from sage.matrix.constructor import matrix
from sage.misc.cachefunc import cached_method
from sage.rings.all import QQ

from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra


class FiniteDimensionalEJAElementSubalgebra(FiniteDimensionalEJASubalgebra):
    def __init__(self, elt, **kwargs):
        superalgebra = elt.parent()

        # TODO: going up to the superalgebra dimension here is
        # overkill.  We should append p vectors as rows to a matrix
        # and continually rref() it until the rank stops going
        # up. When n=10 but the dimension of the algebra is 1, that
        # can save a shitload of time (especially over AA).
        powers = tuple( elt**k for k in range(elt.degree()) )

        super().__init__(superalgebra,
                         powers,
                         associative=True,
                         **kwargs)

        # 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.
        self.rank.set_cache(self.dimension())


    @cached_method
    def one(self):
        """
        Return the multiplicative identity element of this algebra.

        The superclass method computes the identity element, which is
        beyond overkill in this case: the superalgebra identity
        restricted to this algebra is its identity. Note that we can't
        count on the first basis element being the identity -- it might
        have been scaled if we orthonormalized the basis.

        SETUP::

            sage: from mjo.eja.eja_algebra import (HadamardEJA,
            ....:                                  random_eja)

        EXAMPLES::

            sage: J = HadamardEJA(5)
            sage: J.one()
            e0 + e1 + e2 + e3 + e4
            sage: x = sum(J.gens())
            sage: A = x.subalgebra_generated_by(orthonormalize=False)
            sage: A.one()
            f0
            sage: A.one().superalgebra_element()
            e0 + e1 + e2 + e3 + e4

        TESTS:

        The identity element acts like the identity over the rationals::

            sage: set_random_seed()
            sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
            sage: A = x.subalgebra_generated_by()
            sage: x = A.random_element()
            sage: A.one()*x == x and x*A.one() == x
            True

        The identity element acts like the identity over the algebraic
        reals with an orthonormal basis::

            sage: set_random_seed()
            sage: x = random_eja().random_element()
            sage: A = x.subalgebra_generated_by()
            sage: x = A.random_element()
            sage: A.one()*x == x and x*A.one() == x
            True

        The matrix of the unit element's operator is the identity over
        the rationals::

            sage: set_random_seed()
            sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
            sage: A = x.subalgebra_generated_by(orthonormalize=False)
            sage: actual = A.one().operator().matrix()
            sage: expected = matrix.identity(A.base_ring(), A.dimension())
            sage: actual == expected
            True

        The matrix of the unit element's operator is the identity over
        the algebraic reals with an orthonormal basis::

            sage: set_random_seed()
            sage: x = random_eja().random_element()
            sage: A = x.subalgebra_generated_by()
            sage: actual = A.one().operator().matrix()
            sage: expected = matrix.identity(A.base_ring(), A.dimension())
            sage: actual == expected
            True

        """
        if self.dimension() == 0:
            return self.zero()

        return self(self.superalgebra().one())