login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit
SandBox Mapping
  ↑  

SandBoxMorphism
  
last edited 14 years ago by Bill Page

Edit detail for SandBoxMorphism revision 14 of 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Editor: Bill Page
Time: 2008/06/18 19:31:51 GMT-7
Note: new Rep for Morphism 

        

        


spad
)abbrev domain MORPH Morphism
T ==> SetCategory
Morphism(source:T, target:T): with
  domain:%->T
  codomain:%->T
  coerce:(source->target)  -> %
  coerce:%->(source->target)
  coerce:%->OutputForm
  elt:(%,source)->target
 == (source->target) add
  -- Rep == (source->target)
  Rep == SExpression
  domain(p:%):T == source
  codomain(p:%):T == target
  coerce(r:(source->target)):% == per(r pretend SExpression)
  coerce(p:%):(source->target) == (rep(p) pretend (source->target))
  coerce(p:%):OutputForm == p pretend OutputForm
  elt(f,x) == (rep(f) pretend (source->target))(x)


spad
   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/6008188987750019428-25px001.spad using 
      Spad compiler.
   MORPH abbreviates domain Morphism 
   processing macro definition T$ ==> SetCategory 

------------------------------------------------------------------------
   initializing NRLIB MORPH for Morphism 
   compiling into NRLIB MORPH 
   Adding $ modemaps
   Adding source modemaps
   Adding target modemaps
   Adding Rep modemaps
   compiling exported domain : % -> SetCategory
Time: 0.05 SEC.

   compiling exported codomain : % -> SetCategory
Time: 0 SEC.

   compiling exported coerce : source -> target -> %
   Adding SExpression modemaps
      MORPH;coerce;M$;3 is replaced by r 
Time: 0 SEC.

   compiling exported coerce : % -> source -> target
   Adding SExpression modemaps
      MORPH;coerce;$M;4 is replaced by p 
Time: 0.01 SEC.

   Adding OutputForm modemaps
   compiling exported coerce : % -> OutputForm
      MORPH;coerce;$Of;5 is replaced by p 
Time: 0.01 SEC.

   compiling exported elt : (%,source) -> target
   Adding SExpression modemaps
      MORPH;elt;$sourcetarget;6 is replaced by SPADCALLxf 
Time: 0.01 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |Morphism| REDEFINED

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

   Cumulative Statistics for Constructor Morphism
      Time: 0.08 seconds

--------------non extending category----------------------
   Morphism(#1,#2) of category CATEGORY(domain,domain: % -> 
      SetCategory,codomain: % -> SetCategory,coerce: (#1 -> #2) -> %,
      coerce: % -> (#1 -> #2),coerce: % -> OutputForm,elt: (%,#1)
       -> #2)
    has no NIL
   finalizing NRLIB MORPH 
   Processing Morphism for Browser database:
--->-->Morphism((domain (T$ %))): Not documented!!!!
--->-->Morphism((codomain (T$ %))): Not documented!!!!
--->-->Morphism((coerce (% (Mapping target source)))): Not documented!!!!
--->-->Morphism((coerce ((Mapping target source) %))): Not documented!!!!
--->-->Morphism((coerce ((OutputForm) %))): Not documented!!!!
--->-->Morphism((elt (target % source))): Not documented!!!!
--->-->Morphism(constructor): Not documented!!!!
--->-->Morphism: Missing Description

; compiling file "/var/zope2/var/LatexWiki/MORPH.NRLIB/code.lsp" (written 14 OCT 2009 03:13:02 PM):
; compiling (/VERSIONCHECK 2)
; compiling (DEFUN |MORPH;domain;$Sc;1| ...)
; compiling (DEFUN |MORPH;codomain;$Sc;2| ...)
; compiling (PUT (QUOTE |MORPH;coerce;M$;3|) ...)
; compiling (DEFUN |MORPH;coerce;M$;3| ...)
; compiling (PUT (QUOTE |MORPH;coerce;$M;4|) ...)
; compiling (DEFUN |MORPH;coerce;$M;4| ...)
; compiling (PUT (QUOTE |MORPH;coerce;$Of;5|) ...)
; compiling (DEFUN |MORPH;coerce;$Of;5| ...)
; compiling (PUT (QUOTE |MORPH;elt;$sourcetarget;6|) ...)
; compiling (DEFUN |MORPH;elt;$sourcetarget;6| ...)
; compiling (DEFUN |Morphism| ...)
; compiling (DEFUN |Morphism;| ...)
; compiling (MAKEPROP (QUOTE |Morphism|) ...)

