def test_case_2():
# This test case should return True
a = Variable("a")
b = Variable("b")
assumptions = [a, b]
proof = [a, b, Implies(a, Implies(b, a))]
test = Proof(assumptions, proof)
return test.verify()
def test_case_8():
# This test case should return False
a = Variable("a")
b = Variable("b")
assumptions = [a]
proof = [Implies(Implies(Not(a), Not(b)), Implies(b, a)), Implies(b, a)]
test = Proof(assumptions, proof)
return test.verify()
def test_case_5():
# This test case should return True
a = Variable("a")
b = Variable("b")
c = Variable("c")
assumptions = [a, Implies(a, b), Implies(b, c)]
proof = [b, c]
test = Proof(assumptions, proof)
return test.verify()
def test_case_12():
# This test case should return True
a = Variable("a")
b = Variable("b")
c = Variable("c")
assumptions = [Implies(a, b), Implies(b, c)]
proof = prove_implication_transitivity(a, b, c)
test = Proof(assumptions, proof)
return test.verify()
def test_case_1():
# This test case should return False
a = Variable("a")
b = Variable("b")
test = Proof([a, b], [a, b, Implies(a, b)])
return test.verify()
def test_case_11():
# This test case should return True
a = Variable("a")
b = Variable("b")
c = Variable("c")
d = Variable("d")
assumptions = [a, Implies(a, b), Implies(b, c)]
proof = [
# MP
b,
# MP
c,
# Axiom 1
Implies(c, Implies(Not(d), c)),
# MP
Implies(Not(d), c)
]
test = Proof(assumptions, proof)
return test.verify()
def test_case_4():
# This test case should return True
a = Variable("a")
na = Not(a)
nna = Not(na)
nnna = Not(nna)
nnnna = Not(nnna)
assumptions = [nna]
proof = [
# Axiom 1
Implies(nna, Implies(nnnna, nna)),
# Modus ponens on the last step and the assumption
Implies(nnnna, nna),
# Axiom 3
Implies(Implies(nnnna, nna), Implies(na, nnna)),
# Modus ponens on the last 2 steps
Implies(na, nnna),
# Axiom 3
Implies(Implies(na, nnna), Implies(nna, a)),
# Modus ponens on the last 2 steps
Implies(nna, a),
# Modus ponens on the last step and the assumption
a
]
test = Proof(assumptions, proof)
return test.verify()
def test_case_3():
# This test case should return True
# in this test case we prove a -> a
a = Variable("a")
assumptions = []
proof = [
# Axiom 2 with a -> a instead of q
Implies(Implies(a, Implies(Implies(a, a), a)),
Implies(Implies(a, Implies(a, a)), Implies(a, a))),
# Axiom 1 with a -> a instead of q
Implies(a, Implies(Implies(a, a), a)),
# Modus ponens of the first 2 axioms
Implies(Implies(a, Implies(a, a)), Implies(a, a)),
# Axiom 1
Implies(a, Implies(a, a)),
# Modus ponens
Implies(a, a)
]
test = Proof(assumptions, proof)
return test.verify()
def test_case_13():
# This test case should return True
a = Variable("a")
aa = Implies(a, a)
a_aa = Implies(a, aa)
n_a_aa = Not(a_aa)
nn_a_aa = Not(n_a_aa)
na = Not(a)
nna = Not(na)
assumptions = [Implies(a_aa, Implies(Implies(a_aa, a), a))]
proof = [
# Axiom 1
a_aa,
# Axiom 3
Implies(Implies(nn_a_aa, nna), Implies(na, n_a_aa)),
# Axiom 3
Implies(Implies(na, n_a_aa), Implies(a_aa, a)),
# Combine last 2 steps
*prove_implication_transitivity(Implies(nn_a_aa, nna),
Implies(na, n_a_aa), Implies(a_aa, a)),
# It is now proven that:
Implies(Implies(nn_a_aa, nna), Implies(a_aa, a)),
# Axiom 1
Implies(nna, Implies(nn_a_aa, nna)),
# Combine last 2 steps
*prove_implication_transitivity(nna, Implies(nn_a_aa, nna),
Implies(a_aa, a)),
# It is now proven that
Implies(nna, Implies(a_aa, a)),
# Modus ponens for the assumption and the first formula
Implies(Implies(a_aa, a), a),
# Combine the last step with proof[6]
*prove_implication_transitivity(nna, Implies(a_aa, a), a),
# We have proven that
Implies(nna, a)
]
test = Proof(assumptions, proof)
return test.verify()
def prove_implication_transitivity(a, b, c):
ab = Implies(a, b)
bc = Implies(b, c)
ac = Implies(a, c)
ax2_q = Implies(a, bc)
ax2_r = Implies(ab, ac)
return [
# Axiom 1
Implies(Implies(Implies(a, bc), Implies(ab, ac)),
Implies(bc, Implies(Implies(a, bc), Implies(ab, ac)))),
# Axiom 2
Implies(Implies(a, bc), Implies(ab, ac)),
# Modus ponens
Implies(bc, Implies(Implies(a, bc), Implies(ab, ac))),
# Axiom 2
Implies(Implies(bc, Implies(ax2_q, ax2_r)),
Implies(Implies(bc, ax2_q), Implies(bc, ax2_r))),
# Modus ponens
Implies(Implies(bc, ax2_q), Implies(bc, ax2_r)),
# Axiom 1
Implies(bc, Implies(a, bc)),
# Modus ponens
Implies(bc, Implies(ab, ac)),
# Unfold this formula with modus ponens to get a -> c
Implies(ab, ac),
ac
]
def test_case_10():
# This test case should return True
a = Variable("a")
b = Variable("b")
c = Variable("c")
na = Not(a)
nb = Not(b)
assumptions = [
# Proof by contradiction
Implies(Implies(a, b), Implies(Implies(a, nb), na)),
# Proof by contradiction - deduction theorem
Implies(a, b),
Implies(a, nb),
# Implication of inconsistency
Implies(na, Implies(a, c))
]
proof = [Implies(Implies(a, nb), na), na, Implies(a, c)]
test = Proof(assumptions, proof)
return test.verify()
def test_case_9():
# This test case should return True
a = Variable("a")
b = Variable("b")
c = Variable("c")
na = Not(a)
nna = Not(na)
ab = Implies(a, b)
bc = Implies(b, c)
assumptions = [a, Implies(a, nna)]
proof = [
nna,
# Axiom 1
Implies(nna, Implies(Not(Implies(ab, Not(bc))), nna)),
# MP
Implies(Not(Implies(ab, Not(bc))), nna),
# Axiom 3
Implies(Implies(Not(Implies(ab, Not(bc))), nna),
Implies(na, Implies(ab, Not(bc)))),
# MP
Implies(na, Implies(ab, Not(bc))),
# Axiom 2
Implies(Implies(na, Implies(ab, Not(bc))),
Implies(Implies(na, ab), Implies(na, Not(bc)))),
# MP
Implies(Implies(na, ab), Implies(na, Not(bc)))
]
test = Proof(assumptions, proof)
return test.verify()