def test_prenex_basic(self):
a, b, c = (Symbol(x) for x in "abc")
f = Not(And(a, Exists([b], And(a, b)), ForAll([c], Or(a, c))))
prenex = prenex_normal_form(f)
# Two prenex normal forms are possible
my_prenex_1 = Exists([c], ForAll([b], Not(And(a, And(a, b), Or(a,
c)))))
my_prenex_2 = ForAll([b], Exists([c], Not(And(a, And(a, b), Or(a,
c)))))
self.assertTrue(prenex == my_prenex_1 or prenex == my_prenex_2)
def test_substitution_complex_mss(self):
x, y = FreshSymbol(REAL), FreshSymbol(REAL)
# y = 0 /\ (Forall x. x > 3 /\ y < 2)
f = And(Equals(y, Real(0)),
ForAll([x], And(GT(x, Real(3)), LT(y, Real(2)))))
subs = {
y: Real(0),
ForAll([x], And(GT(x, Real(3)), LT(Real(0), Real(2)))): TRUE()
}
f_subs = MSSubstituter(env=self.env).substitute(f, subs).simplify()
self.assertEqual(f_subs, TRUE())
def test_quantifiers(self):
x = Symbol("x")
fa = ForAll([x], And(x, Not(x)))
fe = Exists([x], And(x, Not(x)))
self.assertEqual(fa.to_smtlib(daggify=False),
"(forall ((x Bool)) (and x (not x)))")
self.assertEqual(fe.to_smtlib(daggify=False),
"(exists ((x Bool)) (and x (not x)))")
self.assertEqual(fa.to_smtlib(daggify=True),
"(let ((.def_0 (forall ((x Bool)) (let ((.def_0 (not x))) (let ((.def_1 (and x .def_0))) .def_1))))).def_0)")
self.assertEqual(fe.to_smtlib(daggify=True),
"(let ((.def_0 (exists ((x Bool)) (let ((.def_0 (not x))) (let ((.def_1 (and x .def_0))) .def_1))))).def_0)")
def test_substitution_on_quantifiers(self):
x, y = FreshSymbol(), FreshSymbol()
# y /\ Forall x. x /\ y.
f = And(y, ForAll([x], And(x, y)))
subs = {y: Bool(True)}
f_subs = substitute(f, subs).simplify()
self.assertEqual(f_subs, ForAll([x], x))
subs = {x: Bool(True)}
f_subs = substitute(f, subs).simplify()
self.assertEqual(f_subs, f)
def test_substitution_on_quantifiers(self):
x, y = FreshSymbol(), FreshSymbol()
# y /\ Forall x. x /\ y.
f = And(y, ForAll([x], And(x, y)))
# Symbols within the quantified formula are not free symbols
# and should not be substituted.
subs = {y: TRUE()}
f_subs = substitute(f, subs).simplify()
self.assertEqual(f_subs, ForAll([x], x))
subs = {x: TRUE()}
f_subs = substitute(f, subs).simplify()
self.assertEqual(f_subs, f)
def smart_walk_not(self, formula):
if formula in self.subs:
