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SandBoxSum

Edit detail for SandBoxSum revision 9 of 14

	1 2 3 4 5 6 7 8 9 10 11 12 13 14
Editor: Bill Page Time: 2008/08/28 17:55:09 GMT-7
Note: CoTuple

changed:
-       selectsum(x:%) == rep(x)
-       in1(a:A):%   == per [a]
-       in2(b:B): % == per [b]
       selectsum(x) == rep(x)
       in1(a) == per [a]
       in2(b) == per [b]

changed:
-                   rep(x) case acomp and rep(x).acomp=1$A and rep(y) case bcomp => y
                   x=1 => y
                   y=1 => x

changed:
-                   rep(x) case acomp and x=0 and rep(y) case bcomp => y
-                   rep(x) case bcomp and y=0 and rep(y) case acomp => x
                   x=0 => y
                   y=0 => x

changed:
-          - x ==
          -x ==

changed:
-                   rep(x) case acomp and x=0 and rep(y) case bcomp => - y
-                   rep(x) case bcomp and y=0 and rep(y) case acomp => x
                   x=0 => -y
                   y=0 => x

changed:
-                   rep(x) case acomp and rep(y) case bcomp => true
-                   false
-
                   rep(x) case acomp and rep(y) case bcomp

added:

The CoTuple domain constructor is intended to be the
"Categorical Dual":http://en.wikipedia.org/wiki/Dual_(category_theory)
of the "Tuple":/axiom--test--1/src/algebra/Array1Spad
domain constructor

\begin{spad}
)abbrev domain COT CoTuple
++ This domain is intended to be the categorical dual of a
++ Tuple (comma-delimited homogeneous sequence of values).
++ As such it implements an n-ary disjoint union.
CoTuple(S:Type): CoercibleTo(S) with
    inj: (NonNegativeInteger,S) -> %
      ++ inject(x,n) returns x as an element of the n-th
      ++ component of the CoTuple. CoTuples are 0-based
    ind: % -> NonNegativeInteger
      ++ index(x) returns the component number of x in the CoTuple
    if S has Monoid then Monoid
    if S has AbelianMonoid then AbelianMonoid
    if S has CancellationAbelianMonoid then CancellationAbelianMonoid
    if S has Group then Group
    if S has AbelianGroup then  AbelianGroup
    if S has OrderedAbelianMonoidSup then OrderedAbelianMonoidSup
    if S has SetCategory then SetCategory
    if S has OrderedSet then OrderedSet
  == add
    Rep == Record(ind : NonNegativeInteger, elt : S)
    --declarations
    x,y: %
    s: S
    i: NonNegativeInteger
    p: NonNegativeInteger
    c: PositiveInteger
    d: Integer

    coerce(x:%):S == rep(x).elt
    inj(i,s) == per [i,s]
    ind(x) == rep(x).ind

    if S has SetCategory then
      x = y == (rep(x).ind = rep(y).ind) and (rep(x).elt = rep(y).elt)
      coerce(x : %): OutputForm == sub(coerce(rep(x).elt),coerce(rep(x).ind))
    if S has Monoid then
      -- represent unit of CoTuple(S) as 1$S of component 0
      1 == per [0,1]
      x * y ==
        rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt * rep(y).elt]
        x = 1 => y
        y = 1 => x
        error "not same type"
      x ** p == per [rep(x).ind, rep(x).elt ** p]

    if S has Group then
       inv(x) == per [rep(x).ind, inv(rep(x).elt)]

    if S has AbelianMonoid then
       -- represent zero of Sum(A,B) as 0$S
       0 == per [0,0]
       x + y ==
         rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt + rep(y).elt]
         x = 0 => y
         y = 0 => x
         error "not same type"
       c * x == per [rep(x).ind, c * rep(x).elt]

    if S has AbelianGroup then
      - x == per [rep(x).ind, -rep(x).elt]
      (x - y):% ==
        rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt - rep(y).elt]
        x = 0 => -y
        y = 0 => x
        error "not same type"

