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last edited 7 years ago by Bill Page |
Edit detail for SandBoxSymbolicExpressions revision 1 of 9
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Editor: Bill Page
Time: 2016/12/24 17:12:59 GMT+0 |
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| Note: Still a few problems | ||
changed: - From: - https://github.com/billpage/fricas/blob/symbolic-expressions/src/algebra/sexpr.spad \begin{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 smaller?(x:SIN,y:SIN):Boolean == --output("smaller? ",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 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 -- commutative --symbol? y and commutative?(symbol y) => null? x or list? x or integer? x or float? x -- or symbol? x and commutative?(symbol x) and symbol x < symbol y -- non-commutative --symbol? y => null? x or list? x or integer? x or float? x -- or symbol? x and (commutative?(symbol x) or symbol x < symbol y) false )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 f(y,x) else "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) == --if (t:=terms(x)) case Pair SIN then -- return f(t.left,f(t.right,y)) if (t:=terms(y)) case Pair SIN then 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 IDAX IdentityAxiom ++ Identities IdentityAxiom(): Category == Axiom with rewriteIdentity: (SIN,SIN,SIN)-> Maybe SIN add rewriteIdentity(u:SIN,x:SIN,y:SIN): Maybe SIN == --if x=u then return y if y=u then return x "failed" )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) == 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 -- use equality from Expression(R) (x:% = y:%):Boolean == coerce(x)@F = coerce(y)@F hash(x:%):SingleInteger == hash(rep x)$SIN -- term ordering: -- breadth first < integer (numeric) < float (numeric) < symbol (lexical) equal?(eq:Equation %):Boolean == not smaller?(lhs eq,rhs eq) and not smaller?(lhs eq,rhs eq) -- subst(x:%,e:Equation %):% == per SUBST(rep rhs e,rep lhs e,rep x)$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:%):% == subst(x,[k],[s]) if R has ConvertibleTo SIN then eval(x:%):% == coerce(convert(x)@SIN)@% eval(x:%,a:%,b:%):% == coerce eval(coerce(x)@F, coerce(a)@F, coerce(b)@F) eval(x:%,e:Equation %):% == eval(x,lhs e,rhs e) -- -- need to handle subscripted Symbol retractIfCan(x:%):Maybe(Kernel %) == 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 if not member?(s,['_+,'_-,'_*,'_/])$List(Symbol) then k:Kernel(%) := kernel(operator(s)$CommonOperators, destruct cdr rep x, #cdr(rep x)::NonNegativeInteger) return k return "failed" retract(x:%):Symbol == k:Maybe(Kernel %):=retractIfCan(x) if k case "failed" then error "not a kernel" else return name k -- 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" 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)$K rr2(r:R):SMP == coerce r numer(x:%):SparseMultivariatePolynomial(R,Kernel %) == map(kk2,rr2, numer(coerce x)$F)$LIFT denom(x:%):SparseMultivariatePolynomial(R,Kernel %) == map(kk2,rr2, denom(coerce x)$F)$LIFT (x:SMP / y:SMP):% == coerce(x) / coerce(y) isPlus(x:%):Maybe(List %) == r := isPlus1 rep x if r case Pair SIN then r1:=isPlus per(r.left) r2:=isPlus per(r.right) if r1 case List % then if r2 case List % then concat(r1,r2) else concat(r1, [per(r.right)]) else if r2 case List % then concat([per(r.left)],r2) else [per(r.left),per(r.right)] else "failed" factorials(x:%):% == coerce factorials(coerce x)$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)$Fraction(R) if R has RetractableTo Integer or R has RetractableTo Fraction Integer then retractIfCan(x:%):Maybe(Fraction Integer) == r:Maybe(R):=retractIfCan(x) if R has RetractableTo Fraction Integer then return retractIfCan(r::R)@Maybe(Fraction Integer) if R has RetractableTo Integer then r2:Maybe(Integer):=retractIfCan(r::R) if r2 case Integer then return r2::Integer::Fraction Integer return "failed" if R has SemiGroup then isTimes(x:%):Maybe(List %) == r := isTimes1 rep x if r case Pair SIN then r1:=isTimes per(r.left) r2:=isTimes per(r.right) if r1 case List % then if r2 case List % then concat(r1,r2) else concat(r1, per(r.right)) else if r2 case List % then concat(per(r.left),r2) else [per(r.left),per(r.right)] else "failed" if F has FunctionSpace(Integer) then factor(x:%):% == (factor(numer(coerce x)$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) coerceOutputForm(x:Rep):OutputForm == if null? x then return empty()$OutputForm if atom? x then return (x pretend OutputForm) if list? x and symbol? car x then s := symbol car x -- scripted? if #string(s)>2 and ord string(s).1 = 42 and digit? string(s).2 then return coerce(x pretend Symbol) if s='float then return interpret(x)$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='exp and car(cdr x)=1 then return outputForm '%e if member?(s,['_+,'_-,'_*,'_/,'_^])$List(Symbol) and #cdr(x)>1 then if AxiomList has AssociativeAxiom then return infix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x]) else return paren infix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x]) else if list? car(cdr x) or s ~= '_- then return prefix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x]) else return hconcat([outputForm s, hspace(1), coerceOutputForm car cdr x]) else return hconcat(message("what is:")$OutputForm,x pretend OutputForm) coerce(x:%):OutputForm == coerceOutputForm(rep x) -- +0 is a hack to avoid premature conversion (do we need it?) hack(x:%):Rep == binary(convert('_+),[rep x,0$Rep]) coerce(x:%):F == v:List Symbol := variables x if #v = 0 then return interpret(hack x)$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):% == per x -- 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)$RepeatedSquaring(%) (x:% ^ y:NonNegativeInteger):% == if y = 0 then x^0 else x ^ y::PositiveInteger (x:% ^ y:Integer):% == if y < 0 then 1/(x ^ (-y)::PositiveInteger) else x ^ y::NonNegativeInteger (x:% ^ y:Fraction Integer):% == nthRoot(x,denom(y))^numer(y) -- plus: (Rep,Rep) -> Rep minus: (Rep,Rep) -> Rep uminus: Rep -> Rep times(x:Rep,y:Rep):Rep == --output("times: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(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])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(0,x,y) if AxiomList has AssociativeAxiom_+ then returnIf