Edit detail for SandBoxFreeProduct revision 2 of 4

)abbrev domain FPROD FreeProduct
++ Description:
++ This domain implements the free product of monoids (groups)
++ It is the coproduct in the category of monoids (groups).
++ FreeProduct(A,B) is the monoid (group) whose elements are
++ the reduced words in A and B, under the operation of concatenation
++ followed by reduction:
++   * Remove identity elements (of either A or B)
++   * Replace a1a2 by its product in A and b1b2 by its product in B
++ Ref: http://en.wikipedia.org/wiki/Free_product
FreeProduct(A:Monoid,B:Monoid):Monoid with
    if A has Group and B has Group then Group
    in1: A -> %
    in2: B -> %
    RetractableTo(A)
    RetractableTo(B)
    is1: % -> Boolean
    is2: % -> Boolean
    factors: % -> List %
    if A has Comparable and B has Comparable then Comparable
  == add
    Rep == List Union(A,B)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    One() == per []
    coerce(x:%):OutputForm ==
      r:=rep(x)
      if empty?(r) then
        return coerce(1$InputForm)
      else if #r=1 then
        s:=first r
        if s case A then
          return coerce(s::A)
        else if s case B then
          return overbar(coerce(s::B))
      return blankSeparate([coerce(per [s]) for s in r])

    if A has Comparable and B has Comparable then
      smaller?(x:%,y:%):Boolean ==
        r1:=rep(x); r2:=rep(y)
        for s1 in r1 for s2 in r2 repeat
          if s1 case A then
            if s2 case A then
              if smaller?(s1::A, s2::A) then return true
            if s2 case B then return true
          else if s1 case B then
            if s2 case B then
              if smaller?(s1::B, s2::B) then return true
            if s2 case A then return false
        if #r1 < #r2 then return true
        return false

    (x:% = y:%):Boolean ==
      r1:=rep(x); r2:=rep(y)
      if #r1 ~= #r2 then return false
      for s1 in r1 for s2 in r2 repeat
        if s1 case A then
          if s2 case A then
            if (s1::A) ~= (s2::A) then return false
          if s2 case B then return false
        else if s1 case B then
          if s2 case B then
            if (s1::B) ~= (s2::B) then return false
          if s2 case A then return false
      return true

    in1(x:A):% == if x=1 then 1 else per [[x]]
    in2(y:B):% == if y=1 then 1 else per [[y]]
    coerce(x:A):% == in1(x)
    coerce(x:B):% == in2(x)
    is1(x:%):Boolean == first(rep x) case A
    is2(x:%):Boolean == first(rep x) case B
    retract(x:%):A == if x=1 or not is1(x) then 1 else coerce(first rep x)@A
    retract(x:%):B == if x=1 or not is2(x) then 1 else coerce(first rep x)@B
    factors(x:%):List % == [per [s] for s in rep(x)]

    if A has Group and B has Group then
      inv(x:%):% ==
        if x=1 then return 1
        return per [( _
          s case A => inv(s::A); _
          s case B => inv(s::B)) _
        for s in reverse rep(x)]

    (x:% * y:%):% ==
      if x=1 then return y
      if y=1 then return x
      r1:=rep(x); r2:=rep(y)
      f1:=first(r1,(#r1-1)::NonNegativeInteger); l1:=last r1
      f2:=first r2; l2:=last(r2,(#r2-1)::NonNegativeInteger)
      -- reduction
      if l1 case A and f2 case A then
        return per(f1)*in1((l1::A)*(f2::A))*per(l2)
      if l1 case B and f2 case B then
        return per(f1)*in2((l1::B)*(f2::B))*per(l2)
      return per concat(r1,r2)

    (x:% ^ n:NonNegativeInteger):% ==
      if x=1 then return 1
      if n>0 then return x*x^(n-1)::NonNegativeInteger
      return 1

    (x:% ^ n:PositiveInteger):% ==
      if x=1 then return 1
      if n>1 then return x*x^((n-1)::PositiveInteger)
      return x

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/7597684502788721203-25px001.spad using 
      old system compiler.
   FPROD abbreviates domain FreeProduct 
------------------------------------------------------------------------
   initializing NRLIB FPROD for FreeProduct 
   compiling into NRLIB FPROD 
   compiling local rep : $ -> List Union(A,B)
      FPROD;rep is replaced by x 
Time: 0.05 SEC.

   compiling local per : List Union(A,B) -> $
      FPROD;per is replaced by x 
Time: 0 SEC.

   compiling exported One : () -> $
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.11 SEC.

****** Domain: A already in scope
augmenting A: (Comparable)
****** Domain: B already in scope
augmenting B: (Comparable)
   compiling exported smaller? : ($,$) -> Boolean
Time: 0.01 SEC.

   compiling exported = : ($,$) -> Boolean
Time: 0.01 SEC.

   compiling exported in1 : A -> $
Time: 0 SEC.

   compiling exported in2 : B -> $
Time: 0.004 SEC.

   compiling exported coerce : A -> $
Time: 0 SEC.

   compiling exported coerce : B -> $
Time: 0 SEC.

   compiling exported is1 : $ -> Boolean
Time: 0 SEC.

   compiling exported is2 : $ -> Boolean
Time: 0 SEC.

   compiling exported retract : $ -> A
Time: 0.004 SEC.

   compiling exported retract : $ -> B
Time: 0 SEC.

   compiling exported factors : $ -> List $
Time: 0 SEC.