      d * x == per [rep(x).ind, d* rep(x).elt]

    if S has OrderedAbelianMonoidSup then
      sup(x,y) ==
        rep(x).ind = rep(y).ind => per [rep(x).ind, sup(rep(x).elt,rep(y).elt)]
        rep(x).ind < rep(y).ind => y
        x
    if S has OrderedSet then
      x < y ==
        rep(x).ind = rep(y).ind => rep(x).elt < rep(y).elt
        rep(x).ind < rep(y).ind
\end{spad}

\begin{axiom}
inj(1,10)
inj(1,10) < inj(2,10)
sup(inj(1,99), inj(2,10))$CoTuple(NNI)
\end{axiom}

The Sum domain constructor is intended to be the Categorical Dual of the Product domain constructor

spad

)abbrev domain SUM Sum
++ Description:
++ This domain implements direct union
Sum (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

       selectsum     : % -> Union(acomp:A,bcomp:B)
        ++ selectsum(x) \undocumented
       in1  :   A -> %
        ++ makefirst(a) \undocumented
       in2 :   B -> %
        ++ makesecond(b) \undocumented

  T == add

    --representations
       Rep == Union(acomp:A,bcomp:B)
       import Rep

    --declarations
       x,y: %
       i: NonNegativeInteger
       p: NonNegativeInteger
       a: A
       b: B
       c: PositiveInteger
       d: Integer

    --define
       coerce(x:%):OutputForm ==
         rep(x) case acomp => sub(coerce(rep(x).acomp),"1")
         sub(coerce(rep(x).bcomp),"2")
       x=y == rep(x)=rep(y)
       selectsum(x) == rep(x)
       in1(a) == per [a]
       in2(b) == per [b]

       if A has Monoid and B has Monoid then
          -- represent unit of Sum(A,B) as 1$A (We could use either 1$A or 1$B)
          1 == per [1$A]
          x * y ==
                   rep(x) case acomp and rep(y) case acomp => per [rep(x).acomp * rep(y).acomp]
                   rep(x) case bcomp and rep(y) case bcomp => per [rep(x).bcomp * rep(y).bcomp]
                   -- unit of Sum(A,B)=1$A is unit for B
                   x=1 => y
                   y=1 => x
                   error "not same type"
          x ** p ==
                   rep(x) case acomp => per [rep(x).acomp ** p]
                   per [rep(x).bcomp ** p]

       if A has Finite and B has Finite then
          size == size$A + size$B
          index(n) ==
                      n > size$B => per [index((n::Integer - size$B)::PositiveInteger)$A]
                      per [index(n)$B]
          random() ==
                      random()$Boolean => per [random()$A]
                      per [random()$B]
          lookup(x) ==
                      rep(x) case acomp => (lookup(rep(x).acomp)$A::NonNegativeInteger + size$B)::PositiveInteger
                      lookup(rep(x).bcomp)$B
          hash(x) ==
                      rep(x) case acomp => hash(rep(x).acomp)$A + size$B::SingleInteger
                      hash(rep(x).bcomp)$B

       if A has Group and B has Group then
          inv(x) ==
                    rep(x) case acomp => per [inv(rep(x).acomp)]
                    per [inv(rep(x).bcomp)]

       if A has AbelianMonoid and B has AbelianMonoid then
          -- represent zero of Sum(A,B) as 0$A (We could use either 0$A or 0$B)
          0 == per [0$A]
          x + y ==
                   rep(x) case acomp and rep(y) case acomp => per [rep(x).acomp + rep(y).acomp]
                   rep(x) case bcomp and rep(y) case bcomp => per [rep(x).bcomp + rep(y).bcomp]
                   -- zero of Sum(A,B)=0$A is zero for B
                   x=0 => y
                   y=0 => x
                   error "not same type"
          c * x ==
                   rep(x) case acomp => per [c * rep(x).acomp]
                   per [c* rep(x).bcomp]

