Edit detail for SandBoxCombinat revision 4 of 8

aldor
#include "axiom.as"
macro CCC == CombinatorialClassCategory;
import from Integer;
import from List Integer;
import from OutputForm;
CombinatorialClassCategory: Category == with {        
        list: Integer -> List %;
        count: Integer -> Integer;
        import from NonNegativeInteger;
        default count(i: Integer): Integer == (#list(i))::Integer;
        coerce: % -> OutputForm;
} 
Primitive(n: Integer): CombinatorialClassCategory == add {
        Rep == Integer;
        count(i: Integer): Integer == if i=n then 1 else 0;
        list(i: Integer): List % == if i=n then [per 1] else [];
        coerce(x: %): OutputForm == message(" ")
}
Epsilon: CombinatorialClassCategory == Primitive(0) add;
Atom: CombinatorialClassCategory == Primitive(1) add;
UnionClass(S1: CCC, S2: CCC): CCC == add {
        Rep == Union(s1: S1, s2: S2);
        count(i: Integer): Integer == count(i)$S1 + count(i)$S2;
        import from Rep;
        import from List S1;
        import from List S2;
        list(i: Integer): List % == {
                result: List % := [];
                for a in list(i)$S1 repeat result := cons(per union a, result);
                for b in list(i)$S2 repeat result := cons(per union b, result);
                result;
        }
        coerce(x: %): OutputForm == {
                (rep x) case s1 => ((rep x).s1)::OutputForm;
                ((rep x).s2)::OutputForm;
        }
}
CrossClass(S1: CCC, S2: CCC): with {
        CombinatorialClassCategory
} == add {
        Rep == Record(t1: S1, t2: S2);
        count(i: Integer): Integer == {
                result := 0;
                for k in 1 .. i-1 repeat {
                        result := result + count(k)$S1 * count(i-k)$S2;
                }
                result;
        }
        import from Rep;
        import from List S1;
        import from List S2;
        list(i: Integer): List % == {
                result: List % := [];
                for k in 1..i-1 repeat {
                  for a in list(k)$S1 repeat {
                    for b in list(i-k)$S2 repeat {
                      result := cons(per record(a, b), result);
                    }
                  }
                }
                result;
        }
        coerce(x: %): OutputForm == {
                a := ((rep x).t1)::OutputForm;
                b := ((rep x).t2)::OutputForm;
                c: List OutputForm := [a, b];
                vconcat [hconcat [hspace 1, message("o")], hconcat [hspace 1, message("/"), hspace 1, message("\")], hconcat [a,hspace 3,b]]
        }
}
TreeClass: CombinatorialClassCategory == UnionClass(Atom, CrossClass(TreeClass, TreeClass)
) add;

axiom
[count(i)$TreeClass for i in 1..7]

axiom
)set output tex off
)set output algebra on
(list(5)$TreeClass)::List OutputForm
   (2)
   [        o        ,         o        ,         o        ,         o        ,
           / \                / \                / \
/ \
              o                  o                  o                  o
             / \                / \                / \
/ \
                o                  o            o       o            o
               / \                / \          / \     / \
/ \
                  o              o                                     o
                 / \            / \                                   /
\
            o        ,         o        ,         o        ,         o        ,
           / \                / \                / \
/ \
              o          o         o        o         o          o         o
             / \        / \       / \      / \       / \
/ \       / \
            o                        o              o              o
           / \                      / \            / \            /
\
          o
         / \
            o        ,         o        ,         o        ,         o        ,
           / \                / \                / \
/ \
        o         o          o                  o                  o
       / \       / \        / \                / \
/ \
      o                        o                  o            o       o
     / \                      / \                / \          /
\     / \
                                 o              o
                                / \            / \
            o        ,         o        ]
           / \                / \
          o                  o
         / \                / \
        o                  o
       / \                / \
          o              o
         / \            / \

