/var/LatexWiki/IDAX.NRLIB/IDAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/IDAX.NRLIB/IDAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
IdentityAxiom is now explicitly exposed in frame initial
IdentityAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/IDAX.NRLIB/IDAX
INVAX abbreviates category InverseAxiom
------------------------------------------------------------------------
initializing NRLIB INVAX for InverseAxiom
compiling into NRLIB INVAX
;;;     ***       |InverseAxiom| REDEFINED
Time: 0 SEC.
INVAX- abbreviates domain InverseAxiom&
------------------------------------------------------------------------
initializing NRLIB INVAX- for InverseAxiom&
compiling into NRLIB INVAX-
compiling exported rewriteInverse : (InputForm -> InputForm,InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
compiling exported rewriteInverseInverse : (InputForm -> InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
compiling exported rewriteInverseBinary : (InputForm -> InputForm,(InputForm,InputForm) -> InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |InverseAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor InverseAxiom&
Time: 0 seconds
finalizing NRLIB INVAX-
Processing InverseAxiom& for Browser database:
--------constructor---------
--->-->InverseAxiom&((rewriteInverse ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
--->-->InverseAxiom&((rewriteInverseInverse ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (InputForm)))): Not documented!!!!
--->-->InverseAxiom&((rewriteInverseBinary ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/INVAX-.NRLIB/INVAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/INVAX-.NRLIB/INVAX-.fasl written
; compilation finished in 0:00:00.015
------------------------------------------------------------------------
InverseAxiom& is now explicitly exposed in frame initial
InverseAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/INVAX-.NRLIB/INVAX-
finalizing NRLIB INVAX
Processing InverseAxiom for Browser database:
--------constructor---------
--->-->InverseAxiom((rewriteInverse ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
--->-->InverseAxiom((rewriteInverseInverse ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (InputForm)))): Not documented!!!!
--->-->InverseAxiom((rewriteInverseBinary ((Union (InputForm) failed) (Mapping (InputForm) (InputForm)) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/INVAX.NRLIB/INVAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/INVAX.NRLIB/INVAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
InverseAxiom is now explicitly exposed in frame initial
InverseAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/INVAX.NRLIB/INVAX
DBLAX abbreviates category DoublesAxiom
------------------------------------------------------------------------
initializing NRLIB DBLAX for DoublesAxiom
compiling into NRLIB DBLAX
;;;     ***       |DoublesAxiom| REDEFINED
Time: 0.01 SEC.
DBLAX- abbreviates domain DoublesAxiom&
------------------------------------------------------------------------
initializing NRLIB DBLAX- for DoublesAxiom&
compiling into NRLIB DBLAX-
compiling exported rewriteDoubles : ((InputForm,InputForm) -> InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |DoublesAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor DoublesAxiom&
Time: 0 seconds
finalizing NRLIB DBLAX-
Processing DoublesAxiom& for Browser database:
--------constructor---------
--->-->DoublesAxiom&((rewriteDoubles ((Union (InputForm) failed) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/DBLAX-.NRLIB/DBLAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/DBLAX-.NRLIB/DBLAX-.fasl written
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
DoublesAxiom& is now explicitly exposed in frame initial
DoublesAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/DBLAX-.NRLIB/DBLAX-
finalizing NRLIB DBLAX
Processing DoublesAxiom for Browser database:
--------constructor---------
--->-->DoublesAxiom((rewriteDoubles ((Union (InputForm) failed) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/DBLAX.NRLIB/DBLAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/DBLAX.NRLIB/DBLAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
DoublesAxiom is now explicitly exposed in frame initial
DoublesAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/DBLAX.NRLIB/DBLAX
CANAX abbreviates category CancelsAxiom
------------------------------------------------------------------------
initializing NRLIB CANAX for CancelsAxiom
compiling into NRLIB CANAX
;;;     ***       |CancelsAxiom| REDEFINED
Time: 0 SEC.
CANAX- abbreviates domain CancelsAxiom&
------------------------------------------------------------------------
initializing NRLIB CANAX- for CancelsAxiom&
compiling into NRLIB CANAX-
compiling exported rewriteCancels : (InputForm,InputForm) -> Union(InputForm,failed)
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |CancelsAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor CancelsAxiom&
Time: 0 seconds
finalizing NRLIB CANAX-
Processing CancelsAxiom& for Browser database:
--------constructor---------
--->-->CancelsAxiom&((rewriteCancels ((Union (InputForm) failed) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/CANAX-.NRLIB/CANAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/CANAX-.NRLIB/CANAX-.fasl written
; compilation finished in 0:00:00.011
------------------------------------------------------------------------
CancelsAxiom& is now explicitly exposed in frame initial
CancelsAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/CANAX-.NRLIB/CANAX-
finalizing NRLIB CANAX