****** Domain: A already in scope
augmenting A: (Group)
****** Domain: B already in scope
augmenting B: (Group)
   compiling exported inv : $ -> $
Time: 0.004 SEC.

   compiling exported * : ($,$) -> $
Time: 0.03 SEC.

   compiling exported ^ : ($,NonNegativeInteger) -> $
Time: 0.004 SEC.

   compiling exported ^ : ($,PositiveInteger) -> $
Time: 0.004 SEC.

****** Domain: A already in scope
augmenting A: (Comparable)
****** Domain: B already in scope
augmenting B: (Comparable)
****** Domain: A already in scope
augmenting A: (Group)
****** Domain: B already in scope
augmenting B: (Group)
(time taken in buildFunctor:  4)

;;;     ***       |FreeProduct| REDEFINED

;;;     ***       |FreeProduct| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor FreeProduct
      Time: 0.24 seconds

   finalizing NRLIB FPROD 
   Processing FreeProduct for Browser database:
--->-->FreeProduct((in1 (% A))): Not documented!!!!
--->-->FreeProduct((in2 (% B))): Not documented!!!!
--->-->FreeProduct((is1 ((Boolean) %))): Not documented!!!!
--->-->FreeProduct((is2 ((Boolean) %))): Not documented!!!!
--->-->FreeProduct((factors ((List %) %))): Not documented!!!!
--------constructor---------
; compiling file "/var/zope2/var/LatexWiki/FPROD.NRLIB/FPROD.lsp" (written 23 SEP 2009 12:45:31 AM):
; compiling (/VERSIONCHECK 2)
; compiling (PUT (QUOTE |FPROD;rep|) ...)
; compiling (DEFUN |FPROD;rep| ...)
; compiling (PUT (QUOTE |FPROD;per|) ...)
; compiling (DEFUN |FPROD;per| ...)
; compiling (DEFUN |FPROD;One;$;3| ...)
; compiling (DEFUN |FPROD;coerce;$Of;4| ...)
; compiling (DEFUN |FPROD;smaller?;2$B;5| ...)
; compiling (DEFUN |FPROD;=;2$B;6| ...)
; compiling (DEFUN |FPROD;in1;A$;7| ...)
; compiling (DEFUN |FPROD;in2;B$;8| ...)
; compiling (DEFUN |FPROD;coerce;A$;9| ...)
; compiling (DEFUN |FPROD;coerce;B$;10| ...)
; compiling (DEFUN |FPROD;is1;$B;11| ...)
; compiling (DEFUN |FPROD;is2;$B;12| ...)
; compiling (DEFUN |FPROD;retract;$A;13| ...)
; compiling (DEFUN |FPROD;retract;$B;14| ...)
; compiling (DEFUN |FPROD;factors;$L;15| ...)
; compiling (DEFUN |FPROD;inv;2$;16| ...)
; compiling (DEFUN |FPROD;*;3$;17| ...)
; compiling (DEFUN |FPROD;^;$Nni$;18| ...)
; compiling (DEFUN |FPROD;^;$Pi$;19| ...)
; compiling (DEFUN |FreeProduct| ...)
; compiling (DEFUN |FreeProduct;| ...)
; compiling (MAKEPROP (QUOTE |FreeProduct|) ...)

; /var/zope2/var/LatexWiki/FPROD.NRLIB/FPROD.fasl written
; compilation finished in 0:00:00.221
------------------------------------------------------------------------
   FreeProduct is now explicitly exposed in frame initial 
   FreeProduct will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/FPROD.NRLIB/FPROD

II:=FPROD(INT,INT)

(1)

i1:=in1(2)$II

(2)

i2:=in2(3)$II

(3)

i1

(4)

i2

(5)

i1=i2

(6)

test(i1=i2)

(7)

test(i2=i2)

(8)

i1*i1

(9)

i2*i2

(10)

i1*i2

(11)

i2*i1

(12)

f:=FreeProduct(FreeMonoid Symbol,FreeMonoid Symbol)

(13)

p:=in1('p)$f

(14)

q:=in2('q)$f

(15)

r:=in1('r)$f

(16)

(p*q*r)^2

(17)

(p*q)*(r*p)*(q*r)

(18)

a:FreeMonoid Symbol:='a

(19)

b:FreeMonoid Symbol:='b

(20)

a*b

(21)

p*q

(22)

p*q^2

(23)

g:=MonoidRing(Integer,f)

(24)

m:=(in1('m)$f)::g

(25)

n:=(in2('n)$f)::g

(26)

nm:=m*n-n*m

(27)

nm^2

(28)

g:=FreeProduct(FreeGroup Symbol,FreeGroup Symbol)

(29)

g1:=(in1('g1)$g)

(30)

g2:=(in2('g2)$g)

(31)

g1*g2*g1^(-1)

(32)

h:=MonoidRing(Integer,g)

(33)

h1:=g1::h

(34)

h2:=g2::h

(35)

hh:=h2*h1-h1*h2

(36)

hh^2

(37)

recip(h1)*recip(h2)

(38)

h1*recip(h1)

(39)

h1*h2

(40)

recip(h1*h2)

(41)

h1*h2*recip(h1*h2)

(42)