# Smarties contains a string.
# In the future, we could allow for arbitrary function calls
self.write(self.subs[formula])
else:
arg = formula.arg(0)
if arg.is_iff():
arg = self.get_iff(arg)
if arg.is_exists():
argn = ForAll(arg.quantifier_vars(), Not(arg.arg(0)))
return self.walk(argn)
elif arg.is_forall():
argn = Exists(arg.quantifier_vars(), Not(arg.arg(0)))
return self.walk(argn)
elif arg.is_and():
args = [Not(a) for a in arg.args()]
argn = Or(args)
return self.walk(argn)
elif arg.is_or():
args = [Not(a) for a in arg.args()]
argn = And(args)
return self.walk(argn)
elif arg.is_not():
return self.walk(arg.arg(0))
else:
if (arg.is_not()):
assert (0)
return HRPrinter.super(self, formula)
def test_real(self):
r, s = Symbol("r", REAL), Symbol("s", REAL)
f = ForAll([r], Implies(LT(Real(0), r), LT(s, r)))
with Solver(name='cvc4', logic=LRA) as s:
s.add_assertion(f)
res = s.solve()
self.assertTrue(res)
def test_int(self):
p, q = Symbol("p", INT), Symbol("q", INT)
f = ForAll([p], Implies(LT(Int(0), p), LT(q, p)))
with Solver(name='cvc4', logic=LIA) as s:
s.add_assertion(f)
res = s.solve()
self.assertTrue(res)
def test_yices_quantifier(self):
x = Symbol('x', REAL)
f = ForAll([x], LE(x, Real(0)))
with self.assertRaises(InternalSolverError):
with Solver(name='yices') as s:
s.add_assertion(f)
self.assertFalse(s.solve())
def test_bool(self):
x, y = Symbol("x"), Symbol("y")
f = ForAll([x], Implies(x, y))
with Solver(name='cvc4', logic=LIA) as s:
s.add_assertion(f)
res = s.solve()
self.assertTrue(res)
def _alternation_bool_example(self, qe):
# Alternation of quantifiers
x, y = Symbol("x"), Symbol("y")
f = ForAll([x], Exists([y], Iff(x, Not(y))))
qf = qe.eliminate_quantifiers(f).simplify()
self.assertEqual(qf, TRUE())
def _bool_example(self, qe):
# Bool Example
x, y = Symbol("x"), Symbol("y")
f = ForAll([x], Implies(x,y))
qf = qe.eliminate_quantifiers(f).simplify()
self.assertEqual(qf, y)
def _real_example(self, qe):
# Real Example
r, s = Symbol("r", REAL), Symbol("s", REAL)
f = ForAll([r], Implies(LT(Real(0), r), LT(s, r)))
qf = qe.eliminate_quantifiers(f).simplify()
self.assertEqual(qf, LE(s, Real(0)))
def _alternation_int_example(self, qe):
# Alternation of quantifiers
p, q = Symbol("p", INT), Symbol("q", INT)
f = ForAll([p], Exists([q], Equals(p, Plus(q, Int(1)))))
qf = qe.eliminate_quantifiers(f).simplify()
self.assertEqual(qf, TRUE())
def _alternation_real_example(self, qe):
# Alternation of quantifiers
r, s = Symbol("r", REAL), Symbol("s", REAL)
f = ForAll([r], Exists([s], Equals(r, Plus(s, Real(1)))))
qf = qe.eliminate_quantifiers(f).simplify()
self.assertEqual(qf, TRUE())
def test_quantifiers(self):
x = Symbol("x")
fa = ForAll([x], And(x, Not(x)))
fe = Exists([x], And(x, Not(x)))
self.assertEqual(fa.to_smtlib(daggify=False),
"(forall ((x Bool)) (and x (not x)))")
self.assertEqual(fe.to_smtlib(daggify=False),
"(exists ((x Bool)) (and x (not x)))")
self.assertEqual(
fa.to_smtlib(daggify=True),
"(let ((.def_0 (forall ((x Bool)) (let ((.def_0 (not x))) (let ((.def_1 (and x .def_0))) .def_1))))).def_0)"
)
self.assertEqual(
fe.to_smtlib(daggify=True),
"(let ((.def_0 (exists ((x Bool)) (let ((.def_0 (not x))) (let ((.def_1 (and x .def_0))) .def_1))))).def_0)"
)
def _int_example(self, qe):
# Int Example
p, q = Symbol("p", INT), Symbol("q", INT)
f = ForAll([p], Implies(LT(Int(0), p), LT(q, p)))
qf = qe.eliminate_quantifiers(f).simplify()
