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last edited 6 years ago by test1 |
Edit detail for Cartesian Product revision 4 of 4
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Editor: test1
Time: 2018/04/13 15:35:04 GMT+0 |
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changed: -++ This domain implements cartesian product - -\begin{axiom} -)show Product -\end{axiom} - -\begin{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])$PrintableForm -\end{spad} This domain implements cartesian product, we give example usage here: changed: -lookup(makeprod(2,[2])$X) lookup(construct(2,[2])$X)
This domain implements cartesian product, we give example usage here:
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X:=Product(IntegerMod 3,Set PF 3)
 | (1) |
Type: Type
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size()$X
 | (2) |
Type: NonNegativeInteger?
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[index(i)$X for i in 1..size()$X::PositiveInteger]
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Compiling function G675 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.
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\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.
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\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.
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\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.
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\displaystyle
\left.{\left[ 0, \:{\left\{ 1, \: 2, \: 0 \right\}}\right]}\right]](images/5409320734295793718-16.0px.png "\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.
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\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.
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\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.
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\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.
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\displaystyle
\left.{\left[ 0, \:{\left\{ 1, \: 2, \: 0 \right\}}\right]}\right]") | (3) |
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reduce(_and,[(lookup(index(i)$X)=i)::Boolean for i in 1..size()$X::PositiveInteger])
 | (4) |
Type: Boolean
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lookup(construct(2,[2])$X)
 | (5) |
Type: PositiveInteger?
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[random()$X for i in 1..5]
![\label{eq6}\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]](images/646777928112323046-16.0px.png "\label{eq6}\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]") | (6) |