def _(expr): # Implicitly assumes Mul has more than one arg # Would be allargs(x, Q.prime(x) | Q.composite(x)) except 1 is composite # More advanced prime assumptions will require inequalities, as 1 provides # a corner case. allargs_prime = allargs(x, Q.prime(x), expr) return Implies(allargs_prime, ~Q.prime(expr))
def test_binary_symbols(): assert ITE(x < 1, y, z).binary_symbols == set((y, z)) for f in (Eq, Ne): assert f(x, 1).binary_symbols == set() assert f(x, True).binary_symbols == set([x]) assert f(x, False).binary_symbols == set([x]) assert S.true.binary_symbols == set() assert S.false.binary_symbols == set() assert x.binary_symbols == set([x]) assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == set([x, y]) assert Q.prime(x).binary_symbols == set() assert Q.is_true(x < 1).binary_symbols == set() assert Q.is_true(x).binary_symbols == set([x]) assert Q.is_true(Eq(x, True)).binary_symbols == set([x]) assert Q.prime(x).binary_symbols == set()
def test_binary_symbols(): assert ITE(x < 1, y, z).binary_symbols == set((y, z)) for f in (Eq, Ne): assert f(x, 1).binary_symbols == set() assert f(x, True).binary_symbols == set([x]) assert f(x, False).binary_symbols == set([x]) assert S.true.binary_symbols == set() assert S.false.binary_symbols == set() assert x.binary_symbols == set([x]) assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == set([x, y]) assert Q.prime(x).binary_symbols == set() assert Q.is_true(x < 1).binary_symbols == set() assert Q.is_true(x).binary_symbols == set([x]) assert Q.is_true(Eq(x, True)).binary_symbols == set([x]) assert Q.prime(x).binary_symbols == set()
def test_binary_symbols(): assert ITE(x < 1, y, z).binary_symbols == {y, z} for f in (Eq, Ne): assert f(x, 1).binary_symbols == set() assert f(x, True).binary_symbols == {x} assert f(x, False).binary_symbols == {x} assert S.true.binary_symbols == set() assert S.false.binary_symbols == set() assert x.binary_symbols == {x} assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == {x, y} assert Q.prime(x).binary_symbols == set() assert Q.lt(x, 1).binary_symbols == set() assert Q.is_true(x).binary_symbols == {x} assert Q.eq(x, True).binary_symbols == {x} assert Q.prime(x).binary_symbols == set()
def test_prime(): assert satask(Q.prime(5)) is True assert satask(Q.prime(6)) is False assert satask(Q.prime(-5)) is False assert satask(Q.prime(x*y), Q.integer(x) & Q.integer(y)) is None assert satask(Q.prime(x*y), Q.prime(x) & Q.prime(y)) is False
def test_prime_composite(): assert satask(Q.prime(x), Q.composite(x)) is False assert satask(Q.composite(x), Q.prime(x)) is False assert satask(Q.composite(x), ~Q.prime(x)) is None assert satask(Q.prime(x), ~Q.composite(x)) is None # since 1 is neither prime nor composite the following should hold assert satask(Q.prime(x), Q.integer(x) & Q.positive(x) & ~Q.composite(x)) is None assert satask(Q.prime(2)) is True assert satask(Q.prime(4)) is False assert satask(Q.prime(1)) is False assert satask(Q.composite(1)) is False
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