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spad
)abbrev domain FSUM FreeSum
++ Description:
++ This domain implements the free "product" of abelian monoids (groups)
++ It is the coproduct in the category of abelian monoids (groups).
++ FreeSum(A,B) is the abelian 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 sum in A and b1b2 by its sum in B
++ Ref: http://en.wikipedia.org/wiki/Free_product
FreeSum(A:AbelianMonoid,B:AbelianMonoid):AbelianMonoid with
    if A has AbelianGroup and B has AbelianGroup then AbelianGroup
    in1: A -> %
    in2: B -> %
    RetractableTo(A)
    RetractableTo(B)
    is1: % -> Boolean
    is2: % -> Boolean
    if A has Comparable and B has Comparable then Comparable
    terms: % -> List %
  == add
    Rep ==> List Union(A,B)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %
Zero() == per [] coerce(x:%):OutputForm == r:=rep(x) if empty?(r) then return coerce(0$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 infix('+::OutputForm,[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=0 then 0 else per [[x]] in2(y:B):% == if y=0 then 0 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=0 or not is1(x) then 0 else coerce(first rep x)@A retract(x:%):B == if x=0 or not is2(x) then 0 else coerce(first rep x)@B terms(x:%):List % == [ per [s] for s in rep(x) ]
if A has AbelianGroup and B has AbelianGroup then _-(x:%):% == if x=0 then return 0 return per [( _ s case A => -(s::A); _ s case B => -(s::B)) _ for s in reverse rep(x)] (x:% - y:%):% == x + (-y) (n:Integer * x:%):% == if x=0 then return 0 if n>0 then return (n-1) * x + x if n<0 then return (n+1) * x - x return 0
(x:% + y:%):% == if x=0 then return y if y=0 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)
(n:NonNegativeInteger * x:%):% == if x=0 then return 0 if n>0 then return (n-1)::NonNegativeInteger * x + x return 0 (n:PositiveInteger * x:%):% == if x=0 then return 0 if n>1 then return (n-1)::PositiveInteger * x + x return x
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7481739711252847129-25px001.spad
      using old system compiler.
   FSUM abbreviates domain FreeSum 
------------------------------------------------------------------------
   initializing NRLIB FSUM for FreeSum 
   compiling into NRLIB FSUM 
   processing macro definition Rep ==> List Union(A,B) 
   compiling local rep : $ -> List Union(A,B)
      FSUM;rep is replaced by x 
Time: 0.01 SEC.
compiling local per : List Union(A,B) -> $ FSUM;per is replaced by x Time: 0 SEC.
compiling exported Zero : () -> $ Time: 0 SEC.
compiling exported coerce : $ -> OutputForm Time: 0.03 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: 0.01 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 SEC.
compiling exported is1 : $ -> Boolean Time: 0 SEC.
compiling exported is2 : $ -> Boolean Time: 0.01 SEC.
compiling exported retract : $ -> A Time: 0 SEC.
compiling exported retract : $ -> B Time: 0 SEC.
compiling exported terms : $ -> List $ Time: 0 SEC.
****** Domain: A already in scope augmenting A: (AbelianGroup) ****** Domain: B already in scope augmenting B: (AbelianGroup) compiling exported - : $ -> $ Time: 0 SEC.
compiling exported - : ($,$) -> $ Time: 0 SEC.
compiling exported * : (Integer,$) -> $ Time: 0.01 SEC.
compiling exported + : ($,$) -> $ Time: 0.01 SEC.
compiling exported * : (NonNegativeInteger,$) -> $ Time: 0 SEC.
compiling exported * : (PositiveInteger,$) -> $ Time: 0 SEC.
****** Domain: A already in scope augmenting A: (AbelianGroup) ****** Domain: B already in scope augmenting B: (AbelianGroup) ****** Domain: A already in scope augmenting A: (Comparable) ****** Domain: B already in scope augmenting B: (Comparable) (time taken in buildFunctor: 0)
;;; *** |FreeSum| REDEFINED
;;; *** |FreeSum| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor FreeSum Time: 0.08 seconds
finalizing NRLIB FSUM Processing FreeSum for Browser database: --------constructor--------- --->-->FreeSum((in1 (% A))): Not documented!!!! --->-->FreeSum((in2 (% B))): Not documented!!!! --->-->FreeSum((is1 ((Boolean) %))): Not documented!!!! --->-->FreeSum((is2 ((Boolean) %))): Not documented!!!! --->-->FreeSum((terms ((List %) %))): Not documented!!!! ; compiling file "/var/aw/var/LatexWiki/FSUM.NRLIB/FSUM.lsp" (written 31 JUL 2013 03:43:40 PM):
; /var/aw/var/LatexWiki/FSUM.NRLIB/FSUM.fasl written ; compilation finished in 0:00:00.091 ------------------------------------------------------------------------ FreeSum is now explicitly exposed in frame initial FreeSum will be automatically loaded when needed from /var/aw/var/LatexWiki/FSUM.NRLIB/FSUM