rewriteAssociative(isPlus1,plus,x,y) if AxiomList has CommutativeAxiom_+ then returnIf rewriteCommutative(plus,x,y) if AxiomList has InverseAxiom then returnIf rewriteInverse(uminus,0,x,y) returnIf rewriteInverseBinary(uminus,minus,x,y) if AxiomList has DoublesAxiom then returnIf rewriteDoubles(times,x,y) binary(convert('_+),[x,y])$SIN (x:% + y:%):% == per plus(rep x,rep y) minus(x:Rep,y:Rep):Rep == --output("minus: ",paren [coerce(x)@OutputForm,coerce(y)@OutputForm])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(0,x,y) if AxiomList has CancelsAxiom then returnIf rewriteCancels(x,y) if AxiomList has InverseAxiom then returnIf rewriteInverseBinary(uminus,plus,x,y) binary(convert('_-),[x,y])$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])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(1,x,y) if AxiomList has DividesAxiom then returnIf rewriteDivides(x,y) binary(convert('_/),[x,y])$SIN (x:% / y:%):% == per div(rep x,rep y) elt(f:BasicOperator,x:List %):% == 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)$F variables1(x:Rep):Set Symbol == if null? x then return empty() if atom? x and symbol? x then return set [symbol x] if list? x and symbol? car x then if list? cdr x then return reduce("union", [variables1(i) for i in destruct cdr x]) else return variables1 car cdr x return empty() variables(x:%):List Symbol == members variables1(rep x) -- still need to handle non symbols and scripted symbols kernels1(x:Rep):Set Kernel % == if null? x then return empty() if atom? x and symbol? x then return set [kernel symbol x] if list? x and symbol? car x then if list? cdr x then s := symbol car x if member?(s,['_+,'_-,'_*,'_/])$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 ID Identities Identities():IdentityAxiom == add import Symbolic(Integer,None) x := coerce 'x axioms() == [ convert equation (x+0, 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 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 SEXPR SymbolicExpression SymbolicExpression(R:Comparable) : SymbolicCategory(R) == Symbolic(R, Commutative_+/\Commutative_*/\Distributive/\Associative_+/\Associative_*/\Identities/\Inverse/\Doubles/\Cancels/\Squares/\Divides) )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))$InputFormFunctions1(A),sk) \end{spad} Some Tests - https://github.com/billpage/fricas/blob/symbolic-expressions/src/input/sexpr.input \begin{axiom} s:SymbolicExpression Integer := 1+1+1 eval s s::Integer equal? (s=3) test(s=3) (1/5)::SEXPR INT -- (x,y,z):SymbolicExpression Integer (p,q,r):Expression Integer -- x::Expression Integer x+x = 2*x test % x-x = 0 test % x+y = y+x test % x+(y+z)= (x+y)+z test % x*x = x^2 test % x/x = 1 test % x*y = y*x test % x*(y*z)= (x*y)*z test % x*(y+z) = x*y + x*z test % -- sin(x)+cos(x)+tan(x) asin(x)+acos(x)+atan(x) sin(x)^2 + cos(x)^2 = 1 test map(simplify,%) -- exp(x+1) = %e*exp(x) test map(simplify, %) -- nthRoot(x, y) nthRoot(x,2) sqrt(x) test(sqrt(x)=nthRoot(x,2)) test(nthRoot(x,2)=sqrt(x)) (nthRoot(x,2)=sqrt(x))@Boolean -- variables(sin(x*y)+2*z) kernels(sin(x*y)+2*z) -- x+1/y simplify % simplify p factor(x^2+2*x*y+y^2) expand((x+y)/z) expand(x*(y+z)) -- -- Kurt -- SEI==>SymbolicExpression INT (x,y):SEI (x+y)^2 r0:=(x+y)^2 r0::InputForm rs := simplify r0 factor rs convert(r)$SEI coerce(r)$SEI sum(x,'x=1..10) reduce(+,[n::SEXPR INT for n in 1..10]) x - x (x+y)^x exp(-x) -x exp(-x)::InputForm f:=operator 'f f(x) f(0$SEXPR INT) f(0$EXPR INT) D(x^2,'x) simplify % x^n D(x^n,'x) \end{axiom}
From:
spad
-- Type safe representation rep(x) ==> x @ % pretend Rep per(x) ==> x @ Rep pretend % -- Applicative-style macros Maybe(S) ==> Union(S,"failed") applyIf(f, x) ==> if (%r:=x) case "failed" then "failed" else f(%r) returnIf(x) ==> if not((%r:=x) case "failed") then return %r orReturn(x) ==> if x case "failed" then return "failed"
SIN ==> InputForm Pair SIN ==> Record(left:SIN,right:SIN)
)abbrev package SINF SINFunctions -- This should be built-in to InputForm,i.e. InputForm should satisfy Comparable SINFunctions(): with smaller? : (SIN, SIN) -> Boolean rank: (Symbol, Integer) -> SIN rank: Symbol -> SIN == add rank(x:Symbol, n:Integer):SIN == PUT(x, 'rank, convert(n)@SIN)$Lisp rank(x:Symbol):SIN == GET(x, 'rank)$Lisp --null < list < integer < float < symbol rank < symbol rank smaller?(x:SIN, y:SIN):Boolean == --output("smaller? ", paren [x::OutputForm, y::OutputForm])$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 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 -- commutative --symbol? y and commutative?(symbol y) => null? x or list? x or integer? x or float? x -- or symbol? x and commutative?(symbol x) and symbol x < symbol y -- non-commutative --symbol? y => null? x or list? x or integer? x or float? x -- or symbol? x and (commutative?(symbol x) or symbol x < symbol y) false
)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 f(y, x) else "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) == --if (t:=terms(x)) case Pair SIN then -- return f(t.left, f(t.right, y)) if (t:=terms(y)) case Pair SIN then 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 IDAX IdentityAxiom ++ Identities IdentityAxiom(): Category == Axiom with rewriteIdentity: (SIN,SIN, SIN)-> Maybe SIN add rewriteIdentity(u:SIN, x:SIN, y:SIN): Maybe SIN == --if x=u then return y if y=u then return x "failed"
)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) == 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
-- use equality from Expression(R) (x:% = y:%):Boolean == coerce(x)@F = coerce(y)@F
hash(x:%):SingleInteger == hash(rep x)$SIN -- term ordering: -- breadth first < integer (numeric) < float (numeric) < symbol (lexical)
