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 Topics FrontPage SandBox SandBoxINFSUM <-- You are here. spad)abbrev package UNIFY Unify ++ Author: Kurt Pagani ++ Date Created: Mon Feb 06 18:56:20 CET 2017 ++ License: BSD ++ References: ++ Description: ++ Unify(X) : Exports == Implementation where X: Join(ConvertibleTo InputForm, ExpressionSpace) IF ==> InputForm SEXP ==> SExpression Exports == with sexpr : X -> IF ++ Convert to InputForm occursIn : (IF,IF) -> Boolean ++ Return true if ELT occurs in EXP at any level cons : (IF,IF) -> IF ++ Cons subst1 : (IF,IF,IF) -> IF ++ Substitute A for each occurrence of B in LST. doSubst : (IF,IF) -> IF ++ Perform all substitutions in L on EXP in reverse order. unify : (IF,IF) -> IF Implementation == IF add failed:IF:=convert('failure)$IF sexpr(x:X):IF == convert(x)@IF occursIn(elt,exp) == elt=exp => true atom? exp => false occursIn(elt,car exp) or occursIn(elt,cdr exp) cons(a:IF,b:IF):IF == s:SExpression:=CONS(a,b)$Lisp convert(s)$IF subst1(a:IF,b:IF,lst:IF):IF == null? lst => lst lst=b => a atom? lst => lst b=car lst => cons(a,subst1(a,b,cdr lst)) atom? car lst => cons(car lst, subst1(a,b,cdr lst)) cons(subst1(a,b,car lst),subst1(a,b,cdr lst)) doSubst(exp:IF,l:IF):IF == null? l => exp a:=car l subst1(car a, car cdr a, doSubst(exp,cdr l)) variable?(x:IF):Boolean == not atom? x => false symbol? x => true false addPair(term:IF,variable:IF,u:IF):IF == occursIn(variable,term) => failed cons(convert [term,variable],subst1(term,variable,u)) unify1(termlist1:IF,termlist2:IF,u:IF):IF == termlist1=termlist2 => u null? termlist1 or null? termlist2 => failed variable? termlist1 => addPair(termlist2,termlist1,u) variable? termlist2 => addPair(termlist1,termlist2,u) atom? termlist1 or atom? termlist2 => failed u:=unify1(doSubst(car termlist1,u),doSubst(car termlist2,u),u) u=failed => failed unify1(cdr termlist1,cdr termlist2,u) unify(literal1:IF,literal2:IF):IF == u:=convert([])$IF if car literal1 = car literal2 then unify1(cdr literal1,cdr literal2,u) else failed spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1957568863081842059-25px001.spad using old system compiler. UNIFY abbreviates package Unify ------------------------------------------------------------------------ initializing NRLIB UNIFY for Unify compiling into NRLIB UNIFY ****** Domain: X already in scope compiling exported sexpr : X -> InputForm Time: 0.01 SEC. compiling exported occursIn : (InputForm,InputForm) -> Boolean Time: 0 SEC. compiling exported cons : (InputForm,InputForm) -> InputForm Time: 0 SEC. compiling exported subst1 : (InputForm,InputForm,InputForm) -> InputForm Time: 0 SEC. compiling exported doSubst : (InputForm,InputForm) -> InputForm Time: 0 SEC. compiling local variable? : InputForm -> Boolean Time: 0 SEC. compiling local addPair : (InputForm,InputForm,InputForm) -> InputForm Time: 0 SEC. compiling local unify1 : (InputForm,InputForm,InputForm) -> InputForm Time: 0 SEC. compiling exported unify : (InputForm,InputForm) -> InputForm Time: 0 SEC. (time taken in buildFunctor: 0) ;;; *** |Unify| REDEFINED ;;; *** |Unify| REDEFINED Time: 0 SEC. Cumulative Statistics for Constructor Unify Time: 0.01 seconds --------------non extending category---------------------- .. Unify(#1) of cat (CATEGORY |package| (SIGNATURE |sexpr| ((|InputForm|) |#1|)) (SIGNATURE |occursIn| ((|Boolean|) (|InputForm|) (|InputForm|))) (SIGNATURE |cons| ((|InputForm|) (|InputForm|) (|InputForm|))) (SIGNATURE |subst1| ((|InputForm|) (|InputForm|) (|InputForm|) (|InputForm|))) (SIGNATURE |doSubst| ((|InputForm|) (|InputForm|) (|InputForm|))) (SIGNATURE |unify| ((|InputForm|) (|InputForm|) (|InputForm|)))) has no (|SExpressionCategory| (|String|) (|Symbol|) (|Integer|) (|DoubleFloat|)) finalizing NRLIB UNIFY Processing Unify for Browser database: --------constructor--------- --------(sexpr ((InputForm) X))--------- --->-->Unify((sexpr ((InputForm) X))): Improper first word in comments: Convert "Convert to InputForm" --------(occursIn ((Boolean) (InputForm) (InputForm)))--------- --->-->Unify((occursIn ((Boolean) (InputForm) (InputForm)))): Improper first word in comments: Return "Return \\spad{true} if ELT occurs in EXP at any