       if A has AbelianGroup and B has AbelianGroup then
          -x ==
                 rep(x) case acomp => per [- rep(x).acomp]
                 per [- rep(x).bcomp]
          (x - y):% ==
                   rep(x) case acomp and rep(y) case acomp => per [rep(x).acomp - rep(y).acomp]
                   rep(x) case bcomp and rep(y) case bcomp => per [rep(x).bcomp - rep(y).bcomp]
                   -- zero of Sum(A,B)=0$A is zero for B
                   x=0 => -y
                   y=0 => x
                   error "not same type"

          d * x ==
                   rep(x) case acomp => per [d * rep(x).acomp]
                   per [d* rep(x).bcomp]

       if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then
          sup(x,y) ==
                   rep(x) case acomp and rep(y) case acomp => per [sup(rep(x).acomp,rep(y).acomp)]
                   rep(x) case bcomp and rep(y) case bcomp => per [sup(rep(x).bcomp,rep(y).bcomp)]
                   rep(x) case acomp and rep(y) case bcomp => y
                   x

       if A has OrderedSet and B has OrderedSet then
          x < y ==
                   rep(x) case acomp and rep(y) case acomp => rep(x).acomp < rep(y).acomp
                   rep(x) case bcomp and rep(y) case bcomp => rep(x).bcomp < rep(y).bcomp
                   rep(x) case acomp and rep(y) case bcomp

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/8219446594478290117-25px001.spad using 
      Spad compiler.
   SUM abbreviates domain Sum 
------------------------------------------------------------------------
   initializing NRLIB SUM for Sum 
   compiling into NRLIB SUM 
   compiling local rep : % -> Union(acomp: A,bcomp: B)
      SUM;rep is replaced by G1424 
Time: 0.01 SEC.

   compiling local per : Union(acomp: A,bcomp: B) -> %
      SUM;per is replaced by G1424 
Time: 0 SEC.

   importing Rep
   compiling exported coerce : % -> OutputForm
Time: 0.01 SEC.

   compiling exported = : (%,%) -> Boolean
Time: 0 SEC.

   compiling exported selectsum : % -> Union(acomp: A,bcomp: B)
Time: 0 SEC.

   compiling exported in1 : A -> %
Time: 0 SEC.

   compiling exported in2 : B -> %
Time: 0 SEC.

****** Domain: A already in scope
augmenting A: (Monoid)
****** Domain: B already in scope
augmenting B: (Monoid)
   compiling exported One : () -> %
Time: 0.01 SEC.

   compiling exported * : (%,%) -> %
Time: 0.07 SEC.

   compiling exported ** : (%,NonNegativeInteger) -> %
Time: 0.01 SEC.

****** Domain: A already in scope
augmenting A: (Finite)
****** Domain: B already in scope
augmenting B: (Finite)
   compiling exported size : () -> NonNegativeInteger
Time: 0.01 SEC.

   compiling exported index : PositiveInteger -> %
Time: 0 SEC.

   compiling exported random : () -> %
Time: 0 SEC.

   compiling exported lookup : % -> PositiveInteger
Time: 0.01 SEC.

   compiling exported hash : % -> SingleInteger
Time: 0.01 SEC.

****** Domain: A already in scope
augmenting A: (Group)
****** Domain: B already in scope
augmenting B: (Group)
   compiling exported inv : % -> %
Time: 0.01 SEC.

****** Domain: A already in scope
augmenting A: (AbelianMonoid)
****** Domain: B already in scope
augmenting B: (AbelianMonoid)
   compiling exported Zero : () -> %
Time: 0 SEC.

   compiling exported + : (%,%) -> %
Time: 0.02 SEC.

   compiling exported * : (PositiveInteger,%) -> %
Time: 0.07 SEC.