equal?(eq:Equation %):Boolean == not smaller?(lhs eq,rhs eq) and not smaller?(lhs eq, rhs eq) -- subst(x:%, e:Equation %):% == per SUBST(rep rhs e, rep lhs e, rep x)$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:%):% == subst(x, [k], [s]) if R has ConvertibleTo SIN then eval(x:%):% == coerce(convert(x)@SIN)@% eval(x:%, a:%, b:%):% == coerce eval(coerce(x)@F, coerce(a)@F, coerce(b)@F) eval(x:%, e:Equation %):% == eval(x, lhs e, rhs e) -- -- need to handle subscripted Symbol retractIfCan(x:%):Maybe(Kernel %) == 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 if not member?(s, ['_+, '_-, '_*, '_/])$List(Symbol) then k:Kernel(%) := kernel(operator(s)$CommonOperators, destruct cdr rep x, #cdr(rep x)::NonNegativeInteger) return k return "failed" retract(x:%):Symbol == k:Maybe(Kernel %):=retractIfCan(x) if k case "failed" then error "not a kernel" else return name k -- 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" 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)$K rr2(r:R):SMP == coerce r numer(x:%):SparseMultivariatePolynomial(R, Kernel %) == map(kk2, rr2, numer(coerce x)$F)$LIFT denom(x:%):SparseMultivariatePolynomial(R, Kernel %) == map(kk2, rr2, denom(coerce x)$F)$LIFT (x:SMP / y:SMP):% == coerce(x) / coerce(y) isPlus(x:%):Maybe(List %) == r := isPlus1 rep x if r case Pair SIN then r1:=isPlus per(r.left) r2:=isPlus per(r.right) if r1 case List % then if r2 case List % then concat(r1, r2) else concat(r1, [per(r.right)]) else if r2 case List % then concat([per(r.left)], r2) else [per(r.left), per(r.right)] else "failed"
factorials(x:%):% == coerce factorials(coerce x)$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)$Fraction(R) if R has RetractableTo Integer or R has RetractableTo Fraction Integer then retractIfCan(x:%):Maybe(Fraction Integer) == r:Maybe(R):=retractIfCan(x) if R has RetractableTo Fraction Integer then return retractIfCan(r::R)@Maybe(Fraction Integer) if R has RetractableTo Integer then r2:Maybe(Integer):=retractIfCan(r::R) if r2 case Integer then return r2::Integer::Fraction Integer return "failed"
if R has SemiGroup then isTimes(x:%):Maybe(List %) == r := isTimes1 rep x if r case Pair SIN then r1:=isTimes per(r.left) r2:=isTimes per(r.right) if r1 case List % then if r2 case List % then concat(r1,r2) else concat(r1, per(r.right)) else if r2 case List % then concat(per(r.left), r2) else [per(r.left), per(r.right)] else "failed"
if F has FunctionSpace(Integer) then factor(x:%):% == (factor(numer(coerce x)$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)
coerceOutputForm(x:Rep):OutputForm == if null? x then return empty()$OutputForm if atom? x then return (x pretend OutputForm) if list? x and symbol? car x then s := symbol car x -- scripted? if #string(s)>2 and ord string(s).1 = 42 and digit? string(s).2 then return coerce(x pretend Symbol) if s='float then return interpret(x)$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='exp and car(cdr x)=1 then return outputForm '%e if member?(s, ['_+, '_-, '_*, '_/, '_^])$List(Symbol) and #cdr(x)>1 then if AxiomList has AssociativeAxiom then return infix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x]) else return paren infix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x]) else if list? car(cdr x) or s ~= '_- then return prefix(outputForm s, [coerceOutputForm(i) for i in destruct cdr x]) else return hconcat([outputForm s, hspace(1), coerceOutputForm car cdr x]) else return hconcat(message("what is:")$OutputForm, x pretend OutputForm) coerce(x:%):OutputForm == coerceOutputForm(rep x)
-- +0 is a hack to avoid premature conversion (do we need it?) hack(x:%):Rep == binary(convert('_+),[rep x, 0$Rep])
coerce(x:%):F == v:List Symbol := variables x if #v = 0 then return interpret(hack x)$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):% == per x -- 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)$RepeatedSquaring(%) (x:% ^ y:NonNegativeInteger):% == if y = 0 then x^0 else x ^ y::PositiveInteger (x:% ^ y:Integer):% == if y < 0 then 1/(x ^ (-y)::PositiveInteger) else x ^ y::NonNegativeInteger (x:% ^ y:Fraction Integer):% == nthRoot(x, denom(y))^numer(y) -- plus: (Rep, Rep) -> Rep minus: (Rep, Rep) -> Rep uminus: Rep -> Rep
times(x:Rep,y:Rep):Rep == --output("times: ", paren [coerce(x)@OutputForm, coerce(y)@OutputForm])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(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])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(0, x, y) if AxiomList has AssociativeAxiom_+ then returnIf rewriteAssociative(isPlus1, plus, x, y) if AxiomList has CommutativeAxiom_+ then returnIf rewriteCommutative(plus, x, y) if AxiomList has InverseAxiom then returnIf rewriteInverse(uminus, 0, x, y) returnIf rewriteInverseBinary(uminus, minus, x, y) if AxiomList has DoublesAxiom then returnIf rewriteDoubles(times, x, y) binary(convert('_+), [x, y])$SIN (x:% + y:%):% == per plus(rep x, rep y)
minus(x:Rep,y:Rep):Rep == --output("minus: ", paren [coerce(x)@OutputForm, coerce(y)@OutputForm])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(0, x, y) if AxiomList has CancelsAxiom then returnIf rewriteCancels(x, y) if AxiomList has InverseAxiom then returnIf rewriteInverseBinary(uminus, plus, x, y) binary(convert('_-), [x, y])$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])$OutputPackage if AxiomList has IdentityAxiom then returnIf rewriteIdentity(1, x, y) if AxiomList has DividesAxiom then returnIf rewriteDivides(x, y) binary(convert('_/), [x, y])$SIN (x:% / y:%):% == per div(rep x, rep y)
elt(f:BasicOperator,x:List %):% == 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)$F
variables1(x:Rep):Set Symbol == if null? x then return empty() if atom? x and symbol? x then return set [symbol x] if list? x and symbol? car x then if list? cdr x then return reduce("union",[variables1(i) for i in destruct cdr x]) else return variables1 car cdr x return empty() variables(x:%):List Symbol == members variables1(rep x)
-- still need to handle non symbols and scripted symbols kernels1(x:Rep):Set Kernel % == if null? x then return empty() if atom? x and symbol? x then return set [kernel symbol x] if list? x and symbol? car x then if list? cdr x then s := symbol car x if member?(s,['_+, '_-, '_*, '_/])$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 ID Identities Identities():IdentityAxiom == add import Symbolic(Integer,None) x := coerce 'x axioms() == [ convert equation (x+0, 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 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 SEXPR SymbolicExpression SymbolicExpression(R:Comparable) : SymbolicCategory(R) == Symbolic(R,Commutative_+/\Commutative_*/\Distributive/\Associative_+/\Associative_*/\Identities/\Inverse/\Doubles/\Cancels/\Squares/\Divides)
)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/1802296561408897569-25px001.spad
using old system compiler.