Processing CancelsAxiom for Browser database:
--------constructor---------
--->-->CancelsAxiom((rewriteCancels ((Union (InputForm) failed) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/CANAX.NRLIB/CANAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/CANAX.NRLIB/CANAX.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
CancelsAxiom is now explicitly exposed in frame initial
CancelsAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/CANAX.NRLIB/CANAX
SQAX abbreviates category SquaresAxiom
------------------------------------------------------------------------
initializing NRLIB SQAX for SquaresAxiom
compiling into NRLIB SQAX
;;;     ***       |SquaresAxiom| REDEFINED
Time: 0 SEC.
SQAX- abbreviates domain SquaresAxiom&
------------------------------------------------------------------------
initializing NRLIB SQAX- for SquaresAxiom&
compiling into NRLIB SQAX-
compiling exported rewriteSquares : ((InputForm,InputForm) -> InputForm,InputForm,InputForm) -> Union(InputForm,failed)
Time: 0.01 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |SquaresAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor SquaresAxiom&
Time: 0.01 seconds
finalizing NRLIB SQAX-
Processing SquaresAxiom& for Browser database:
--------constructor---------
--->-->SquaresAxiom&((rewriteSquares ((Union (InputForm) failed) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/SQAX-.NRLIB/SQAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/SQAX-.NRLIB/SQAX-.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
SquaresAxiom& is now explicitly exposed in frame initial
SquaresAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/SQAX-.NRLIB/SQAX-
finalizing NRLIB SQAX
Processing SquaresAxiom for Browser database:
--------constructor---------
--->-->SquaresAxiom((rewriteSquares ((Union (InputForm) failed) (Mapping (InputForm) (InputForm) (InputForm)) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/SQAX.NRLIB/SQAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/SQAX.NRLIB/SQAX.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
SquaresAxiom is now explicitly exposed in frame initial
SquaresAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/SQAX.NRLIB/SQAX
DIVAX abbreviates category DividesAxiom
------------------------------------------------------------------------
initializing NRLIB DIVAX for DividesAxiom
compiling into NRLIB DIVAX
;;;     ***       |DividesAxiom| REDEFINED
Time: 0 SEC.
DIVAX- abbreviates domain DividesAxiom&
------------------------------------------------------------------------
initializing NRLIB DIVAX- for DividesAxiom&
compiling into NRLIB DIVAX-
compiling exported rewriteDivides : (InputForm,InputForm) -> Union(InputForm,failed)
Time: 0.01 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |DividesAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor DividesAxiom&
Time: 0.01 seconds
finalizing NRLIB DIVAX-
Processing DividesAxiom& for Browser database:
--------constructor---------
--->-->DividesAxiom&((rewriteDivides ((Union (InputForm) failed) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/DIVAX-.NRLIB/DIVAX-.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/DIVAX-.NRLIB/DIVAX-.fasl written
; compilation finished in 0:00:00.015
------------------------------------------------------------------------
DividesAxiom& is now explicitly exposed in frame initial
DividesAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/DIVAX-.NRLIB/DIVAX-
finalizing NRLIB DIVAX
Processing DividesAxiom for Browser database:
--------constructor---------
--->-->DividesAxiom((rewriteDivides ((Union (InputForm) failed) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/DIVAX.NRLIB/DIVAX.lsp" (written 04 APR 2022 08:12:22 PM):
; /var/aw/var/LatexWiki/DIVAX.NRLIB/DIVAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
DividesAxiom is now explicitly exposed in frame initial
DividesAxiom will be automatically loaded when needed from
/var/aw/var/LatexWiki/DIVAX.NRLIB/DIVAX
SYMB abbreviates domain Symbolic
------------------------------------------------------------------------
initializing NRLIB SYMB for Symbolic
compiling into NRLIB SYMB
compiling exported smaller? : ($,$) -> Boolean
Time: 0.04 SEC.
****** Domain: AxiomList already in scope
augmenting AxiomList: (Axiom)
compiling exported axioms : () -> List Equation InputForm
Time: 0 SEC.
compiling exported axioms : () -> List Equation InputForm
SYMB;axioms;L;3 is replaced by
Time: 0 SEC.
compiling exported belong? : BasicOperator -> Boolean
SYMB;belong?;BoB;4 is replaced by QUOTET
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (AbelianSemiGroup)
compiling exported Zero : () -> $Time: 0 SEC. compiling exported zero? :$ -> Boolean
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (SemiGroup)
compiling exported One : () -> $Time: 0 SEC. compiling exported one? :$ -> Boolean
Time: 0 SEC.
compiling exported hash : $-> SingleInteger Time: 0 SEC. compiling exported equal? : Equation$ -> Boolean
Time: 0 SEC.
compiling exported = : ($,$) -> Boolean
Time: 0 SEC.
compiling exported subst : ($,Equation$) -> $Time: 0 SEC. compiling exported subst : ($,List Kernel $,List$) -> $Time: 0.03 SEC. compiling exported subst : ($,List Equation $) ->$
Time: 0 SEC.
compiling exported eval : ($,Kernel$,$) ->$
Time: 0.01 SEC.
compiling exported eval : ($,List Kernel$,List $) ->$
Time: 0.01 SEC.
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
compiling local eval : $->$
>> Apparent user error:
eval  is local and exported

## Some Tests

fricas
-- left and right identities
RID has IDAXR
RID is not a valid type.

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