def search(self, state, id, prevs):
prev_ths = [state.get_proof_item(prev).th for prev in prevs]
thy = state.thy
if len(prevs) == 0:
return []
results = []
for name, th in thy.get_data("theorems").items():
if 'hint_forward' not in thy.get_attributes(name):
continue
instsp = (dict(), dict())
As, C = th.prop.strip_implies()
if len(prevs) != len(As):
continue
if set(term.get_vars(As)) != set(term.get_vars(As + [C])):
continue
if not term.get_consts(As):
continue
try:
for pat, prev in zip(As, prev_ths):
matcher.first_order_match_incr(pat, prev.concl, instsp)
except matcher.MatchException:
continue
# All matches succeed
t = logic.subst_norm(th.prop, instsp)
_, new_fact = t.strip_implies()
results.append({"theorem": name, "_fact": [new_fact]})
return sorted(results, key=lambda d: d['theorem'])
def is_pattern_list(ts, matched_vars):
"""Test whether a list of ts can be matched."""
if len(ts) == 0:
return True
elif len(ts) == 1:
return is_pattern(ts[0], matched_vars)
else:
if is_pattern(ts[0], matched_vars):
all_vars = list(set(matched_vars + term.get_vars(ts[0])))
return is_pattern_list(ts[1:], all_vars)
elif is_pattern_list(ts[1:], matched_vars):
all_vars = list(set(matched_vars + term.get_vars(ts[1:])))
return is_pattern(ts[0], all_vars)
else:
return False
def solve_core(s, t, debug=False):
# First strip foralls from t.
t = norm_term(t)
new_names = logic.get_forall_names(t, svar=False)
_, As, C = logic.strip_all_implies(t, new_names, svar=False)
def print_debug(*args):
if debug:
print(*args)
var_names = [v.name for v in term.get_vars(As + [C])]
assms = dict()
to_real = dict()
for A in As:
try:
z3_A = convert(A, var_names, assms, to_real, s.ctx)
print_debug('A', z3_A)
s.add(z3_A)
except Z3Exception as e:
print_debug(e)
try:
z3_C = convert(C, var_names, assms, to_real, s.ctx)
print_debug('C', z3_C)
s.add(z3.Not(z3_C))
except Z3Exception as e:
print_debug(e)
for nm, A in assms.items():
print_debug('A', A)
s.add(A)
return s
def testGetVars(self):
test_data = [
(a, [a]),
(f(a), [a, f]),
(f(c), [f]),
([a, f(c)], [a, f]),
]
for t, res in test_data:
self.assertEqual(term.get_vars(t), res)
def search(self, state, id, prevs):
goal_th = state.get_proof_item(id).th
prev_ths = [state.get_proof_item(prev).th for prev in prevs]
thy = state.thy
results = []
for name, th in thy.get_data("theorems").items():
if 'hint_backward' not in thy.get_attributes(name):
continue
instsp = (dict(), dict())
As, C = th.assums, th.concl
# Only process those theorems where C and the matched As
# contain all of the variables.
if set(term.get_vars(As[:len(prevs)] + [C])) != set(
term.get_vars(As + [C])):
continue
# When there is no assumptions to match, only process those
# theorems where C contains at least a constant (skip falseE,
# induction theorems, etc).
if len(prevs) == 0 and term.get_consts(C) == []:
continue
try:
if matcher.is_pattern(C, []):
matcher.first_order_match_incr(C, goal_th.prop, instsp)
for pat, prev in zip(As, prev_ths):
matcher.first_order_match_incr(pat, prev.prop, instsp)
else:
for pat, prev in zip(As, prev_ths):
matcher.first_order_match_incr(pat, prev.prop, instsp)
matcher.first_order_match_incr(C, goal_th.prop, instsp)
except matcher.MatchException:
continue
# All matches succeed
t = logic.subst_norm(th.prop, instsp)
As, C = t.strip_implies()
results.append({"theorem": name, "_goal": As[len(prevs):]})
return sorted(results, key=lambda d: d['theorem'])
def add_induct_predicate(name, T, props):
"""Add the given inductive predicate.
The inductive predicate is specified by the name and type of
the predicate, and a list of introduction rules, where each
introduction rule must be given a name.
"""
exts = TheoryExtension()
exts.add_extension(AxConstant(name, T))
for th_name, prop in props:
exts.add_extension(Theorem(th_name, Thm([], prop)))
exts.add_extension(Attribute(th_name, "hint_backward"))
# Case rule
Targs, _ = T.strip_type()
vars = []
for i, Targ in enumerate(Targs):
vars.append(Var("_a" + str(i + 1), Targ))
P = Var("P", boolT)
pred = Const(name, T)
assum0 = pred(*vars)
assums = []
for th_name, prop in props:
As, C = prop.strip_implies()
assert C.head == pred, "add_induct_predicate: wrong form of prop."
