expr mk_expr_placeholder(optional<expr> const & type, expr_placeholder_kind k) {
name n(to_prefix(k), next_placeholder_id());
if (type)
return mk_local(n, *type);
else
return mk_constant(n);
}
json serialize_decl(name const & short_name, name const & long_name, environment const & env, options const & o) {
declaration const & d = env.get(long_name);
type_context_old tc(env);
auto fmter = mk_pretty_formatter_factory()(env, o, tc);
expr type = d.get_type();
if (LEAN_COMPLETE_CONSUME_IMPLICIT) {
while (true) {
if (!is_pi(type))
break;
if (!binding_info(type).is_implicit() && !binding_info(type).is_inst_implicit())
break;
std::string q("?");
q += binding_name(type).to_string();
expr m = mk_constant(name(q.c_str()));
type = instantiate(binding_body(type), m);
}
}
json completion;
completion["text"] = short_name.to_string();
interactive_report_type(env, o, type, completion);
add_source_info(env, long_name, completion);
if (auto doc = get_doc_string(env, long_name))
completion["doc"] = *doc;
return completion;
}
static optional<pair<expr, expr>> mk_op(environment const & env, old_local_context & ctx, type_checker_ptr & tc,
name const & op, unsigned nunivs, unsigned nargs, std::initializer_list<expr> const & explicit_args,
constraint_seq & cs, tag g) {
levels lvls;
for (unsigned i = 0; i < nunivs; i++)
lvls = levels(mk_meta_univ(mk_fresh_name()), lvls);
expr c = mk_constant(op, lvls);
expr op_type = instantiate_type_univ_params(env.get(op), lvls);
buffer<expr> args;
for (unsigned i = 0; i < nargs; i++) {
if (!is_pi(op_type))
return optional<pair<expr, expr>>();
expr arg = ctx.mk_meta(some_expr(binding_domain(op_type)), g);
args.push_back(arg);
op_type = instantiate(binding_body(op_type), arg);
}
expr r = mk_app(c, args, g);
for (expr const & explicit_arg : explicit_args) {
if (!is_pi(op_type))
return optional<pair<expr, expr>>();
r = mk_app(r, explicit_arg);
expr type = tc->infer(explicit_arg, cs);
justification j = mk_app_justification(r, op_type, explicit_arg, type);
if (!tc->is_def_eq(binding_domain(op_type), type, j, cs))
return optional<pair<expr, expr>>();
op_type = instantiate(binding_body(op_type), explicit_arg);
}
return some(mk_pair(r, op_type));
}
// 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 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();
}
static simp_rule_sets add_core(type_checker & tc, simp_rule_sets const & s, name const & cname) {
declaration const & d = tc.env().get(cname);
buffer<level> us;
unsigned num_univs = d.get_num_univ_params();
for (unsigned i = 0; i < num_univs; i++) {
us.push_back(mk_meta_univ(name(*g_prefix, i)));
}
levels ls = to_list(us);
expr e = instantiate_type_univ_params(d, ls);
expr h = mk_constant(cname, ls);
return add_core(tc, s, cname, ls, e, h);
}
void initialize_string() {
g_string_macro = new name("string_macro");
g_string_opcode = new std::string("Str");
g_nat = new expr(Const(get_nat_name()));
g_char = new expr(Const(get_char_name()));
g_char_of_nat = new expr(Const(get_char_of_nat_name()));
g_string = new expr(Const(get_string_name()));
g_empty = new expr(Const(get_string_empty_name()));
g_str = new expr(Const(get_string_str_name()));
g_fin_mk = new expr(Const(get_fin_mk_name()));
g_list_char = new expr(mk_app(mk_constant(get_list_name(), {mk_level_one()}), *g_char));
g_list_cons = new expr(mk_constant(get_list_cons_name(), {mk_level_one()}));
g_list_nil_char = new expr(mk_app(mk_constant(get_list_nil_name(), {mk_level_one()}), *g_char));
register_macro_deserializer(*g_string_opcode,
[](deserializer & d, unsigned num, expr const *) {
