from sage.matrix.constructor import matrix
from sage.structure.category_object import normalize_names

from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement


class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
    """
    The subalgebra of an EJA generated by a single element.
    """
    @staticmethod
    def __classcall_private__(cls, elt):
        superalgebra = elt.parent()

        # First compute the vector subspace spanned by the powers of
        # the given element.
        V = superalgebra.vector_space()
        eja_basis = [superalgebra.one()]
        basis_vectors = [superalgebra.one().vector()]
        W = V.span_of_basis(basis_vectors)
        for exponent in range(1, V.dimension()):
            new_power = elt**exponent
            basis_vectors.append( new_power.vector() )
            try:
                W = V.span_of_basis(basis_vectors)
                eja_basis.append( new_power )
            except ValueError:
                # Vectors weren't independent; bail and keep the
                # last subspace that worked.
                break

        # Make the basis hashable for UniqueRepresentation.
        eja_basis = tuple(eja_basis)

        # Now figure out the entries of the right-multiplication
        # matrix for the successive basis elements b0, b1,... of
        # that subspace.
        F = superalgebra.base_ring()
        mult_table = []
        for b_right in eja_basis:
                b_right_rows = []
                # The first row of the right-multiplication matrix by
                # b1 is what we get if we apply that matrix to b1. The
                # second row of the right multiplication matrix by b1
                # is what we get when we apply that matrix to b2...
                #
                # IMPORTANT: this assumes that all vectors are COLUMN
                # vectors, unlike our superclass (which uses row vectors).
                for b_left in eja_basis:
                    # Multiply in the original EJA, but then get the
                    # coordinates from the subalgebra in terms of its
                    # basis.
                    this_row = W.coordinates((b_left*b_right).vector())
                    b_right_rows.append(this_row)
                b_right_matrix = matrix(F, b_right_rows)
                mult_table.append(b_right_matrix)

        for m in mult_table:
            m.set_immutable()
        mult_table = tuple(mult_table)

        # The rank is the highest possible degree of a minimal
        # polynomial, and is bounded above by the dimension. We know
        # in this case that there's an element whose minimal
        # polynomial has the same degree as the space's dimension
        # (remember how we constructed the space?), so that must be
        # its rank too.
        rank = W.dimension()

        # EJAs are power-associative, and this algebra is nothin but
        # powers.
        assume_associative=True

        # TODO: Un-hard-code this. It should be possible to get the "next"
        # name based on the parent's generator names.
        names = 'f'
        names = normalize_names(W.dimension(), names)

        cat = superalgebra.category().Associative()

        # TODO: compute this and actually specify it.
        natural_basis = None

        fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls)
        return fdeja.__classcall__(cls,
                                   F,
                                   mult_table,
                                   rank,
                                   eja_basis,
                                   W,
                                   assume_associative=assume_associative,
                                   names=names,
                                   category=cat,
                                   natural_basis=natural_basis)

    def __init__(self,
                 field,
                 mult_table,
                 rank,
                 eja_basis,
                 vector_space,
                 assume_associative=True,
                 names='f',
                 category=None,
                 natural_basis=None):

        self._superalgebra = eja_basis[0].parent()
        self._vector_space = vector_space
        self._eja_basis = eja_basis

        fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
        fdeja.__init__(field,
                       mult_table,
                       rank,
                       assume_associative=assume_associative,
                       names=names,
                       category=category,
                       natural_basis=natural_basis)


    def vector_space(self):
        """
        SETUP::

            sage: from mjo.eja.eja_algebra import RealSymmetricEJA
            sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra

        EXAMPLES::

            sage: J = RealSymmetricEJA(3)
            sage: x = sum( i*J.gens()[i] for i in range(6) )
            sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
            sage: K.vector_space()
            Vector space of degree 6 and dimension 3 over Rational Field
            User basis matrix:
            [ 1  0  0  1  0  1]
            [ 0  1  2  3  4  5]
            [ 5 11 14 26 34 45]
            sage: (x^0).vector()
            (1, 0, 0, 1, 0, 1)
            sage: (x^1).vector()
            (0, 1, 2, 3, 4, 5)
            sage: (x^2).vector()
            (5, 11, 14, 26, 34, 45)

        """
        return self._vector_space


    class Element(FiniteDimensionalEuclideanJordanAlgebraElement):
        def __init__(self, A, elt=None):
            """
            SETUP::

                sage: from mjo.eja.eja_algebra import RealSymmetricEJA
                sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra

            EXAMPLES::

                sage: J = RealSymmetricEJA(3)
                sage: x = sum( i*J.gens()[i] for i in range(6) )
                sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
                sage: [ K(x^k) for k in range(J.rank()) ]
                [f0, f1, f2]

            ::

            """
            if elt in A._superalgebra:
                    # Try to convert a parent algebra element into a
                    # subalgebra element...
                try:
                    coords = A.vector_space().coordinates(elt.vector())
                    elt = A(coords)
                except AttributeError:
                    # Catches a missing method in elt.vector()
                    pass

            FiniteDimensionalEuclideanJordanAlgebraElement.__init__(self,
                                                                    A,
                                                                    elt)