def test_remove_assumption(debug=False):
from propositions.some_proofs import prove_and_commutativity
for oldp in [
DISJUNCTION_COMMUTATIVITY_PROOF,
prove_and_commutativity(), DISJUNCTION_RIGHT_ASSOCIATIVITY_PROOF,
DISJUNCTION_ROTATION_PROOF, TRI_AND_PROOF
]:
p = oldp
assert p.is_valid(), offending_line(p)
while (True):
rb = None
for r in p.rules:
if r != MP and len(r.assumptions) > 0:
rb = r
break
if rb is None:
break
pr = prove_from_encoding(rb)
p = inline_proof(p, pr)
assert p.is_valid(), offending_line(p)
if debug:
print("Testing remove_assumption on:", p)
pp = remove_assumption(p)
if debug:
print("Got:", pp)
assert pp.statement.assumptions == p.statement.assumptions[:-1]
assert pp.statement.conclusion == Formula('->',
p.statement.assumptions[-1],
p.statement.conclusion)
assert pp.rules.issubset(p.rules.union({MP, I0, I1, D}))
assert pp.is_valid(), offending_line(pp)
def test_prove_from_opposites(debug=False):
assumptions = (Formula.parse('(~~p->~~q)'), Formula.parse('p'),
Formula.parse('~q'))
rules = {MP, I1, N, NI}
pf = Proof(InferenceRule(assumptions, Formula.parse('(q->p)')), rules, [
Proof.Line(Formula.parse('p')),
Proof.Line(Formula.parse('(p->(q->p))'), I1, []),
Proof.Line(Formula.parse('(q->p)'), MP, [0, 1])
])
pnf = Proof(InferenceRule(assumptions, Formula.parse('~(q->p)')), rules, [
Proof.Line(Formula.parse('(~~p->~~q)')),
Proof.Line(Formula.parse('((~~p->~~q)->(~q->~p))'), N, []),
Proof.Line(Formula.parse('(~q->~p)'), MP, [0, 1]),
Proof.Line(Formula.parse('((~q->~p)->(p->q))'), N, []),
Proof.Line(Formula.parse('(p->q)'), MP, [2, 3]),
Proof.Line(Formula.parse('p')),
Proof.Line(Formula.parse('q'), MP, [5, 4]),
Proof.Line(Formula.parse('~q')),
Proof.Line(Formula.parse('~p'), MP, [7, 2]),
Proof.Line(Formula.parse('(q->(~p->~(q->p)))'), NI, []),
Proof.Line(Formula.parse('(~p->~(q->p))'), MP, [6, 9]),
Proof.Line(Formula.parse('~(q->p)'), MP, [8, 10])
])
g = Formula.parse('(p->r)')
if debug:
print("Testing prove_from_opposites with assumptions", assumptions,
"and conclusion", g)
p = prove_from_opposites(pf, pnf, g)
assert p.statement.conclusion == g
assert p.statement.assumptions == assumptions
assert p.rules == rules.union({I2})
assert p.is_valid(), offending_line(p)
def __test_prove_inference(prover, rule, rules, debug):
return test_inference_proof(prover, rule, rules, debug) # My own util
if debug:
print('Testing', prover.__qualname__)
proof = prover()
if proof.statement != rule:
print('*' * 25)
print('Error: proof.statement != rule')
print('proof.statement: ', proof.statement)
print('rule: ', rule)
print('*' * 25)
assert proof.statement == rule
assert proof.rules.issubset(rules), \
"got " + str(proof.rules) + ", expected " + str(rules)
if not proof.is_valid() or not (proof):
print('=' * 35)
print("INVALID Proof:")
print(proof)
invalid_lines = []
for line_number in range(0, len(proof.lines)):
if not proof.is_line_valid(line_number):
invalid_lines.append(line_number)
if len(invalid_lines) > 0:
print('')
print('=' * 35)
sys.exit(1)
assert proof.is_valid(), offending_line(proof)
print('=' * 35)
print("VALID Proof:")
print(proof)
print('=' * 35)
def __test_prove_inference(prover, rule, rules, debug):
