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++ This domain implements cartesian product

fricas
)show Product
Product(A: SetCategory,B: SetCategory) is a domain constructor Abbreviation for Product is PRODUCT This constructor is not exposed in this frame. ------------------------------- Operations -------------------------------- ?=? : (%,%) -> Boolean coerce : % -> OutputForm construct : (A,B) -> % first : % -> A hash : % -> SingleInteger latex : % -> String second : % -> B ?~=? : (%,%) -> Boolean ?*? : (Integer,%) -> % if A has ABELGRP and B has ABELGRP ?*? : (NonNegativeInteger,%) -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?*? : (PositiveInteger,%) -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?*? : (%,%) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID ?+? : (%,%) -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?-? : (%,%) -> % if A has ABELGRP and B has ABELGRP -? : % -> % if A has ABELGRP and B has ABELGRP ?/? : (%,%) -> % if A has GROUP and B has GROUP ?<? : (%,%) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET ?<=? : (%,%) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET ?>? : (%,%) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET ?>=? : (%,%) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET 1 : () -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID 0 : () -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS ?^? : (%,Integer) -> % if A has GROUP and B has GROUP ?^? : (%,NonNegativeInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID ?^? : (%,PositiveInteger) -> % if A has GROUP and B has GROUP or A has MONOID and B has MONOID commutator : (%,%) -> % if A has GROUP and B has GROUP conjugate : (%,%) -> % if A has GROUP and B has GROUP convert : % -> InputForm if A has FINITE and B has FINITE enumerate : () -> List(%) if A has FINITE and B has FINITE hashUpdate! : (HashState,%) -> HashState index : PositiveInteger -> % if A has FINITE and B has FINITE inv : % -> % if A has GROUP and B has GROUP lookup : % -> PositiveInteger if A has FINITE and B has FINITE max : (%,%) -> % if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET min : (%,%) -> % if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET one? : % -> Boolean if A has GROUP and B has GROUP or A has MONOID and B has MONOID opposite? : (%,%) -> Boolean if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS random : () -> % if A has FINITE and B has FINITE recip : % -> Union(%,"failed") if A has GROUP and B has GROUP or A has MONOID and B has MONOID sample : () -> % if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has GROUP and B has GROUP or A has MONOID and B has MONOID or A has OAMONS and B has OAMONS size : () -> NonNegativeInteger if A has FINITE and B has FINITE smaller? : (%,%) -> Boolean if A has OAMONS and B has OAMONS or A has ORDSET and B has ORDSET subtractIfCan : (%,%) -> Union(%,"failed") if A has ABELGRP and B has ABELGRP or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS sup : (%,%) -> % if A has OAMONS and B has OAMONS zero? : % -> Boolean if A has ABELGRP and B has ABELGRP or A has ABELMON and B has ABELMON or A has CABMON and B has CABMON or A has OAMONS and B has OAMONS

spad
)abbrev domain PRODUCT Product
Product (A:SetCategory,B:SetCategory) : C == T
 where
  C == SetCategory  with
       if A has Finite and B has Finite then Finite
       if A has Monoid and B has Monoid then Monoid
       if A has AbelianMonoid and B has AbelianMonoid then AbelianMonoid
       if A has CancellationAbelianMonoid and
          B has CancellationAbelianMonoid then CancellationAbelianMonoid
       if A has Group  and B has Group  then  Group
       if A has AbelianGroup and B has AbelianGroup then  AbelianGroup
       if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup
                                             then OrderedAbelianMonoidSup
       if A has OrderedSet and B has OrderedSet then  OrderedSet
makeprod : (A,B) -> % ++ makeprod(a,b) \undocumented selectfirst : % -> A ++ selectfirst(x) \undocumented selectsecond : % -> B ++ selectsecond(x) \undocumented
T == add
--representations Rep := Record(acomp:A,bcomp:B)
--declarations x,y: % i: NonNegativeInteger p: NonNegativeInteger a: A b: B d: Integer
--define coerce(x):OutputForm == paren [(x.acomp)::OutputForm, (x.bcomp)::OutputForm] x=y == x.acomp = y.acomp => x.bcomp = y.bcomp false makeprod(a:A,b:B) :% == [a,b]
selectfirst(x:%) : A == x.acomp
selectsecond (x:%) : B == x.bcomp
if A has Monoid and B has Monoid then 1 == [1$A,1$B] x * y == [x.acomp * y.acomp,x.bcomp * y.bcomp] x ** p == [x.acomp ** p ,x.bcomp ** p]
if A has Finite and B has Finite then size == size$A * size$B index(n) == [index((((n::Integer-1) quo size$B )+1)::PositiveInteger)$A, index((((n::Integer-1) rem size$B )+1)::PositiveInteger)$B] random() == [random()$A,random()$B] lookup(x) == ((lookup(x.acomp)$A::Integer-1) * size$B::Integer + lookup(x.bcomp)$B::Integer)::PositiveInteger hash(x) == hash(x.acomp)$A * size$B::SingleInteger + hash(x.bcomp)$B
if A has Group and B has Group then inv(x) == [inv(x.acomp),inv(x.bcomp)]
if A has AbelianMonoid and B has AbelianMonoid then 0 == [0$A,0$B]
x + y == [x.acomp + y.acomp,x.bcomp + y.bcomp]
c:NonNegativeInteger * x == [c * x.acomp,c*x.bcomp]
if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid then subtractIfCan(x, y) : Union(%,"failed") == (na:= subtractIfCan(x.acomp, y.acomp)) case "failed" => "failed" (nb:= subtractIfCan(x.bcomp, y.bcomp)) case "failed" => "failed" [na::A,nb::B]
if A has AbelianGroup and B has AbelianGroup then - x == [- x.acomp,-x.bcomp] (x - y):% == [x.acomp - y.acomp,x.bcomp - y.bcomp] d * x == [d * x.acomp,d * x.bcomp]
if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then sup(x,y) == [sup(x.acomp,y.acomp),sup(x.bcomp,y.bcomp)]
if A has OrderedSet and B has OrderedSet then x < y == xa:= x.acomp ; ya:= y.acomp xa < ya => true xb:= x.bcomp ; yb:= y.bcomp xa = ya => (xb < yb) false
-- coerce(x:%):Symbol == -- PrintableForm() -- formList([x.acomp::Expression,x.bcomp::Expression])$PrintableForm
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8594176356433254296-25px002.spad
      using old system compiler.
   PRODUCT abbreviates domain Product 
******** Spad syntax error detected ********
Expected: BACKTAB
The prior line was:
50> x * y == [x.acomp * y.acomp,x.bcomp * y.bcomp]
The current line is:
51> x ** p == [x.acomp ** p ,x.bcomp ** p]
The number of valid tokens is 1. The prior token was #S(TOKEN :SYMBOL |x| :TYPE IDENTIFIER :NONBLANK 51) The current token is #S(TOKEN :SYMBOL POWER :TYPE KEYWORD :NONBLANK 51)

