def test_overloading():
"""Test that |, & are overloaded as expected."""
assert A & B == And(A, B)
assert A | B == Or(A, B)
assert (A & B) | C == Or(And(A, B), C)
assert A >> B == Implies(A, B)
assert A << B == Implies(B, A)
assert ~A == Not(A)
assert A ^ B == Xor(A, B)
def test_relational_logic_symbols():
# See issue sympy/sympy#6204
assert (x < y) & (z < t) == And(x < y, z < t)
assert (x < y) | (z < t) == Or(x < y, z < t)
assert ~(x < y) == Not(x < y)
assert (x < y) >> (z < t) == Implies(x < y, z < t)
assert (x < y) << (z < t) == Implies(z < t, x < y)
assert (x < y) ^ (z < t) == Xor(x < y, z < t)
assert isinstance((x < y) & (z < t), And)
assert isinstance((x < y) | (z < t), Or)
assert isinstance(~(x < y), GreaterThan)
assert isinstance((x < y) >> (z < t), Implies)
assert isinstance((x < y) << (z < t), Implies)
assert isinstance((x < y) ^ (z < t), (Or, Xor))
def test_to_nnf():
assert to_nnf(true) is true
assert to_nnf(false) is false
assert to_nnf(a) == a
assert to_nnf(~a) == ~a
class Foo(BooleanFunction):
pass
pytest.raises(ValueError, lambda: to_nnf(~Foo(a)))
assert to_nnf(a | ~a | b) is true
assert to_nnf(a & ~a & b) is false
assert to_nnf(a >> b) == ~a | b
assert to_nnf(Implies(a, b, evaluate=False)) == ~a | b
assert to_nnf(a >> (c >> ~b)) == (~b | ~c) | ~a
assert to_nnf(Equivalent(a, b)) == (a | ~b) & (b | ~a)
assert to_nnf(Equivalent(a, b, c)) == (~a | b) & (~b | c) & (~c | a)
assert to_nnf(Equivalent(a, b, c, d)) == (~a | b) & (~b | c) & (~c | d) & (~d | a)
assert to_nnf(a ^ b ^ c) == (a | b | c) & (~a | ~b | c) & (a | ~b | ~c) & (~a | b | ~c)
assert to_nnf(ITE(a, b, c)) == (~a | b) & (a | c)
assert to_nnf(~(a | b | c)) == ~a & ~b & ~c
assert to_nnf(~(a & b & c)) == ~a | ~b | ~c
assert to_nnf(~(a >> b)) == a & ~b
assert to_nnf(~(Equivalent(a, b, c))) == (a | b | c) & (~a | ~b | ~c)
assert to_nnf(~(a ^ b ^ c)) == (~a | b | c) & (a | ~b | c) & (a | b | ~c) & (~a | ~b | ~c)
assert to_nnf(~(ITE(a, b, c))) == (~a | ~b) & (a | ~c)
assert to_nnf((a >> b) ^ (b >> a)) == (a & ~b) | (~a & b)
assert to_nnf((a >> b) ^ (b >> a), False) == (~a | ~b | a | b) & ((a & ~b) | (~a & b))
def test_simplification():
"""Test working of simplification methods."""