SINF abbreviates package SINFunctions
------------------------------------------------------------------------
initializing NRLIB SINF for SINFunctions
compiling into NRLIB SINF
compiling exported rank : (Symbol,
compiling exported rank : Symbol -> InputForm
Time: 0.01 SEC.
compiling exported smaller? : (InputForm,
(time taken in buildFunctor: 0)
;;; *** |SINFunctions| REDEFINED
;;; *** |SINFunctions| REDEFINED
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Cumulative Statistics for Constructor SINFunctions
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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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/SINF.NRLIB/SINF.fasl written
; compilation finished in 0:00:00.042
------------------------------------------------------------------------
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
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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 24 DEC 2016 05:12:59 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
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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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/AXIOM.NRLIB/AXIOM.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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
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COMAX- abbreviates domain CommutativeAxiom&
------------------------------------------------------------------------
initializing NRLIB COMAX- for CommutativeAxiom&
compiling into NRLIB COMAX-
compiling exported rewriteCommutative : ((InputForm,
(time taken in buildFunctor: 0)
;;; *** |CommutativeAxiom&| REDEFINED
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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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/COMAX-.NRLIB/COMAX-.fasl written
; compilation finished in 0:00:00.005
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/COMAX.NRLIB/COMAX.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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
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finalizing NRLIB COMAX+
Processing CommutativeAxiom+ for Browser database:
--->-->CommutativeAxiom+(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX+.NRLIB/COMAX+.lsp" (written 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/COMAX+.NRLIB/COMAX+.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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
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finalizing NRLIB COMAX1
Processing CommutativeAxiom* for Browser database:
--->-->CommutativeAxiom*(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX1.NRLIB/COMAX1.lsp" (written 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/COMAX1.NRLIB/COMAX1.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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
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finalizing NRLIB COMAX2
Processing CommutativeAxiom** for Browser database:
--->-->CommutativeAxiom**(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX2.NRLIB/COMAX2.lsp" (written 24 DEC 2016 05:12:59 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
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finalizing NRLIB COMAX3
Processing CommutativeAxiom/\ for Browser database:
--->-->CommutativeAxiom/\(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMAX3.NRLIB/COMAX3.lsp" (written 24 DEC 2016 05:12:59 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,
(time taken in buildFunctor: 0)
;;; *** |AntiCommutativeAxiom&| REDEFINED
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Cumulative Statistics for Constructor AntiCommutativeAxiom&
Time: 0.01 seconds
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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/ALTAX-.NRLIB/ALTAX-.fasl written
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 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,
(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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/DISAX-.NRLIB/DISAX-.fasl written
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/DISAX.NRLIB/DISAX.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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,
(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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/ASSAX-.NRLIB/ASSAX-.fasl written
; compilation finished in 0:00:00.007
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/ASSAX.NRLIB/ASSAX.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 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: 0.01 SEC.
finalizing NRLIB ASSAXS
Processing AssociativeAxiom* for Browser database:
--->-->AssociativeAxiom*(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ASSAXS.NRLIB/ASSAXS.lsp" (written 24 DEC 2016 05:12:59 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
IDAX abbreviates category IdentityAxiom
------------------------------------------------------------------------
initializing NRLIB IDAX for IdentityAxiom
compiling into NRLIB IDAX
;;; *** |IdentityAxiom| REDEFINED
Time: 0 SEC.
IDAX- abbreviates domain IdentityAxiom&
------------------------------------------------------------------------
initializing NRLIB IDAX- for IdentityAxiom&
compiling into NRLIB IDAX-
compiling exported rewriteIdentity : (InputForm,
(time taken in buildFunctor: 0)
;;; *** |IdentityAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor IdentityAxiom&
Time: 0 seconds
finalizing NRLIB IDAX-
Processing IdentityAxiom& for Browser database:
--------constructor---------
--->-->IdentityAxiom&((rewriteIdentity ((Union (InputForm) failed) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/IDAX-.NRLIB/IDAX-.lsp" (written 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/IDAX-.NRLIB/IDAX-.fasl written
; compilation finished in 0:00:00.005
------------------------------------------------------------------------
IdentityAxiom& is now explicitly exposed in frame initial
IdentityAxiom& will be automatically loaded when needed from
/var/aw/var/LatexWiki/IDAX-.NRLIB/IDAX-
finalizing NRLIB IDAX
Processing IdentityAxiom for Browser database:
--------constructor---------
--->-->IdentityAxiom((rewriteIdentity ((Union (InputForm) failed) (InputForm) (InputForm) (InputForm)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/IDAX.NRLIB/IDAX.lsp" (written 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/IDAX.NRLIB/IDAX.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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,
compiling exported rewriteInverseInverse : (InputForm -> InputForm,
compiling exported rewriteInverseBinary : (InputForm -> InputForm,
(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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/INVAX-.NRLIB/INVAX-.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 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 SEC.
DBLAX- abbreviates domain DoublesAxiom&
------------------------------------------------------------------------
initializing NRLIB DBLAX- for DoublesAxiom&
compiling into NRLIB DBLAX-
compiling exported rewriteDoubles : ((InputForm,
(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 24 DEC 2016 05:12:59 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 24 DEC 2016 05:12:59 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,
(time taken in buildFunctor: 0)
;;; *** |CancelsAxiom&| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor CancelsAxiom&
Time: 0.01 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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/CANAX-.NRLIB/CANAX-.fasl written
; compilation finished in 0:00:00.010
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/CANAX.NRLIB/CANAX.fasl written
; compilation finished in 0:00:00.002
------------------------------------------------------------------------
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,
(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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/SQAX-.NRLIB/SQAX-.fasl written
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 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,
(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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/DIVAX-.NRLIB/DIVAX-.fasl written
; compilation finished in 0:00:00.014
------------------------------------------------------------------------
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 24 DEC 2016 05:12:59 PM):
; /var/aw/var/LatexWiki/DIVAX.NRLIB/DIVAX.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
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? : ($,
****** Domain: AxiomList already in scope
augmenting AxiomList: (Axiom)
compiling exported axioms : () -> List Equation InputForm
Time: 0.01 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 = : ($,
compiling exported hash : $ -> SingleInteger
Time: 0.01 SEC.
compiling exported equal? : Equation $ -> Boolean
Time: 0 SEC.
compiling exported subst : ($,
compiling exported subst : ($,
compiling exported subst : ($,
compiling exported eval : ($,
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
compiling exported eval : $ -> $
Time: 0.01 SEC.
compiling exported eval : ($,
compiling exported eval : ($,
compiling exported retractIfCan : $ -> Union(Kernel $,
compiling exported retract : $ -> Symbol
Time: 0.01 SEC.
compiling local isPlus1 : InputForm -> Union(Record(left: InputForm,
compiling local isTimes1 : InputForm -> Union(Record(left: InputForm,
****** Domain: R already in scope
augmenting R: (IntegralDomain)
compiling local kk2 : Kernel Expression R -> SparseMultivariatePolynomial(R,
compiling local rr2 : R -> SparseMultivariatePolynomial(R,
compiling exported numer : $ -> SparseMultivariatePolynomial(R,
compiling exported denom : $ -> SparseMultivariatePolynomial(R,
compiling exported / : (SparseMultivariatePolynomial(R,
compiling exported isPlus : $ -> Union(List $,
compiling exported factorials : $ -> $
Time: 0 SEC.
compiling exported factorials : ($,
compiling exported expand : $ -> $
Time: 0.01 SEC.
