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