if debug:
print('Testing', prover.__qualname__)
proof = prover()
assert proof.statement == rule
assert proof.rules.issubset(rules), \
"got " + str(proof.rules) + ", expected " + str(rules)
assert proof.is_valid(), offending_line(proof)
def test_prove_tautology(debug=False):
for f, m in [('(p->p)', {
'p': True
}), ('(p->p)', {
'p': False
}), ('(p->p)', {}), ('((~q->~p)->(p->q))', {
'p': True,
'q': False
}), ('((~q->~p)->(p->q))', {
'p': False
}), ('((~q->~p)->(p->q))', {})]:
f = Formula.parse(f)
if debug:
print("Testing prove_tautology on formula", f, "and model", m)
p = prove_tautology(f, frozendict(m))
assert p.statement.conclusion == f
assert p.statement.assumptions == tuple(formulae_capturing_model(m))
assert p.rules == AXIOMATIC_SYSTEM
assert p.is_valid(), offending_line(p)
for t in [
'((~q->~p)->(p->q))',
'(~~p->p)',
'(p->~~p)',
'((~p->~q)->((p->~q)->~q))',
#'((p1->(p2->(p3->p4)))->(p3->(p2->(p1->p4))))',
'((p2->(p3->p4))->(p3->(p2->p4)))',
#'(((((p->q)->(~r->~s))->r)->t)->((t->p)->(s->p)))',
#'(((((r->q)->(~r->~q))->r)->t)->((t->r)->(q->r)))',
'(~~~~x13->~~x13)'
]:
t = Formula.parse(t)
if debug:
print("Testing prove_tautology on formula", t)
p = prove_tautology(t)
if debug:
if len(p.lines) < 20:
print("Proof is", p)
else:
print("Proof has", len(p.lines), "lines.")
assert len(p.statement.assumptions) == 0
assert p.statement.conclusion == t
assert p.rules == AXIOMATIC_SYSTEM
assert p.is_valid(), offending_line(p)
def test_prove_in_model_full(debug=False):
for (f, m, a, cp) in [ ('x', {'x':True}, ['x'], ''),
('x',{'x':False}, ['~x'], '~'),
('~x', {'x':False}, ['~x'], ''),
('~x', {'x':True}, ['x'], '~'),
('x', {'x':True, 'z5':False}, ['x', '~z5'], ''),
('(p->~p)', {'p':True}, ['p'], '~'),
('(p->~p)', {'p':False}, ['~p'], ''),
('(p->q)', {'p':True, 'q':True}, ['p','q'], ''),
('(p->q)', {'p':True, 'q':False}, ['p','~q'], '~'),
('(p->q)', {'p':False, 'q':True}, ['~p','q'], ''),
('(p->q)', {'p':False, 'q':False}, ['~p', '~q'], ''),
('~~~~y7', {'y7':True}, ['y7'], ''),
('~~~~y7', {'y7':False}, ['~y7'], '~'),
('~(~p->~q)', {'p':True, 'q':True}, ['p','q'], '~'),
('~(~p->~q)', {'p':False, 'q':True}, ['~p','q'], ''),
('((p1->p2)->(p3->p4))',
{'p1':True, 'p2':True, 'p3':True, 'p4':True},
['p1','p2','p3','p4'], ''),
('((p1->p2)->(p3->p4))',
{'p1':True, 'p2':True, 'p3':True, 'p4':False},
['p1','p2','p3','~p4'], '~'),
('((p1->p2)->(p3->p4))',
{'p1':True, 'p2':False, 'p3':True, 'p4':False},
['p1','~p2','p3','~p4'], ''),
('(~(~x->~~y)->(z->(~x->~~~z)))',
{'z':True, 'x':False, 'y':False},
['~x','~y','z'], '~'),
('T', {}, [], ''),
('F', {}, [], '~'),
('(p|q)', {'p': True, 'q': True}, ['p', 'q'], ''),
('(p|q)', {'p': True, 'q': False}, ['p', '~q'], ''),
('(p|q)', {'p': False, 'q': True}, ['~p', 'q'], ''),
('(p|q)',
{'p': False, 'q': False}, ['~p', '~q'], '~'),
('(p&q)', {'p': True, 'q': True}, ['p', 'q'], ''),
('(p&q)', {'p': True, 'q': False}, ['p', '~q'], '~'),
('(p&q)', {'p': False, 'q': True}, ['~p', 'q'], '~'),
('(p&q)',
{'p': False, 'q': False}, ['~p', '~q'], '~'),
('~(~(q|p)&(r->~(s|q)))',
{'p': False, 'q': False, 'r': False, 's': False},
['~p', '~q', '~r', '~s'], '~'),
('~(~(q|p)&(r->~(s|q)))',