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
compiling exported coerce : SparseMultivariatePolynomial(R,
compiling exported coerce : Polynomial R -> $
Time: 0 SEC.
compiling exported coerce : Fraction R -> $
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
compiling exported retractIfCan : $ -> Union(Fraction Integer,
****** Domain: R already in scope
augmenting R: (RetractableTo (Fraction (Integer)))
compiling exported retractIfCan : $ -> Union(Fraction Integer,
****** Domain: R already in scope
augmenting R: (SemiGroup)
compiling exported isTimes : $ -> Union(List $,
****** Domain: (Expression R) already in scope
augmenting (Expression R): (FunctionSpace (Integer))
compiling exported factor : $ -> $
Time: 0.05 SEC.
compiling exported coerce : Pi -> $
Time: 0.01 SEC.
compiling exported coerce : Symbol -> $
Time: 0 SEC.
compiling exported coerce : Kernel $ -> $
Time: 0 SEC.
compiling local coerceOutputForm : Rep -> OutputForm
****** Domain: AxiomList already in scope
augmenting AxiomList: (AssociativeAxiom)
Time: 0.07 SEC.
compiling exported coerce : $ -> OutputForm
Time: 0 SEC.
compiling local hack : $ -> Rep
Time: 0 SEC.
compiling exported coerce : $ -> Expression R
Time: 0.01 SEC.
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
compiling exported coerce : Expression R -> $
Time: 0.01 SEC.
compiling exported coerce : R -> $
Time: 0 SEC.
compiling exported convert : $ -> InputForm
SYMB;convert;$If;48 is replaced by x
Time: 0 SEC.
compiling exported coerce : InputForm -> $
SYMB;coerce;If$;49 is replaced by x
Time: 0 SEC.
compiling exported factorial : $ -> $
Time: 0.01 SEC.
compiling exported binomial : ($,
compiling exported permutation : ($,
compiling local sum : ($,
compiling local sum : ($,
compiling exported summation : ($,
compiling exported summation : ($,
compiling exported product : ($,
compiling exported product : ($,
compiling local power : (Rep,
compiling exported ^ : ($,
compiling exported ^ : ($,
compiling exported ^ : ($,
compiling exported ^ : ($,
compiling exported ^ : ($,
compiling local times : (Rep,
compiling exported * : ($,
compiling exported * : (PositiveInteger,
compiling exported * : (NonNegativeInteger,
compiling exported * : (Integer,
compiling local plus : (Rep,
compiling exported + : ($,
compiling local minus : (Rep,
compiling exported - : ($,
compiling local uminus : Rep -> Rep
****** Domain: AxiomList already in scope
augmenting AxiomList: (InverseAxiom)
Time: 0.02 SEC.
compiling exported - : $ -> $
Time: 0 SEC.
compiling local div : (Rep,
compiling exported / : ($,
compiling exported elt : (BasicOperator,
compiling exported acos : $ -> $
Time: 0.01 SEC.
compiling exported acosh : $ -> $
Time: 0.02 SEC.
compiling exported acot : $ -> $
Time: 0.01 SEC.
compiling exported acoth : $ -> $
Time: 0.02 SEC.
compiling exported acsc : $ -> $
Time: 0.01 SEC.
compiling exported acsch : $ -> $
Time: 0.02 SEC.
compiling exported asec : $ -> $
Time: 0.01 SEC.
compiling exported asech : $ -> $
Time: 0.02 SEC.
compiling exported asin : $ -> $
Time: 0.01 SEC.
compiling exported asinh : $ -> $
Time: 0.02 SEC.
compiling exported atan : $ -> $
Time: 0.01 SEC.
compiling exported atanh : $ -> $
Time: 0.01 SEC.
compiling exported cos : $ -> $
Time: 0.02 SEC.
compiling exported cosh : $ -> $
Time: 0.02 SEC.
compiling exported cot : $ -> $
Time: 0.01 SEC.
compiling exported coth : $ -> $
Time: 0.01 SEC.
compiling exported csc : $ -> $
Time: 0.02 SEC.
compiling exported csch : $ -> $
Time: 0.02 SEC.
compiling exported exp : $ -> $
Time: 0.01 SEC.
compiling exported log : $ -> $
Time: 0.02 SEC.
compiling exported nthRoot : ($,
compiling exported nthRoot : ($,
compiling exported sec : $ -> $
Time: 0.02 SEC.
compiling exported sech : $ -> $
Time: 0.01 SEC.
compiling exported sin : $ -> $
Time: 0.02 SEC.
compiling exported sinh : $ -> $
Time: 0.02 SEC.
compiling exported sqrt : $ -> $
Time: 0.01 SEC.
compiling exported tan : $ -> $
Time: 0.01 SEC.
compiling exported tanh : $ -> $
Time: 0.02 SEC.
compiling exported retract : $ -> R
Time: 0 SEC.
compiling local variables1 : Rep -> Set Symbol
Time: 0.02 SEC.
compiling exported variables : $ -> List Symbol
Time: 0.01 SEC.
compiling local kernels1 : Rep -> Set Kernel $
Time: 0.02 SEC.
compiling exported kernels : $ -> List Kernel $
Time: 0 SEC.
****** Domain: (Expression R) already in scope
augmenting (Expression R): (TranscendentalFunctionCategory)
****** Domain: R already in scope
augmenting R: (GcdDomain)
compiling exported simplify : $ -> $
Time: 0.02 SEC.
compiling exported convert : Equation $ -> Equation InputForm
Time: 0.01 SEC.
compiling exported convert : Equation InputForm -> Equation $
Time: 0.01 SEC.
****** Domain: $ already in scope
augmenting $: (RetractableTo (Integer))
****** Domain: R already in scope
augmenting R: (IntegralDomain)
****** Domain: $ already in scope
augmenting $: (TranscendentalFunctionCategory)
****** Domain: R already in scope
augmenting R: (GcdDomain)
****** Domain: R already in scope
augmenting R: (LinearlyExplicitOver (Integer))
****** Domain: $ already in scope
augmenting $: (FunctionSpace (Integer))
****** Domain: $ already in scope
augmenting $: (RetractableTo (Integer))
****** Domain: $ already in scope
augmenting $: (Ring)
****** Domain: R already in scope
augmenting R: (AbelianGroup)
****** Domain: R already in scope
augmenting R: (AbelianSemiGroup)
****** Domain: R already in scope
augmenting R: (CharacteristicNonZero)
****** Domain: R already in scope
augmenting R: (CommutativeRing)
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Float)))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Integer)))
****** Domain: R already in scope
augmenting R: (Group)
****** Domain: R already in scope
augmenting R: (IntegralDomain)
****** Domain: R already in scope
augmenting R: (PatternMatchable (Float))
****** Domain: R already in scope
augmenting R: (PatternMatchable (Integer))
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
****** Domain: R already in scope
augmenting R: (Ring)
****** Domain: R already in scope
augmenting R: (SemiGroup)
(time taken in buildFunctor: 130)
;;; *** |Symbolic| REDEFINED
;;; *** |Symbolic| REDEFINED
Time: 0.17 SEC.
