def get_proof_term(self, t):
def strip_conj_all(t):
if t.is_conj():
return strip_conj_all(t.arg1) + strip_conj_all(t.arg)
else:
return [t]
conj_terms = term_ord.sorted_terms(strip_conj_all(t))
goal = Eq(t, And(*conj_terms))
return imp_conj_iff(goal)
def add_invariant(self):
"""Add the invariant for the system in GCL."""
s = Var("s", gcl.stateT)
invC = Const("inv", TFun(gcl.stateT, BoolType))
inv_rhs = And(
*[gcl.convert_term(self.var_map, s, t) for _, t in self.invs])
prop = Eq(invC(s), inv_rhs)
exts = [
extension.Constant("inv", TFun(gcl.stateT, BoolType)),
extension.Theorem("inv_def", Thm([], prop))
]
theory.thy.unchecked_extend(exts)
def convert_cnf_to_HOL(cnf_file):
disjs = read_cnf_file(cnf_file)
tms = []
for disj in disjs:
lits = []
for var, stat in disj:
if stat:
lits.append(Var(var, BoolType))
else:
lits.append(Not(Var(var, BoolType)))
tms.append(Or(*lits))
return And(*tms)
def testPrintUnicode(self):
test_data = [
(And(A, B), "A ∧ B"),
(Or(A, B), "A ∨ B"),
(Implies(A, B), "A ⟶ B"),
(Lambda(a, P(a)), "λa. P a"),
(Forall(a, P(a)), "∀a. P a"),
(Exists(a, P(a)), "∃a. P a"),
(Not(A), "¬A"),
(Lambda(m, m + 2), "λm::nat. m + 2"),
(Lambda(m, m + n), "λm. m + n"),
]
with global_setting(unicode=True):
for t, s in test_data:
self.assertEqual(printer.print_term(t), s)
def testParseThm(self):
A = Var('A', BoolType)
B = Var('B', BoolType)
C = Var('C', BoolType)
test_data = [
("|- A", Thm([], A)),
("|- A & B", Thm([], And(A, B))),
("A |- B", Thm([A], B)),
("A, B |- C", Thm([A, B], C)),
]
context.set_context('logic',
vars={
'A': 'bool',
'B': 'bool',
'C': 'bool'
})
for s, th in test_data:
self.assertEqual(parser.parse_thm(s), th)
def nnf(fm):
"""Negation normal form of a formula."""
if fm.is_conj():
return And(nnf(fm.arg1), nnf(fm.arg))
elif fm.is_disj():
return Or(nnf(fm.arg1), nnf(fm.arg))
elif fm.is_implies():
return Or(nnf(Not(fm.arg1)), nnf(fm.arg))
elif fm.is_equals():
return Or(And(nnf(fm.arg1), nnf(fm.arg)),
And(nnf(Not(fm.arg1)), nnf(Not(fm.arg))))
elif fm.is_not():
p = fm.arg
if p.is_not():
return nnf(p.arg)
elif p.is_conj():
return Or(nnf(Not(p.arg1)), nnf(Not(p.arg)))
elif p.is_disj():
return And(nnf(Not(p.arg1)), nnf(Not(p.arg)))
elif p.is_implies():
return And(nnf(p.arg1), nnf(Not(p.arg)))
elif p.is_equals():
return Or(And(nnf(p.arg1), nnf(Not(p.arg))),
And(nnf(Not(p.arg1)), nnf(p.arg)))
elif p.is_forall():
assert p.arg.is_abs()
return term.exists(p.arg.var_T)(Abs(p.arg.var_name, p.arg.var_T,
nnf(Not(p.arg.body))))
elif p.is_exists():
assert p.arg.is_abs()
return term.forall(p.arg.var_T)(Abs(p.arg.var_name, p.arg.var_T,
nnf(Not(p.arg.body))))
else:
return fm
elif fm.is_forall() or fm.is_exists():
assert fm.arg.is_abs()
return fm.fun(Abs(fm.arg.var_name, fm.arg.var_T, nnf(fm.arg.body)))
else:
return fm
def testPrintLogical(self):
test_data = [
# Variables
(SVar("P", BoolType), "?P"),
(a, "a"),
# Equality and implies
(Eq(a, b), "a = b"),
(Implies(A, B), "A --> B"),
(Implies(A, B, C), "A --> B --> C"),
(Implies(Implies(A, B), C), "(A --> B) --> C"),
(Implies(A, Eq(a, b)), "A --> a = b"),