if (num != 0)
throw corrupted_stream_exception();
std::string v = d.read_string();
return mk_string_macro(v);
});
}
expr gexpr::to_expr(type_context & ctx) const {
if (m_univ_poly) {
declaration const & fdecl = ctx.env().get(const_name(m_expr));
buffer<level> ls_buffer;
unsigned num_univ_ps = fdecl.get_num_univ_params();
for (unsigned i = 0; i < num_univ_ps; i++)
ls_buffer.push_back(ctx.mk_uvar());
levels ls = to_list(ls_buffer.begin(), ls_buffer.end());
return mk_constant(const_name(m_expr), ls);
} else {
return m_expr;
}
}
expr copy(expr const & a) {
switch (a.kind()) {
case expr_kind::Var: return mk_var(var_idx(a));
case expr_kind::Constant: return mk_constant(const_name(a));
case expr_kind::Type: return mk_type(ty_level(a));
case expr_kind::Value: return mk_value(static_cast<expr_value*>(a.raw())->m_val);
case expr_kind::App: return mk_app(num_args(a), begin_args(a));
case expr_kind::Eq: return mk_eq(eq_lhs(a), eq_rhs(a));
case expr_kind::Lambda: return mk_lambda(abst_name(a), abst_domain(a), abst_body(a));
case expr_kind::Pi: return mk_pi(abst_name(a), abst_domain(a), abst_body(a));
case expr_kind::Let: return mk_let(let_name(a), let_type(a), let_value(a), let_body(a));
case expr_kind::MetaVar: return mk_metavar(metavar_idx(a), metavar_ctx(a));
}
lean_unreachable();
}
optional<constraints> try_instance(name const & inst) {
environment const & env = m_C->env();
if (auto decl = env.find(inst)) {
name_generator & ngen = m_C->m_ngen;
buffer<level> ls_buffer;
unsigned num_univ_ps = decl->get_num_univ_params();
for (unsigned i = 0; i < num_univ_ps; i++)
ls_buffer.push_back(mk_meta_univ(ngen.next()));
levels ls = to_list(ls_buffer.begin(), ls_buffer.end());
expr inst_cnst = copy_tag(m_meta, mk_constant(inst, ls));
expr inst_type = instantiate_type_univ_params(*decl, ls);
return try_instance(inst_cnst, inst_type);
} else {
return optional<constraints>();
}
}
environment mk_rec_on(environment const & env, name const & n) {
if (!inductive::is_inductive_decl(env, n))
throw exception(sstream() << "error in 'rec_on' generation, '" << n << "' is not an inductive datatype");
name rec_on_name(n, "rec_on");
name_generator ngen;
declaration rec_decl = env.get(inductive::get_elim_name(n));
buffer<expr> locals;
expr rec_type = rec_decl.get_type();
while (is_pi(rec_type)) {
expr local = mk_local(ngen.next(), binding_name(rec_type), binding_domain(rec_type), binding_info(rec_type));
rec_type = instantiate(binding_body(rec_type), local);
locals.push_back(local);
}
// locals order
// A C minor_premises indices major-premise
// new_locals order
// A C indices major-premise minor-premises
buffer<expr> new_locals;
unsigned idx_major_sz = *inductive::get_num_indices(env, n) + 1;
unsigned minor_sz = *inductive::get_num_minor_premises(env, n);
unsigned AC_sz = locals.size() - minor_sz - idx_major_sz;
for (unsigned i = 0; i < AC_sz; i++)
new_locals.push_back(locals[i]);
for (unsigned i = 0; i < idx_major_sz; i++)
new_locals.push_back(locals[AC_sz + minor_sz + i]);
unsigned rec_on_major_idx = new_locals.size() - 1;
for (unsigned i = 0; i < minor_sz; i++)
new_locals.push_back(locals[AC_sz + i]);
expr rec_on_type = Pi(new_locals, rec_type);
levels ls = param_names_to_levels(rec_decl.get_univ_params());
expr rec = mk_constant(rec_decl.get_name(), ls);
expr rec_on_val = Fun(new_locals, mk_app(rec, locals));
bool use_conv_opt = true;
environment new_env = module::add(env,
check(env, mk_definition(env, rec_on_name, rec_decl.get_univ_params(),
rec_on_type, rec_on_val, use_conv_opt)));