self.assertValid(Iff(qf, LE(q, Int(0))))
def export_goals(self, formula):
ex = self.export_expr
if formula.is_not() and self.extract_universal(formula.args()[0])[0]:
formula = formula.arg(0)
uvars, inner = self.extract_universal(formula)
if inner.is_equals() or inner.is_iff():
goal = formula
elif inner.is_implies():
goal = formula
if (not inner.arg(1).is_equals()) and (
not inner.arg(1).is_iff()):
ForAll(
uvars,
Implies(inner.arg(0), Iff(inner.arg(1), Bool(True))))
else:
goal = ForAll(uvars, Iff(inner, Bool(True)))
yield SExpression(['prove', ex(goal)])
def test_real(self):
r, s = Symbol("r", REAL), Symbol("s", REAL)
f = ForAll([r], Implies(LT(Real(0), r), LT(s, r)))
with Solver(name='cvc4', logic=LRA) as s:
s.add_assertion(f)
try:
res = s.solve()
self.assertTrue(res)
except SolverReturnedUnknownResultError:
pass
def test_int(self):
p, q = Symbol("p", INT), Symbol("q", INT)
f = ForAll([p], Implies(LT(Int(0), p), LT(q, p)))
with Solver(name='cvc4', logic=LIA) as s:
s.add_assertion(f)
try:
res = s.solve()
self.assertTrue(res)
except SolverReturnedUnknownResultError:
pass
def test_bool(self):
x, y = Symbol("x"), Symbol("y")
f = ForAll([x], Implies(x,y))
with Solver(name='cvc4', logic=LIA) as s:
s.add_assertion(f)
try:
res = s.solve()
self.assertTrue(res)
except SolverReturnedUnknownResultError:
pass
def test_substitution_complex(self):
x, y = Symbol("x", REAL), Symbol("y", REAL)
# y = 0 /\ (Forall x. x > 3 /\ y < 2)
f = And(Equals(y, Real(0)),
ForAll([x], And(GT(x, Real(3)), LT(y, Real(2)))))
subs = {
y: Real(0),
ForAll([x], And(GT(x, Real(3)), LT(y, Real(2)))): TRUE()
}
f_subs = substitute(f, subs).simplify()
if self.env.SubstituterClass == MGSubstituter:
self.assertEqual(f_subs, TRUE())
else:
# In the MSS the y=0 substitution is performed first,
# therefore, the overall quantified expression does not
# match the one defined in the substitution map.
# See test_substitution_complex_mss for a positive example.
self.assertEqual(f_subs, ForAll([x], GT(x, Real(3))))
def export_rule_def(self, defun):
name, args, rettype, body = defun
ftype = FunctionType(rettype, [a.symbol_type() for a in args])
fsymb = Symbol(name, ftype)
yield self.export_func(fsymb)
eqop = Iff if rettype.is_bool_type() else Equals
for rule in self.export_rules(
ForAll(args, eqop(Function(fsymb, args), body))):
yield rule
def test_quantifiers(self):
x = Symbol("x")
fa = ForAll([x], And(x, Not(x)))
fe = Exists([x], And(x, Not(x)))
fa_string = self.print_to_string(fa)
fe_string = self.print_to_string(fe)
self.assertEqual(fa_string, "(forall ((x Bool)) (and x (not x)))")
self.assertEqual(fe_string, "(exists ((x Bool)) (and x (not x)))")
def test_substitution_complex(self):
x, y = FreshSymbol(REAL), FreshSymbol(REAL)
# y = 0 /\ (Forall x. x > 3 /\ y < 2)
f = And(Equals(y, Real(0)),
ForAll([x], And(GT(x, Real(3)), LT(y, Real(2)))))
if "MSS" in str(self.env.SubstituterClass):
subs = {
y: Real(0),
ForAll([x], And(GT(x, Real(3)), LT(Real(0), Real(2)))): TRUE()
}
else:
assert "MGS" in str(self.env.SubstituterClass)
subs = {
y: Real(0),
ForAll([x], And(GT(x, Real(3)), LT(y, Real(2)))): TRUE()
}
f_subs = substitute(f, subs).simplify()
self.assertEqual(f_subs, TRUE())
def test_substitution_term(self):
x, y = FreshSymbol(REAL), FreshSymbol(REAL)
# y = 0 /\ Forall x. x > 3
f = And(Equals(y, Real(0)), ForAll([x], GT(x, Real(3))))
subs = {GT(x, Real(3)): TRUE()}
f_subs = substitute(f, subs)