level" --------(cons ((InputForm) (InputForm) (InputForm)))--------- --------(subst1 ((InputForm) (InputForm) (InputForm) (InputForm)))--------- --->-->Unify((subst1 ((InputForm) (InputForm) (InputForm) (InputForm)))): Improper first word in comments: Substitute "Substitute A for each occurrence of \\spad{B} in \\spad{LST}." --------(doSubst ((InputForm) (InputForm) (InputForm)))--------- --->-->Unify((doSubst ((InputForm) (InputForm) (InputForm)))): Improper first word in comments: Perform "Perform all substitutions in \\spad{L} on EXP in reverse order." --->-->Unify((unify ((InputForm) (InputForm) (InputForm)))): Not documented!!!! ; compiling file "/var/aw/var/LatexWiki/UNIFY.NRLIB/UNIFY.lsp" (written 28 JUL 2018 08:58:06 PM): ; /var/aw/var/LatexWiki/UNIFY.NRLIB/UNIFY.fasl written ; compilation finished in 0:00:00.049 ------------------------------------------------------------------------ Unify is now explicitly exposed in frame initial Unify will be automatically loaded when needed from /var/aw/var/LatexWiki/UNIFY.NRLIB/UNIFY fricas)set break resume  fricas)set output algebra on  fricas)set output tex off X ==> Expression Integer Type: Void fricasIF ==> InputForm Type: Void fricasUFX ==> Unify(X) Type: Void fricasREC ==> Record(inp:X,res:X) Type: Void fricasr1:=[sum((-1)^(k?+1)/k?,k?=1..oo),log(2)]$REC oo k? + 1 --+ (- 1) (5) [inp = > -----------, res = log(2)] --+ k? k? = 1 Type: Record(inp: Expression(Integer),res: Expression(Integer)) fricasr2:=[sum((-1)^(k?+1)/(2*k?-1),k?=1..oo),%pi/4]$REC oo k? + 1 --+ (- 1) %pi (6) [inp = > -----------, res = ---] --+ 2 k? - 1 4 k? = 1 Type: Record(inp: Expression(Integer),res: Expression(Integer)) fricasr3:=[sum(1/k?^p?,k?=1..oo),riemannZeta(p?)]$REC oo --+ 1 (7) [inp = > ----, res = riemannZeta(p?)] --+ p? k? = 1 k? Type: Record(inp: Expression(Integer),res: Expression(Integer)) fricasr4:=[sum((-1)^(k?+1)/k?^p?,k?=1..oo),(1-2^(1-p?))*riemannZeta(p?)]$REC (8) oo k? + 1 --+ (- 1) [inp = > -----------, --+ p? k? = 1 k? - p? + 1 res = - riemannZeta(p?)2 + riemannZeta(p?)] Type: Record(inp: Expression(Integer),res: Expression(Integer)) fricasr5:=[sum(1/k?^2,k?=1..oo),%pi^2/6]$REC oo 2 --+ 1 %pi (9) [inp = > ---, res = ----] --+ 2 6 k? = 1 k? Type: Record(inp: Expression(Integer),res: Expression(Integer)) fricasmatch(a,b) == r:=unify(a,b)$UFX r='failure::IF => [] dr:=destruct r ddr:=[destruct s for s in dr] eq:=[symbol(s.1)::X=symbol(s.2)::X for s in ddr] Type: Void fricaslookup(e,r) == test(e=r.inp) => r.res ei:=e::IF test(ei=(r.inp)::IF) => r.res m:=match(ei,(r.inp)::IF) subst(r.res,m) Type: Void fricasi1:=sum((-1)^(n+1)/n,n=1..oo) oo k? + 1 --+ (- 1) (12) > ----------- --+ k? k? = 1 Type: Expression(Integer) fricasi2:=sum((-1)^(m+1)/m,m=1..N) N m + 1 --+ (- 1) (13) > ---------- --+ m m = 1 Type: Expression(Integer) fricasi3:=sum((-1)^(m^2+1)/m,m=1..N) 2 N m + 1 --+ (- 1) (14) > ----------- --+ m m = 1 Type: Expression(Integer) fricaslookup(i2,r1) fricasCompiling function match with type (InputForm, InputForm) -> List( Equation(Expression(Integer))) fricasCompiling function lookup with type (Expression(Integer), Record(inp : Expression(Integer),res: Expression(Integer))) -> Expression( Integer) (15) log(2) Type: Expression(Integer) fricasii3:=i3::IF (16) (%defsum (/ (^ - 1 (+ (^ %S 2) 1)) %S) %S m 1 N) Type: InputForm fricasunify(ii3,(r1.inp)::IF)\$UFX (17) failure Type: InputForm fricasi4:=sum((-1)^(s+1)/(2*s-1),s=1..oo) oo k? + 1 --+ (- 1) (18) > ----------- --+ 2 k? - 1 k? = 1 Type: Expression(Integer) fricaslookup(i4,r2) --> %pi/4 %pi (19) --- 4 Type: Expression(Integer) fricasi5:=sum(1/n^z,n=1..oo) oo --+ 1 (20) > -- --+ z n = 1 n Type: Expression(Integer) fricaslookup(i5,r3) --> riemannZeta(z) (21) riemannZeta(z) Type: Expression(Integer) fricasi6:=sum((-1)^(t+1)/t^s,t=1..oo) oo t + 1 --+ (- 1) (22) > ---------- --+ s t = 1 t Type: Expression(Integer) fricaslookup(i6,r4) --> - s + 1 (23) - riemannZeta(s)2 + riemannZeta(s) Type: Expression(Integer) fricasi7:=sum(1/m^2,m=1..oo) oo --+ 1 (24) > --- --+ 2 k? = 1 k? Type: Union(Expression(Integer),...) fricaslookup(i7,r5) --> %pi^2/6 2 %pi (25) ---- 6 Type: Expression(Integer)

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