fricas
II:=FSUM(INT,INT)

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

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

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

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

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

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

\label{eq10}\overline 6(10)
Type: FreeSum?(Integer,Integer)
fricas
i1+i2

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

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

fricas
fs:=FreeSum(FreeAbelianMonoid Symbol,FreeAbelianMonoid Symbol)

\label{eq13}\hbox{\axiomType{FreeSum}\ } (\hbox{\axiomType{FreeAbelianMonoid}\ } (\hbox{\axiomType{Symbol}\ }) , \hbox{\axiomType{FreeAbelianMonoid}\ } (\hbox{\axiomType{Symbol}\ }))(13)
Type: Type
fricas
p:=in1('p)$fs

\label{eq14}p(14)
Type: FreeSum?(FreeAbelianMonoid?(Symbol),FreeAbelianMonoid?(Symbol))
fricas
q:=in2('q)$fs

\label{eq15}\overline q(15)
Type: FreeSum?(FreeAbelianMonoid?(Symbol),FreeAbelianMonoid?(Symbol))
fricas
r:=in1('r)$fs

\label{eq16}r(16)
Type: FreeSum?(FreeAbelianMonoid?(Symbol),FreeAbelianMonoid?(Symbol))
fricas
2*(p+q+r)

\label{eq17}p +{\overline q}+ p + r +{\overline q}+ r(17)
Type: FreeSum?(FreeAbelianMonoid?(Symbol),FreeAbelianMonoid?(Symbol))
fricas
(p+q)+(r+p)+(q+r)

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

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

\label{eq20}b(20)
Type: FreeAbelianMonoid?(Symbol)
fricas
a+b

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

\label{eq22}p +{\overline q}(22)
Type: FreeSum?(FreeAbelianMonoid?(Symbol),FreeAbelianMonoid?(Symbol))
fricas
p+2*q

\label{eq23}p +{\overline{2 \  q}}(23)
Type: FreeSum?(FreeAbelianMonoid?(Symbol),FreeAbelianMonoid?(Symbol))

fricas
gs:=FreeSum(FreeAbelianGroup Symbol,FreeAbelianGroup Symbol)

\label{eq24}\hbox{\axiomType{FreeSum}\ } (\hbox{\axiomType{FreeAbelianGroup}\ } (\hbox{\axiomType{Symbol}\ }) , \hbox{\axiomType{FreeAbelianGroup}\ } (\hbox{\axiomType{Symbol}\ }))(24)
Type: Type
fricas
g1:=(in1('g1)$gs)

\label{eq25}g 1(25)
Type: FreeSum?(FreeAbelianGroup?(Symbol),FreeAbelianGroup?(Symbol))
fricas
g2:=(in2('g2)$gs)

\label{eq26}\overline g 2(26)
Type: FreeSum?(FreeAbelianGroup?(Symbol),FreeAbelianGroup?(Symbol))
fricas
g1+g2+ -1*g1

\label{eq27}g 1 +{\overline g 2}- g 1(27)
Type: FreeSum?(FreeAbelianGroup?(Symbol),FreeAbelianGroup?(Symbol))

FreeRing = free algebra




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