// If restricted is true, we don't use (e <-> true) rewrite
list<expr_pair> apply(expr const & e, expr const & H, bool restrited) {
expr c, Hdec, A, arg1, arg2;
if (is_relation(e)) {
return mk_singleton(e, H);
} else if (is_standard(m_env) && is_not(m_env, e, arg1)) {
expr new_e = mk_iff(arg1, mk_false());
expr new_H = mk_app(mk_constant(get_iff_false_intro_name()), arg1, H);
return mk_singleton(new_e, new_H);
} else if (is_standard(m_env) && is_and(e, arg1, arg2)) {
// TODO(Leo): we can extend this trick to any type that has only one constructor
expr H1 = mk_app(mk_constant(get_and_elim_left_name()), arg1, arg2, H);
expr H2 = mk_app(mk_constant(get_and_elim_right_name()), arg1, arg2, H);
auto r1 = apply(arg1, H1, restrited);
auto r2 = apply(arg2, H2, restrited);
return append(r1, r2);
} else if (is_pi(e)) {
// TODO(dhs): keep name?
expr local = m_tctx.mk_tmp_local(binding_domain(e), binding_info(e));
expr new_e = instantiate(binding_body(e), local);
expr new_H = mk_app(H, local);
auto r = apply(new_e, new_H, restrited);
unsigned len = length(r);
if (len == 0) {
return r;
} else if (len == 1 && head(r).first == new_e && head(r).second == new_H) {
return mk_singleton(e, H);
} else {
return lift(local, r);
}
} else if (is_standard(m_env) && is_ite(e, c, Hdec, A, arg1, arg2) && is_prop(e)) {
// TODO(Leo): support HoTT mode if users request
expr not_c = mk_app(mk_constant(get_not_name()), c);
expr Hc = m_tctx.mk_tmp_local(c);
expr Hnc = m_tctx.mk_tmp_local(not_c);
expr H1 = mk_app({mk_constant(get_implies_of_if_pos_name()),
c, arg1, arg2, Hdec, e, Hc});
expr H2 = mk_app({mk_constant(get_implies_of_if_neg_name()),
c, arg1, arg2, Hdec, e, Hnc});
auto r1 = lift(Hc, apply(arg1, H1, restrited));
auto r2 = lift(Hnc, apply(arg2, H2, restrited));
return append(r1, r2);
} else if (!restrited) {
expr new_e = m_tctx.whnf(e);
if (new_e != e) {
if (auto r = apply(new_e, H, true))
return r;
}
if (is_standard(m_env) && is_prop(e)) {
expr new_e = mk_iff(e, mk_true());
expr new_H = mk_app(mk_constant(get_iff_true_intro_name()), e, H);
return mk_singleton(new_e, new_H);
} else {
return list<expr_pair>();
}
} else {
return list<expr_pair>();
}
}
action_result assert_cc_action(hypothesis_idx hidx) {
if (!get_config().m_cc)
return action_result::failed();
congruence_closure & cc = get_cc();
if (has_expr_metavar(curr_state().get_hypothesis_decl(hidx).get_type()))
return action_result::failed();
cc.add(hidx);
// cc.display();
if (cc.is_inconsistent()) {
try {
app_builder & b = get_app_builder();
expr false_proof = *cc.get_inconsistency_proof();
trace_action("contradiction by congruence closure");
return action_result(b.mk_false_rec(curr_state().get_target(), false_proof));
} catch (app_builder_exception &) {
return action_result::failed();
}
} else {
expr const & target = curr_state().get_target();
name R; expr lhs, rhs;
if (is_relation_app(target, R, lhs, rhs) && cc.is_eqv(R, lhs, rhs)) {
expr proof = *cc.get_eqv_proof(R, lhs, rhs);
trace_action("equivalence by congruence closure");
return action_result(proof);
} else if (is_prop(target) && !is_false(target) && cc.proved(target)) {
expr proof = *cc.get_proof(target);
trace_action("equivalent to true by congruence closure");
return action_result(proof);
} else {
return action_result::new_branch();
}
}
}
action_result by_contradiction_action() {
state & s = curr_state();
expr target = whnf(s.get_target());
if (!is_prop(target)) return action_result::failed();
if (blast::is_false(target)) return action_result::failed();
expr not_target;
if (is_not(target, not_target)) {
s.set_target(mk_arrow(not_target, mk_constant(get_false_name())));
return intros_action(1);
}
blast_tmp_type_context tmp_tctx;
optional<expr> target_decidable = tmp_tctx->mk_class_instance(mk_app(mk_constant(get_decidable_name()), target));
