def testMultZeroRight(self):
"""Proof of n * 0 = 0 by induction."""
thy = basic.load_theory('nat')
n = Var("n", nat.natT)
state = ProofState.init_state(thy, [n], [], Term.mk_equals(nat.times(n, nat.zero), nat.zero))
state.apply_induction(0, "nat_induct", "n")
state.rewrite_goal(0, "times_def_1")
state.introduction(1, names=["n"])
state.rewrite_goal((1, 2), "times_def_2")
state.rewrite_goal((1, 2), "plus_def_1")
self.assertEqual(state.check_proof(no_gaps=True), Thm.mk_equals(nat.times(n, nat.zero), nat.zero))
def testMultZeroRight(self):
"""Proof of n * 0 = 0 by induction."""
thy = basic.load_theory('nat')
n = Var("n", nat.natT)
eq = Term.mk_equals
prf = Proof()
prf.add_item(0, "theorem", args="nat_induct")
prf.add_item(1,
"substitution",
args={
"P": Term.mk_abs(n,
eq(nat.times(n, nat.zero),
nat.zero)),
"x": n
},
prevs=[0])
prf.add_item(2, "beta_norm", prevs=[1])
prf.add_item(3, "theorem", args="times_def_1")
prf.add_item(4, "substitution", args={"n": nat.zero}, prevs=[3])
prf.add_item(5, "implies_elim", prevs=[2, 4])
prf.add_item(6, "assume", args=eq(nat.times(n, nat.zero), nat.zero))
prf.add_item(7, "theorem", args="times_def_2")
prf.add_item(8,
"substitution",
args={
"m": n,
"n": nat.zero
},
prevs=[7])
prf.add_item(9, "theorem", args="plus_def_1")
prf.add_item(10,
"substitution",
args={"n": nat.times(n, nat.zero)},
prevs=[9])
prf.add_item(11, "transitive", prevs=[8, 10])
prf.add_item(12, "transitive", prevs=[11, 6])
prf.add_item(13,
"implies_intr",
args=eq(nat.times(n, nat.zero), nat.zero),
prevs=[12])
prf.add_item(14, "forall_intr", args=n, prevs=[13])
prf.add_item(15, "implies_elim", prevs=[5, 14])
th = Thm.mk_equals(nat.times(n, nat.zero), nat.zero)
self.assertEqual(thy.check_proof(prf), th)
def testPrintUnicode(self):
test_data = [
(conj(A, B), "A ∧ B"),
(disj(A, B), "A ∨ B"),
(imp(A, B), "A ⟶ B"),
(abs(a, P(a)), "λa. P a"),
(all(a, P(a)), "∀a. P a"),
(exists(a, P(a)), "∃a. P a"),
(neg(A), "¬A"),
(nat.plus(m, n), "m + n"),
(nat.times(m, n), "m * n"),
]
for t, s in test_data:
self.assertEqual(printer.print_term(thy, t, unicode=True), s)
def testPrintArithmetic(self):
test_data = [
(nat.plus(m, n), "m + n"),
(nat.plus(nat.plus(m, n), p), "m + n + p"),
(nat.plus(m, nat.plus(n, p)), "m + (n + p)"),
(nat.times(m, n), "m * n"),
(nat.times(nat.times(m, n), p), "m * n * p"),
(nat.times(m, nat.times(n, p)), "m * (n * p)"),
(nat.plus(m, nat.times(n, p)), "m + n * p"),
(nat.times(m, nat.plus(n, p)), "m * (n + p)"),
(nat.zero, "0"),
(nat.plus(nat.zero, nat.zero), "0 + 0"),
(nat.times(m, nat.zero), "m * 0"),
]
for t, s in test_data:
self.assertEqual(printer.print_term(thy, t), s)
def times(self, lhs, rhs):
return nat.times(lhs, rhs)
def times_expr(self, e1, e2):
return nat.times(e1, e2)