****** Domain: A already in scope
augmenting A: (AbelianGroup)
****** Domain: B already in scope
augmenting B: (AbelianGroup)
   compiling exported - : % -> %
Time: 0.02 SEC.

   compiling exported - : (%,%) -> %
Time: 0.01 SEC.

   compiling exported * : (Integer,%) -> %
Time: 0.01 SEC.

****** Domain: A already in scope
augmenting A: (OrderedAbelianMonoidSup)
****** Domain: B already in scope
augmenting B: (OrderedAbelianMonoidSup)
   compiling exported sup : (%,%) -> %
Time: 0.02 SEC.

****** Domain: A already in scope
augmenting A: (OrderedSet)
****** Domain: B already in scope
augmenting B: (OrderedSet)
   compiling exported < : (%,%) -> Boolean
Time: 0.02 SEC.

****** Domain: A already in scope
augmenting A: (AbelianGroup)
****** Domain: B already in scope
augmenting B: (AbelianGroup)
****** Domain: A already in scope
augmenting A: (Finite)
****** Domain: B already in scope
augmenting B: (Finite)
****** Domain: A already in scope
augmenting A: (Group)
****** Domain: B already in scope
augmenting B: (Group)
****** Domain: A already in scope
augmenting A: (OrderedAbelianMonoidSup)
****** Domain: B already in scope
augmenting B: (OrderedAbelianMonoidSup)
****** Domain: A already in scope
augmenting A: (AbelianGroup)
****** Domain: B already in scope
augmenting B: (AbelianGroup)
****** Domain: A already in scope
augmenting A: (AbelianGroup)
****** Domain: B already in scope
augmenting B: (AbelianGroup)
****** Domain: A already in scope
augmenting A: (AbelianGroup)
****** Domain: B already in scope
augmenting B: (AbelianGroup)
****** Domain: A already in scope
augmenting A: (Group)
****** Domain: B already in scope
augmenting B: (Group)
****** Domain: A already in scope
augmenting A: (OrderedAbelianMonoidSup)
****** Domain: B already in scope
augmenting B: (OrderedAbelianMonoidSup)
(time taken in buildFunctor:  2)

;;;     ***       |Sum| REDEFINED

;;;     ***       |Sum| REDEFINED
Time: 0.10 SEC.

   Warnings: 
      [1] coerce:  acomp has no value
      [2] coerce:  bcomp has no value
      [3] *:  acomp has no value
      [4] *:  bcomp has no value
      [5] **:  acomp has no value
      [6] **:  bcomp has no value
      [7] lookup:  acomp has no value
      [8] lookup:  bcomp has no value
      [9] hash:  acomp has no value
      [10] hash:  bcomp has no value
      [11] inv:  acomp has no value
      [12] inv:  bcomp has no value
      [13] +:  acomp has no value
      [14] +:  bcomp has no value
      [15] -:  acomp has no value
      [16] -:  bcomp has no value
      [17] sup:  acomp has no value
      [18] sup:  bcomp has no value
      [19] <:  acomp has no value
      [20] <:  bcomp has no value

   Cumulative Statistics for Constructor Sum
      Time: 0.42 seconds

   finalizing NRLIB SUM 
   Processing Sum for Browser database:
     -- selectsum : % -> Union(acomp: A,bcomp: B)
--->-->Sum((selectsum ((Union (: acomp A) (: bcomp B)) %))): Unexpected HT command: \spad
"\\spad{selectsum(x)} \\undocumented"
     -- in1 : A -> %
--->-->Sum((in1 (% A))): Improper first word in comments: makefirst
"makefirst(a) \\undocumented"
     -- in2 : B -> %
--->-->Sum((in2 (% B))): Improper first word in comments: makesecond
"makesecond(\\spad{b}) \\undocumented"
     -- constructor
------------------------------------------------------------------------
   Sum is now explicitly exposed in frame initial 
   Sum will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/SUM.NRLIB/code.o

axiom

size()$Sum(PF 7,PF 13)

(1)

Type: NonNegativeInteger?

axiom

size()$Sum(PF 7,Product(PF 3,PF 13))

(2)

Type: NonNegativeInteger?

axiom

in1(10)$Sum(Integer,Integer)

(3)

Type: Sum(Integer,Integer)

axiom

in1(10)$Sum(Integer,Integer) < in2(10)$Sum(Integer,Integer)

(4)

Type: Boolean

axiom

sup(in1(99)$Sum(NNI,NNI), in2(10)$Sum(NNI,NNI))

(5)

Type: Sum(NonNegativeInteger?,NonNegativeInteger?)

