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spad
)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
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1109760127526461248-25px001.spad
      using old system compiler.
   FPROD abbreviates domain FreeProduct 
------------------------------------------------------------------------
   initializing NRLIB FPROD for FreeProduct 
   compiling into NRLIB FPROD 
   processing macro definition Rep ==> List Union(A,B) 
   compiling local rep : $ -> List Union(A,B)
      FPROD;rep is replaced by x 
Time: 0.01 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: 0.02 SEC.
****** Domain: A already in scope augmenting A: (Comparable) ****** Domain: B already in scope augmenting B: (Comparable) compiling exported smaller? : ($,$) -> Boolean Time: 0 SEC.
compiling exported = : ($,$) -> Boolean Time: 0 SEC.
compiling exported in1 : A -> $ Time: 0 SEC.
compiling exported in2 : B -> $ Time: 0 SEC.
compiling exported coerce : A -> $ Time: 0 SEC.
compiling exported coerce : B -> $ Time: 0.01 SEC.
compiling exported is1 : $ -> Boolean Time: 0 SEC.
compiling exported is2 : $ -> Boolean Time: 0 SEC.
compiling exported retract : $ -> A Time: 0 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 SEC.
compiling exported * : ($,$) -> $ Time: 0.01 SEC.
compiling exported ^ : ($,NonNegativeInteger) -> $ Time: 0.01 SEC.
compiling exported ^ : ($,PositiveInteger) -> $ Time: 0 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: 0)
;;; *** |FreeProduct| REDEFINED
;;; *** |FreeProduct| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor FreeProduct Time: 0.06 seconds
finalizing NRLIB FPROD Processing FreeProduct for Browser database: --------constructor--------- --->-->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!!!! ; compiling file "/var/aw/var/LatexWiki/FPROD.NRLIB/FPROD.lsp" (written 25 APR 2013 07:08:59 PM):
; /var/aw/var/LatexWiki/FPROD.NRLIB/FPROD.fasl written ; compilation finished in 0:00:00.080 ------------------------------------------------------------------------ FreeProduct is now explicitly exposed in frame initial FreeProduct will be automatically loaded when needed from /var/aw/var/LatexWiki/FPROD.NRLIB/FPROD

fricas
II:=FPROD(INT,INT)

\label{eq1}\hbox{\axiomType{FreeProduct}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{Integer}\ })(1)
Type: Type
fricas
i1:=in1(2)$II

\label{eq2}2(2)
Type: FreeProduct?(Integer,Integer)
fricas
i2:=in2(3)$II

\label{eq3}\overline 3(3)
Type: FreeProduct?(Integer,Integer)
fricas
i1

\label{eq4}2(4)
Type: FreeProduct?(Integer,Integer)
fricas
i2

\label{eq5}\overline 3(5)
Type: FreeProduct?(Integer,Integer)
fricas
i1=i2

\label{eq6}2 ={\overline 3}(6)
Type: Equation(FreeProduct?(Integer,Integer))
fricas
test(i1=i2)

\label{eq7} \mbox{\rm false} (7)
Type: Boolean
fricas
test(i2=i2)

\label{eq8} \mbox{\rm true} (8)
Type: Boolean
fricas
i1*i1

\label{eq9}4(9)
Type: FreeProduct?(Integer,Integer)
fricas
i2*i2

\label{eq10}\overline 9(10)
Type: FreeProduct?(Integer,Integer)
fricas
i1*i2

\label{eq11}2 \ {\overline 3}(11)
Type: FreeProduct?(Integer,Integer)
fricas
i2*i1

\label{eq12}{\overline 3}\  2(12)
Type: FreeProduct?(Integer,Integer)

fricas
f:=FreeProduct(FreeMonoid Symbol,FreeMonoid Symbol)

\label{eq13}\hbox{\axiomType{FreeProduct}\ } (\hbox{\axiomType{FreeMonoid}\ } (\hbox{\axiomType{Symbol}\ }) , \hbox{\axiomType{FreeMonoid}\ } (\hbox{\axiomType{Symbol}\ }))(13)
Type: Type
fricas
p:=in1('p)$f

\label{eq14}p(14)
Type: FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol))
fricas
q:=in2('q)$f

\label{eq15}\overline q(15)
Type: FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol))
fricas
r:=in1('r)$f

\label{eq16}r(16)
Type: FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol))
fricas
(p*q*r)^2

\label{eq17}p \ {\overline q}\ {r \  p}\ {\overline q}\  r(17)
Type: FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol))
fricas
(p*q)*(r*p)*(q*r)

\label{eq18}p \ {\overline q}\ {r \  p}\ {\overline q}\  r(18)
Type: FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol))
fricas
a:FreeMonoid Symbol:='a