set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
assert Not(SOPform([x, y, z], set2)) == Not(Or(And(Not(x), Not(z)), And(x, z)))
assert POSform([x, y, z], set1 + set2) is true
assert SOPform([x, y, z], set1 + set2) is true
assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(Not(w), z), And(y, z)))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
# test simplification
ans = And(A, Or(B, C))
assert simplify_logic(A & (B | C)) == ans
assert simplify_logic((A & B) | (A & C)) == ans
assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
assert simplify_logic(Equivalent(A, B)) == \
Or(And(A, B), And(Not(A), Not(B)))
assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
== And(Equality(A, 3), Or(B, C))
e = And(A, x**2 - x)
assert simplify_logic(e) == And(A, x*(x - 1))
assert simplify_logic(e, deep=False) == e
pytest.raises(ValueError, lambda: simplify_logic(A & (B | C), form='spam'))
e = x & y ^ z | (z ^ x)
res = [(x & ~z) | (z & ~x) | (z & ~y), (x & ~y) | (x & ~z) | (z & ~x)]
assert simplify_logic(e) in res
assert SOPform([z, y, x], [[0, 0, 1], [0, 1, 1],
[1, 0, 0], [1, 0, 1], [1, 1, 0]]) == res[1]
# check input
ans = SOPform([x, y], [[1, 0]])
assert SOPform([x, y], [[1, 0]]) == ans
assert POSform([x, y], [[1, 0]]) == ans
pytest.raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
assert SOPform([x], [[1]], [[0]]) is true
assert SOPform([x], [[0]], [[1]]) is true
assert SOPform([x], [], []) is false
pytest.raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
assert POSform([x], [[1]], [[0]]) is true
assert POSform([x], [[0]], [[1]]) is true
assert POSform([x], [], []) is false
# check working of simplify
assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
assert simplify(And(x, Not(x))) is false
assert simplify(Or(x, Not(x))) is true
def test_Implies():
pytest.raises(ValueError, lambda: Implies(A, B, C))
assert Implies(True, True) is true
assert Implies(True, False) is false
assert Implies(False, True) is true
assert Implies(False, False) is true
assert Implies(0, A) is true
assert Implies(1, 1) is true
assert Implies(1, 0) is false
assert A >> B == B << A
assert (A < 1) >> (A >= 1) == (A >= 1)
assert (A < 1) >> (Integer(1) > A) is true
assert A >> A is true
assert ((A < 1) >> (B >= 1)) == Implies(A < 1, B >= 1, evaluate=False)
def test_Implies():
pytest.raises(ValueError, lambda: Implies(a, b, c))
assert Implies(True, True) is true
assert Implies(True, False) is false
assert Implies(False, True) is true
assert Implies(False, False) is true
assert Implies(0, a) is true
assert Implies(1, 1) is true
assert Implies(1, 0) is false
assert a >> b == b << a
assert (a < 1) >> (a >= 1) == (a >= 1)
assert (a < 1) >> (1 > a) is true
assert (a < 1) >> (Integer(1) > a) is true
assert a >> a is true
assert ((a < 1) >> (b >= 1)) == Implies(a < 1, b >= 1, evaluate=False)
def test_satisfiable_non_symbols():
class Zero(Boolean):
pass
assumptions = Zero(x*y)
facts = Implies(Zero(x*y), Zero(x) | Zero(y))
query = ~Zero(x) & ~Zero(y)
refutations = [
{Zero(x): True, Zero(x*y): True},
{Zero(y): True, Zero(x*y): True},
{Zero(x): True, Zero(y): True, Zero(x*y): True},
{Zero(x): True, Zero(y): False, Zero(x*y): True},
{Zero(x): False, Zero(y): True, Zero(x*y): True}]
assert not satisfiable(And(assumptions, facts, query), algorithm='dpll')
assert satisfiable(And(assumptions, facts, ~query), algorithm='dpll') in refutations