The CoTuple? domain constructor is intended to be the Categorical Dual of the Tuple domain constructor

spad

)abbrev domain COT CoTuple
++ This domain is intended to be the categorical dual of a
++ Tuple (comma-delimited homogeneous sequence of values).
++ As such it implements an n-ary disjoint union.
CoTuple(S:Type): CoercibleTo(S) with
    inj: (NonNegativeInteger,S) -> %
      ++ inject(x,n) returns x as an element of the n-th
      ++ component of the CoTuple. CoTuples are 0-based
    ind: % -> NonNegativeInteger
      ++ index(x) returns the component number of x in the CoTuple
    if S has Monoid then Monoid
    if S has AbelianMonoid then AbelianMonoid
    if S has CancellationAbelianMonoid then CancellationAbelianMonoid
    if S has Group then Group
    if S has AbelianGroup then  AbelianGroup
    if S has OrderedAbelianMonoidSup then OrderedAbelianMonoidSup
    if S has SetCategory then SetCategory
    if S has OrderedSet then OrderedSet
  == add
    Rep == Record(ind : NonNegativeInteger, elt : S)
    --declarations
    x,y: %
    s: S
    i: NonNegativeInteger
    p: NonNegativeInteger
    c: PositiveInteger
    d: Integer

    coerce(x:%):S == rep(x).elt
    inj(i,s) == per [i,s]
    ind(x) == rep(x).ind

    if S has SetCategory then
      x = y == (rep(x).ind = rep(y).ind) and (rep(x).elt = rep(y).elt)
      coerce(x : %): OutputForm == sub(coerce(rep(x).elt),coerce(rep(x).ind))
    if S has Monoid then
      -- represent unit of CoTuple(S) as 1$S of component 0
      1 == per [0,1]
      x * y ==
        rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt * rep(y).elt]
        x = 1 => y
        y = 1 => x
        error "not same type"
      x ** p == per [rep(x).ind, rep(x).elt ** p]

    if S has Group then
       inv(x) == per [rep(x).ind, inv(rep(x).elt)]

    if S has AbelianMonoid then
       -- represent zero of Sum(A,B) as 0$S
       0 == per [0,0]
       x + y ==
         rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt + rep(y).elt]
         x = 0 => y
         y = 0 => x
         error "not same type"
       c * x == per [rep(x).ind, c * rep(x).elt]

    if S has AbelianGroup then
      - x == per [rep(x).ind, -rep(x).elt]
      (x - y):% ==
        rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt - rep(y).elt]
        x = 0 => -y
        y = 0 => x
        error "not same type"

      d * x == per [rep(x).ind, d* rep(x).elt]

    if S has OrderedAbelianMonoidSup then
      sup(x,y) ==
        rep(x).ind = rep(y).ind => per [rep(x).ind, sup(rep(x).elt,rep(y).elt)]
        rep(x).ind < rep(y).ind => y
        x
    if S has OrderedSet then
      x < y ==
        rep(x).ind = rep(y).ind => rep(x).elt < rep(y).elt
        rep(x).ind < rep(y).ind

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/2899346089849641949-25px003.spad using 
      Spad compiler.
   COT abbreviates domain CoTuple 
------------------------------------------------------------------------
   initializing NRLIB COT for CoTuple 
   compiling into NRLIB COT 
   compiling local rep : % -> Record(ind: NonNegativeInteger,elt: S)
      COT;rep is replaced by G1625 
Time: 0 SEC.

   compiling local per : Record(ind: NonNegativeInteger,elt: S) -> %
      COT;per is replaced by G1625 
Time: 0 SEC.

   compiling exported coerce : % -> S
Time: 0.01 SEC.

   compiling exported inj : (NonNegativeInteger,S) -> %
Time: 0 SEC.

   compiling exported ind : % -> NonNegativeInteger
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (SetCategory)
   compiling exported = : (%,%) -> Boolean
Time: 0.01 SEC.