axiom
)cd ~/combinat/src
   The current FriCAS default directory is /var/lib/zope/combinat/src
)re ../lib/combinat.input
)lib cscombinatversion.ao
   Reading /var/lib/zope/combinat/src/cscombinatversion.asy
   LibraryInformationCombinat is now explicitly exposed in frame
      initial
   LibraryInformationCombinat will be automatically loaded when needed
      from /var/lib/zope/combinat/src/cscombinatversion
)lib csaxcompat.ao
   Reading /var/lib/zope/combinat/src/csaxcompat.asy
   ACSymbol is now explicitly exposed in frame initial
   ACSymbol will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACLabelType is now explicitly exposed in frame initial
   ACLabelType will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   Array is now explicitly exposed in frame initial
   Array will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   OutputType is now explicitly exposed in frame initial
   OutputType will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   TotallyOrderedType is now explicitly exposed in frame initial
   TotallyOrderedType will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACCharacter is now explicitly exposed in frame initial
   ACCharacter will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACFraction is now explicitly exposed in frame initial
   ACFraction will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACIntegerTools is now explicitly exposed in frame initial
   ACIntegerTools will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACInteger is now explicitly exposed in frame initial
   ACInteger will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACList is now explicitly exposed in frame initial
   ACList will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACMachineInteger is now explicitly exposed in frame initial
   ACMachineInteger will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACPrimitiveArray is now explicitly exposed in frame initial
   ACPrimitiveArray will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
   ACString is now explicitly exposed in frame initial
   ACString will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat
)lib csaxcompat2.ao
   Reading /var/lib/zope/combinat/src/csaxcompat2.asy
   ExpressionTreeExpt is now explicitly exposed in frame initial
   ExpressionTreeExpt will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreeLeaf is now explicitly exposed in frame initial
   ExpressionTreeLeaf will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreePlus is now explicitly exposed in frame initial
   ExpressionTreePlus will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreePrefix is now explicitly exposed in frame initial
   ExpressionTreePrefix will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreeTimes is now explicitly exposed in frame initial
   ExpressionTreeTimes will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTree is now explicitly exposed in frame initial
   ExpressionTree will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   GeneratorException is now explicitly exposed in frame initial
   GeneratorException will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   Generator is now explicitly exposed in frame initial
   Generator will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreeOperator is now explicitly exposed in frame initial
   ExpressionTreeOperator will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   Partial is now explicitly exposed in frame initial
   Partial will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionType is now explicitly exposed in frame initial
   ExpressionType will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   GeneratorExceptionType is now explicitly exposed in frame initial
   GeneratorExceptionType will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
Won't parse: (Type)->NIL
)lib csistruc.ao
   Reading /var/lib/zope/combinat/src/csistruc.asy
   IndexedFreeAdditiveCombinationType is now explicitly exposed in
      frame initial
   IndexedFreeAdditiveCombinationType will be automatically loaded when
      needed from /var/lib/zope/combinat/src/csistruc
   IndexedFreeArithmeticType is now explicitly exposed in frame initial
   IndexedFreeArithmeticType will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csistruc
   SparseAdditiveArray is now explicitly exposed in frame initial
   SparseAdditiveArray will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csistruc
   SparseFiniteMonoidRing is now explicitly exposed in frame initial
   SparseFiniteMonoidRing will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csistruc
)lib csdistpoly.ao
   Reading /var/lib/zope/combinat/src/csdistpoly.asy
   SparseDistributedPolynomial is now explicitly exposed in frame
      initial
   SparseDistributedPolynomial will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csdistpoly
)lib csstream.ao
   Reading /var/lib/zope/combinat/src/csstream.asy
   DataStream is now explicitly exposed in frame initial
   DataStream will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csstream
)lib csseries.ao
   Reading /var/lib/zope/combinat/src/csseries.asy
   FormalPowerSeries is now explicitly exposed in frame initial
   FormalPowerSeries will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csseries
   FormalPowerSeriesCategory is now explicitly exposed in frame initial
   FormalPowerSeriesCategory will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csseries
   SeriesOrder is now explicitly exposed in frame initial
   SeriesOrder will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csseries
)lib csidxpp.ao
   Reading /var/lib/zope/combinat/src/csidxpp.asy
   SparseIndexedPowerProduct is now explicitly exposed in frame initial
   SparseIndexedPowerProduct will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csidxpp
)lib cssiprimes.ao
   Reading /var/lib/zope/combinat/src/cssiprimes.asy
   SmallIntegerPrimes is now explicitly exposed in frame initial
   SmallIntegerPrimes will be automatically loaded when needed from
      /var/lib/zope/combinat/src/cssiprimes
)lib cssitools.ao
   Reading /var/lib/zope/combinat/src/cssitools.asy
   SmallIntegerTools is now explicitly exposed in frame initial
   SmallIntegerTools will be automatically loaded when needed from
      /var/lib/zope/combinat/src/cssitools
)lib csgseries.ao
   Reading /var/lib/zope/combinat/src/csgseries.asy
   CycleIndexSeries is now explicitly exposed in frame initial
