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Test species in axiom/FriCAS.

The species project is written in Aldor. Thus we need to load the libraries explicitly:

fricas
)cd /var/lib/zope/combinat/src
The current FriCAS default directory is /var/aw/var/LatexWiki
fricas
)re ../lib/combinat.input
The file ../lib/combinat.input is needed but does not exist.

Let's define a binary tree.

aldor
#includeDir "/var/lib/zope/combinat/include"
#libraryDir "/var/lib/zope/combinat/lib"
#include "combinat"
macro { E == EmptySetSpecies; X == SingletonSpecies; + == Plus; * == Times; }
A(L: LabelType): CombinatorialSpecies L == (E + X*A*A)(L) add;
aldor
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5200085684951427974-25px002.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   The )library system command was not called after compilation.

aldor
#includeDir "/var/lib/zope/combinat/include"
#libraryDir "/var/lib/zope/combinat/lib"
#assert MacrosCombinat
#assert Axiom
#include "combinat"
macro {
        SPECIES == (L: LabelType) -> CombinatorialSpecies L;
        V == CycleIndexVariable;
        NonNegativeMachineInteger == I;
        T == SparseIndexedPowerProduct(V, NonNegativeMachineInteger);
        P == SparseDistributedPolynomial(Q, V, T);
}
LinearOrder(L: LabelType): with {
CombinatorialSpecies L;
coerce: % -> List L;
} == List L add {
        Rep == List L;
        import from Rep;
coerce(x: %): List L == rep x; local lists(l: List L): Generator List L == generate { empty? l => yield l; current := l; c := first current; for u in lists(rest l) repeat yield cons(c, u); assert(not empty? current); while not empty?(tmp := rest current) repeat { c := first tmp; setrest!(current, rest tmp); -- remove c from l for u in lists l repeat yield cons(c, u); setrest!(current, tmp); -- put c back into l current := tmp; } } structures(s: SetSpecies L): Generator % == generate { for l in lists(s :: List L) repeat yield per l; } local LinearOrderIsomorphismType: IsomorphismTypeCategory L == add { isomorphismTypes(s: MultiSet L): Generator % == never; (x:%) = (y:%): Boolean == never; (tw: TextWriter) << (x: %): TextWriter == never; }
IsomorphismType: IsomorphismTypeCategory L == LinearOrderIsomorphismType; generatingSeries: ExponentialGeneratingSeries == { (stream(1$Q)$DataStream(Q)) :: ExponentialGeneratingSeries; } isomorphismTypeGeneratingSeries: OrdinaryGeneratingSeries == { (stream(1$Z)$DataStream(Z)) :: OrdinaryGeneratingSeries; } local cisGenerator: Generator P == generate { import from I, T, P; x1: V := 1::V; for n: I in 0.. repeat yield power(x1, n) :: P; } cycleIndexSeries: CycleIndexSeries == cisGenerator :: CycleIndexSeries; import from String; expression: SpeciesExpression == leaf("LinearOrder"); }
Cycle(L: LabelType): with { CombinatorialSpecies L; coerce: % -> List L; cycle: List L -> %; } == List L add { Rep == List L; import from I, Rep;
local cisCycle(ao: I): Generator P == generate { macro PrimePowerProduct == SparseIndexedPowerProduct(I, I);
local multiply(k: PrimePowerProduct): I == { r: I := 1; for ep in k repeat {(e, p) := ep; r := r * p^e} r; }
local eulerPhi(t: SparseIndexedPowerProduct(I, I)): I == { phi: I := 1; for ep in t repeat { (e, p) := ep; phi := phi * p^(e-1) * (p-1) } phi; }
local cisCoefficient(n: I): P == BugWorkaround( PrimePowerProduct has with { divisors: % -> Generator %; /: (%, %) -> %; } ){ import from Z, V, SmallIntegerTools; nn: PrimePowerProduct := factor n; p: P := 0; for m in divisors nn repeat { k: PrimePowerProduct := nn/m; q: Q := (eulerPhi(k) :: Z) / (n :: Z); xk: V := multiply(k) :: V; t: T := power(xk, multiply m); p := [q, t]$P + p; } p; } yield 0$P; for n:I in 1.. repeat yield cisCoefficient(n); } coerce(x: %): List L == rep x; cycle(l: List L): % == per l; structures(s: SetSpecies L): Generator % == generate { import from LinearOrder L; if not empty? s then { l: List L := s :: List L; u := first l; for t in structures(set rest l)$LinearOrder(L) repeat { yield per cons(u, t :: List L); } } }
local CycleIsomorphismType: IsomorphismTypeCategory L == add { isomorphismTypes(s: MultiSet L): Generator % == never; (x:%) = (y:%): Boolean == never; (tw: TextWriter) << (x: %): TextWriter == never; } IsomorphismType: IsomorphismTypeCategory L == CycleIsomorphismType; local cycleOrder(): SeriesOrder == 1 :: SeriesOrder; egsCycle(ao: I): Generator Q == generate { import from Z, Q; yield 0; for n:I in 1.. repeat yield inv(n :: Z); } generatingSeries: ExponentialGeneratingSeries == new(egsCycle, cycleOrder); ogsCycle(ao: I): Generator Z == generate {yield 0$Z; yield 1$Z}; isomorphismTypeGeneratingSeries: OrdinaryGeneratingSeries == { new(ogsCycle, cycleOrder); } cycleIndexSeries: CycleIndexSeries == new(cisCycle, cycleOrder); import from String; expression: SpeciesExpression == leaf("Cycle"); }
aldor
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6373670188426008557-25px003.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   The )library system command was not called after compilation.