(Eq(Implies(A, B), Implies(B, C)), "(A --> B) <--> (B --> C)"),
(Eq(A, Eq(B, C)), "A <--> B <--> C"),
(Eq(Eq(A, B), C), "(A <--> B) <--> C"),
# Conjunction and disjunction
(And(A, B), "A & B"),
(Or(A, B), "A | B"),
(And(A, And(B, C)), "A & B & C"),
(And(And(A, B), C), "(A & B) & C"),
(Or(A, Or(B, C)), "A | B | C"),
(Or(Or(A, B), C), "(A | B) | C"),
(Or(And(A, B), C), "A & B | C"),
(And(Or(A, B), C), "(A | B) & C"),
(Or(A, And(B, C)), "A | B & C"),
(And(A, Or(B, C)), "A & (B | C)"),
(Or(And(A, B), And(B, C)), "A & B | B & C"),
(And(Or(A, B), Or(B, C)), "(A | B) & (B | C)"),
# Negation
(Not(A), "~A"),
(Not(Not(A)), "~~A"),
# Constants
(true, "true"),
(false, "false"),
# Mixed
(Implies(And(A, B), C), "A & B --> C"),
(Implies(A, Or(B, C)), "A --> B | C"),
(And(A, Implies(B, C)), "A & (B --> C)"),
(Or(Implies(A, B), C), "(A --> B) | C"),
(Not(And(A, B)), "~(A & B)"),
(Not(Implies(A, B)), "~(A --> B)"),
(Not(Eq(A, B)), "~(A <--> B)"),
(Eq(Not(A), B), "~A <--> B"),
(Eq(Not(A), Not(B)), "~A <--> ~B"),
(Implies(A, Eq(B, C)), "A --> B <--> C"),
(Eq(Implies(A, B), C), "(A --> B) <--> C"),
# Abstraction
(Lambda(a, And(P(a), Q(a))), "%a. P a & Q a"),
# Quantifiers
(Forall(a, P(a)), "!a. P a"),
(Forall(a, Forall(b, And(P(a), P(b)))), "!a. !b. P a & P b"),
(Forall(a, And(P(a), Q(a))), "!a. P a & Q a"),
(And(Forall(a, P(a)), Q(a)), "(!a1. P a1) & Q a"),
(Forall(a, Implies(P(a), Q(a))), "!a. P a --> Q a"),
(Implies(Forall(a, P(a)), Q(a)), "(!a1. P a1) --> Q a"),
(Implies(Forall(a, P(a)),
Forall(a, Q(a))), "(!a. P a) --> (!a. Q a)"),
(Implies(Exists(a, P(a)),
Exists(a, Q(a))), "(?a. P a) --> (?a. Q a)"),
(Eq(A, Forall(a, P(a))), "A <--> (!a. P a)"),
(Exists(a, P(a)), "?a. P a"),
(Exists(a, Forall(b, R(a, b))), "?a. !b. R a b"),
(logic.mk_exists1(a, P(a)), "?!a. P a"),
(logic.mk_the(a, P(a)), "THE a. P a"),
(logic.mk_some(a, P(a)), "SOME a. P a"),
(Forall(a, Exists(b, R(a, b))), "!a. ?b. R a b"),
# If
(mk_if(A, a, b), "if A then a else b"),
(Eq(mk_if(A, a, b), a), "(if A then a else b) = a"),
(mk_if(A, P, Q), "if A then P else Q"),
]
with global_setting(unicode=False):
for t, s in test_data:
self.assertEqual(printer.print_term(t), s)
def testStripConj(self):
test_data = [(a, [a]), (And(a, b, a), [a, b, a])]
for t, res in test_data:
self.assertEqual(t.strip_conj(), res)
def testConj(self):
test_data = [([], term.true), ([a], a), ([a, b], term.conj(a, b)),
([a, b, a], term.conj(a, term.conj(b, a)))]
for ts, res in test_data:
self.assertEqual(And(*ts), res)
def get_proof_term(self, t):
if not t.is_conj():
return refl(t)
d_pos = dict()
d_neg = dict()
qu = deque([ProofTerm.assume(t)])
# collect each conjunct's proof term in conjuntion
while qu:
pt = qu.popleft()
if pt.prop.is_conj():
conj1, conj2 = pt.prop.arg1, pt.prop.arg
pt_conj1, pt_conj2 = apply_theorem('conjD1',
pt), apply_theorem(
'conjD2', pt)
if conj1 == false:
th = ProofTerm.theorem("falseE")
inst = matcher.first_order_match(th.prop.arg, t)
pt_false_implies_conj = th.substitution(inst)
return ProofTerm.equal_intr(pt_conj1.implies_intr(t),
pt_false_implies_conj)
elif conj2 == false:
th = ProofTerm.theorem("falseE")