   CycleIndexSeries will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csgseries
   CycleIndexVariable is now explicitly exposed in frame initial
   CycleIndexVariable will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csgseries
   ExponentialGeneratingSeries is now explicitly exposed in frame
      initial
   ExponentialGeneratingSeries will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csgseries
   OrdinaryGeneratingSeries is now explicitly exposed in frame initial
   OrdinaryGeneratingSeries will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csgseries
)lib csmultinom.ao
   Reading /var/lib/zope/combinat/src/csmultinom.asy
   MultinomialTools is now explicitly exposed in frame initial
   MultinomialTools will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csmultinom
)lib csspexpr.ao
   Reading /var/lib/zope/combinat/src/csspexpr.asy
   SpeciesExpression is now explicitly exposed in frame initial
   SpeciesExpression will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspexpr
)lib csspecies.ao
   Reading /var/lib/zope/combinat/src/csspecies.asy
   Augment is now explicitly exposed in frame initial
   Augment will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   CharacteristicSpecies is now explicitly exposed in frame initial
   CharacteristicSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Combination is now explicitly exposed in frame initial
   Combination will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Compose is now explicitly exposed in frame initial
   Compose will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Derivative is now explicitly exposed in frame initial
   Derivative will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   DropEmpty is now explicitly exposed in frame initial
   DropEmpty will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   EmptySetSpecies is now explicitly exposed in frame initial
   EmptySetSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   FunctorialCompose is now explicitly exposed in frame initial
   FunctorialCompose will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiCompose is now explicitly exposed in frame initial
   MultiCompose will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiDerivative is now explicitly exposed in frame initial
   MultiDerivative will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiPlus is now explicitly exposed in frame initial
   MultiPlus will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiSetCombination is now explicitly exposed in frame initial
   MultiSetCombination will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiSetPartition is now explicitly exposed in frame initial
   MultiSetPartition will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiSet is now explicitly exposed in frame initial
   MultiSet will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiSubset is now explicitly exposed in frame initial
   MultiSubset will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   MultiTimes is now explicitly exposed in frame initial
   MultiTimes will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Multiple is now explicitly exposed in frame initial
   Multiple will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   CombinatorialSpecies is now explicitly exposed in frame initial
   CombinatorialSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   NonEmpty is now explicitly exposed in frame initial
   NonEmpty will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Plus is now explicitly exposed in frame initial
   Plus will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   RestrictedSpecies is now explicitly exposed in frame initial
   RestrictedSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   SetSpecies is now explicitly exposed in frame initial
   SetSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   IsomorphismTypeCategory is now explicitly exposed in frame initial
   IsomorphismTypeCategory will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csspecies
   SingletonSpecies is now explicitly exposed in frame initial
   SingletonSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   ACIsomorphismType is now explicitly exposed in frame initial
   ACIsomorphismType will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Subset is now explicitly exposed in frame initial
   Subset will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Times is now explicitly exposed in frame initial
   Times will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
   Partition is now explicitly exposed in frame initial
   Partition will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csspecies
)lib csexamples.ao
   Reading /var/lib/zope/combinat/src/csexamples.asy
   Combination is already explicitly exposed in frame initial
   Combination will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csexamples
   Generator is already explicitly exposed in frame initial
   Generator will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csexamples
   SetSpecies is already explicitly exposed in frame initial
   SetSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csexamples
   ACBinaryTree is now explicitly exposed in frame initial
   ACBinaryTree will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csexamples
Won't parse: (Type)->NIL
Won't parse: (ACLabelType)->NIL
)lib csinterp.ao
   Reading /var/lib/zope/combinat/src/csinterp.asy
   Interpret is now explicitly exposed in frame initial
   Interpret will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csinterp
   InterpretingTools is now explicitly exposed in frame initial
   InterpretingTools will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csinterp
   LabelSpecies is now explicitly exposed in frame initial
   LabelSpecies will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csinterp
)lib csparse.ao
   Reading /var/lib/zope/combinat/src/csparse.asy
   MyParser is now explicitly exposed in frame initial
   MyParser will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csparse

axiom
lab: SetSpecies ACINT := set [i::ACINT for i in 1..4]
   (3)  [1,2,3,4]

axiom
structures(lab)$Partition ACINT
   (4)  [<generator>]

aldor
#include "combinat"
macro {
        E == EmptySetSpecies;
        X == SingletonSpecies;
        + == Plus;
        * == Times;
}
A(L: LabelType): CombinatorialSpecies L == (E + X*A*A)(L) add;