r"""
Coalgebras with basis
"""
#*****************************************************************************
#  Copyright (C) 2008 Teresa Gomez-Diaz (CNRS) <Teresa.Gomez-Diaz@univ-mlv.fr>
#  Copyright (C) 2008-2011 Nicolas M. Thiery <nthiery at users.sf.net>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#                  http://www.gnu.org/licenses/
#******************************************************************************

from sage.misc.abstract_method import abstract_method
from sage.misc.lazy_attribute import lazy_attribute
from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring
from sage.categories.all import ModulesWithBasis, tensor, Hom
from sage.categories.super_modules import SuperModulesCategory

class CoalgebrasWithBasis(CategoryWithAxiom_over_base_ring):
    """
    The category of coalgebras with a distinguished basis.

    EXAMPLES::

        sage: CoalgebrasWithBasis(ZZ)
        Category of coalgebras with basis over Integer Ring
        sage: sorted(CoalgebrasWithBasis(ZZ).super_categories(), key=str)
        [Category of coalgebras over Integer Ring,
         Category of modules with basis over Integer Ring]

    TESTS::

        sage: TestSuite(CoalgebrasWithBasis(ZZ)).run()
    """

    class ParentMethods:

        @abstract_method(optional = True)
        def coproduct_on_basis(self, i):
            """
            The coproduct of the algebra on the basis (optional).

            INPUT:

            - ``i`` -- the indices of an element of the basis of ``self``

            Returns the coproduct of the corresponding basis elements
            If implemented, the coproduct of the algebra is defined
            from it by linearity.

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: (a, b) = A._group.gens()
                sage: A.coproduct_on_basis(a)
                B[(1,2,3)] # B[(1,2,3)]
            """

        @lazy_attribute
        def coproduct(self):
            """
            If :meth:`coproduct_on_basis` is available, construct the
            coproduct morphism from ``self`` to ``self`` `\otimes`
            ``self`` by extending it by linearity. Otherwise, use
            :meth:`~Coalgebras.Realizations.ParentMethods.coproduct_by_coercion`,
            if available.

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: [a,b] = A.algebra_generators()
                sage: a, A.coproduct(a)
                (B[(1,2,3)], B[(1,2,3)] # B[(1,2,3)])
                sage: b, A.coproduct(b)
                (B[(1,3)], B[(1,3)] # B[(1,3)])

            """
            if self.coproduct_on_basis is not NotImplemented:
                # TODO: if self is a hopf algebra, then one would want
                # to create a morphism of algebras with basis instead
                # should there be a method self.coproduct_homset_category?
                return Hom(self, tensor([self, self]), ModulesWithBasis(self.base_ring()))(on_basis = self.coproduct_on_basis)
            elif hasattr(self, "coproduct_by_coercion"):
                return self.coproduct_by_coercion

        @abstract_method(optional = True)
        def counit_on_basis(self, i):
            """
            The counit of the algebra on the basis (optional).

            INPUT:

            - ``i`` -- the indices of an element of the basis of ``self``

            Returns the counit of the corresponding basis elements
            If implemented, the counit of the algebra is defined
            from it by linearity.

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: (a, b) = A._group.gens()
                sage: A.counit_on_basis(a)
                1
            """

        @lazy_attribute
        def counit(self):
            """
            If :meth:`counit_on_basis` is available, construct the
            counit morphism from ``self`` to ``self`` `\otimes`
            ``self`` by extending it by linearity

            EXAMPLES::

                sage: A = HopfAlgebrasWithBasis(QQ).example(); A
                An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                sage: [a,b] = A.algebra_generators()
                sage: a, A.counit(a)
                (B[(1,2,3)], 1)
                sage: b, A.counit(b)
                (B[(1,3)], 1)

            """
            if self.counit_on_basis is not NotImplemented:
                return self.module_morphism(self.counit_on_basis,codomain=self.base_ring())
            elif hasattr(self, "counit_by_coercion"):
                return self.counit_by_coercion

    class ElementMethods:
        def coproduct_iterated(self, n=1):
            r"""
            Apply ``n`` coproducts to ``self``.

            .. TODO::

                Remove dependency on ``modules_with_basis`` methods.

            EXAMPLES::

                sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
                sage: Psi[2,2].coproduct_iterated(0)
                Psi[2, 2]
                sage: Psi[2,2].coproduct_iterated(2)
                Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[2] # Psi[2]
                 + Psi[] # Psi[2, 2] # Psi[] + 2*Psi[2] # Psi[] # Psi[2]
                 + 2*Psi[2] # Psi[2] # Psi[] + Psi[2, 2] # Psi[] # Psi[]

            TESTS::

                sage: p = SymmetricFunctions(QQ).p()
                sage: p[5,2,2].coproduct_iterated()
                p[] # p[5, 2, 2] + 2*p[2] # p[5, 2] + p[2, 2] # p[5]
                 + p[5] # p[2, 2] + 2*p[5, 2] # p[2] + p[5, 2, 2] # p[]
                sage: p([]).coproduct_iterated(3)
                p[] # p[] # p[] # p[]

            ::

                sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
                sage: Psi[2,2].coproduct_iterated(0)
                Psi[2, 2]
                sage: Psi[2,2].coproduct_iterated(3)
                Psi[] # Psi[] # Psi[] # Psi[2, 2] + 2*Psi[] # Psi[] # Psi[2] # Psi[2]
                 + Psi[] # Psi[] # Psi[2, 2] # Psi[] + 2*Psi[] # Psi[2] # Psi[] # Psi[2]
                 + 2*Psi[] # Psi[2] # Psi[2] # Psi[] + Psi[] # Psi[2, 2] # Psi[] # Psi[]
                 + 2*Psi[2] # Psi[] # Psi[] # Psi[2] + 2*Psi[2] # Psi[] # Psi[2] # Psi[]
                 + 2*Psi[2] # Psi[2] # Psi[] # Psi[] + Psi[2, 2] # Psi[] # Psi[] # Psi[]

            ::

                sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
                sage: m[[1,3],[2]].coproduct_iterated(2)
                m{} # m{} # m{{1, 3}, {2}} + m{} # m{{1}} # m{{1, 2}}
                 + m{} # m{{1, 2}} # m{{1}} + m{} # m{{1, 3}, {2}} # m{}
                 + m{{1}} # m{} # m{{1, 2}} + m{{1}} # m{{1, 2}} # m{}
                 + m{{1, 2}} # m{} # m{{1}} + m{{1, 2}} # m{{1}} # m{}
                 + m{{1, 3}, {2}} # m{} # m{}
                sage: m[[]].coproduct_iterated(3), m[[1,3],[2]].coproduct_iterated(0)
                (m{} # m{} # m{} # m{}, m{{1, 3}, {2}})
            """
            if n < 0:
                raise ValueError("cannot take fewer than 0 coproduct iterations: %s < 0" % str(n))
            if n == 0:
                return self
            if n == 1:
                return self.coproduct()
            from sage.functions.all import floor, ceil
            from sage.rings.all import Integer

            # Use coassociativity of `\Delta` to perform many coproducts simultaneously.
            fn = floor(Integer(n-1)/2); cn = ceil(Integer(n-1)/2)
            split = lambda a,b: tensor([a.coproduct_iterated(fn), b.coproduct_iterated(cn)])
            return self.coproduct().apply_multilinear_morphism(split)

    class Super(SuperModulesCategory):
        pass