assert not satisfiable(And(assumptions, facts, query), algorithm='dpll2')
assert satisfiable(And(assumptions, facts, ~query), algorithm='dpll2') in refutations
def test_operators():
# Mostly test __and__, __rand__, and so on
assert True & A == (A & True) == A
assert False & A == (A & False) == false
assert A & B == And(A, B)
assert True | A == (A | True) == true
assert False | A == (A | False) == A
assert A | B == Or(A, B)
assert ~A == Not(A)
assert True >> A == (A << True) == A
assert False >> A == (A << False) == true
assert (A >> True) == (True << A) == true
assert (A >> False) == (False << A) == ~A
assert A >> B == B << A == Implies(A, B)
assert True ^ A == A ^ True == ~A
assert False ^ A == (A ^ False) == A
assert A ^ B == Xor(A, B)
def test_count_ops_non_visual():
def count(val):
return count_ops(val, visual=False)
assert count(x) == 0
assert count(x) is not Integer(0)
assert count(x + y) == 1
assert count(x + y) is not Integer(1)
assert count(x + y*x + 2*y) == 4
assert count({x + y: x}) == 1
assert count({x + y: 2 + x}) is not Integer(1)
assert count(Or(x, y)) == 1
assert count(And(x, y)) == 1
assert count(Not(x)) == 1
assert count(Nor(x, y)) == 2
assert count(Nand(x, y)) == 2
assert count(Xor(x, y)) == 1
assert count(Implies(x, y)) == 1
assert count(Equivalent(x, y)) == 1
assert count(ITE(x, y, z)) == 1
assert count(ITE(True, x, y)) == 0
def test_true_false():
# pylint: disable=singleton-comparison,comparison-with-itself
assert true is true
assert false is false
assert true is not True
assert false is not False
assert true
assert not false
assert true == True # noqa: E712
assert false == False # noqa: E712
assert not true == False # noqa: E712
assert not false == True # noqa: E712
assert not true == false
assert hash(true) == hash(True)
assert hash(false) == hash(False)
assert len({true, True}) == len({false, False}) == 1
assert int(true) == 1
assert int(false) == 0
assert isinstance(true, BooleanAtom)
assert isinstance(false, BooleanAtom)
assert not isinstance(true, bool)
assert not isinstance(false, bool)
assert ~true is false
assert Not(True) is false
assert ~false is true
assert Not(False) is true
for T, F in itertools.product([True, true], [False, false]):
assert And(T, F) is false
assert And(F, T) is false
assert And(F, F) is false
assert And(T, T) is true
assert And(T, x) == x
assert And(F, x) is false
if not (T is True and F is False):
assert T & F is false
assert F & T is false
if F is not False:
assert F & F is false
if T is not True:
assert T & T is true
assert Or(T, F) is true
assert Or(F, T) is true
assert Or(F, F) is false
assert Or(T, T) is true
assert Or(T, x) is true
assert Or(F, x) == x
if not (T is True and F is False):
assert T | F is true
assert F | T is true
if F is not False:
assert F | F is false
if T is not True:
assert T | T is true
assert Xor(T, F) is true
assert Xor(F, T) is true
assert Xor(F, F) is false
assert Xor(T, T) is false
assert Xor(T, x) == ~x
assert Xor(F, x) == x
if not (T is True and F is False):
assert T ^ F is true
assert F ^ T is true
if F is not False:
assert F ^ F is false
if T is not True:
assert T ^ T is false
assert Nand(T, F) is true
assert Nand(F, T) is true
assert Nand(F, F) is true
assert Nand(T, T) is false
assert Nand(T, x) == ~x
assert Nand(F, x) is true
assert Nor(T, F) is false
assert Nor(F, T) is false
assert Nor(F, F) is true
assert Nor(T, T) is false