{'p': False, 'q': False, 'r': True, 's': True},
['~p', '~q', 'r', 's'], '')
]:
c = Formula.parse(cp+f)
f = Formula.parse(f)
a = (Formula.parse(v) for v in a)
if debug:
print('Testing prove_in_model_full on formula',f, 'in model', m)
p = prove_in_model_full(f, frozendict(m))
assert p.statement == InferenceRule(a, c)
assert p.rules == AXIOMATIC_SYSTEM_FULL
assert p.is_valid(), offending_line(p)
def test_combine_proofs(debug=False):
# test1
p = Formula('p')
lp = Proof.Line(p)
q = Formula('q')
nq = Formula('~',q)
lnq = Proof.Line(nq)
pfp = Proof(InferenceRule([p,nq],p), AXIOMATIC_SYSTEM, [lp])
pfnq = Proof(InferenceRule([p,nq],nq), AXIOMATIC_SYSTEM, [lnq])
h1 = Formula.parse('~(p->q)')
#test2
np = Formula('~',p)
lnp = Proof.Line(np)
lq = Proof.Line(q)
nnq = Formula('~',nq)
rip = Formula('->',Formula('r'),p)
linesp = [lp,
Proof.Line(Formula('->',p,rip),I1,[]),
Proof.Line(rip, MP, [0,1])]
linesq = [lq,
Proof.Line(Formula('->',q,nnq),NN,[]),
Proof.Line(nnq, MP, [0,1])]
pfp2 = Proof(InferenceRule([p,q],rip), {MP,NN,I1, NI}, linesp)
pfnq2 = Proof(InferenceRule([p,q],nnq), {MP,NN,I1, NI}, linesq)
h2 = Formula.parse('~((r->p)->~q)')
# Variant for test1.5
pfp15 = Proof(InferenceRule([p,nq],rip), AXIOMATIC_SYSTEM, linesp)
h15 = Formula.parse('~((r->p)->q)')
# test3
pp3 = Formula.parse('(x->y)')
pq3 = Formula.parse('(~x->y)')
y = Formula('y')
pfp3 = Proof(InferenceRule([y],pp3), {MP, R, I0, I1},
[Proof.Line(y),
Proof.Line(Formula.parse('(y->(x->y))'),I1,[]),
Proof.Line(Formula.parse('(x->y)'),MP,[0,1])])
pfq3 = Proof(InferenceRule([y],pq3), {MP, R, I0, I1},
[Proof.Line(y),
Proof.Line(Formula.parse('(y->(~x->y))'),I1,[]),
Proof.Line(Formula.parse('(~x->y)'),MP,[0,1])])
h3 = y
for pp, pnq, h, r in [(pfp,pfnq,h1,NI),(pfp15,pfnq, h15, NI),
(pfp2,pfnq2, h2, NI), (pfp3,pfq3,h3,R)]:
if debug:
print('Testing combine_proof of', h,'from', pp.statement, 'and',
pnq.statement, 'using rule', r)
pnpiq = combine_proofs(pp, pnq, h, r)
assert pnpiq.rules == pp.rules
assert pnpiq.statement.conclusion == h
assert pnpiq.statement.assumptions == pp.statement.assumptions
assert pnpiq.is_valid(), offending_line(pnpiq)
def test_model_or_inconsistency(debug=False):
for s in [{'p'}, {'p', '~p'}, {'(p->p)'}, {'~(p->p)'},
{'(x->y)', 'x', '~y'}, {'(x->y)', 'x', '~z'}]:
s = {Formula.parse(f) for f in s}
if debug:
print("Testing model_or_inconsistency on", s)
r = model_or_inconsistency(s)
if type(r) is Proof:
assert r.statement.conclusion == Formula.parse('~(p->p)')
assert set(r.statement.assumptions) == s
assert r.rules == AXIOMATIC_SYSTEM
assert r.is_valid(), offending_line(r)
else:
assert is_model(r)
for f in s:
assert evaluate(f, r)
def test_reduce_assumption(debug=False):
for f, m, v in [ ('(y->x)', {'x':True}, 'y'),
('(p->p)', {}, 'p'),
('(p->(r->q))', {'p':True, 'q':True}, 'r')]:
f = Formula.parse(f)
m[v]=True
pt = prove_in_model(f, frozendict(m))
m[v]=False
pf = prove_in_model(f, frozendict(m))
if debug:
print("testing reduce assumption on", pt.statement, "and",
pf.statement)
p = reduce_assumption(pt,pf)
assert p.statement.conclusion == pf.statement.conclusion
assert p.statement.assumptions[:] == pt.statement.assumptions[:-1]
assert p.rules == AXIOMATIC_SYSTEM