Warnings:
[1] smaller?: pretendRep -- should replace by @
[2] Zero: pretend$ -- should replace by @
[3] One: pretend$ -- should replace by @
[4] hash: pretendRep -- should replace by @
[5] subst: pretendRep -- should replace by @
[6] subst: pretend$ -- should replace by @
[7] subst: $$ has no value
[8] retractIfCan: pretendRep -- should replace by @
[9] isPlus: pretendRep -- should replace by @
[10] isPlus: left has no value
[11] isPlus: pretend$ -- should replace by @
[12] isPlus: right has no value
[13] coerce: pretend$ -- should replace by @
[14] isTimes: pretendRep -- should replace by @
[15] isTimes: left has no value
[16] isTimes: pretend$ -- should replace by @
[17] isTimes: right has no value
[18] coerce: pretendRep -- should replace by @
[19] hack: pretendRep -- should replace by @
[20] convert: pretendRep -- should replace by @
[21] factorial: pretendRep -- should replace by @
[22] factorial: pretend$ -- should replace by @
[23] sum: pretendRep -- should replace by @
[24] sum: pretend$ -- should replace by @
[25] summation: pretendRep -- should replace by @
[26] summation: pretend$ -- should replace by @
[27] product: pretendRep -- should replace by @
[28] product: pretend$ -- should replace by @
[29] ^: pretendRep -- should replace by @
[30] ^: pretend$ -- should replace by @
[31] *: pretendRep -- should replace by @
[32] *: pretend$ -- should replace by @
[33] +: pretendRep -- should replace by @
[34] +: pretend$ -- should replace by @
[35] -: pretendRep -- should replace by @
[36] -: pretend$ -- should replace by @
[37] /: pretendRep -- should replace by @
[38] /: pretend$ -- should replace by @
[39] elt: pretendRep -- should replace by @
[40] elt: pretend$ -- should replace by @
[41] acos: pretendRep -- should replace by @
[42] acos: pretend$ -- should replace by @
[43] acosh: pretendRep -- should replace by @
[44] acosh: pretend$ -- should replace by @
[45] acot: pretendRep -- should replace by @
[46] acot: pretend$ -- should replace by @
[47] acoth: pretendRep -- should replace by @
[48] acoth: pretend$ -- should replace by @
[49] acsc: pretendRep -- should replace by @
[50] acsc: pretend$ -- should replace by @
[51] acsch: pretendRep -- should replace by @
[52] acsch: pretend$ -- should replace by @
[53] asec: pretendRep -- should replace by @
[54] asec: pretend$ -- should replace by @
[55] asech: pretendRep -- should replace by @
[56] asech: pretend$ -- should replace by @
[57] asin: pretendRep -- should replace by @
[58] asin: pretend$ -- should replace by @
[59] asinh: pretendRep -- should replace by @
[60] asinh: pretend$ -- should replace by @
[61] atan: pretendRep -- should replace by @
[62] atan: pretend$ -- should replace by @
[63] atanh: pretendRep -- should replace by @
[64] atanh: pretend$ -- should replace by @
[65] cos: pretendRep -- should replace by @
[66] cos: pretend$ -- should replace by @
[67] cosh: pretendRep -- should replace by @
[68] cosh: pretend$ -- should replace by @
[69] cot: pretendRep -- should replace by @
[70] cot: pretend$ -- should replace by @
[71] coth: pretendRep -- should replace by @
[72] coth: pretend$ -- should replace by @
[73] csc: pretendRep -- should replace by @
[74] csc: pretend$ -- should replace by @
[75] csch: pretendRep -- should replace by @
[76] csch: pretend$ -- should replace by @
[77] exp: pretendRep -- should replace by @
[78] exp: pretend$ -- should replace by @
[79] log: pretendRep -- should replace by @
[80] log: pretend$ -- should replace by @
[81] nthRoot: pretendRep -- should replace by @
[82] sec: pretendRep -- should replace by @
[83] sec: pretend$ -- should replace by @
[84] sech: pretendRep -- should replace by @
[85] sech: pretend$ -- should replace by @
[86] sin: pretendRep -- should replace by @
[87] sin: pretend$ -- should replace by @
[88] sinh: pretendRep -- should replace by @
[89] sinh: pretend$ -- should replace by @
[90] sqrt: pretendRep -- should replace by @
[91] sqrt: pretend$ -- should replace by @
[92] tan: pretendRep -- should replace by @
[93] tan: pretend$ -- should replace by @
[94] tanh: pretendRep -- should replace by @
[95] tanh: pretend$ -- should replace by @
[96] variables: pretendRep -- should replace by @
[97] kernels: pretendRep -- should replace by @
[98] simplify: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce ($ $)) (SIGNATURE number? ((Boolean) $)) (SIGNATURE simplifyPower ($ $ (Integer))) (IF (has R (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (IF (has R (RetractableTo (Integer))) (ATTRIBUTE (RetractableTo (AlgebraicNumber))) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
Cumulative Statistics for Constructor Symbolic
Time: 1.24 seconds
finalizing NRLIB SYMB
Processing Symbolic for Browser database:
--------constructor---------
; compiling file "/var/aw/var/LatexWiki/SYMB.NRLIB/SYMB.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/SYMB.NRLIB/SYMB.fasl written
; compilation finished in 0:00:00.428
------------------------------------------------------------------------
Symbolic is now explicitly exposed in frame initial
Symbolic will be automatically loaded when needed from
/var/aw/var/LatexWiki/SYMB.NRLIB/SYMB
COM+ abbreviates domain Commutative+
------------------------------------------------------------------------
initializing NRLIB COM+ for Commutative+
compiling into NRLIB COM+
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Commutative+| REDEFINED
;;; *** |Commutative+| REDEFINED
Time: 0.01 SEC.
Cumulative Statistics for Constructor Commutative+
Time: 0.02 seconds
finalizing NRLIB COM+
Processing Commutative+ for Browser database:
--->-->Commutative+(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COM+.NRLIB/COM+.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/COM+.NRLIB/COM+.fasl written
; compilation finished in 0:00:00.009
------------------------------------------------------------------------
Commutative+ is now explicitly exposed in frame initial
Commutative+ will be automatically loaded when needed from
/var/aw/var/LatexWiki/COM+.NRLIB/COM+
COMS abbreviates domain Commutative*
------------------------------------------------------------------------
initializing NRLIB COMS for Commutative*
compiling into NRLIB COMS
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Commutative*| REDEFINED
;;; *** |Commutative*| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Commutative*
Time: 0.02 seconds
finalizing NRLIB COMS
Processing Commutative* for Browser database:
--->-->Commutative*(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COMS.NRLIB/COMS.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/COMS.NRLIB/COMS.fasl written
; compilation finished in 0:00:00.009
------------------------------------------------------------------------
Commutative* is now explicitly exposed in frame initial
Commutative* will be automatically loaded when needed from
/var/aw/var/LatexWiki/COMS.NRLIB/COMS
ALT abbreviates domain AntiCommutative
------------------------------------------------------------------------
initializing NRLIB ALT for AntiCommutative
compiling into NRLIB ALT
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |AntiCommutative| REDEFINED
;;; *** |AntiCommutative| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor AntiCommutative
Time: 0.03 seconds
finalizing NRLIB ALT
Processing AntiCommutative for Browser database:
--->-->AntiCommutative(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ALT.NRLIB/ALT.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/ALT.NRLIB/ALT.fasl written
; compilation finished in 0:00:00.011
------------------------------------------------------------------------
AntiCommutative is now explicitly exposed in frame initial
AntiCommutative will be automatically loaded when needed from
/var/aw/var/LatexWiki/ALT.NRLIB/ALT
DIS abbreviates domain Distributive
------------------------------------------------------------------------
initializing NRLIB DIS for Distributive
compiling into NRLIB DIS
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Distributive| REDEFINED
;;; *** |Distributive| REDEFINED
Time: 0.01 SEC.