assert Nor(T, x) is false
assert Nor(F, x) == ~x
assert Implies(T, F) is false
assert Implies(F, T) is true
assert Implies(F, F) is true
assert Implies(T, T) is true
assert Implies(T, x) == x
assert Implies(F, x) is true
assert Implies(x, T) is true
assert Implies(x, F) == ~x
if not (T is True and F is False):
assert T >> F is false
assert F << T is false
assert F >> T is true
assert T << F is true
if F is not False:
assert F >> F is true
assert F << F is true
if T is not True:
assert T >> T is true
assert T << T is true
assert Equivalent(T, F) is false
assert Equivalent(F, T) is false
assert Equivalent(F, F) is true
assert Equivalent(T, T) is true
assert Equivalent(T, x) == x
assert Equivalent(F, x) == ~x
assert Equivalent(x, T) == x
assert Equivalent(x, F) == ~x
assert ITE(T, T, T) is true
assert ITE(T, T, F) is true
assert ITE(T, F, T) is false
assert ITE(T, F, F) is false
assert ITE(F, T, T) is true
assert ITE(F, T, F) is false
assert ITE(F, F, T) is true
assert ITE(F, F, F) is false
def test_overloading():
assert a & b == And(a, b)
assert a | b == Or(a, b)
assert a >> b == Implies(a, b)
assert ~a == Not(a)
assert a ^ b == Xor(a, b)
def test_count_ops_visual():
ADD, MUL, POW, SIN, COS, AND, D, G = symbols(
'Add Mul Pow sin cos And Derivative Integral'.upper())
DIV, SUB, NEG = symbols('DIV SUB NEG')
NOT, OR, AND, XOR, IMPLIES, EQUIVALENT, _ITE, BASIC, TUPLE = symbols(
'Not Or And Xor Implies Equivalent ITE Basic Tuple'.upper())
def count(val):
return count_ops(val, visual=True)
assert count(7) is Integer(0)
assert count(-1) == NEG
assert count(-2) == NEG
assert count(Rational(2, 3)) == DIV
assert count(pi / 3) == DIV
assert count(-pi / 3) == DIV + NEG
assert count(I - 1) == SUB
assert count(1 - I) == SUB
assert count(1 - 2 * I) == SUB + MUL
assert count(x) is Integer(0)
assert count(-x) == NEG
assert count(-2 * x / 3) == NEG + DIV + MUL
assert count(1 / x) == DIV
assert count(1 / (x * y)) == DIV + MUL
assert count(-1 / x) == NEG + DIV
assert count(-2 / x) == NEG + DIV
assert count(x / y) == DIV
assert count(-x / y) == NEG + DIV
assert count(x**2) == POW
assert count(-x**2) == POW + NEG
assert count(-2 * x**2) == POW + MUL + NEG
assert count(x + pi / 3) == ADD + DIV
assert count(x + Rational(1, 3)) == ADD + DIV
assert count(x + y) == ADD
assert count(x - y) == SUB
assert count(y - x) == SUB
assert count(-1 / (x - y)) == DIV + NEG + SUB
assert count(-1 / (y - x)) == DIV + NEG + SUB
assert count(1 + x**y) == ADD + POW
assert count(1 + x + y) == 2 * ADD
assert count(1 + x + y + z) == 3 * ADD
assert count(1 + x**y + 2 * x * y + y**2) == 3 * ADD + 2 * POW + 2 * MUL
assert count(2 * z + y + x + 1) == 3 * ADD + MUL
assert count(2 * z + y**17 + x + 1) == 3 * ADD + MUL + POW
assert count(2 * z + y**17 + x + sin(x)) == 3 * ADD + POW + MUL + SIN
assert count(2 * z + y**17 + x +
sin(x**2)) == 3 * ADD + MUL + 2 * POW + SIN
assert count(2 * z + y**17 + x + sin(x**2) +
exp(cos(x))) == 4 * ADD + MUL + 3 * POW + COS + SIN
assert count(Derivative(x, x)) == D
assert count(Integral(x, x) + 2 * x / (1 + x)) == G + DIV + MUL + 2 * ADD
assert count(Basic()) is Integer(0)
assert count({x + 1: sin(x)}) == ADD + SIN
assert count([x + 1, sin(x) + y, None]) == ADD + SIN + ADD
assert count({x + 1: sin(x), y: cos(x) + 1}) == SIN + COS + 2 * ADD
assert count({}) is Integer(0)
assert count([x + 1, sin(x) * y, None]) == SIN + ADD + MUL
assert count([]) is Integer(0)