new_env = set_reducible(new_env, rec_on_name, reducible_status::Reducible);
new_env = add_unfold_hint(new_env, rec_on_name, rec_on_major_idx);
new_env = add_aux_recursor(new_env, rec_on_name);
return add_protected(new_env, rec_on_name);
}
expr visit_cases_on(name const & fn, buffer<expr> & args) {
name const & I_name = fn.get_prefix();
if (is_inductive_predicate(env(), I_name))
throw exception(sstream() << "code generation failed, inductive predicate '" << I_name << "' is not supported");
bool is_builtin = is_vm_builtin_function(fn);
buffer<name> cnames;
get_intro_rule_names(env(), I_name, cnames);
lean_assert(args.size() >= cnames.size() + 1);
if (args.size() > cnames.size() + 1)
distribute_extra_args_over_minors(I_name, cnames, args);
lean_assert(args.size() == cnames.size() + 1);
/* Process major premise */
args[0] = visit(args[0]);
unsigned num_reachable = 0;
optional<expr> reachable_case;
/* Process minor premises */
for (unsigned i = 0; i < cnames.size(); i++) {
buffer<bool> rel_fields;
get_constructor_info(cnames[i], rel_fields);
auto p = visit_minor_premise(args[i+1], rel_fields);
expr new_minor = p.first;
if (i == 0 && has_trivial_structure(I_name, rel_fields)) {
/* Optimization for an inductive datatype that has a single constructor with only one relevant field */
return beta_reduce(mk_app(new_minor, args[0]));
}
args[i+1] = new_minor;
if (!p.second) {
num_reachable++;
reachable_case = p.first;
}
}
if (num_reachable == 0) {
return mk_unreachable_expr();
} else if (num_reachable == 1 && !is_builtin) {
/* Use _cases.1 */
return mk_app(mk_cases(1), args[0], *reachable_case);
} else if (is_builtin) {
return mk_app(mk_constant(fn), args);
} else {
return mk_app(mk_cases(cnames.size()), args);
}
}
expr extract(expr const & e) {
lean_assert(is_nested_declaration(e));
expr const & d = visit(get_nested_declaration_arg(e));
name new_name = mk_name_for(e);
name new_real_name = get_namespace(m_env) + new_name;
collected_locals locals;
collect_locals(d, locals);
buffer<name> uparams;
collect_univ_params(d).to_buffer(uparams);
expr new_value = Fun(locals.get_collected(), d);
expr new_type = m_tc.infer(new_value).first;
level_param_names new_ps = to_list(uparams);
levels ls = param_names_to_levels(new_ps);
m_env = module::add(m_env, check(m_env, mk_definition(m_env, new_real_name, new_ps,
new_type, new_value)));
if (new_name != new_real_name)
m_env = add_expr_alias_rec(m_env, new_name, new_real_name);
decl_attributes const & attrs = get_nested_declaration_attributes(e);
m_env = attrs.apply(m_env, m_ios, new_real_name, get_namespace(m_env));
return mk_app(mk_constant(new_real_name, ls), locals.get_collected());
}
pair<environment, expr> operator()(name const & c, expr const & type, expr const & value, bool is_lemma, optional<bool> const & is_meta) {
lean_assert(!is_lemma || is_meta);
lean_assert(!is_lemma || *is_meta == false);
expr new_type = collect(m_ctx.instantiate_mvars(type));
expr new_value = collect(m_ctx.instantiate_mvars(value));
buffer<expr> norm_params;
collect_and_normalize_dependencies(norm_params);
new_type = replace_locals(new_type, m_params, norm_params);
new_value = replace_locals(new_value, m_params, norm_params);
expr def_type = m_ctx.mk_pi(norm_params, new_type);
expr def_value = m_ctx.mk_lambda(norm_params, new_value);
environment const & env = m_ctx.env();
declaration d;
if (is_lemma) {
d = mk_theorem(c, to_list(m_level_params), def_type, def_value);
} else if (is_meta) {
bool use_self_opt = true;
d = mk_definition(env, c, to_list(m_level_params), def_type, def_value, use_self_opt, !*is_meta);
