def testAuto(self):
test_data = [
# No conditions
("sqrt 2 >= 1"),
("~((1::real) = 2)"),
("(4::nat) > 2"),
# Interval condition
("x Mem real_closed_interval 0 1 --> 1 - x ^ (2::nat) >= 0"),
("x Mem real_open_interval 0 1 --> 1 - x ^ (2::nat) > 0"),
("x Mem real_closed_interval 0 (sqrt 2) --> 2 - x ^ (2::nat) >= 0"
),
("x Mem real_closed_interval 0 (pi / 2) --> sqrt 2 * cos x >= 0"),
("x Mem real_closed_interval 0 (2 ^ (1 / 2)) --> 2 - x ^ (2::nat) >= 0"
),
]
vars = {'x': 'real'}
for expr in test_data:
test_macro(self,
'interval_arith',
'auto',
vars=vars,
args=expr,
res=expr)
def testRealContinuousOn(self):
test_data = [
"real_continuous_on (%x. x) (real_closed_interval 0 1)",
"real_continuous_on (%x. x * x) (real_closed_interval 0 1)",
"real_continuous_on (%x. -x) (real_closed_interval 0 1)",
"real_continuous_on (%x. x ^ (2::nat)) (real_closed_interval 0 1)",
"real_continuous_on (%x. x ^ (3::nat)) (real_closed_interval 0 1)",
"real_continuous_on (%x. (x + 1) ^ (3::nat)) (real_closed_interval 0 1)",
"real_continuous_on (%x. exp x) (real_closed_interval 0 1)",
"real_continuous_on (%x. exp (x ^ (2::nat))) (real_closed_interval 0 1)",
"real_continuous_on (%x. exp (exp x)) (real_closed_interval 0 1)",
"real_continuous_on (%x. sin x) (real_closed_interval 0 1)",
"real_continuous_on (%x. cos x) (real_closed_interval 0 1)",
"real_continuous_on (%x. sin x * cos x) (real_closed_interval 0 1)",
"real_continuous_on (%x. sin (cos x)) (real_closed_interval 0 1)",
"real_continuous_on (%x. 1 / x) (real_closed_interval 1 2)",
"real_continuous_on (%x. 1 / (x ^ (2::nat))) (real_closed_interval 1 2)",
"real_continuous_on (%x. 1 / (x ^ (2::nat))) (real_closed_interval (-2) (-1))",
"real_continuous_on (%x. 1 / (x ^ (2::nat) + 1)) (real_closed_interval (-1) 1)",
"real_continuous_on (%x. abs x) (real_closed_interval (-1) 1)",
"real_continuous_on (%x. log x) (real_closed_interval (exp (-1)) (exp 1))",
"real_continuous_on (%x. log (x ^ (2::nat) + 1)) (real_closed_interval (-1) 1)",
"real_continuous_on (%x. sqrt x) (real_closed_interval 0 1)",
"real_continuous_on (%x. sqrt (1 - x ^ (2::nat))) (real_closed_interval 0 1)",
"real_continuous_on (%x. sqrt (2 - x ^ (2::nat))) (real_closed_interval 0 (sqrt 2))",
"real_continuous_on (%x. x ^ (-(2::real))) (real_closed_interval 1 2)",
"real_continuous_on (%x. (3 * x + 1) ^ (-(2::real))) (real_closed_interval 0 1)",
"real_continuous_on (%x. x ^ (1 / 2)) (real_closed_interval 0 1)",
"real_continuous_on (%x::real. 2 ^ x) (real_closed_interval 0 1)",
"real_continuous_on (%x. (1 + -(x ^ (2::nat))) ^ -(1 / 2)) (real_open_interval (1 / 2 * 2 ^ (1 / 2)) 1)",
]
for expr in test_data:
test_macro(self, 'interval_arith', 'auto', args=expr, res=expr)
def testRealIntegrableOn(self):
test_data = [
"real_integrable_on (%x. x) (real_closed_interval 0 1)",
