trs/src/rules/fractions.py

from itertools import combinations, product

from .utils import least_common_multiple, partition
from ..node import ExpressionLeaf as L, Scope, negate, OP_DIV, OP_ADD, \
        OP_MUL, OP_POW, nary_node, negate
from ..possibilities import Possibility as P, MESSAGES
from ..translate import _


def match_constant_division(node):
    """
    a / 0  ->  Division by zero
    a / 1  ->  a
    0 / a  ->  0
    a / a  ->  1
    """
    assert node.is_op(OP_DIV)

    p = []
    nominator, denominator = node

    # a / 0
    if denominator == 0:
        raise ZeroDivisionError('Division by zero: %s.' % node)

    # a / 1
    if denominator == 1:
        p.append(P(node, division_by_one, (nominator,)))

    # 0 / a
    if nominator == 0:
        p.append(P(node, division_of_zero, (denominator,)))

    # a / a
    if nominator == denominator:
        p.append(P(node, division_by_self, (nominator,)))

    return p


def division_by_one(root, args):
    """
    a / 1  ->  a
    """
    return args[0]


MESSAGES[division_by_one] = _('Division by 1 yields the nominator.')


def division_of_zero(root, args):
    """
    0 / a  ->  0
    """
    return L(0)


MESSAGES[division_of_zero] = _('Division of 0 by {1} reduces to 0.')


def division_by_self(root, args):
    """
    a / a  ->  1
    """
    return L(1)


MESSAGES[division_by_self] = _('Division of {1} by itself reduces to 1.')


def match_add_constant_fractions(node):
    """
    1 / 2 + 3 / 4  ->  2 / 4 + 3 / 4  # Equalize denominators
    2 / 2 - 3 / 4  ->  4 / 4 - 3 / 4
    2 / 4 + 3 / 4  ->  5 / 4          # Equal denominators, so nominators can
                                      # be added
    2 / 4 - 3 / 4  ->  -1 / 4
    1 / 2 + 3 / 4  ->  4 / 8 + 6 / 8  # Equalize denominators by multiplying
                                      # them with eachother

    """
    assert node.is_op(OP_ADD)

    p = []
    scope = Scope(node)

    fractions = filter(lambda node: node.is_op(OP_DIV), scope)

    for a, b in combinations(fractions, 2):
        na, da = a
        nb, db = b

        if da == db:
            # Equal denominators, add nominators to create a single fraction
            p.append(P(node, add_nominators, (a, b)))
        elif da.is_numeric() and db.is_numeric():
            # Denominators are both numeric, rewrite both fractions to the
            # least common multiple of their denominators. Later, the
            # nominators will be added
            denom = least_common_multiple(da.value, db.value)
            p.append(P(node, equalize_denominators, (scope, a, b, denom)))

            # Also, add the (non-recommended) possibility to multiply the
            # denominators
            p.append(P(node, equalize_denominators, (scope, a, b,
                                                     da.value * db.value)))

    return p


def equalize_denominators(root, args):
    """
    1 / 2 + 3 / 4  ->  2 / 4 + 3 / 4
    1 / 2 - 3 / 4  ->  2 / 4 - 3 / 4
    a / 2 + b / 4  ->  2a / 4 + b / 4
    """
    scope, denom = args[::3]

    for fraction in args[1:3]:
        n, d = fraction
        mult = denom / d.value

        if mult != 1:
            if n.is_numeric():
                nom = L(n.value * mult)
            else:
                nom = L(mult) * n

            scope.replace(fraction, negate(nom / L(d.value * mult), n.negated))

    return scope.as_nary_node()


MESSAGES[equalize_denominators] = _('Equalize the denominators of divisions'
    ' {2} and {3} to {4}.')


def add_nominators(root, args):
    """
    a / b + c / b      ->  (a + c) / b
    a / b - c / b      ->  (a - c) / b
    -(a / b) + c / b   ->  -((a + c) / b)
    -(a / b) - c / b   ->  (c - a) / -b
    """
    # TODO: is 'add' Appropriate when rewriting to "(a + (-c)) / b"?
    ab, cb = args
    a, b = ab
    scope = Scope(root)