# Since 'x' is quantified, we cannot replace the term
# therefore the substitution does not yield any result.
self.assertEqual(f_subs, f)
def test_qe_z3(self):
qe = QuantifierEliminator(name='z3')
self._bool_example(qe)
self._real_example(qe)
self._int_example(qe)
self._alternation_bool_example(qe)
self._alternation_real_example(qe)
self._alternation_int_example(qe)
self._std_examples(qe, LRA)
self._std_examples(qe, LIA)
# Additional test for raising error on back conversion of
# quantified formulae
p, q = Symbol("p", INT), Symbol("q", INT)
f = ForAll([p], Exists([q], Equals(ToReal(p),
Plus(ToReal(q), ToReal(Int(1))))))
with self.assertRaises(NotImplementedError):
qe.eliminate_quantifiers(f).simplify()
def test_examples_solving(self):
for example in get_example_formulae():
if example.logic != pysmt.logics.BOOL:
continue
fv = example.expr.get_free_variables()
f = Exists(fv, example.expr)
g = qelim(f, solver_name="shannon").simplify()
if example.is_sat:
self.assertTrue(g.is_true())
else:
self.assertTrue(g.is_false())
f = ForAll(fv, example.expr)
g = qelim(f, solver_name="shannon").simplify()
if example.is_valid:
self.assertTrue(g.is_true())
else:
self.assertTrue(g.is_false())
def action_noop(self):
f = dict()
for s in self._states:
if s in self._pre2nex:
n = self._pre2nex[s]
s_type = s.symbol_type()
args = []
if s_type.is_function_type():
i = 0
for paramt in s_type.param_types:
i += 1
paramt_name = str(i) + ":" + str(paramt)
args.append(Symbol(paramt_name, paramt))
lhs = Function(s, args)
rhs = Function(n, args)
eq = ForAll(args, EqualsOrIff(lhs, rhs))
f[n] = eq
return f
def get_full_example_formulae(environment=None):
"""Return a list of Examples using the given environment."""
if environment is None:
environment = get_env()
with environment:
x = Symbol("x", BOOL)
y = Symbol("y", BOOL)
p = Symbol("p", INT)
q = Symbol("q", INT)
r = Symbol("r", REAL)
s = Symbol("s", REAL)
aii = Symbol("aii", ARRAY_INT_INT)
ari = Symbol("ari", ArrayType(REAL, INT))
arb = Symbol("arb", ArrayType(REAL, BV8))
abb = Symbol("abb", ArrayType(BV8, BV8))
nested_a = Symbol("a_arb_aii",
ArrayType(ArrayType(REAL, BV8), ARRAY_INT_INT))
rf = Symbol("rf", FunctionType(REAL, [REAL, REAL]))
rg = Symbol("rg", FunctionType(REAL, [REAL]))
ih = Symbol("ih", FunctionType(INT, [REAL, INT]))
ig = Symbol("ig", FunctionType(INT, [INT]))
bf = Symbol("bf", FunctionType(BOOL, [BOOL]))
bg = Symbol("bg", FunctionType(BOOL, [BOOL]))
bv8 = Symbol("bv1", BV8)
bv16 = Symbol("bv2", BV16)
result = [
# Formula, is_valid, is_sat, is_qf
Example(hr="(x & y)",
expr=And(x, y),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
Example(hr="(x <-> y)",
expr=Iff(x, y),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
Example(hr="((x | y) & (! (x | y)))",
expr=And(Or(x, y), Not(Or(x, y))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BOOL),
Example(hr="(x & (! y))",
expr=And(x, Not(y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
Example(hr="(False -> True)",
expr=Implies(FALSE(), TRUE()),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
#
# LIA
#
Example(hr="((q < p) & (x -> y))",
expr=And(GT(p, q), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_IDL),
Example(hr="(((p + q) = 5) & (q < p))",
expr=And(Equals(Plus(p, q), Int(5)), GT(p, q)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="((q <= p) | (p <= q))",
expr=Or(GE(p, q), LE(p, q)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_IDL),
Example(hr="(! (p < (q * 2)))",
