MathAction SandBox Direct Product

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    - SandBox Direct Product <-- You are here.

The direct product of two functions over the same domain is another function over that domain whose co-domain is the product of the co-domains their co-domains.

Axiom makes this a little awkward because it currently does not properly support non-homogeneous tuples (Tuple ANY) as the output of functions (i.e. co-domains that are general products). We use Axiom's Record type below. It is also possible to write this using non-homogeneous Lists or DirectProducts?.

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(1) -> )abbrev package MAPHAC3X MapPackInternalHac3x
++ Description: Mapping hack for Addition
++ The function \spad{fDirPrd(f,g,x)} is \spad{(fx,gx)} which is needed below
MapPackInternalHac3x(A: SetCategory, B: SetCategory, C: SetCategory):
   with fDirPrd:  (A->B, A->C,A) -> Record(a:B,b:C) ==
      add
         fDirPrd(g,h,x)  == [(g x), (h x)]

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5523093780900071215-25px.001.spad
      using old system compiler.
   MAPHAC3X abbreviates package MapPackInternalHac3x 
------------------------------------------------------------------------
   initializing NRLIB MAPHAC3X for MapPackInternalHac3x 
   compiling into NRLIB MAPHAC3X 
   compiling exported fDirPrd : (A -> B,A -> C,A) -> Record(a: B,b: C)
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |MapPackInternalHac3x| REDEFINED

;;;     ***       |MapPackInternalHac3x| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor MapPackInternalHac3x
      Time: 0 seconds

   finalizing NRLIB MAPHAC3X 
   Processing MapPackInternalHac3x for Browser database:
--------constructor---------
--->-->MapPackInternalHac3x((fDirPrd ((Record (: a B) (: b C)) (Mapping B A) (Mapping C A) A))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/MAPHAC3X.NRLIB/MAPHAC3X.lsp" (written 28 NOV 2024 07:44:40 PM):

; wrote /var/aw/var/LatexWiki/MAPHAC3X.NRLIB/MAPHAC3X.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
   MapPackInternalHac3x is now explicitly exposed in frame initial 
   MapPackInternalHac3x will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MAPHAC3X.NRLIB/MAPHAC3X

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)abbrev package MAPPK3X MappingPackage3x
++ Description: Functional Direct Product
++ Given functions f:A->B and g:A->C, X returns the function ++ \spad{(fx,gx):A->BxC}
++ \spad<b>&X g</b> is the function \spad{h} such that \spad{h x = (f x , g x)}.
MappingPackage3x(A:SetCategory, B:SetCategory, C:SetCategory):
   with X: (A->B, A->C) -> (A->Record(a:B,b:C)) ==
      add
         import MapPackInternalHac3x(A, B, C)
         fab:  A -> B
         fac:  A -> C
         X(fab,fac) == fDirPrd(fab,fac,#1)

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6114962443719445214-25px.002.spad
      using old system compiler.
   MAPPK3X abbreviates package MappingPackage3x 
------------------------------------------------------------------------
   initializing NRLIB MAPPK3X for MappingPackage3x 
   compiling into NRLIB MAPPK3X 
   importing MapPackInternalHac3x(A,B,C)
   compiling exported X : (A -> B,A -> C) -> A -> Record(a: B,b: C)
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |MappingPackage3x| REDEFINED

;;;     ***       |MappingPackage3x| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor MappingPackage3x
      Time: 0 seconds

   finalizing NRLIB MAPPK3X 
   Processing MappingPackage3x for Browser database:
--------constructor---------
--->-->MappingPackage3x((X ((Mapping (Record (: a B) (: b C)) A) (Mapping B A) (Mapping C A)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/MAPPK3X.NRLIB/MAPPK3X.lsp" (written 28 NOV 2024 07:44:40 PM):

; wrote /var/aw/var/LatexWiki/MAPPK3X.NRLIB/MAPPK3X.fasl
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
   MappingPackage3x is now explicitly exposed in frame initial 
   MappingPackage3x will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MAPPK3X.NRLIB/MAPPK3X

Test the product with two functions f and g.