inst = matcher.first_order_match(th.prop.arg, t)
pt_false_implies_conj = th.substitution(inst)
return ProofTerm.equal_intr(pt_conj2.implies_intr(t),
pt_false_implies_conj)
if conj1.is_conj():
qu.appendleft(pt_conj1)
else:
if conj1.is_not():
d_neg[conj1] = pt_conj1
else:
d_pos[conj1] = pt_conj1
if conj2.is_conj():
qu.appendleft(pt_conj2)
else:
if conj2.is_not():
d_neg[conj2] = pt_conj2
else:
d_pos[conj2] = pt_conj2
else:
if pt.prop.is_not():
d_neg[pt.prop] = pt
else:
d_pos[pt.prop] = pt
# first check if there are opposite terms in conjunctions, if there exists, return a false proof term
for key in d_pos:
if Not(key) in d_neg:
pos_pt, neg_pt = d_pos[key], d_neg[Not(key)]
pt_conj_pos_neg = apply_theorem("conjI", pos_pt, neg_pt)
pt_conj_implies_false = pt_conj_pos_neg.on_prop(
rewr_conv("conj_pos_neg")).implies_intr(t)
th = ProofTerm.theorem("falseE")
inst = matcher.first_order_match(th.prop.arg, t)
pt_false_implies_conj = th.substitution(inst)
return ProofTerm.equal_intr(pt_conj_implies_false,
pt_false_implies_conj)
d_pos.update(d_neg)
d = d_pos
def right_assoc(ts):
l = len(ts)
if l == 1:
return d[ts[0]]
elif l == 2:
return apply_theorem('conjI', d[ts[0]], d[ts[1]])
else:
return apply_theorem('conjI', d[ts[0]], right_assoc(ts[1:]))
# pt_right = functools.reduce(lambda x, y: apply_theorem('conjI', x, d[y]), sorted(d.keys()), d[sorted(d.keys())[0]])
if true not in d:
sorted_keys = term_ord.sorted_terms(d.keys())
else:
d_keys_without_true = term_ord.sorted_terms(
[k for k in d if k != true])
sorted_keys = [true] + d_keys_without_true
sorted_keys_num = len(sorted_keys)
pt_right = functools.reduce(lambda x, y: apply_theorem('conjI', d[sorted_keys[sorted_keys_num - y - 2]], x), \
range(sorted_keys_num - 1), d[sorted_keys[-1]])
# pt_right = right_assoc(sorted_keys)
# order implies original
dd = dict()
norm_conj = And(*sorted_keys)
norm_conj_pt = ProofTerm.assume(norm_conj)
for k in sorted_keys:
if k != sorted_keys[-1]:
dd[k] = apply_theorem('conjD1', norm_conj_pt)
norm_conj_pt = apply_theorem('conjD2', norm_conj_pt)
else:
dd[k] = norm_conj_pt
def traverse(t):
if not t.is_conj():
return dd[t]
else:
return apply_theorem('conjI', traverse(t.arg1),
traverse(t.arg))
pt_left = traverse(t)
pt_final = ProofTerm.equal_intr(pt_right.implies_intr(t),
pt_left.implies_intr(norm_conj))
if true in d:
return pt_final.on_rhs(rewr_conv("conj_true_left"),
top_sweep_conv(sort_disj()))
else:
return pt_final.on_rhs(top_sweep_conv(sort_disj()))
def conj(self, s, t):
return And(s, t)
def testParseProofRule(self):
A = Var('A', BoolType)
B = Var('B', BoolType)
test_data = [
({
'id': "0",
'rule': "theorem",
'args': "conjD1",
'prevs': [],
'th': ""
}, ProofItem(0, "theorem", args="conjD1", prevs=[])),
({
'id': "2",
'rule': "implies_elim",
'args': "",
'prevs': ["1", "0"],
'th': ""
}, ProofItem(2, "implies_elim", prevs=[1, 0])),
({
'id': "5",
'rule': "substitution",
'args': "{A: B, B: A}",
'prevs': ["4"],
'th': ""
}, ProofItem(5, "substitution", args=Inst(A=B, B=A), prevs=[4])),
({
'id': "8",
'rule': "implies_intr",
'args': "conj A B",