    # Replace the left node with the new expression
    scope.replace(ab, (a + cb[0].negate(cb.negated)) / b)

    # Remove the right node
    scope.remove(cb)

    return scope.as_nary_node()


# TODO: convert this to a lambda. Example: 22 / 77 - 28 / 77. the "-" is above
# the "28/77" division.
MESSAGES[add_nominators] = _('Add the nominators of {1} and {2}.')


def match_expand_and_add_fractions(node):
    """
    a * b / c + d * b / c      ->  (a + d) * (b / c)
    a * b / c + (- d * b / c)  ->  (a + (-d)) * (b / c)
    """
    # TODO: is 'add' Appropriate when rewriting to "(a + (-d)) / * (b / c)"?
    assert node.is_op(OP_MUL)

    p = []

    return p


def match_multiply_fractions(node):
    """
    a / b * (c / d)  ->  ac / (bd)
    a * (b / c)      ->  ab / c
    """
    assert node.is_op(OP_MUL)

    p = []
    scope = Scope(node)
    fractions, others = partition(lambda n: n.is_op(OP_DIV), scope)

    for ab, cd in combinations(fractions, 2):
        p.append(P(node, multiply_fractions, (scope, ab, cd)))

    for a, bc in product(others, fractions):
        p.append(P(node, multiply_with_fraction, (scope, a, bc)))

    return p


def multiply_fractions(root, args):
    """
    a / b * (c / d)  ->  ac / (bd)
    """
    scope, ab, cd = args
    a, b = ab
    c, d = cd

    scope.replace(ab, a * c / (b * d))
    scope.remove(cd)

    return scope.as_nary_node()


MESSAGES[multiply_fractions] = _('Multiply fractions {2} and {3}.')


def multiply_with_fraction(root, args):
    """
    a * (b / c)  ->  ab / c
    """
    scope, a, bc = args
    b, c = bc

    scope.replace(a, a * b / c)
    scope.remove(bc)

    return scope.as_nary_node()


MESSAGES[multiply_with_fraction] = _('Multiply {2} with fraction {3}.')


def match_divide_fractions(node):
    """
    Reduce divisions of fractions to a single fraction.

    Examples:
    a / b / c        ->  a / (bc)
    a / (b / c)      ->  ac / b
    """
    assert node.is_op(OP_DIV)

    nom, denom = node
    p = []

    if nom.is_op(OP_DIV):
        p.append(P(node, divide_fraction, tuple(nom) + (denom,)))

    if denom.is_op(OP_DIV):
        p.append(P(node, divide_by_fraction, (nom,) + tuple(denom)))

    return p


def divide_fraction(root, args):
    """
    a / b / c        ->  a / (bc)
    """
    a, b, c = args

    return a / (b * c)


MESSAGES[divide_fraction] = _('Move {3} to denominator of fraction {1} / {2}.')


def divide_by_fraction(root, args):
    """
    a / (b / c)      ->  ac / b
    """
    a, b, c = args

    return a * c / b


MESSAGES[divide_by_fraction] = \
        _('Move {3} to nominator of fraction {1} / {2}.')


#def match_extract_divided_fractions(node):
#    """
#    Reduce divisions of fractions to a single fraction.
#
#    Examples:
#    a / b / c        ->  a / bc
#    a / (b / c)      ->  ac / b
#    # IMPLICIT: a / b / (c / d)  ->*  ad / bd  ->  validation test!
#    """
#    assert node.is_op(OP_DIV)
#
#    nom, denom = node
#    n_scope, d_scope = fraction_scopes(node)
#    is_division = lambda n: n.is_op(OP_DIV)
#    n_fractions, n_others = partition(is_division, n_scope)
#    d_fractions, d_others = partition(is_division, d_scope)
#
#
#    return []


def fraction_scopes(node):
    """
    Get the multiplication scopes of the nominator and denominator of a
    fraction.
    """
    assert node.is_op(OP_DIV)

    nominator, denominator = node

    if nominator.is_op(OP_MUL):
        n_scope = list(Scope(nominator))
    else:
        n_scope = [nominator]

    if denominator.is_op(OP_MUL):
        d_scope = list(Scope(denominator))
    else:
        d_scope = [denominator]

    return n_scope, d_scope


def match_equal_fraction_parts(node):
    """
    Divide nominator and denominator by the same part.