; /var/zope2/var/LatexWiki/MORPH.NRLIB/code.fasl written
; compilation finished in 0:00:00.174
------------------------------------------------------------------------
   Morphism is now explicitly exposed in frame initial 
   Morphism will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MORPH.NRLIB/code.fasl


The naive represetation of Rep == (source->target) gives the
following error.
  Error: Caught fatal error [memory may be damaged]?
  ...
  Error signalled by EVALHOOK.
  Backtrace: system:universal-error-handler > evalhook > lambda > lambda-closure > block > apply > APPLY




But using SExpression seems to work.

axiom
)show Morphism(Float,Integer)

 Morphism(Float,Integer) is a domain constructor.
 Abbreviation for Morphism is MORPH
 This constructor is exposed in this frame.
 Issue )edit /var/zope2/var/LatexWiki/6008188987750019428-25px001.spad to see algebra source
code for MORPH 

------------------------------- Operations --------------------------------

 codomain : % -> SetCategory           coerce : % -> (Float -> Integer)
 coerce : % -> OutputForm              coerce : (Float -> Integer) -> %
 domain : % -> SetCategory             ?.? : (%,Float) -> Integer

f:Morphism(Float,Integer)


Type: Void


axiom
f:=(x:Float):Integer +->wholePart(x)


LatexWiki Image(1)


Type: Morphism(Float,Integer)


axiom
f(1.1)


LatexWiki Image(2)


Type: PositiveInteger?


axiom
(f::(Float->Integer))(1.1)


LatexWiki Image(3)


Type: PositiveInteger?


axiom
domain f


LatexWiki Image(4)


Type: SetCategory?


axiom
codomain f


LatexWiki Image(5)


Type: SetCategory?


axiom
(f::(domain(f)->codomain(f)))(1.1)


LatexWiki Image(6)


Type: PositiveInteger?


axiom
g:Morphism(Integer,String):=(i:Integer):String +-> i::String


LatexWiki Image(7)


Type: Morphism(Integer,String)


axiom
g(1985)


LatexWiki Image(8)


Type: String


Next let's study how this sort of thing is accomplished in a similar domain.

spad
)abbrev domain HOM Hom
++ Based on Automorphism by: Manuel Bronstein
++ Description:
++       Hom(R,S) is the domain of homomorphisms - Mapping(S,R).
Hom(R:SetCategory,S:SetCategory): Join(SetCategory, Eltable(R, S)) with
      morphism: (R -> S) -> %
        ++ morphism(f) returns the non-invertible morphism given by f.
   == add
      -- Why doesn't the obvious representation just (R->S) work?
      Rep == SExpression
      morphism(f) == per(f pretend SExpression)
      elt(f, r)   == (rep(f) pretend (R->S))(r)

      --Rep == List(R -> S)
      --morphism(f) == per([f #1])
      --elt(f, r)   == (rep(f)(1))(r)

      --Rep := ((R, Integer) -> S)
      --morphism(f)                     == (#2; f(#1))
      --elt(f, r) == (g := f pretend ((R, Integer) -> S); g(r, 1))

      f = g                           == EQ(f, g)$Lisp
      coerce(f:%):OutputForm          == message("theMap ...")


spad
   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/8077618115899383420-25px003.spad using 
      Spad compiler.
   HOM abbreviates domain Hom 
------------------------------------------------------------------------
   initializing NRLIB HOM for Hom 
   compiling into NRLIB HOM 
   Adding $ modemaps
   Adding R modemaps
   Adding S modemaps
   Adding Rep modemaps
   compiling exported morphism : R -> S -> %
   Adding SExpression modemaps
      HOM;morphism;M$;1 is replaced by f 
Time: 0.03 SEC.

   compiling exported elt : (%,R) -> S
   Adding SExpression modemaps
      HOM;elt;$RS;2 is replaced by SPADCALLrf 
Time: 0.01 SEC.

   Adding Boolean modemaps
   compiling exported = : (%,%) -> Boolean
      HOM;=;2$B;3 is replaced by EQ 
Time: 0.01 SEC.

   Adding OutputForm modemaps
   compiling exported coerce : % -> OutputForm
   Adding String modemaps
Time: 0.02 SEC.

(time taken in buildFunctor:  10)

;;;     ***       |Hom| REDEFINED

;;;     ***       |Hom| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor Hom
      Time: 0.08 seconds

   finalizing NRLIB HOM 
   Processing Hom for Browser database:
     -- morphism : (R -> S) -> %
     -- constructor
; compiling file "/var/zope2/var/LatexWiki/HOM.NRLIB/code.lsp" (written 14 OCT 2009 03:13:21 PM):
; compiling (/VERSIONCHECK 2)
; compiling (PUT (QUOTE |HOM;morphism;M$;1|) ...)
; compiling (DEFUN |HOM;morphism;M$;1| ...)
; compiling (PUT (QUOTE |HOM;elt;$RS;2|) ...)
; compiling (DEFUN |HOM;elt;$RS;2| ...)
; compiling (PUT (QUOTE |HOM;=;2$B;3|) ...)
; compiling (DEFUN |HOM;=;2$B;3| ...)
; compiling (DEFUN |HOM;coerce;$Of;4| ...)
; compiling (DEFUN |Hom| ...)
; compiling (DEFUN |Hom;| ...)
; compiling (MAKEPROP (QUOTE |Hom|) ...)

; /var/zope2/var/LatexWiki/HOM.NRLIB/code.fasl written
; compilation finished in 0:00:00.095
------------------------------------------------------------------------
   Hom is now explicitly exposed in frame initial 
   Hom will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/HOM.NRLIB/code.fasl


axiom
p:=morphism((x:Integer):Float +-> x+1.0)$Hom(Integer,Float)


LatexWiki Image(9)


Type: Hom(Integer,Float)


axiom
p(2)


LatexWiki Image(10)


Type: Float


axiom
p(-1)


LatexWiki Image(11)


Type: Float