fricas
labels: SetSpecies ACINT := set [i::ACINT for i in 1..3]
There are no library operations named SetSpecies Use HyperDoc Browse or issue )what op SetSpecies to learn if there is any operation containing " SetSpecies " in its name.
Cannot find a definition or applicable library operation named SetSpecies with argument type(s) Variable(ACINT)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
I := ACMachineInteger

\label{eq1}\hbox{\axiomType{ACMachineInteger}\ }(1)
Type: Variable(ACMachineInteger?)
fricas
Z := ACInteger

\label{eq2}\hbox{\axiomType{ACInteger}\ }(2)
Type: Variable(ACInteger?)
fricas
Q := ACFraction Z
There are no library operations named ACFraction Use HyperDoc Browse or issue )what op ACFraction to learn if there is any operation containing " ACFraction " in its name.
Cannot find a definition or applicable library operation named ACFraction with argument type(s) Variable(ACInteger)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Let's count how many structures of a certain size exist. This is encoded in the generating series. Note that this is an exponential generating series.

fricas
es: ExponentialGeneratingSeries := generatingSeries()$X;
ExponentialGeneratingSeries is not a valid type.

AldorCombinat? can also count the number of isomorphism types of structures. That is encoded in the isomorphism type series.

fricas
os: OrdinaryGeneratingSeries := isomorphismTypeGeneratingSeries()$X;
OrdinaryGeneratingSeries is not a valid type.

The output of the next command apparently causes a "Proxy Error" on the mathaction server. This can be avoided by changing the output from LaTeX to ASCII mode:

fricas
)set output tex off
 
fricas
)set output algebra on
coeffs3: ACList P := [coefficient(cs, i) for i in 0..5]
There are no library operations named ACList Use HyperDoc Browse or issue )what op ACList to learn if there is any operation containing " ACList " in its name.
Cannot find a definition or applicable library operation named ACList with argument type(s) Variable(P)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

P is the polynomial domain in infinitely many variables with rational coefficients.

fricas
p: P := 1$P
P is not a valid type.

Let's do some more advanced stuff with Aldor-Combinat . See the functorial composition testpage for more details.

Here we compute the cycle index series of simple graphs.

fricas
e: S := cycleIndexSeries() $ SetSpecies(Z);
S is not a valid type.

Suspicious juxtaposition --Bill Page, Tue, 17 Jun 2008 16:08:35 -0700 reply
What is the intention of BugWorkaround in this code:
  local cisCoefficient(n: I): P == BugWorkaround(
    PrimePowerProduct has with {
            divisors: % -> Generator %;
            /: (%, %) -> %;
    }
  ){
        import from Z, V, SmallIntegerTools;
        ...

It results in the message:

  L94 C2] #2 (Warning) Suspicious juxtaposition.  Check for missing `;'.




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