def test_invertible(): assert ask(Q.invertible(X), Q.invertible(X)) assert ask(Q.invertible(Y)) is False assert ask(Q.invertible(X * Y), Q.invertible(X)) is False assert ask(Q.invertible(X * Z), Q.invertible(X)) is None assert ask(Q.invertible(X * Z), Q.invertible(X) & Q.invertible(Z)) is True assert ask(Q.invertible(X.T)) is None assert ask(Q.invertible(X.T), Q.invertible(X)) is True assert ask(Q.invertible(X.I)) is True assert ask(Q.invertible(Identity(3))) is True assert ask(Q.invertible(ZeroMatrix(3, 3))) is False assert ask(Q.invertible(OneMatrix(1, 1))) is True assert ask(Q.invertible(OneMatrix(3, 3))) is False assert ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))
def _(expr): allargs_square = allargs(x, Q.square(x), expr) allargs_invertible = allargs(x, Q.invertible(x), expr) return Implies(allargs_square, Equivalent(Q.invertible(expr), allargs_invertible))
def test_square(): assert ask(Q.square(X)) assert not ask(Q.square(Y)) assert ask(Q.square(Y * Y.T))
def get_known_facts(x=None): """ Facts between unary predicates. Parameters ========== x : Symbol, optional Placeholder symbol for unary facts. Default is ``Symbol('x')``. Returns ======= fact : Known facts in conjugated normal form. """ if x is None: x = Symbol('x') fact = And( # primitive predicates for extended real exclude each other. Exclusive(Q.negative_infinite(x), Q.negative(x), Q.zero(x), Q.positive(x), Q.positive_infinite(x)), # build complex plane Exclusive(Q.real(x), Q.imaginary(x)), Implies(Q.real(x) | Q.imaginary(x), Q.complex(x)), # other subsets of complex Exclusive(Q.transcendental(x), Q.algebraic(x)), Equivalent(Q.real(x), Q.rational(x) | Q.irrational(x)), Exclusive(Q.irrational(x), Q.rational(x)), Implies(Q.rational(x), Q.algebraic(x)), # integers Exclusive(Q.even(x), Q.odd(x)), Implies(Q.integer(x), Q.rational(x)), Implies(Q.zero(x), Q.even(x)), Exclusive(Q.composite(x), Q.prime(x)), Implies(Q.composite(x) | Q.prime(x), Q.integer(x) & Q.positive(x)), Implies(Q.even(x) & Q.positive(x) & ~Q.prime(x), Q.composite(x)), # hermitian and antihermitian Implies(Q.real(x), Q.hermitian(x)), Implies(Q.imaginary(x), Q.antihermitian(x)), Implies(Q.zero(x), Q.hermitian(x) | Q.antihermitian(x)), # define finity and infinity, and build extended real line Exclusive(Q.infinite(x), Q.finite(x)), Implies(Q.complex(x), Q.finite(x)), Implies( Q.negative_infinite(x) | Q.positive_infinite(x), Q.infinite(x)), # commutativity Implies(Q.finite(x) | Q.infinite(x), Q.commutative(x)), # matrices Implies(Q.orthogonal(x), Q.positive_definite(x)), Implies(Q.orthogonal(x), Q.unitary(x)), Implies(Q.unitary(x) & Q.real_elements(x), Q.orthogonal(x)), Implies(Q.unitary(x), Q.normal(x)), Implies(Q.unitary(x), Q.invertible(x)), Implies(Q.normal(x), Q.square(x)), Implies(Q.diagonal(x), Q.normal(x)), Implies(Q.positive_definite(x), Q.invertible(x)), Implies(Q.diagonal(x), Q.upper_triangular(x)), Implies(Q.diagonal(x), Q.lower_triangular(x)), Implies(Q.lower_triangular(x), Q.triangular(x)), Implies(Q.upper_triangular(x), Q.triangular(x)), Implies(Q.triangular(x), Q.upper_triangular(x) | Q.lower_triangular(x)), Implies(Q.upper_triangular(x) & Q.lower_triangular(x), Q.diagonal(x)), Implies(Q.diagonal(x), Q.symmetric(x)), Implies(Q.unit_triangular(x), Q.triangular(x)), Implies(Q.invertible(x), Q.fullrank(x)), Implies(Q.invertible(x), Q.square(x)), Implies(Q.symmetric(x), Q.square(x)), Implies(Q.fullrank(x) & Q.square(x), Q.invertible(x)), Equivalent(Q.invertible(x), ~Q.singular(x)), Implies(Q.integer_elements(x), Q.real_elements(x)), Implies(Q.real_elements(x), Q.complex_elements(x)), ) return fact