} else {
bool use_self_opt = true;
d = mk_definition_inferring_trusted(env, c, to_list(m_level_params), def_type, def_value, use_self_opt);
}
environment new_env = module::add(env, check(env, d, true));
buffer<level> ls;
for (name const & n : m_level_params) {
if (level const * l = m_univ_meta_to_param_inv.find(n))
ls.push_back(*l);
else
ls.push_back(mk_param_univ(n));
}
buffer<expr> ps;
for (expr const & x : m_params) {
if (expr const * m = m_meta_to_param_inv.find(mlocal_name(x)))
ps.push_back(*m);
else
ps.push_back(x);
}
expr r = mk_app(mk_constant(c, to_list(ls)), ps);
return mk_pair(new_env, r);
}
expr visit_constructor(name const & fn, buffer<expr> const & args) {
bool is_builtin = is_vm_builtin_function(fn);
name I_name = *inductive::is_intro_rule(env(), fn);
unsigned nparams = *inductive::get_num_params(env(), I_name);
unsigned cidx = get_constructor_idx(env(), fn);
buffer<bool> rel_fields;
get_constructor_info(fn, rel_fields);
lean_assert(args.size() == nparams + rel_fields.size());
buffer<expr> new_args;
for (unsigned i = 0; i < rel_fields.size(); i++) {
if (rel_fields[i]) {
new_args.push_back(visit(args[nparams + i]));
}
}
if (has_trivial_structure(I_name, rel_fields)) {
lean_assert(new_args.size() == 1);
return new_args[0];
} else if (is_builtin) {
return mk_app(mk_constant(fn), new_args);
} else {
return mk_app(mk_cnstr(cidx), new_args);
}
}
optional<expr> mk_hset_instance(type_checker & tc, io_state const & ios, list<expr> const & ctx, expr const & type) {
expr trunc_index = mk_app(mk_constant(get_is_trunc_trunc_index_of_nat_name()), mk_constant(get_nat_zero_name()));
level lvl = sort_level(tc.ensure_type(type).first);
expr is_hset = mk_app(mk_constant(get_is_trunc_name(), {lvl}), trunc_index, type);
return mk_class_instance(tc.env(), ios, ctx, tc.mk_fresh_name(), is_hset);
}
struct pnode *make_constant_list(int value1, int value2)
{
return mk_list(mk_constant(value1), mk_list(mk_constant(value2), NULL));
}
void initialize_expr() {
g_dummy = new expr(mk_constant("__expr_for_default_constructor__"));
g_default_name = new name("a");
g_Type1 = new expr(mk_sort(mk_level_one()));
g_Prop = new expr(mk_sort(mk_level_zero()));
}
static expr mk_proj(unsigned idx) {
return mk_constant(name(*g_proj, idx));
}
static expr mk_cases(unsigned n) {
return mk_constant(name(*g_cases, n));
}
expr update_constant(expr const & e, levels const & new_levels) {
if (!is_eqp(const_levels(e), new_levels))
return mk_constant(const_name(e), new_levels, e.get_tag());
else
return e;
}
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;
}
tactic contradiction_tactic() {
auto fn = [=](environment const & env, io_state const & ios, proof_state const & s) {
goals const & gs = s.get_goals();
if (empty(gs)) {
throw_no_goal_if_enabled(s);
return optional<proof_state>();
}
goal const & g = head(gs);
expr const & t = g.get_type();
substitution subst = s.get_subst();
auto tc = mk_type_checker(env);
auto conserv_tc = mk_type_checker(env, UnfoldReducible);
buffer<expr> hyps;
g.get_hyps(hyps);
for (expr const & h : hyps) {
expr h_type = mlocal_type(h);
h_type = tc->whnf(h_type).first;
expr lhs, rhs, arg;
if (is_false(env, h_type)) {
assign(subst, g, mk_false_rec(*tc, h, t));
return some_proof_state(proof_state(s, tail(gs), subst));
} else if (is_not(env, h_type, arg)) {
optional<expr> h_pos;
for (expr const & h_prime : hyps) {
constraint_seq cs;
if (conserv_tc->is_def_eq(arg, mlocal_type(h_prime), justification(), cs) && !cs) {
h_pos = h_prime;
break;
}
}
if (h_pos) {
