def test_prove_from_skeleton_proof(debug=False):
from propositions.tautology import prove_tautology
for formula,skeleton,substitution_map in [
('(R(c)->R(c))', '(z23->z23)', {'z23':Formula.parse('R(c)')}),
('(x=0->x=0)', '(z24->z24)', {'z24':Formula.parse('x=0')}),
('(Ex[x=0]->Ex[x=0])', '(z25->z25)', {'z25':Formula.parse('Ex[x=0]')}),
('((~y=1->~x=1)->(x=1->y=1))', '((~z26->~z27)->(z27->z26))',
{'z26':Formula.parse('y=1'), 'z27':Formula.parse('x=1')}),
('(~~Ex[y=2]->Ex[y=2])', '(~~z28->z28)',
{'z28':Formula.parse('Ex[y=2]')}),
('(Ex[Ey[x=y]]->~~Ex[Ey[x=y]])', '(z29->~~z29)',
{'z29':Formula.parse('Ex[Ey[x=y]]')}),
('((~Ex[(x=2->x=3)]->~R(y,4))->((Ex[(x=2->x=3)]->~R(y,4))->~R(y,4)))',
'((~z30->~z31)->((z30->~z31)->~z31))',
{'z30':Formula.parse('Ex[(x=2->x=3)]'),
'z31':Formula.parse('R(y,4)')}),
('((Ey[~x=y]->(y=3->y=74))->(y=3->(Ey[~x=y]->y=74)))',
'((z32->(z33->z34))->(z33->(z32->z34)))',
{'z32':Formula.parse('Ey[~x=y]'), 'z33':Formula.parse('y=3'),
'z34':Formula.parse('y=74')}),
('(~~~~Q()->~~Q())', '(~~~~z35->~~z35)', {'z35':Formula.parse('Q()')})]:
if debug:
print('Testing proving', formula, 'from proof of', skeleton)
formula = Formula.parse(formula)
skeleton = PropositionalFormula.parse(skeleton)
skeleton_proof = prove_tautology(skeleton)
assert skeleton_proof.is_valid(), 'Bug in prove_tautology!'
proof = prove_from_skeleton_proof(formula, skeleton_proof,
frozendict(substitution_map))
assert proof.assumptions == PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS
assert proof.conclusion == formula
assert proof.is_valid()
def test_prove_tautology(debug=False):
for tautology in [
'(R(c)->R(c))', '(x=0->x=0)', '(Ex[x=0]->Ex[x=0])',
'((~y=1->~x=1)->(x=1->y=1))', '(~~Ex[y=2]->Ex[y=2])',
'(Ex[Ey[x=y]]->~~Ex[Ey[x=y]])',
'((~Ex[(x=2->x=3)]->~R(y,4))->' +
'((Ex[(x=2->x=3)]->~R(y,4))->~R(y,4)))',
'((Ey[~x=y]->(y=3->y=74))->(y=3->(Ey[~x=y]->y=74)))',
'(~~~~Q()->~~Q())']:
if debug:
print('Testing proving the tautology', tautology)
tautology = Formula.parse(tautology)
proof = prove_tautology(tautology)
assert proof.assumptions == PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS
assert proof.conclusion == tautology
for line in proof.lines:
assert type(line) in {Proof.AssumptionLine, Proof.MPLine}
assert proof.is_valid()