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

****** Domain: S already in scope
augmenting S: (Monoid)
   compiling exported One : () -> %
Time: 0 SEC.

   compiling exported * : (%,%) -> %
Time: 0.02 SEC.

   compiling exported ** : (%,NonNegativeInteger) -> %
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (Group)
   compiling exported inv : % -> %
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (AbelianMonoid)
   compiling exported Zero : () -> %
Time: 0.01 SEC.

   compiling exported + : (%,%) -> %
Time: 0.01 SEC.

   compiling exported * : (PositiveInteger,%) -> %
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (AbelianGroup)
   compiling exported - : % -> %
Time: 0 SEC.

   compiling exported - : (%,%) -> %
Time: 0.02 SEC.

   compiling exported * : (Integer,%) -> %
Time: 0 SEC.

****** Domain: S already in scope
augmenting S: (OrderedAbelianMonoidSup)
   compiling exported sup : (%,%) -> %
Time: 0.01 SEC.

****** Domain: S already in scope
augmenting S: (OrderedSet)
   compiling exported < : (%,%) -> Boolean
Time: 0.02 SEC.

****** Domain: S already in scope
augmenting S: (AbelianMonoid)
****** Domain: S already in scope
augmenting S: (AbelianGroup)
****** Domain: S already in scope
augmenting S: (AbelianMonoid)
****** Domain: S already in scope
augmenting S: (CancellationAbelianMonoid)
****** Domain: S already in scope
augmenting S: (Group)
****** Domain: S already in scope
augmenting S: (Monoid)
****** Domain: S already in scope
augmenting S: (OrderedAbelianMonoidSup)
****** Domain: S already in scope
augmenting S: (OrderedSet)
****** Domain: S already in scope
augmenting S: (SetCategory)
(time taken in buildFunctor:  1)

;;;     ***       |CoTuple| REDEFINED

;;;     ***       |CoTuple| REDEFINED
Time: 0.10 SEC.

   Cumulative Statistics for Constructor CoTuple
      Time: 0.23 seconds

   finalizing NRLIB COT 
   Processing CoTuple for Browser database:
     -- inj : (NonNegativeInteger,S) -> %
--->-->CoTuple((inj (% (NonNegativeInteger) S))): Improper first word in comments: inject
"inject(\\spad{x},{}\\spad{n}) returns \\spad{x} as an element of the \\spad{n}-th component of the CoTuple. CoTuples are 0-based"
     -- ind : % -> NonNegativeInteger
--->-->CoTuple((ind ((NonNegativeInteger) %))): Improper first word in comments: index
"index(\\spad{x}) returns the component number of \\spad{x} in the CoTuple"
     -- constructor
--->-->CoTuple(constructor): Unexpected HT command: \indented
"\\indented{1}{This domain is intended to be the categorical dual of a} Tuple (comma-delimited homogeneous sequence of values). As such it implements an \\spad{n}-ary disjoint union."
------------------------------------------------------------------------
   CoTuple is now explicitly exposed in frame initial 
   CoTuple will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/COT.NRLIB/code.o

axiom

inj(1,10)

(6)

Type: CoTuple? PositiveInteger?

axiom

inj(1,10) < inj(2,10)

(7)

Type: Boolean

axiom

sup(inj(1,99), inj(2,10))$CoTuple(NNI)

(8)

Type: CoTuple? NonNegativeInteger?