def prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Task 9.11.2
# create the proof, translate lines, assumption and conclusion
# creating lines:
predicate_line_formulas = [
Formula.from_propositional_skeleton(line.formula, substitution_map)
for line in skeleton_proof.lines
]
real_predicate_lines = []
for predicate_formula, skeleton_line in zip(predicate_line_formulas,
skeleton_proof.lines):
# if line was proven using MP
if skeleton_line.rule == MP:
# if the line is MP create a predicate MP line
real_predicate_lines.append(
Proof.MPLine(predicate_formula, skeleton_line.assumptions[0],
skeleton_line.assumptions[1]))
else: # line must be from an axiom
line_to_specialization_map = PropositionalInferenceRule.formula_specialization_map(
skeleton_line.rule.conclusion, skeleton_line.formula)
instantiation_map = axiom_specialization_map_to_schema_instantiation_map(
line_to_specialization_map, substitution_map)
real_predicate_lines.append(
Proof.AssumptionLine(
predicate_formula,
PROPOSITIONAL_AXIOM_TO_SCHEMA[skeleton_line.rule],
instantiation_map))
# create the final proof
return Proof(PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS, formula,
real_predicate_lines)
def prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Task 9.11.2
assumptions = PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS
conclusion = formula
lines = []
for skeleton_line in skeleton_proof.lines:
line_formula = Formula.from_propositional_skeleton(
skeleton_line.formula, substitution_map)
if PropositionalProof.Line.is_assumption(skeleton_line):
assumption_schema = Schema(line_formula, set())
line = Proof.AssumptionLine(line_formula, assumption_schema,
dict())
else:
if skeleton_line.rule == MP:
ante_line_num, cond_line_num = [
skeleton_line.assumptions[i] for i in [0, 1]
]
line = Proof.MPLine(line_formula, ante_line_num, cond_line_num)
else:
schema_rule = PROPOSITIONAL_AXIOM_TO_SCHEMA[skeleton_line.rule]
prop_spec_map = PropositionalInferenceRule.formula_specialization_map(
skeleton_line.rule.conclusion, skeleton_line.formula)
inst_map = axiom_specialization_map_to_schema_instantiation_map(
prop_spec_map, substitution_map)
line = Proof.AssumptionLine(line_formula, schema_rule,
inst_map)
lines.append(line)
return Proof(assumptions, conclusion, lines)
def prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Task 9.11.2
lines = list()
for line in skeleton_proof.lines:
if len(line.assumptions) == 0:
new_line = Proof.AssumptionLine(
Formula.from_propositional_skeleton(line.formula,
substitution_map),
PROPOSITIONAL_AXIOM_TO_SCHEMA[line.rule],
axiom_specialization_map_to_schema_instantiation_map(
PropositionalInferenceRule.formula_specialization_map(
line.rule.conclusion, line.formula), substitution_map))
else:
new_line = Proof.MPLine(
Formula.from_propositional_skeleton(line.formula,
substitution_map),
line.assumptions[0], line.assumptions[1])
lines.append(new_line)
return Proof(PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS, formula, lines)
def prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Setup the proof builder
builder = Proof.ProofBuilder() \
.with_assumptions(PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS) \
.with_conclusion(Formula.from_propositional_skeleton(skeleton_proof.statement.conclusion, substitution_map))
for line in skeleton_proof.lines:
# Which rule type do we got
if line.rule == MP:
builder.add_mp_line(Formula.from_propositional_skeleton(line.formula, substitution_map),
line.assumptions[0], line.assumptions[1])
else:
builder.add_claim_line(Formula.from_propositional_skeleton(line.formula, substitution_map),
PROPOSITIONAL_AXIOM_TO_SCHEMA[line.rule],
axiom_specialization_map_to_schema_instantiation_map(
PropositionalInferenceRule.formula_specialization_map(line.rule.conclusion,
line.formula),
substitution_map))
return builder.build()