"real_integrable_on (%x. sqrt x) (real_closed_interval 0 1)",
]
for expr in test_data:
test_macro(self, 'interval_arith', 'auto', args=expr, res=expr)
def testNatLessMacro(self):
test_data = [
(0, 1),
(3, 5),
]
for m, n in test_data:
goal = "(%d::nat) < %d" % (m, n)
test_macro(self, 'nat', 'nat_const_less', args=goal, res=goal)
def testNatLessEqMacro(self):
test_data = [
(3, 5),
(4, 4),
]
for m, n in test_data:
goal = "(%d::nat) <= %d" % (m, n)
test_macro(self, 'nat', 'nat_const_less_eq', args=goal, res=goal)
def testRealNormMacroFailed(self):
test_data = [
("x * y * z = x * y + z"),
("1 + x + y = x + y + 2"),
]
vars = {'x': 'real', 'y': 'real', 'z': 'real'}
for expr in test_data:
test_macro(self, 'real', 'real_norm', vars=vars, args=expr, failed=AssertionError, eval_only=True)
def testNatNormMacro(self):
test_data = [
("x * (y * z) = y * (z * x)"),
("Suc (Suc (Suc x)) + y = x + Suc (Suc (Suc y))"),
("x + y + (y + z) = y * 2 + (x + z)"),
]
vars = {"x": 'nat', "y": 'nat', "z": 'nat'}
for expr in test_data:
test_macro(self, 'nat', 'nat_norm', vars=vars, args=expr, res=expr)
def testRealNormMacro(self):
test_data = [
("x * (y * z) = y * (z * x)"),
("-x = -1 * x"),
("x - y = x + -1 * y"),
]
vars = {'x': 'real', 'y': 'real', 'z': 'real'}
for expr in test_data:
test_macro(self, 'real', 'real_norm', vars=vars, args=expr, res=expr, eval_only=True)
def testSolve(self):
test_data = [("A --> B --> A & B"), ("A & B --> B & A"),
("A | B --> B | A"), ("A & B & C --> (A & B) & C"),
("A | B | C --> (A | B) | C")]
vars = {'A': 'bool', 'B': 'bool'}
for expr in test_data:
test_macro(self,
'logic_base',
'auto',
vars=vars,
args=expr,
res=expr)
def testRealIncreasing(self):
test_data = [
"real_derivative (%x. x) x >= 0",
"real_derivative (%x. 3 * x + 1) x >= 0",
]
for expr in test_data:
test_macro(self,
'interval_arith',
'auto',
vars={'x': 'real'},
args=expr,
res=expr)
def testIneq(self):
test_data = [
("x Mem real_open_interval 0 (pi / 2) --> sin x > 0"),
("x Mem real_open_interval 0 (pi / 2) --> exp(-x) + sin x > 0"),
("x Mem real_open_interval 0 (pi / 2) --> ~(exp(-x) + sin x = 0)"),
]
for expr in test_data:
test_macro(self,
'interval_arith',
'auto',
vars={'x': 'real'},
args=expr,
res=expr)
def testZ3MacroFail(self):
if not z3wrapper.z3_loaded:
return
test_macro(self,
'logic_base',
'z3',
vars={
'P': 'bool',
'Q': 'bool'
},
args='P --> Q',
eval_only=True,
failed=AssertionError)
def testRealContinuousOnFail(self):
test_data = [
"real_continuous_on (%x. 1 / x) (real_closed_interval (-1) 1)",
"real_continuous_on (%x. 1 / x) (real_closed_interval 0 1)",
"real_continuous_on (%x. log x) (real_closed_interval 0 1)",
"real_continuous_on (%x. sqrt x) (real_closed_interval (-1) 1)",
"real_continuous_on (%x. sqrt (1 - x ^ (2::nat))) (real_closed_interval 0 (sqrt 2))",
"real_continuous_on (%x. sqrt (2 - x ^ (2::nat))) (real_closed_interval 0 2)",