\label{eq19}a(19)
Type: FreeMonoid?(Symbol)
fricas
b:FreeMonoid Symbol:='b

\label{eq20}b(20)
Type: FreeMonoid?(Symbol)
fricas
a*b

\label{eq21}a \  b(21)
Type: FreeMonoid?(Symbol)
fricas
p*q

\label{eq22}p \ {\overline q}(22)
Type: FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol))
fricas
p*q^2

\label{eq23}p \ {\overline{{q}^{2}}}(23)
Type: FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol))
fricas
g:=MonoidRing(Integer,f)

\label{eq24}\hbox{\axiomType{MonoidRing}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{FreeProduct}\ } (\hbox{\axiomType{FreeMonoid}\ } (\hbox{\axiomType{Symbol}\ }) , \hbox{\axiomType{FreeMonoid}\ } (\hbox{\axiomType{Symbol}\ })))(24)
Type: Type
fricas
m:=(in1('m)$f)::g

\label{eq25}m(25)
Type: MonoidRing?(Integer,FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol)))
fricas
n:=(in2('n)$f)::g

\label{eq26}\overline n(26)
Type: MonoidRing?(Integer,FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol)))
fricas
nm:=m*n-n*m

\label{eq27}-{\left({\overline n}\  m \right)}+{\left(m \ {\overline n}\right)}(27)
Type: MonoidRing?(Integer,FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol)))
fricas
nm^2

\label{eq28}{\left(m \ {\overline n}\  m \ {\overline n}\right)}-{\left(m \ {\overline{{n}^{2}}}\  m \right)}-{\left({\overline n}\ {{m}^{2}}\ {\overline n}\right)}+{\left({\overline n}\  m \ {\overline n}\  m \right)}(28)
Type: MonoidRing?(Integer,FreeProduct?(FreeMonoid?(Symbol),FreeMonoid?(Symbol)))

fricas
g:=FreeProduct(FreeGroup Symbol,FreeGroup Symbol)

\label{eq29}\hbox{\axiomType{FreeProduct}\ } (\hbox{\axiomType{FreeGroup}\ } (\hbox{\axiomType{Symbol}\ }) , \hbox{\axiomType{FreeGroup}\ } (\hbox{\axiomType{Symbol}\ }))(29)
Type: Type
fricas
g1:=(in1('g1)$g)

\label{eq30}g 1(30)
Type: FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol))
fricas
g2:=(in2('g2)$g)

\label{eq31}\overline g 2(31)
Type: FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol))
fricas
g1*g2*g1^(-1)

\label{eq32}g 1 \ {\overline g 2}\ {{g 1}^{- 1}}(32)
Type: FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol))
fricas
h:=MonoidRing(Integer,g)

\label{eq33}\hbox{\axiomType{MonoidRing}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{FreeProduct}\ } (\hbox{\axiomType{FreeGroup}\ } (\hbox{\axiomType{Symbol}\ }) , \hbox{\axiomType{FreeGroup}\ } (\hbox{\axiomType{Symbol}\ })))(33)
Type: Type
fricas
h1:=g1::h

\label{eq34}g 1(34)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))
fricas
h2:=g2::h

\label{eq35}\overline g 2(35)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))
fricas
hh:=h2*h1-h1*h2

\label{eq36}-{\left(g 1 \ {\overline g 2}\right)}+{\left({\overline g 2}\  g 1 \right)}(36)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))
fricas
hh^2

\label{eq37}{\left({\overline g 2}\  g 1 \ {\overline g 2}\  g 1 \right)}-{\left({\overline g 2}\ {{g 1}^{2}}\ {\overline g 2}\right)}-{\left(g 1 \ {\overline{{g 2}^{2}}}\  g 1 \right)}+{\left(g 1 \ {\overline g 2}\  g 1 \ {\overline g 2}\right)}(37)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))
fricas
recip(h1)*recip(h2)

\label{eq38}{{g 1}^{- 1}}\ {\overline{{g 2}^{- 1}}}(38)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))
fricas
h1*recip(h1)

\label{eq39}1(39)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))
fricas
h1*h2

\label{eq40}g 1 \ {\overline g 2}(40)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))
fricas
recip(h1*h2)

\label{eq41}{\overline{{g 2}^{- 1}}}\ {{g 1}^{- 1}}(41)
Type: Union(MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol))),...)
fricas
h1*h2*recip(h1*h2)

\label{eq42}1(42)
Type: MonoidRing?(Integer,FreeProduct?(FreeGroup?(Symbol),FreeGroup?(Symbol)))

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