fricas
X:=Product(IntegerMod 3,Set PF 3)

\label{eq1}\hbox{\axiomType{Product}\ } (\hbox{\axiomType{IntegerMod}\ } (3) , \hbox{\axiomType{Set}\ } (\hbox{\axiomType{PrimeField}\ } (3)))(1)
Type: Type
fricas
size()$X

\label{eq2}24(2)
fricas
[index(i)$X for i in 1..size()$X::PositiveInteger]
fricas
Compiling function G694 with type NonNegativeInteger -> Boolean

\label{eq3}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 1, \:{\left\{ \right\}}\right]}, \:{\left[ 1, \:{\left\{ 1 \right\}}\right]}, \:{\left[ 1, \:{\left\{ 2 \right\}}\right]}, \:{\left[ 1, \:{\left\{ 1, \: 2 \right\}}\right]}, \:{\left[ 1, \:{\left\{ 0 \right\}}\right]}, \:{\left[ 1, \:{\left\{ 1, \: 0 \right\}}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 1, \:{\left\{ 2, \: 0 \right\}}\right]}, \:{\left[ 1, \:{\left\{ 1, \: 2, \: 0 \right\}}\right]}, \:{\left[ 2, \:{\left\{ \right\}}\right]}, \:{\left[ 2, \:{\left\{ 1 \right\}}\right]}, \:{\left[ 2, \:{\left\{ 2 \right\}}\right]}, \:{\left[ 2, \:{\left\{ 1, \: 2 \right\}}\right]}, \right.
\
\
\displaystyle
\left.\:{\left[ 2, \:{\left\{ 0 \right\}}\right]}, \:{\left[ 2, \:{\left\{ 1, \: 0 \right\}}\right]}, \:{\left[ 2, \:{\left\{ 2, \: 0 \right\}}\right]}, \:{\left[ 2, \:{\left\{ 1, \: 2, \: 0 \right\}}\right]}, \:{\left[ 0, \:{\left\{ \right\}}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \:{\left\{ 1 \right\}}\right]}, \:{\left[ 0, \:{\left\{ 2 \right\}}\right]}, \:{\left[ 0, \:{\left\{ 1, \: 2 \right\}}\right]}, \:{\left[ 0, \:{\left\{ 0 \right\}}\right]}, \:{\left[ 0, \:{\left\{ 1, \: 0 \right\}}\right]}, \:{\left[ 0, \:{\left\{ 2, \: 0 \right\}}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \:{\left\{ 1, \: 2, \: 0 \right\}}\right]}\right] (3)
Type: List(Product(IntegerMod?(3),Set(PrimeField?(3))))
fricas
reduce(_and,[(lookup(index(i)$X)=i)::Boolean for i in 1..size()$X::PositiveInteger])

\label{eq4} \mbox{\rm true} (4)
Type: Boolean
fricas
lookup(makeprod(2,[2])$X)
The function makeprod is not implemented in Product(IntegerMod(3), Set(PrimeField(3))) . [random()$X for i in 1..5]

\label{eq5}\left[{\left[ 0, \:{\left\{ 2, \: 0 \right\}}\right]}, \:{\left[ 2, \:{\left\{ 1 \right\}}\right]}, \:{\left[ 0, \:{\left\{ 1, \: 2, \: 0 \right\}}\right]}, \:{\left[ 1, \:{\left\{ 1 \right\}}\right]}, \:{\left[ 2, \:{\left\{ 1, \: 2 \right\}}\right]}\right](5)
Type: List(Product(IntegerMod?(3),Set(PrimeField?(3))))




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