assign(subst, g, mk_absurd(*tc, t, *h_pos, h));
return some_proof_state(proof_state(s, tail(gs), subst));
}
} else if (is_eq(h_type, lhs, rhs)) {
lhs = tc->whnf(lhs).first;
rhs = tc->whnf(rhs).first;
optional<name> lhs_c = is_constructor_app(env, lhs);
optional<name> rhs_c = is_constructor_app(env, rhs);
if (lhs_c && rhs_c && *lhs_c != *rhs_c) {
if (optional<name> I_name = inductive::is_intro_rule(env, *lhs_c)) {
name no_confusion(*I_name, "no_confusion");
try {
expr I = tc->whnf(tc->infer(lhs).first).first;
buffer<expr> args;
expr I_fn = get_app_args(I, args);
if (is_constant(I_fn)) {
level t_lvl = sort_level(tc->ensure_type(t).first);
expr V = mk_app(mk_app(mk_constant(no_confusion, cons(t_lvl, const_levels(I_fn))), args),
t, lhs, rhs, h);
if (auto r = lift_down_if_hott(*tc, V)) {
check_term(*tc, *r);
assign(subst, g, *r);
return some_proof_state(proof_state(s, tail(gs), subst));
}
}
} catch (kernel_exception & ex) {
regular(env, ios) << ex << "\n";
}
}
}
}
}
return none_proof_state();
};
return tactic01(fn);
}
void add_congr_core(environment const & env, simp_rule_sets & s, name const & n) {
declaration const & d = env.get(n);
type_checker tc(env);
buffer<level> us;
unsigned num_univs = d.get_num_univ_params();
for (unsigned i = 0; i < num_univs; i++) {
us.push_back(mk_meta_univ(name(*g_prefix, i)));
}
levels ls = to_list(us);
expr pr = mk_constant(n, ls);
expr e = instantiate_type_univ_params(d, ls);
buffer<bool> explicit_args;
buffer<expr> metas;
unsigned idx = 0;
while (is_pi(e)) {
expr mvar = mk_metavar(name(*g_prefix, idx), binding_domain(e));
idx++;
explicit_args.push_back(is_explicit(binding_info(e)));
metas.push_back(mvar);
e = instantiate(binding_body(e), mvar);
pr = mk_app(pr, mvar);
}
expr rel, lhs, rhs;
if (!is_simp_relation(env, e, rel, lhs, rhs) || !is_constant(rel)) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' resulting type is not of the form t ~ s, where '~' is a transitive and reflexive relation");
}
name_set found_mvars;
buffer<expr> lhs_args, rhs_args;
expr const & lhs_fn = get_app_args(lhs, lhs_args);
expr const & rhs_fn = get_app_args(rhs, rhs_args);
if (is_constant(lhs_fn)) {
if (!is_constant(rhs_fn) || const_name(lhs_fn) != const_name(rhs_fn) || lhs_args.size() != rhs_args.size()) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' resulting type is not of the form (" << const_name(lhs_fn) << " ...) "
<< "~ (" << const_name(lhs_fn) << " ...), where ~ is '" << const_name(rel) << "'");
}
for (expr const & lhs_arg : lhs_args) {
if (is_sort(lhs_arg))
continue;
if (!is_metavar(lhs_arg) || found_mvars.contains(mlocal_name(lhs_arg))) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' the left-hand-side of the congruence resulting type must be of the form ("
<< const_name(lhs_fn) << " x_1 ... x_n), where each x_i is a distinct variable or a sort");
}
found_mvars.insert(mlocal_name(lhs_arg));
}
} else if (is_binding(lhs)) {
if (lhs.kind() != rhs.kind()) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' kinds of the left-hand-side and right-hand-side of "
<< "the congruence resulting type do not match");
}
if (!is_valid_congr_rule_binding_lhs(lhs, found_mvars)) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' left-hand-side of the congruence resulting type must "
<< "be of the form (fun/Pi (x : A), B x)");
}
} else {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' left-hand-side is not an application nor a binding");
}
buffer<expr> congr_hyps;
lean_assert(metas.size() == explicit_args.size());
for (unsigned i = 0; i < metas.size(); i++) {