expr=Not(LT(p, Times(q, Int(2)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="(p < (p - (5 - 2)))",
expr=GT(Minus(p, Minus(Int(5), Int(2))), p),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_IDL),
Example(hr="((x ? 7 : ((p + -1) * 3)) = q)",
expr=Equals(
Ite(x, Int(7), Times(Plus(p, Int(-1)), Int(3))), q),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="(p < (q + 1))",
expr=LT(p, Plus(q, Int(1))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
#
# LRA
#
Example(hr="((s < r) & (x -> y))",
expr=And(GT(r, s), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_RDL),
Example(hr="(((r + s) = 28/5) & (s < r))",
expr=And(Equals(Plus(r, s), Real(Fraction("5.6"))),
GT(r, s)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="((s <= r) | (r <= s))",
expr=Or(GE(r, s), LE(r, s)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_RDL),
Example(hr="(! ((r * 2.0) < (s * 2.0)))",
expr=Not(LT(Div(r, Real((1, 2))), Times(s, Real(2)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="(! (r < (r - (5.0 - 2.0))))",
expr=Not(GT(Minus(r, Minus(Real(5), Real(2))), r)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_RDL),
Example(hr="((x ? 7.0 : ((s + -1.0) * 3.0)) = r)",
expr=Equals(
Ite(x, Real(7), Times(Plus(s, Real(-1)), Real(3))), r),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
#
# EUF
#
Example(hr="(bf(x) <-> bg(x))",
expr=Iff(Function(bf, (x, )), Function(bg, (x, ))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UF),
Example(hr="(rf(5.0, rg(r)) = 0.0)",
expr=Equals(Function(rf, (Real(5), Function(rg, (r, )))),
Real(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UFLRA),
Example(hr="((rg(r) = (5.0 + 2.0)) <-> (rg(r) = 7.0))",
expr=Iff(Equals(Function(rg, [r]), Plus(Real(5), Real(2))),
Equals(Function(rg, [r]), Real(7))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLRA),
Example(
hr="((r = (s + 1.0)) & (rg(s) = 5.0) & (rg((r - 1.0)) = 7.0))",
expr=And([
Equals(r, Plus(s, Real(1))),
Equals(Function(rg, [s]), Real(5)),
Equals(Function(rg, [Minus(r, Real(1))]), Real(7))
]),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_UFLRA),
#
# BV
#
Example(hr="((1_32 & 0_32) = 0_32)",
expr=Equals(BVAnd(BVOne(32), BVZero(32)), BVZero(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((! 2_3) = 5_3)",
expr=Equals(BVNot(BV("010")), BV("101")),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((7_3 xor 0_3) = 0_3)",
expr=Equals(BVXor(BV("111"), BV("000")), BV("000")),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((bv1::bv1) u< 0_16)",
expr=BVULT(BVConcat(bv8, bv8), BVZero(16)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(1_32[0:7] = 1_8)",
expr=Equals(BVExtract(BVOne(32), end=7), BVOne(8)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(0_8 u< (((bv1 + 1_8) * 5_8) u/ 5_8))",
expr=BVUGT(
BVUDiv(BVMul(BVAdd(bv8, BVOne(8)), BV(5, width=8)),
BV(5, width=8)), BVZero(8)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(0_16 u<= bv2)",
expr=BVUGE(bv16, BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(0_16 s<= bv2)",
expr=BVSGE(bv16, BVZero(16)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(
hr="((0_32 u< (5_32 u% 2_32)) & ((5_32 u% 2_32) u<= 1_32))",
expr=And(
BVUGT(BVURem(BV(5, width=32), BV(2, width=32)),
BVZero(32)),
BVULE(BVURem(BV(5, width=32), BV(2, width=32)),
BVOne(32))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((((1_32 + (- 1_32)) << 1_32) >> 1_32) = 1_32)",
expr=Equals(