'prevs': ["7"],
'th': ""
}, ProofItem(8, "implies_intr", args=And(A, B), prevs=[7])),
({
'id': "0",
'rule': "sorry",
'args': "",
'prevs': [],
'th': "conj A B |- conj B A"
}, ProofItem(0, "sorry", th=Thm([And(A, B)], And(B, A)))),
({
'id': "1",
'rule': "",
'args': "",
'prevs': [],
'th': ""
}, ProofItem(1, "")),
({
'id': "5",
'rule': "apply_theorem_for",
'args': "disjI1, {A: B, B: A}",
'prevs': [4],
'th': ""
},
ProofItem(5,
"apply_theorem_for",
args=("disjI1", Inst(A=B, B=A)),
prevs=[4])),
]
context.set_context('logic_base', vars={'A': 'bool', 'B': 'bool'})
for s, res in test_data:
self.assertEqual(parser.parse_proof_rule(s), res)
def conj_cond(self, b1, b2):
return And(b1, b2)
def get_extension(self):
assert self.error is None, "get_extension"
res = []
# Add to type and term signature.
res.append(extension.TConst(self.name, len(self.args)))
for constr in self.constrs:
res.append(
extension.Constant(constr['name'],
constr['type'],
ref_name=constr['cname']))
# Add non-equality theorems.
for constr1, constr2 in itertools.combinations(self.constrs, 2):
# For each A x_1 ... x_m and B y_1 ... y_n, get the theorem
# ~ A x_1 ... x_m = B y_1 ... y_n.
argT1, _ = constr1['type'].strip_type()
argT2, _ = constr2['type'].strip_type()
lhs_vars = [Var(nm, T) for nm, T in zip(constr1['args'], argT1)]
rhs_vars = [Var(nm, T) for nm, T in zip(constr2['args'], argT2)]
A = Const(constr1['name'], constr1['type'])
B = Const(constr2['name'], constr2['type'])
lhs = A(*lhs_vars)
rhs = B(*rhs_vars)
neq = Not(Eq(lhs, rhs))
th_name = "%s_%s_%s_neq" % (self.name, constr1['name'],
constr2['name'])
res.append(extension.Theorem(th_name, Thm([], neq)))
# Add injectivity theorems.
for constr in self.constrs:
# For each A x_1 ... x_m with m > 0, get the theorem
# A x_1 ... x_m = A x_1' ... x_m' --> x_1 = x_1' & ... & x_m = x_m'
if constr['args']:
argT, _ = constr['type'].strip_type()
lhs_vars = [Var(nm, T) for nm, T in zip(constr['args'], argT)]
rhs_vars = [
Var(nm + "1", T) for nm, T in zip(constr['args'], argT)
]
A = Const(constr['name'], constr['type'])
assum = Eq(A(*lhs_vars), A(*rhs_vars))
concls = [
Eq(var1, var2) for var1, var2 in zip(lhs_vars, rhs_vars)
]
concl = And(*concls)
th_name = "%s_%s_inject" % (self.name, constr['name'])
res.append(
extension.Theorem(th_name, Thm([], Implies(assum, concl))))
# Add the inductive theorem.
tvars = [TVar(targ) for targ in self.args]
T = TConst(self.name, *tvars)
var_P = Var("P", TFun(T, BoolType))
ind_assums = []
for constr in self.constrs:
A = Const(constr['name'], constr['type'])
argT, _ = constr['type'].strip_type()
args = [Var(nm, T2) for nm, T2 in zip(constr['args'], argT)]
C = var_P(A(*args))
As = [
var_P(Var(nm, T2)) for nm, T2 in zip(constr['args'], argT)
if T2 == T
]
ind_assum = Implies(*(As + [C]))
for arg in reversed(args):
ind_assum = Forall(arg, ind_assum)
ind_assums.append(ind_assum)
ind_concl = var_P(Var("x", T))
th_name = self.name + "_induct"
res.append(
extension.Theorem(th_name,
Thm([], Implies(*(ind_assums + [ind_concl])))))
res.append(extension.Attribute(th_name, "var_induct"))
return res