eq_assums = [
Term.mk_equals(var, arg) for var, arg in zip(vars, C.args)
]
assum = Term.mk_implies(*(eq_assums + As), P)
prop_vars = term.get_vars(prop)
for var in reversed(term.get_vars(prop)):
assum = Term.mk_all(var, assum)
assums.append(assum)
prop = Term.mk_implies(*([assum0] + assums + [P]))
exts.add_extension(Theorem(name + "_cases", Thm([], prop)))
return exts
def add_theorem(self, name, th, prf):
"""Add a theorem with proof to the file."""
assert len(th.hyps) == 0, "add_theorem"
vars = dict((v.name, str(v.T)) for v in get_vars(th.prop))
data = {
"name": name,
"ty": "thm",
"prop": printer.print_term(self.thy, th.prop),
"vars": vars,
"num_gaps": 0,
"proof": sum([self.export_proof_json(item) for item in prf.items], []),
}
self.content.append(data)
def get_proof_term(self, thy, goal, *, args=None, prevs=None):
if isinstance(args, tuple):
th_name, instsp = args
else:
th_name = args
instsp = None
assert isinstance(th_name, str), "rule: theorem name must be a string"
if prevs is None:
prevs = []
th = thy.get_theorem(th_name)
As, C = th.assums, th.concl
if instsp is None:
instsp = (dict(), dict())
if matcher.is_pattern(C, []):
matcher.first_order_match_incr(C, goal.prop, instsp)
for pat, prev in zip(As, prevs):
matcher.first_order_match_incr(pat, prev.prop, instsp)
else:
for pat, prev in zip(As, prevs):
matcher.first_order_match_incr(pat, prev.prop, instsp)
matcher.first_order_match_incr(C, goal.prop, instsp)
As, _ = logic.subst_norm(th.prop, instsp).strip_implies()
pts = prevs + [
ProofTerm.sorry(Thm(goal.hyps, A)) for A in As[len(prevs):]
]
if set(term.get_vars(th.assums)) != set(term.get_vars(th.prop)) or \
not matcher.is_pattern_list(th.assums, []):
tyinst, inst = instsp
return apply_theorem(thy, th_name, *pts, tyinst=tyinst, inst=inst)
else:
return apply_theorem(thy, th_name, *pts)
def search(self, state, id, prevs):
cur_th = state.get_proof_item(id).th
if len(cur_th.hyps) > 0:
return []
results = []
for name, th in state.thy.get_data("theorems").items():
if 'var_induct' not in state.thy.get_attributes(name):
continue
var_T = th.concl.arg.T
vars = [v for v in term.get_vars(cur_th.prop) if v.T == var_T]
if len(vars) == 1:
results.append({'theorem': name, 'var': vars[0].name})
elif len(vars) > 1:
results.append({'theorem': name})
return results
def get_proof_term(self, goal, *, args=None, prevs=None):
assert isinstance(prevs, list) and len(prevs) >= 1, "apply_prev"
pt, prev_pts = prevs[0], prevs[1:]
# First, obtain the patterns
new_names = logic.get_forall_names(pt.prop)
new_vars, As, C = logic.strip_all_implies(pt.prop, new_names)
assert len(prev_pts) <= len(As), "apply_prev: too many prev_pts"
if args is None:
inst = Inst()
else:
inst = args
inst = matcher.first_order_match(C, goal.prop, inst)
for idx, prev_pt in enumerate(prev_pts):
inst = matcher.first_order_match(As[idx], prev_pt.prop, inst)
unmatched_vars = [v for v in new_names if v not in inst]
if unmatched_vars:
raise theory.ParameterQueryException(
list("param_" + name for name in unmatched_vars))
pt = pt.subst_type(inst.tyinst)
for new_name in new_names:
pt = pt.forall_elim(inst[new_name])
if pt.prop.beta_norm() != pt.prop:
pt = pt.on_prop(beta_norm_conv())
inst_As, inst_C = pt.prop.strip_implies()
inst_arg = [inst[new_name] for new_name in new_names]
new_goals = [
ProofTerm.sorry(Thm(goal.hyps, A)) for A in inst_As[len(prev_pts):]
]
if set(new_names).issubset({v.name for v in term.get_vars(As)}) and \
matcher.is_pattern_list(As, []):
return ProofTerm('apply_fact', args=None, prevs=prevs + new_goals)
else:
return ProofTerm('apply_fact_for',
args=inst_arg,
prevs=prevs + new_goals)
def match(pat, t, instsp, bd_vars):
tyinst, inst = instsp
# print("Match", pat, "with", t)
if pat.head.is_var() and pat.head.name.startswith('_'):
# Case where the head of the function is a variable.
if pat.head.name not in inst:
# If the variable is not instantiated, check that the
# arguments are distinct bound variables, and all bound
# variables appearing in t also appear as an argument.
# If all conditions hold, assign appropriately.
t_vars = term.get_vars(t)
if any(v not in bd_vars for v in pat.args):
raise MatchException
if len(set(pat.args)) != len(pat.args):
raise MatchException
if any(v in t_vars and v not in pat.args for v in bd_vars):
raise MatchException
pat_T = TFun(*([v.T for v in pat.args] + [t.get_type()]))
try:
pat.head.T.match_incr(pat_T, tyinst, internal_only=True)
except TypeMatchException:
raise MatchException
inst_t = t
for v in reversed(pat.args):
if inst_t.is_comb(
) and inst_t.arg == v and v not in term.get_vars(
inst_t.fun):
inst_t = inst_t.fun
else:
inst_t = Term.mk_abs(v, inst_t)
inst[pat.head.name] = inst_t
else:
# If the variable is already instantiated, apply the
# instantiation, simplify, and match again.
pat2 = inst[pat.head.name](*pat.args).beta_norm()
match(pat2, t.beta_norm(), instsp, bd_vars)
elif pat.ty != t.ty:
# In all other cases, top-level structure of the term
# must agree.
raise MatchException
elif pat.is_var():
# The case where pat come from a bound variable.
if pat.name != t.name:
raise MatchException
elif pat.is_const():
# When pat is a constant, t must also be a constant with
# the same name and matching type.
if pat.name != t.name:
raise MatchException
else:
try:
pat.T.match_incr(t.T, tyinst, internal_only=True)
except TypeMatchException:
raise MatchException
elif pat.is_comb():
# In the combination case (where the head is not a variable),
# match fun and arg.
if is_pattern(pat.fun, list(instsp[1].keys())):
match(pat.fun, t.fun, instsp, bd_vars)
match(pat.arg, t.arg, instsp, bd_vars)
else:
match(pat.arg, t.arg, instsp, bd_vars)
match(pat.fun, t.fun, instsp, bd_vars)
elif pat.is_abs():
# When pat is a lambda term, t must also be a lambda term.
# Replace bound variable by a variable, then match the body.
try:
pat.var_T.match_incr(t.var_T, tyinst, internal_only=True)
except TypeMatchException:
raise MatchException
T = pat.var_T.subst(tyinst)
v = Var(pat.var_name, T)
pat_body = pat.subst_type(tyinst).subst_bound(v)
t_body = t.subst_bound(v)
match(pat_body, t_body, instsp, bd_vars + [v])
elif pat.is_bound():
raise MatchException
else:
raise TypeError
def solve(goal, pts=None):
"""The main automation function.
If the function succeeds, it should return a proof term whose
proposition is the goal.
"""
if debug_auto:
print("Solve:", goal, [str(pt.prop) for pt in pts])
if pts is None:
pts = []
elif isinstance(pts, tuple):
pts = list(pts)
# First handle the case where goal matches one of the conditions.
for pt in pts:
if goal == pt.prop:
return pt
# Next, consider the situation where one of the assumptions is
# a conjunction or a disjunction.
for i, pt in enumerate(pts):
if pt.prop.is_conj():
pt1 = apply_theorem('conjD1', pt)
pt2 = apply_theorem('conjD2', pt)
return solve(goal, [pt1, pt2] + pts[:i] + pts[i + 1:])
if pt.prop.is_disj():
a1, a2 = pt.prop.args
assume_pt1 = ProofTerm.assume(a1)
assume_pt2 = ProofTerm.assume(a2)
pt1 = solve(goal, [assume_pt1] + pts[:i] + pts[i + 1:])
pt1 = pt1.implies_intr(a1)
pt2 = solve(goal, [assume_pt2] + pts[:i] + pts[i + 1:])
pt2 = pt2.implies_intr(a2)
return apply_theorem('disjE', pt, pt1, pt2)
# Handle various logical connectives.
if goal.is_conj():
a1, a2 = goal.args
pt1 = solve(a1, pts)
pt2 = solve(a2, pts)
return apply_theorem('conjI', pt1, pt2)
if goal.is_disj():
a1, a2 = goal.args
try:
pt1 = solve(a1, pts)
return apply_theorem('disjI1', pt1, concl=goal)
except TacticException:
pt2 = solve(a2, pts)
return apply_theorem('disjI2', pt2, concl=goal)
if goal.is_implies():
a1, a2 = goal.args
assume_pt = ProofTerm.assume(a1)
return solve(a2, [assume_pt] + pts).implies_intr(a1)
if goal.is_forall():
var_names = [
v.name for v in term.get_vars([goal] + [pt.prop for pt in pts])
]
nm = name.get_variant_name(goal.arg.var_name, var_names)
v = Var(nm, goal.arg.var_T)
t = goal.arg.subst_bound(v)
return solve(t, pts).forall_intr(v)
# Normalize goal
eq_pt = norm(goal, pts)
goal = eq_pt.rhs
if goal.is_conj():
pt = solve(goal, pts)
return eq_pt.symmetric().equal_elim(pt)
res_pt = None
if not pts and goal in solve_record:
res_pt = solve_record[goal]
# Call registered functions
elif goal.is_not() and goal.arg.head in global_autos_neg:
for f in global_autos_neg[goal.arg.head]:
try:
res_pt = f(goal, pts)
break
except TacticException:
pass
elif goal.head in global_autos:
for f in global_autos[goal.head]:
try:
res_pt = f(goal, pts)
break
except TacticException:
pass
if res_pt is not None:
if not pts:
solve_record[goal] = res_pt
return eq_pt.symmetric().equal_elim(res_pt)
else:
raise TacticException('Cannot solve %s' % goal)