BVLShr(BVLShl(BVAdd(BVOne(32), BVNeg(BVOne(32))), 1),
1), BVOne(32)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((1_32 - 1_32) = 0_32)",
expr=Equals(BVSub(BVOne(32), BVOne(32)), BVZero(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# Rotations
Example(hr="(((1_32 ROL 1) ROR 1) = 1_32)",
expr=Equals(BVRor(BVRol(BVOne(32), 1), 1), BVOne(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# Extensions
Example(hr="((0_5 ZEXT 11) = (0_1 SEXT 15))",
expr=Equals(BVZExt(BVZero(5), 11), BVSExt(BVZero(1), 15)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 - bv2) = 0_16)",
expr=Equals(BVSub(bv16, bv16), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 - bv2)[0:7] = bv1)",
expr=Equals(BVExtract(BVSub(bv16, bv16), 0, 7), bv8),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2[0:7] bvcomp bv1) = 1_1)",
expr=Equals(BVComp(BVExtract(bv16, 0, 7), bv8), BVOne(1)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 bvcomp bv2) = 0_1)",
expr=Equals(BVComp(bv16, bv16), BVZero(1)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 s< bv2)",
expr=BVSLT(bv16, bv16),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 s< 0_16)",
expr=BVSLT(bv16, BVZero(16)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 s< 0_16) | (0_16 s<= bv2))",
expr=Or(BVSGT(BVZero(16), bv16), BVSGE(bv16, BVZero(16))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 u< bv2)",
expr=BVULT(bv16, bv16),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 u< 0_16)",
expr=BVULT(bv16, BVZero(16)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 | 0_16) = bv2)",
expr=Equals(BVOr(bv16, BVZero(16)), bv16),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 & 0_16) = 0_16)",
expr=Equals(BVAnd(bv16, BVZero(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((0_16 s< bv2) & ((bv2 s/ 65535_16) s< 0_16))",
expr=And(BVSLT(BVZero(16), bv16),
BVSLT(BVSDiv(bv16, SBV(-1, 16)), BVZero(16))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((0_16 s< bv2) & ((bv2 s% 1_16) s< 0_16))",
expr=And(BVSLT(BVZero(16), bv16),
BVSLT(BVSRem(bv16, BVOne(16)), BVZero(16))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 u% 1_16) = 0_16)",
expr=Equals(BVURem(bv16, BVOne(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 s% 1_16) = 0_16)",
expr=Equals(BVSRem(bv16, BVOne(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 s% (- 1_16)) = 0_16)",
expr=Equals(BVSRem(bv16, BVNeg(BVOne(16))), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 a>> 0_16) = bv2)",
expr=Equals(BVAShr(bv16, BVZero(16)), bv16),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((0_16 s<= bv2) & ((bv2 a>> 1_16) = (bv2 >> 1_16)))",
expr=And(
BVSLE(BVZero(16), bv16),
Equals(BVAShr(bv16, BVOne(16)),
BVLShr(bv16, BVOne(16)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
#
# Quantification
#
Example(hr="(forall y . (x -> y))",
expr=ForAll([y], Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.BOOL),
Example(hr="(forall p, q . ((p + q) = 0))",
expr=ForAll([p, q], Equals(Plus(p, q), Int(0))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LIA),
Example(
hr="(forall r, s . (((0.0 < r) & (0.0 < s)) -> ((r - s) < r)))",
expr=ForAll([r, s],
Implies(And(GT(r, Real(0)), GT(s, Real(0))),
(LT(Minus(r, s), r)))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LRA),
Example(hr="(exists x, y . (x -> y))",
expr=Exists([x, y], Implies(x, y)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.BOOL),
Example(hr="(exists p, q . ((p + q) = 0))",