    Examples:
    ab / (ac)  ->  b / c
    ab / a     ->  b / 1
    a / (ab)   ->  1 / b

    If the same root appears in both nominator and denominator, extrct it so
    that it can be reduced to a single power by power division rules.
    a ^ p * b / a ^ q  ->  a ^ p / a ^ q * b / 1
    a ^ p * b / a      ->  a ^ p / a * b / 1
    a * b / a ^ q      ->  a / a ^ q * b / 1
    """
    assert node.is_op(OP_DIV)

    nominator, denominator = node
    n_scope, d_scope = fraction_scopes(node)
    p = []

    # Look for matching parts in scopes
    for i, n in enumerate(n_scope):
        for j, d in enumerate(d_scope):
            if n.equals(d, ignore_negation=True):
                p.append(P(node, divide_fraction_parts,
                           (negate(n, 0), n_scope, d_scope, i, j)))

            if n.is_op(OP_POW):
                a = n[0]

                if d == a or (d.is_op(OP_POW) and d[0] == a):
                    # a ^ p * b / a  ->  a ^ p / a * b
                    p.append(P(node, extract_divided_roots,
                               (a, n_scope, d_scope, i, j)))
            elif d.is_op(OP_POW) and n == d[0]:
                # a * b / a ^ q  ->  a / a ^ q * b
                p.append(P(node, extract_divided_roots,
                           (d[0], n_scope, d_scope, i, j)))

    return p


def remove_from_scopes(n_scope, d_scope, i, j):
    a_n, a_d = n_scope[i], d_scope[j]

    del n_scope[i]
    del d_scope[j]

    if not n_scope:
        # Last element of nominator scope, replace by 1
        nom = L(1)
    elif len(n_scope) == 1:
        # Only one element left, no multiplication
        nom = n_scope[0]
    else:
        # Still a multiplication
        nom = nary_node('*', n_scope)

    if not d_scope:
        denom = L(1)
    elif len(n_scope) == 1:
        denom = d_scope[0]
    else:
        denom = nary_node('*', d_scope)

    return a_n, a_d, nom, denom


def divide_fraction_parts(root, args):
    """
    Divide nominator and denominator by the same part.

    Examples:
    ab / (ac)  ->  b / c
    ab / a     ->  b / 1
    a / (ab)   ->  1 / b
    -ab / a     ->  -b / 1
    """
    a, n_scope, d_scope, i, j = args
    n, d = root
    a_n, a_d, nom, denom = remove_from_scopes(n_scope, d_scope, i, j)

    # Move negation of removed part to nominator and denominator
    return nom.negate(n.negated + a_n.negated) \
           / denom.negate(d.negated + a_d.negated)


MESSAGES[divide_fraction_parts] = \
        _('Divide nominator and denominator in {0} by {1}.')


def extract_divided_roots(root, args):
    """
    a ^ p * b / a ^ q  ->  a ^ p / a ^ q * b / 1
    a ^ p * b / a      ->  a ^ p / a * b / 1
    a * b / a ^ q      ->  a / a ^ q * b / 1
    """
    a, n_scope, d_scope, i, j = args
    n, d = root
    ap, aq, nom, denom = remove_from_scopes(n_scope, d_scope, i, j)

    return ap / aq * nom.negate(n.negated) / denom.negate(d.negated)


MESSAGES[extract_divided_roots] = \
        _('Extract the root {1} from nominator and denominator in {0}.')