Cumulative Statistics for Constructor Distributive
Time: 0.03 seconds
finalizing NRLIB DIS
Processing Distributive for Browser database:
--->-->Distributive(): Missing Description
; compiling file "/var/aw/var/LatexWiki/DIS.NRLIB/DIS.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/DIS.NRLIB/DIS.fasl written
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
Distributive is now explicitly exposed in frame initial
Distributive will be automatically loaded when needed from
/var/aw/var/LatexWiki/DIS.NRLIB/DIS
ASS+ abbreviates domain Associative+
------------------------------------------------------------------------
initializing NRLIB ASS+ for Associative+
compiling into NRLIB ASS+
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Associative+| REDEFINED
;;; *** |Associative+| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Associative+
Time: 0.03 seconds
finalizing NRLIB ASS+
Processing Associative+ for Browser database:
--->-->Associative+(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ASS+.NRLIB/ASS+.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/ASS+.NRLIB/ASS+.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
Associative+ is now explicitly exposed in frame initial
Associative+ will be automatically loaded when needed from
/var/aw/var/LatexWiki/ASS+.NRLIB/ASS+
ASSS abbreviates domain Associative*
------------------------------------------------------------------------
initializing NRLIB ASSS for Associative*
compiling into NRLIB ASSS
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Associative*| REDEFINED
;;; *** |Associative*| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Associative*
Time: 0.04 seconds
finalizing NRLIB ASSS
Processing Associative* for Browser database:
--->-->Associative*(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ASSS.NRLIB/ASSS.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/ASSS.NRLIB/ASSS.fasl written
; compilation finished in 0:00:00.014
------------------------------------------------------------------------
Associative* is now explicitly exposed in frame initial
Associative* will be automatically loaded when needed from
/var/aw/var/LatexWiki/ASSS.NRLIB/ASSS
ID abbreviates domain Identities
------------------------------------------------------------------------
initializing NRLIB ID for Identities
compiling into NRLIB ID
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Identities| REDEFINED
;;; *** |Identities| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Identities
Time: 0.02 seconds
finalizing NRLIB ID
Processing Identities for Browser database:
--->-->Identities(): Missing Description
; compiling file "/var/aw/var/LatexWiki/ID.NRLIB/ID.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/ID.NRLIB/ID.fasl written
; compilation finished in 0:00:00.012
------------------------------------------------------------------------
Identities is now explicitly exposed in frame initial
Identities will be automatically loaded when needed from
/var/aw/var/LatexWiki/ID.NRLIB/ID
INV abbreviates domain Inverse
------------------------------------------------------------------------
initializing NRLIB INV for Inverse
compiling into NRLIB INV
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Inverse| REDEFINED
;;; *** |Inverse| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Inverse
Time: 0.03 seconds
finalizing NRLIB INV
Processing Inverse for Browser database:
--->-->Inverse(): Missing Description
; compiling file "/var/aw/var/LatexWiki/INV.NRLIB/INV.lsp" (written 24 DEC 2016 05:13:01 PM):
; /var/aw/var/LatexWiki/INV.NRLIB/INV.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
Inverse is now explicitly exposed in frame initial
Inverse will be automatically loaded when needed from
/var/aw/var/LatexWiki/INV.NRLIB/INV
DBL abbreviates domain Doubles
------------------------------------------------------------------------
initializing NRLIB DBL for Doubles
compiling into NRLIB DBL
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Doubles| REDEFINED
;;; *** |Doubles| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Doubles
Time: 0.01 seconds
finalizing NRLIB DBL
Processing Doubles for Browser database:
--->-->Doubles(): Missing Description
; compiling file "/var/aw/var/LatexWiki/DBL.NRLIB/DBL.lsp" (written 24 DEC 2016 05:13:02 PM):
; /var/aw/var/LatexWiki/DBL.NRLIB/DBL.fasl written
; compilation finished in 0:00:00.009
------------------------------------------------------------------------
Doubles is now explicitly exposed in frame initial
Doubles will be automatically loaded when needed from
/var/aw/var/LatexWiki/DBL.NRLIB/DBL
CAN abbreviates domain Cancels
------------------------------------------------------------------------
initializing NRLIB CAN for Cancels
compiling into NRLIB CAN
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Cancels| REDEFINED
;;; *** |Cancels| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Cancels
Time: 0.02 seconds
finalizing NRLIB CAN
Processing Cancels for Browser database:
--->-->Cancels(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CAN.NRLIB/CAN.lsp" (written 24 DEC 2016 05:13:02 PM):
; /var/aw/var/LatexWiki/CAN.NRLIB/CAN.fasl written
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
Cancels is now explicitly exposed in frame initial
Cancels will be automatically loaded when needed from
/var/aw/var/LatexWiki/CAN.NRLIB/CAN
SQ abbreviates domain Squares
------------------------------------------------------------------------
initializing NRLIB SQ for Squares
compiling into NRLIB SQ
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Squares| REDEFINED
;;; *** |Squares| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Squares
Time: 0.02 seconds
finalizing NRLIB SQ
Processing Squares for Browser database:
--->-->Squares(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SQ.NRLIB/SQ.lsp" (written 24 DEC 2016 05:13:02 PM):
; /var/aw/var/LatexWiki/SQ.NRLIB/SQ.fasl written
; compilation finished in 0:00:00.009
------------------------------------------------------------------------
Squares is now explicitly exposed in frame initial
Squares will be automatically loaded when needed from
/var/aw/var/LatexWiki/SQ.NRLIB/SQ
DIV abbreviates domain Divides
------------------------------------------------------------------------
initializing NRLIB DIV for Divides
compiling into NRLIB DIV
importing Symbolic(Integer,
(time taken in buildFunctor: 0)
;;; *** |Divides| REDEFINED
;;; *** |Divides| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor Divides
Time: 0.02 seconds
finalizing NRLIB DIV
Processing Divides for Browser database:
--->-->Divides(): Missing Description
; compiling file "/var/aw/var/LatexWiki/DIV.NRLIB/DIV.lsp" (written 24 DEC 2016 05:13:02 PM):
; /var/aw/var/LatexWiki/DIV.NRLIB/DIV.fasl written
; compilation finished in 0:00:00.009
------------------------------------------------------------------------
Divides is now explicitly exposed in frame initial
Divides will be automatically loaded when needed from
/var/aw/var/LatexWiki/DIV.NRLIB/DIV
AND abbreviates domain /\
------------------------------------------------------------------------
initializing NRLIB AND for /\
compiling into NRLIB AND
compiling exported axioms : () -> List Equation InputForm
Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |/\\| REDEFINED
;;; *** |/\\| REDEFINED
Time: 0.01 SEC.