]
for expr in test_data:
test_macro(self,
'interval_arith',
'auto',
args=expr,
failed=TacticException)
def testZ3Macro(self):
if not z3wrapper.z3_loaded:
return
test_macro(
self,
'real',
'z3',
vars={
'S': 'real set',
'T': 'real set',
'x': 'real'
},
assms=['S Int T = empty_set'],
args=
'(if x Mem S then (1::real) else 0) + (if x Mem T then 1 else 0) = (if x Mem (S Un T) then 1 else 0)',
res=
'(if x Mem S then (1::real) else 0) + (if x Mem T then 1 else 0) = (if x Mem (S Un T) then 1 else 0)',
eval_only=True)
def testNatIneqMacro(self):
test_data = [
(0, 1),
(1, 0),
(0, 2),
(2, 0),
(1, 2),
(2, 1),
(1, 3),
(3, 1),
(2, 3),
(3, 2),
(10, 13),
(17, 19),
(22, 24),
]
for m, n in test_data:
goal = "~(%d::nat) = %d" % (m, n)
test_macro(self, 'nat', 'nat_const_ineq', args=goal, res=goal)
def testRealDifferentiable(self):
test_data = [
"real_differentiable (%x. x) (atreal x)",
"real_differentiable (%x. x * x) (atreal x)",
"real_differentiable (%x. -x) (atreal x)",
"real_differentiable (%x. x ^ (2::nat)) (atreal x)",
"real_differentiable (%x. x ^ (3::nat)) (atreal x)",
"real_differentiable (%x. (x + 1) ^ (3::nat)) (atreal x)",
"real_differentiable (%x. exp x) (atreal x)",
"real_differentiable (%x. exp (x ^ (2::nat))) (atreal x)",
"real_differentiable (%x. exp (exp x)) (atreal x)",
"real_differentiable (%x. sin x) (atreal x)",
"real_differentiable (%x. cos x) (atreal x)",
"real_differentiable (%x. sin x * cos x) (atreal x)",
"real_differentiable (%x. sin (cos x)) (atreal x)",
"x Mem real_open_interval 0 1 --> real_differentiable (%x. 1 / x) (atreal x)",
"x Mem real_open_interval 0 1 --> real_differentiable (%x. 1 / (x ^ (2::nat))) (atreal x)",
"x Mem real_open_interval (-1) 0 --> real_differentiable (%x. 1 / (x ^ (2::nat))) (atreal x)",
"x Mem real_open_interval (-1) 1 --> real_differentiable (%x. 1 / (x ^ (2::nat) + 1)) (atreal x)",
"x Mem real_open_interval 0 (exp 1) --> real_differentiable (%x. log x) (atreal x)",
"x Mem real_open_interval (-1) 1 --> real_differentiable (%x. log (x ^ (2::nat) + 1)) (atreal x)",
"x Mem real_open_interval 0 1 --> real_differentiable (%x. sqrt x) (atreal x)",
"x Mem real_open_interval 0 1 --> real_differentiable (%x. sqrt (1 - x ^ (2::nat))) (atreal x)",
"x Mem real_open_interval 0 (sqrt 2) --> real_differentiable (%x. sqrt (2 - x ^ (2::nat))) (atreal x)",
"x Mem real_open_interval 0 1 --> real_differentiable (%x. x ^ (-(2::real))) (atreal x)",
"x Mem real_open_interval 0 1 --> real_differentiable (%x. (3 * x + 1) ^ (-(2::real))) (atreal x)",
"x Mem real_open_interval 0 1 --> real_differentiable (%x. x ^ (1 / 2)) (atreal x)",
"x Mem real_open_interval (-1) 1 --> real_differentiable (%x::real. 2 ^ x) (atreal x)",
]
for expr in test_data:
test_macro(self,
'interval_arith',
'auto',
vars={'x': 'real'},
args=expr,
res=expr)