assert p.is_valid(), offending_line(p)
def test_proof_or_counterexample(debug=False):
for f in [ 'x', '(y->y)', '((x->y)->(x->y))', '((x->y)->z)',
'((~p->~q)->((p->~q)->~q))', '((~p->~r)->((p->~q)->~q))',
'((~p->~q)->((~p->q)->p))', '((q->~p)->((~~~p->r)->(q->r)))',
'((q->p)->((~q->p)->p))', '((p->~q)->(q->~p))',
'((p->q)->(~p->~q))']:
if debug:
print("Testing proof_or_counterexample on", f)
f = Formula.parse(f)
p = proof_or_counterexample(f)
if type(p) is Proof:
assert len(p.statement.assumptions) == 0
assert p.statement.conclusion == f
assert p.rules == AXIOMATIC_SYSTEM
assert p.is_valid(), offending_line(p)
else:
assert not evaluate(f,p)
def test_prove_from_encoding(debug=True):
from propositions.some_proofs import prove_and_commutativity
from propositions.tautology import encode_as_formula
for p in [
DISJUNCTION_COMMUTATIVITY_PROOF,
prove_and_commutativity(), DISJUNCTION_RIGHT_ASSOCIATIVITY_PROOF
]:
for r in p.rules:
if (debug):
print("\nTesting prove_from_encoding on:", r)
pp = prove_from_encoding(r)
if (debug):
print("Got:", pp)
assert pp.statement == r
newrule = InferenceRule([], encode_as_formula(r))
assert pp.rules == {newrule, MP}
assert pp.is_valid(), offending_line(pp)
def test_prove_sound_inference(debug=False):
for a,c in [ ([], '(p->p)'),
(['p'], 'p'),
(['p'], '~~p'),
(['~~p'], 'p'),
(['p'], '(q->p)'),
(['(p->q)'], '(~q->~p)'),
(['p', 'q'], '~(p->~q)'),
(['(p->q)', '(q->r)'], '(p->r)'),
(['p','(p->q)', '(q->r)', '~r'],'x'),
(['p', '~p'], 'q')]:
r = InferenceRule((Formula.parse(f) for f in a), Formula.parse(c))
if debug:
print("Testing prove_sound_inference on", r)
p = prove_sound_inference(r)
assert p.statement == r
assert p.rules == AXIOMATIC_SYSTEM
assert p.is_valid(), offending_line(p)
def test_prove_by_way_of_contradiction(debug=False):
assumptions = (Formula.parse('(~r->p)'), Formula.parse('~p'),
Formula.parse('~r'))
p = Proof(InferenceRule(assumptions, Formula.parse('~(p->p)')), {MP, NI}, [
Proof.Line(Formula.parse('~r')),
Proof.Line(Formula.parse('(~r->p)')),
Proof.Line(Formula.parse('p'), MP, [0, 1]),
Proof.Line(Formula.parse('(p->(~p->~(p->p)))'), NI, []),
Proof.Line(Formula.parse('(~p->~(p->p))'), MP, [2, 3]),
Proof.Line(Formula.parse('~p')),
Proof.Line(Formula.parse('~(p->p)'), MP, [5, 4])
])
if debug:
print("Testing prove_by_way_of_contradiction on proof of", p.statement)
p = prove_by_way_of_contradiction(p)
assert p.statement.conclusion == Formula.parse('r')
assert p.statement.assumptions == assumptions[:-1]
assert p.rules == {MP, I0, I1, D, N, NI}
assert p.is_valid(), offending_line(p)
def test_prove_corollary(debug=False):
x = Formula('x')
y = Formula('y')
pf = Proof(InferenceRule([x], x), AXIOMATIC_SYSTEM, [Proof.Line(x)])
g1 = Formula.parse('~~x')
r1 = NN
g2 = Formula.parse('(y->~~x)')
r2 = I1
g3 = Formula.parse('((~y->~~x)->~~x)')
r3 = R
for g, r in [(g1, r1), (g2, r2), (g3, r3)]:
if debug:
print('Testing prove_corollary of', g, 'from proof of',
pf.statement, 'using rule', r)
pp = prove_corollary(pf, g, r)
assert pp.rules == pf.rules
assert pp.statement.assumptions == pf.statement.assumptions
assert pp.statement.conclusion == g
assert pp.is_valid(), offending_line(pp)
pf = pp