Cumulative Statistics for Constructor /\
Time: 0.02 seconds
finalizing NRLIB AND
Processing /\ for Browser database:
--------constructor---------
; compiling file "/var/aw/var/LatexWiki/AND.NRLIB/AND.lsp" (written 24 DEC 2016 05:13:02 PM):
; /var/aw/var/LatexWiki/AND.NRLIB/AND.fasl written
; compilation finished in 0:00:00.013
------------------------------------------------------------------------
/\ is now explicitly exposed in frame initial
/\ will be automatically loaded when needed from
/var/aw/var/LatexWiki/AND.NRLIB/AND
SEXPR abbreviates domain SymbolicExpression
------------------------------------------------------------------------
initializing NRLIB SEXPR for SymbolicExpression
compiling into NRLIB SEXPR
****** Domain: $ already in scope
augmenting $: (RetractableTo (Integer))
****** Domain: R already in scope
augmenting R: (IntegralDomain)
****** Domain: $ already in scope
augmenting $: (TranscendentalFunctionCategory)
****** Domain: R already in scope
augmenting R: (GcdDomain)
****** Domain: R already in scope
augmenting R: (LinearlyExplicitOver (Integer))
****** Domain: $ already in scope
augmenting $: (FunctionSpace (Integer))
****** Domain: $ already in scope
augmenting $: (RetractableTo (Integer))
****** Domain: $ already in scope
augmenting $: (Ring)
****** Domain: R already in scope
augmenting R: (AbelianGroup)
****** Domain: R already in scope
augmenting R: (AbelianSemiGroup)
****** Domain: R already in scope
augmenting R: (CharacteristicNonZero)
****** Domain: R already in scope
augmenting R: (CommutativeRing)
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Float)))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Integer)))
****** Domain: R already in scope
augmenting R: (Group)
****** Domain: R already in scope
augmenting R: (IntegralDomain)
****** Domain: R already in scope
augmenting R: (PatternMatchable (Float))
****** Domain: R already in scope
augmenting R: (PatternMatchable (Integer))
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
****** Domain: R already in scope
augmenting R: (Ring)
****** Domain: R already in scope
augmenting R: (SemiGroup)
(time taken in buildFunctor: 120)
;;; *** |SymbolicExpression| REDEFINED
;;; *** |SymbolicExpression| REDEFINED
Time: 0.14 SEC.
Cumulative Statistics for Constructor SymbolicExpression
Time: 0.14 seconds
finalizing NRLIB SEXPR
Processing SymbolicExpression for Browser database:
--->-->SymbolicExpression(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SEXPR.NRLIB/SEXPR.lsp" (written 24 DEC 2016 05:13:02 PM):
; /var/aw/var/LatexWiki/SEXPR.NRLIB/SEXPR.fasl written
; compilation finished in 0:00:00.020
------------------------------------------------------------------------
SymbolicExpression is now explicitly exposed in frame initial
SymbolicExpression will be automatically loaded when needed from
/var/aw/var/LatexWiki/SEXPR.NRLIB/SEXPR
SYMPKG1 abbreviates package SymbolicFunctions1
------------------------------------------------------------------------
initializing NRLIB SYMPKG1 for SymbolicFunctions1
compiling into NRLIB SYMPKG1
****** Domain: R already in scope
****** Domain: S already in scope
compiling local hack : B -> InputForm
Time: 0.01 SEC.
compiling exported convert : B -> A
Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |SymbolicFunctions1| REDEFINED
;;; *** |SymbolicFunctions1| REDEFINED
Time: 0 SEC.
Cumulative Statistics for Constructor SymbolicFunctions1
Time: 0.02 seconds
finalizing NRLIB SYMPKG1
Processing SymbolicFunctions1 for Browser database:
--->-->SymbolicFunctions1(constructor): Not documented!!!!
--->-->SymbolicFunctions1((convert (A B))): Not documented!!!!
--->-->SymbolicFunctions1(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SYMPKG1.NRLIB/SYMPKG1.lsp" (written 24 DEC 2016 05:13:02 PM):
; /var/aw/var/LatexWiki/SYMPKG1.NRLIB/SYMPKG1.fasl written
; compilation finished in 0:00:00.020
------------------------------------------------------------------------
SymbolicFunctions1 is now explicitly exposed in frame initial
SymbolicFunctions1 will be automatically loaded when needed from
/var/aw/var/LatexWiki/SYMPKG1.NRLIB/SYMPKG1Some Tests
fricas
s:SymbolicExpression Integer := 1+1+1
 | (1) |
Type: SymbolicExpression?(Integer)
fricas
eval s
 | (2) |
Type: SymbolicExpression?(Integer)
fricas
s::Integer
 | (3) |
Type: Integer
fricas
equal? (s=3)
 | (4) |
Type: Boolean
fricas
test(s=3)
 | (5) |
Type: Boolean
fricas
(1/5)::SEXPR INT
 | (6) |
Type: SymbolicExpression?(Integer)
fricas
-- (x,y, z):SymbolicExpression Integer
Type: Void
fricas
(p,q, r):Expression Integer
Type: Void
fricas
-- x::Expression Integer
 | (7) |
Type: Expression(Integer)
fricas
x+x = 2*x
 | (8) |
Type: Equation(SymbolicExpression?(Integer))
fricas
test %
 | (9) |
Type: Boolean
fricas
x-x = 0
 | (10) |
Type: Equation(SymbolicExpression?(Integer))
fricas
test %
 | (11) |
Type: Boolean
fricas
x+y = y+x
 | (12) |
Type: Equation(SymbolicExpression?(Integer))
fricas
test %
 | (13) |
Type: Boolean
fricas
x+(y+z)= (x+y)+z
 | (14) |
Type: Equation(SymbolicExpression?(Integer))
fricas
test %
 | (15) |
Type: Boolean
fricas
x*x = x^2
 | (16) |
Type: Equation(SymbolicExpression?(Integer))
fricas
test %
 | (17) |
Type: Boolean
fricas
x/x = 1
 | (18) |
Type: Equation(SymbolicExpression?(Integer))
fricas
test %
 | (19) |
Type: Boolean
fricas
x*y = y*x
 | (20) |
Type: Equation(SymbolicExpression?(Integer))
fricas
test %
 | (21) |
Type: Boolean
fricas
x*(y*z)= (x*y)*z
>> System error: Control stack exhausted (no more space for function call frames). This is probably due to heavily nested or infinitely recursive function calls,or a tail call that SBCL cannot or has not optimized away.
PROCEED WITH CAUTION.