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fricas
(1) -> <spad>
fricas
)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>
fricas
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 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: 0.02 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 01 DEC 2024 04:10:50 PM):
; wrote /var/aw/var/LatexWiki/UNIFY.NRLIB/UNIFY.fasl ; compilation finished in 0:00:00.036 ------------------------------------------------------------------------ Unify is now explicitly exposed in frame initial Unify will be automatically loaded when needed from /var/aw/var/LatexWiki/UNIFY.NRLIB/UNIFY

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)set break resume
 
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)set output algebra on
 
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)set output tex off
X ==> Expression Integer
Type: Void
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IF ==> InputForm
Type: Void
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UFX ==> Unify(X)
Type: Void
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REC ==> Record(inp:X,res:X)
Type: Void
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r1:=[sum((-1)^(k?+1)/k?,k?=1..oo),log(2)]$REC
oo k? + 1 --+ (- 1) (5) [inp = > -----------, res = log(2)] --+ k? k? = 1
Type: Record(inp: Expression(Integer),res: Expression(Integer))
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r2:=[sum((-1)^(k?+1)/(2*k?-1),k?=1..oo),%pi/4]$REC
oo k? + 1 --+ (- 1) %pi (6) [inp = > -----------, res = ---] --+ 2 k? - 1 4 k? = 1
Type: Record(inp: Expression(Integer),res: Expression(Integer))
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r3:=[sum(1/k?^p?,k?=1..oo),riemannZeta(p?)]$REC
oo --+ 1 (7) [inp = > ----, res = riemannZeta(p?)] --+ p? k? = 1 k?
Type: Record(inp: Expression(Integer),res: Expression(Integer))
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r4:=[sum((-1)^(k?+1)/k?^p?,k?=1..oo),(1-2^(1-p?))*riemannZeta(p?)]$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))
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r5:=[sum(1/k?^2,k?=1..oo),%pi^2/6]$REC
oo 2 --+ 1 %pi (9) [inp = > ---, res = ----] --+ 2 6 k? = 1 k?
Type: Record(inp: Expression(Integer),res: Expression(Integer))
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match(a,b) ==
  r:=unify(a,b)$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
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lookup(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
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i1:=sum((-1)^(n+1)/n,n=1..oo)
oo k? + 1 --+ (- 1) (12) > ----------- --+ k? k? = 1
Type: Expression(Integer)
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i2:=sum((-1)^(m+1)/m,m=1..N)
N m + 1 --+ (- 1) (13) > ---------- --+ m m = 1
Type: Expression(Integer)
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i3:=sum((-1)^(m^2+1)/m,m=1..N)
2 N m + 1 --+ (- 1) (14) > ----------- --+ m m = 1
Type: Expression(Integer)
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lookup(i2,r1)
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Compiling function match with type (InputForm, InputForm) -> List(
      Equation(Expression(Integer)))
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Compiling function lookup with type (Expression(Integer), Record(inp
      : Expression(Integer),res: Expression(Integer))) -> Expression(
      Integer) 
(15) log(2)
Type: Expression(Integer)
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ii3:=i3::IF
(16) (sum (/ (^ - 1 (+ (^ %S 2) 1)) %S) (equation %S (SEGMENT 1 N)))
Type: InputForm
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unify(ii3,(r1.inp)::IF)$UFX
(17) failure
Type: InputForm
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i4:=sum((-1)^(s+1)/(2*s-1),s=1..oo)
oo k? + 1 --+ (- 1) (18) > ----------- --+ 2 k? - 1 k? = 1
Type: Expression(Integer)
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lookup(i4,r2) --> %pi/4
%pi (19) --- 4
Type: Expression(Integer)
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i5:=sum(1/n^z,n=1..oo)
oo --+ 1 (20) > -- --+ z n = 1 n
Type: Expression(Integer)
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lookup(i5,r3) --> riemannZeta(z) 
(21) riemannZeta(z)
Type: Expression(Integer)
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i6:=sum((-1)^(t+1)/t^s,t=1..oo)
oo t + 1 --+ (- 1) (22) > ---------- --+ s t = 1 t
Type: Expression(Integer)
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lookup(i6,r4) -->
- s + 1 (23) - riemannZeta(s)2 + riemannZeta(s)
Type: Expression(Integer)
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i7:=sum(1/m^2,m=1..oo)
oo --+ 1 (24) > --- --+ 2 k? = 1 k?
Type: Union(Expression(Integer),...)
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lookup(i7,r5) --> %pi^2/6
2 %pi (25) ---- 6
Type: Expression(Integer)




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