expr=Exists([p, q], Equals(Plus(p, q), Int(0))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LIA),
Example(hr="(exists r . (forall s . (r < (r - s))))",
expr=Exists([r], ForAll([s], GT(Minus(r, s), r))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LRA),
Example(hr="(forall r . (exists s . (r < (r - s))))",
expr=ForAll([r], Exists([s], GT(Minus(r, s), r))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LRA),
Example(hr="(x & (forall r . ((r + s) = 5.0)))",
expr=And(x, ForAll([r], Equals(Plus(r, s), Real(5)))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LRA),
Example(hr="(exists x . ((x <-> (5.0 < s)) & (s < 3.0)))",
expr=Exists([x],
(And(Iff(x, GT(s, Real(5))), LT(s, Real(3))))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.LRA),
#
# UFLIRA
#
Example(hr="((p < ih(r, q)) & (x -> y))",
expr=And(GT(Function(ih, (r, q)), p), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"(((p - 3) = q) -> ((p < ih(r, (q + 3))) | (ih(r, p) <= p)))",
expr=Implies(
Equals(Minus(p, Int(3)), q),
Or(GT(Function(ih, (r, Plus(q, Int(3)))), p),
LE(Function(ih, (r, p)), p))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"(((ToReal((p - 3)) = r) & (ToReal(q) = r)) -> ((p < ih(ToReal((p - 3)), (q + 3))) | (ih(r, p) <= p)))",
expr=Implies(
And(Equals(ToReal(Minus(p, Int(3))), r),
Equals(ToReal(q), r)),
Or(
GT(
Function(
ih,
(ToReal(Minus(p, Int(3))), Plus(q, Int(3)))),
p), LE(Function(ih, (r, p)), p))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"(! (((ToReal((p - 3)) = r) & (ToReal(q) = r)) -> ((p < ih(ToReal((p - 3)), (q + 3))) | (ih(r, p) <= p))))",
expr=Not(
Implies(
And(Equals(ToReal(Minus(p, Int(3))), r),
Equals(ToReal(q), r)),
Or(
GT(
Function(ih, (ToReal(Minus(
p, Int(3))), Plus(q, Int(3)))), p),
LE(Function(ih, (r, p)), p)))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"""("Did you know that any string works? #yolo" & "10" & "|#somesolverskeepthe||" & " ")""",
expr=And(Symbol("Did you know that any string works? #yolo"),
Symbol("10"), Symbol("|#somesolverskeepthe||"),
Symbol(" ")),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
#
# Arrays
#
Example(hr="((q = 0) -> (aii[0 := 0] = aii[0 := q]))",
expr=Implies(
Equals(q, Int(0)),
Equals(Store(aii, Int(0), Int(0)),
Store(aii, Int(0), q))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_ALIA),
Example(hr="(aii[0 := 0][0] = 0)",
expr=Equals(Select(Store(aii, Int(0), Int(0)), Int(0)),
Int(0)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_ALIA),
Example(hr="((Array{Int, Int}(0)[1 := 1] = aii) & (aii[1] = 0))",
expr=And(Equals(Array(INT, Int(0), {Int(1): Int(1)}), aii),
Equals(Select(aii, Int(1)), Int(0))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_ALIA*")),
Example(hr="((Array{Int, Int}(0)[1 := 3] = aii) & (aii[1] = 3))",
expr=And(Equals(Array(INT, Int(0), {Int(1): Int(3)}), aii),
Equals(Select(aii, Int(1)), Int(3))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.get_logic_by_name("QF_ALIA*")),
Example(hr="((Array{Real, Int}(10) = ari) & (ari[6/5] = 0))",
expr=And(Equals(Array(REAL, Int(10)), ari),
Equals(Select(ari, Real((6, 5))), Int(0))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(
hr=
"((Array{Real, Int}(0)[1.0 := 10][2.0 := 20][3.0 := 30][4.0 := 40] = ari) & (! ((ari[0.0] = 0) & (ari[1.0] = 10) & (ari[2.0] = 20) & (ari[3.0] = 30) & (ari[4.0] = 40))))",
expr=And(
Equals(
Array(
REAL, Int(0), {
Real(1): Int(10),
Real(2): Int(20),
Real(3): Int(30),
Real(4): Int(40)
}), ari),
Not(