if (!target_decidable) return action_result::failed();
expr href = s.mk_hypothesis(get_app_builder().mk_not(target));
auto pcell = new by_contradiction_proof_step_cell(href);
s.push_proof_step(pcell);
s.set_target(mk_constant(get_false_name()));
trace_action("by_contradiction");
return action_result::new_branch();
}
environment mk_projections(environment const & env, name const & n, buffer<name> const & proj_names,
implicit_infer_kind infer_k, bool inst_implicit) {
// Given an inductive datatype C A (where A represent parameters)
// intro : Pi A (x_1 : B_1[A]) (x_2 : B_2[A, x_1]) ..., C A
//
// we generate projections of the form
// proj_i A (c : C A) : B_i[A, (proj_1 A n), ..., (proj_{i-1} A n)]
// C.rec A (fun (x : C A), B_i[A, ...]) (fun (x_1 ... x_n), x_i) c
auto p = get_nparam_intro_rule(env, n);
name_generator ngen;
unsigned nparams = p.first;
inductive::intro_rule intro = p.second;
expr intro_type = inductive::intro_rule_type(intro);
name rec_name = inductive::get_elim_name(n);
declaration ind_decl = env.get(n);
if (env.impredicative() && is_prop(ind_decl.get_type()))
throw exception(sstream() << "projection generation, '" << n << "' is a proposition");
declaration rec_decl = env.get(rec_name);
level_param_names lvl_params = ind_decl.get_univ_params();
levels lvls = param_names_to_levels(lvl_params);
buffer<expr> params; // datatype parameters
for (unsigned i = 0; i < nparams; i++) {
if (!is_pi(intro_type))
throw_ill_formed(n);
expr param = mk_local(ngen.next(), binding_name(intro_type), binding_domain(intro_type), binder_info());
intro_type = instantiate(binding_body(intro_type), param);
params.push_back(param);
}
expr C_A = mk_app(mk_constant(n, lvls), params);
binder_info c_bi = inst_implicit ? mk_inst_implicit_binder_info() : binder_info();
expr c = mk_local(ngen.next(), name("c"), C_A, c_bi);
buffer<expr> intro_type_args; // arguments that are not parameters
expr it = intro_type;
while (is_pi(it)) {
expr local = mk_local(ngen.next(), binding_name(it), binding_domain(it), binding_info(it));
intro_type_args.push_back(local);
it = instantiate(binding_body(it), local);
}
buffer<expr> projs; // projections generated so far
unsigned i = 0;
environment new_env = env;
for (name const & proj_name : proj_names) {
if (!is_pi(intro_type))
throw exception(sstream() << "generating projection '" << proj_name << "', '"
<< n << "' does not have sufficient data");
expr result_type = binding_domain(intro_type);
buffer<expr> proj_args;
proj_args.append(params);
proj_args.push_back(c);
expr type_former = Fun(c, result_type);
expr minor_premise = Fun(intro_type_args, mk_var(intro_type_args.size() - i - 1));
expr major_premise = c;
type_checker tc(new_env);
level l = sort_level(tc.ensure_sort(tc.infer(result_type).first).first);
levels rec_lvls = append(to_list(l), lvls);
expr rec = mk_constant(rec_name, rec_lvls);
buffer<expr> rec_args;
rec_args.append(params);
rec_args.push_back(type_former);
rec_args.push_back(minor_premise);
rec_args.push_back(major_premise);
expr rec_app = mk_app(rec, rec_args);
expr proj_type = Pi(proj_args, result_type);
proj_type = infer_implicit_params(proj_type, nparams, infer_k);
expr proj_val = Fun(proj_args, rec_app);
bool opaque = false;
bool use_conv_opt = false;
declaration new_d = mk_definition(env, proj_name, lvl_params, proj_type, proj_val,
opaque, rec_decl.get_module_idx(), use_conv_opt);
new_env = module::add(new_env, check(new_env, new_d));
new_env = set_reducible(new_env, proj_name, reducible_status::Reducible);
new_env = add_unfold_c_hint(new_env, proj_name, nparams);
new_env = save_projection_info(new_env, proj_name, inductive::intro_rule_name(intro), nparams, i, inst_implicit);
expr proj = mk_app(mk_app(mk_constant(proj_name, lvls), params), c);
intro_type = instantiate(binding_body(intro_type), proj);
i++;
}
return new_env;
}