assert count(Basic()) == 0
assert count(Basic(Basic(), Basic(x, x + y))) == ADD + 2 * BASIC
assert count(Basic(x, x + y)) == ADD + BASIC
assert count(Or(x, y)) == OR
assert count(And(x, y)) == AND
assert count(And(x**y, z)) == AND + POW
assert count(Or(x, Or(y, And(z, a)))) == AND + OR
assert count(Nor(x, y)) == NOT + OR
assert count(Nand(x, y)) == NOT + AND
assert count(Xor(x, y)) == XOR
assert count(Implies(x, y)) == IMPLIES
assert count(Equivalent(x, y)) == EQUIVALENT
assert count(ITE(x, y, z)) == _ITE
assert count([Or(x, y), And(x, y), Basic(x + y)]) == ADD + AND + BASIC + OR
assert count(Basic(Tuple(x))) == BASIC + TUPLE
# It checks that TUPLE is counted as an operation.
assert count(Eq(x + y, 2)) == ADD
def test_true_false():
assert true is true
assert false is false
assert true is not True
assert false is not False
assert true
assert not false
assert true == True # noqa: E712
assert false == False # noqa: E712
assert not (true == False) # noqa: E712
assert not (false == True) # noqa: E712
assert not (true == false)
assert hash(true) == hash(True)
assert hash(false) == hash(False)
assert len({true, True}) == len({false, False}) == 1
assert isinstance(true, BooleanAtom)
assert isinstance(false, BooleanAtom)
# We don't want to subclass from bool, because bool subclasses from
# int. But operators like &, |, ^, <<, >>, and ~ act differently on 0 and
# 1 then we want them to on true and false. See the docstrings of the
# various And, Or, etc. functions for examples.
assert not isinstance(true, bool)
assert not isinstance(false, bool)
# Note: using 'is' comparison is important here. We want these to return
# true and false, not True and False
assert Not(true) is false
assert Not(True) is false
assert Not(false) is true
assert Not(False) is true
assert ~true is false
assert ~false is true
for T, F in itertools.product([True, true], [False, false]):
assert And(T, F) is false
assert And(F, T) is false
assert And(F, F) is false
assert And(T, T) is true
assert And(T, x) == x
assert And(F, x) is false
if not (T is True and F is False):
assert T & F is false
assert F & T is false
if F is not False:
assert F & F is false
if T is not True:
assert T & T is true
assert Or(T, F) is true
assert Or(F, T) is true
assert Or(F, F) is false
assert Or(T, T) is true
assert Or(T, x) is true
assert Or(F, x) == x
if not (T is True and F is False):
assert T | F is true
assert F | T is true
if F is not False:
assert F | F is false
if T is not True:
assert T | T is true
assert Xor(T, F) is true
assert Xor(F, T) is true
assert Xor(F, F) is false
assert Xor(T, T) is false
assert Xor(T, x) == ~x
assert Xor(F, x) == x
if not (T is True and F is False):
assert T ^ F is true
assert F ^ T is true
if F is not False:
assert F ^ F is false
if T is not True:
assert T ^ T is false
assert Nand(T, F) is true
assert Nand(F, T) is true
assert Nand(F, F) is true
assert Nand(T, T) is false
assert Nand(T, x) == ~x
assert Nand(F, x) is true
assert Nor(T, F) is false
assert Nor(F, T) is false
assert Nor(F, F) is true
assert Nor(T, T) is false
assert Nor(T, x) is false
assert Nor(F, x) == ~x
assert Implies(T, F) is false
assert Implies(F, T) is true
assert Implies(F, F) is true
assert Implies(T, T) is true
assert Implies(T, x) == x
assert Implies(F, x) is true
assert Implies(x, T) is true
assert Implies(x, F) == ~x
if not (T is True and F is False):
assert T >> F is false
assert F << T is false
assert F >> T is true
assert T << F is true
if F is not False:
assert F >> F is true
assert F << F is true
if T is not True:
assert T >> T is true
assert T << T is true
assert Equivalent(T, F) is false
assert Equivalent(F, T) is false
assert Equivalent(F, F) is true
assert Equivalent(T, T) is true
assert Equivalent(T, x) == x
assert Equivalent(F, x) == ~x
assert Equivalent(x, T) == x
assert Equivalent(x, F) == ~x
assert ITE(T, T, T) is true
assert ITE(T, T, F) is true
assert ITE(T, F, T) is false
assert ITE(T, F, F) is false
assert ITE(F, T, T) is true
assert ITE(F, T, F) is false
assert ITE(F, F, T) is true
assert ITE(F, F, F) is false
def test_eliminate_implications():
assert eliminate_implications(Implies(A, B, evaluate=False)) == (~A) | B
assert eliminate_implications(A >> (C >> Not(B))) == Or(
Or(Not(B), Not(C)), Not(A))
assert eliminate_implications(Equivalent(A, B, C, D)) == \
(~A | B) & (~B | C) & (~C | D) & (~D | A)