And(Equals(Select(ari, Real(0)), Int(0)),
Equals(Select(ari, Real(1)), Int(10)),
Equals(Select(ari, Real(2)), Int(20)),
Equals(Select(ari, Real(3)), Int(30)),
Equals(Select(ari, Real(4)), Int(40))))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(
hr=
"((Array{Real, Int}(0)[1.0 := 10][2.0 := 20][3.0 := 30][4.0 := 40][5.0 := 50] = ari) & (! ((ari[0.0] = 0) & (ari[1.0] = 10) & (ari[2.0] = 20) & (ari[3.0] = 30) & (ari[4.0] = 40) & (ari[5.0] = 50))))",
expr=And(
Equals(
Array(
REAL, Int(0), {
Real(1): Int(10),
Real(2): Int(20),
Real(3): Int(30),
Real(4): Int(40),
Real(5): Int(50)
}), ari),
Not(
And(Equals(Select(ari, Real(0)), Int(0)),
Equals(Select(ari, Real(1)), Int(10)),
Equals(Select(ari, Real(2)), Int(20)),
Equals(Select(ari, Real(3)), Int(30)),
Equals(Select(ari, Real(4)), Int(40)),
Equals(Select(ari, Real(5)), Int(50))))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(
hr=
"((a_arb_aii = Array{Array{Real, BV{8}}, Array{Int, Int}}(Array{Int, Int}(7))) -> (a_arb_aii[arb][42] = 7))",
expr=Implies(
Equals(nested_a,
Array(ArrayType(REAL, BV8), Array(INT, Int(7)))),
Equals(Select(Select(nested_a, arb), Int(42)), Int(7))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(hr="(abb[bv1 := y_][bv1 := z_] = abb[bv1 := z_])",
expr=Equals(
Store(Store(abb, bv8, Symbol("y_", BV8)), bv8,
Symbol("z_", BV8)),
Store(abb, bv8, Symbol("z_", BV8))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_ABV),
Example(hr="((r / s) = (r * s))",
expr=Equals(Div(r, s), Times(r, s)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="(2.0 = (r * r))",
expr=Equals(Real(2), Times(r, r)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="((p ^ 2) = 0)",
expr=Equals(Pow(p, Int(2)), Int(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NIA),
Example(hr="((r ^ 2.0) = 0.0)",
expr=Equals(Pow(r, Real(2)), Real(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="((r * r * r) = 25.0)",
expr=Equals(Times(r, r, r), Real(25)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="((5.0 * r * 5.0) = 25.0)",
expr=Equals(Times(Real(5), r, Real(5)), Real(25)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="((p * p * p) = 25)",
expr=Equals(Times(p, p, p), Int(25)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_NIA),
Example(hr="((5 * p * 5) = 25)",
expr=Equals(Times(Int(5), p, Int(5)), Int(25)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="(((1 - 1) * p * 1) = 0)",
expr=Equals(Times(Minus(Int(1), Int(1)), p, Int(1)),
Int(0)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_LIA),
# Huge Fractions:
Example(
hr=
"((r * 1606938044258990275541962092341162602522202993782792835301376/7) = -20480000000000000000000000.0)",
expr=Equals(Times(r, Real(Fraction(2**200, 7))),
Real(-200**11)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="(((r + 5.0 + s) * (s + 2.0 + r)) = 0.0)",
expr=Equals(
Times(Plus(r, Real(5), s), Plus(s, Real(2), r)),
Real(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(
hr=
"(((p + 5 + q) * (p - (q - 5))) = ((p * p) + (10 * p) + 25 + (-1 * q * q)))",
expr=Equals(
Times(Plus(p, Int(5), q), Minus(p, Minus(q, Int(5)))),
Plus(Times(p, p), Times(Int(10), p), Int(25),
Times(Int(-1), q, q))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_NIA),
]
return result
def test_prenex_negated_forall(self):
a, b = (Symbol(x) for x in "ab")
f = Implies(ForAll([b], Implies(a, b)), b)
prenex = prenex_normal_form(f)
self.assertTrue(prenex.is_exists())
self.assertValid(Iff(f, prenex), logic=BOOL)