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f := (x:INT):INT +-> 3*x

$\label{eq1}\mbox{theMap (...)}$

(1)

Type: (Integer -> Integer)

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g := (x:INT):INT +-> x^3

$\label{eq2}\mbox{theMap (...)}$

(2)

Type: (Integer -> Integer)

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fg:=(f &X g)

$\label{eq3}\mbox{theMap (...)}$

(3)

Type: (Integer -> Record(a: Integer,b: Integer))

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fg(3)

$\label{eq4}\left[{a = 9}, \:{b ={27}}\right]$

(4)

Type: Record(a: Integer,b: Integer)

Since the result of the functional direct product is a function whose co-domain is a Record, for functional composition we need to write a function whose domain is a Record, i.e. so that it takes as input the data output output by another function.

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s := (x:Record(a:INT,b:INT)):INT+->(x.a+x.b)

$\label{eq5}\mbox{theMap (...)}$

(5)

Type: (Record(a: Integer,b: Integer) -> Integer)

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sX := (s * (f &X g))

$\label{eq6}\mbox{theMap (...)}$

(6)

Type: (Integer -> Integer)

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sX(9)

$\label{eq7}756$

(7)

Type: PositiveInteger?

From William Sit, July 1, 2005 7:46:00 -5:00

The Cartesian Product of two domains is implemented in Product. The DirectProduct domain is when the component domains are all the same. You worked too hard. The "product" of two functions can be defined on the fly naturally:

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h(x:INT):Product(INT,INT)== [f x,g x]

   Function declaration h : Integer -> Product(Integer,Integer) has 
      been added to workspace.

Type: Void

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h(3)

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Compiling function h with type Integer -> Product(Integer,Integer)

$\label{eq8}\left[ 9, \:{27}\right]$

(8)

Type: Product(Integer,Integer)

Or, if you really want to make a package out of this:

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)abbrev package FUNCPROD FunctionalProduct
++ Description: Allows making one function out of two components
++ The function \spad{product(f,g)} is the function whose components are f and g
FunctionalProduct(A: SetCategory, B: SetCategory, C: SetCategory): Package == Code where
   D ==> Product(B,C)
   Package ==> with 
      product:  (A->B, A->C) -> (A->D)
   Code ==> add
      product(g,h) == [g #1, h #1]$D

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3755633970012740168-25px.006.spad
      using old system compiler.
   FUNCPROD abbreviates package FunctionalProduct 
------------------------------------------------------------------------
   initializing NRLIB FUNCPROD for FunctionalProduct 
   compiling into NRLIB FUNCPROD 
   compiling exported product : (A -> B,A -> C) -> A -> Product(B,C)
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |FunctionalProduct| REDEFINED

;;;     ***       |FunctionalProduct| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor FunctionalProduct
      Time: 0 seconds

   finalizing NRLIB FUNCPROD 
   Processing FunctionalProduct for Browser database:
--------constructor---------
--->-->FunctionalProduct((product ((Mapping (Product B C) A) (Mapping B A) (Mapping C A)))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/FUNCPROD.NRLIB/FUNCPROD.lsp" (written 28 NOV 2024 07:44:40 PM):

; wrote /var/aw/var/LatexWiki/FUNCPROD.NRLIB/FUNCPROD.fasl
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
   FunctionalProduct is now explicitly exposed in frame initial 
   FunctionalProduct will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/FUNCPROD.NRLIB/FUNCPROD

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k:=product(f,g)

$\label{eq9}\mbox{theMap (...)}$

(9)

Type: (Integer -> Product(Integer,Integer))

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k(3)

$\label{eq10}\left[ 9, \:{27}\right]$

(10)

Type: Product(Integer,Integer)

Product is exactly what I was looking for! --Bill Page, Fri, 01 Jul 2005 11:10:11 -0500 reply

William Sit wrote:

The Cartesian Product of two domains is implemented in Product. The DirectProduct? domain is when the component domains are all the same. You worked too hard. The "product" of two functions can be defined on the fly naturally ...

I have not been able to find any documentation about this domain in the Axiom Book. Did I just miss it?

So now I want to know how the domain Product relates to tuples. Really from the point of view of category theory we want a tuple to be a member of an n-ary Product.

Still the use of Product for 2-tuples is quite straightforward. For example:

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)expose Product

   Product is now explicitly exposed in frame initial 
g:Product(INT,Float)->Product(Float,INT)

   Compiled code for h has been cleared.

   Cannot process mapping declaration on g since it already has a 
      value.

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