expr const & mvar = metas[i];
if (explicit_args[i] && !found_mvars.contains(mlocal_name(mvar))) {
buffer<expr> locals;
expr type = mlocal_type(mvar);
while (is_pi(type)) {
expr local = mk_local(tc.mk_fresh_name(), binding_domain(type));
locals.push_back(local);
type = instantiate(binding_body(type), local);
}
expr h_rel, h_lhs, h_rhs;
if (!is_simp_relation(env, type, h_rel, h_lhs, h_rhs) || !is_constant(h_rel))
continue;
unsigned j = 0;
for (expr const & local : locals) {
j++;
if (!only_found_mvars(mlocal_type(local), found_mvars)) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' argument #" << j << " of parameter #" << (i+1) << " contains "
<< "unresolved parameters");
}
}
if (!only_found_mvars(h_lhs, found_mvars)) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' argument #" << (i+1) << " is not a valid hypothesis, the left-hand-side contains "
<< "unresolved parameters");
}
if (!is_valid_congr_hyp_rhs(h_rhs, found_mvars)) {
throw exception(sstream() << "invalid congruence rule, '" << n
<< "' argument #" << (i+1) << " is not a valid hypothesis, the right-hand-side must be "
<< "of the form (m l_1 ... l_n) where m is parameter that was not "
<< "'assigned/resolved' yet and l_i's are locals");
}
found_mvars.insert(mlocal_name(mvar));
congr_hyps.push_back(mvar);
}
}
congr_rule rule(n, ls, to_list(metas), lhs, rhs, pr, to_list(congr_hyps));
s.insert(const_name(rel), rule);
}
environment mk_no_confusion(environment const & env, name const & n) {
optional<environment> env1 = mk_no_confusion_type(env, n);
if (!env1)
return env;
environment new_env = *env1;
type_checker tc(new_env);
inductive::inductive_decls decls = *inductive::is_inductive_decl(new_env, n);
unsigned nparams = std::get<1>(decls);
name_generator ngen;
declaration no_confusion_type_decl = new_env.get(name{n, "no_confusion_type"});
declaration cases_decl = new_env.get(name(n, "cases_on"));
level_param_names lps = no_confusion_type_decl.get_univ_params();
levels ls = param_names_to_levels(lps);
expr no_confusion_type_type = instantiate_type_univ_params(no_confusion_type_decl, ls);
name eq_name("eq");
name heq_name("heq");
name eq_refl_name{"eq", "refl"};
name heq_refl_name{"heq", "refl"};
buffer<expr> args;
expr type = no_confusion_type_type;
type = to_telescope(ngen, type, args, some(mk_implicit_binder_info()));
lean_assert(args.size() >= nparams + 3);
unsigned nindices = args.size() - nparams - 3; // 3 is for P v1 v2
expr range = mk_app(mk_constant(no_confusion_type_decl.get_name(), ls), args);
expr P = args[args.size()-3];
expr v1 = args[args.size()-2];
expr v2 = args[args.size()-1];
expr v_type = mlocal_type(v1);
level v_lvl = sort_level(tc.ensure_type(v_type).first);
expr eq_v = mk_app(mk_constant(eq_name, to_list(v_lvl)), v_type);
expr H12 = mk_local(ngen.next(), "H12", mk_app(eq_v, v1, v2), binder_info());
args.push_back(H12);
name no_confusion_name{n, "no_confusion"};
expr no_confusion_ty = Pi(args, range);
// The gen proof is of the form
// (fun H11 : v1 = v1, cases_on Params (fun Indices v1, no_confusion_type Params Indices P v1 v1) Indices v1
// <for-each case>
// (fun H : (equations -> P), H (refl) ... (refl))
// ...
// )
// H11 is for creating the generalization
expr H11 = mk_local(ngen.next(), "H11", mk_app(eq_v, v1, v1), binder_info());
// Create the type former (fun Indices v1, no_confusion_type Params Indices P v1 v1)
buffer<expr> type_former_args;
for (unsigned i = nparams; i < nparams + nindices; i++)
type_former_args.push_back(args[i]);
type_former_args.push_back(v1);
buffer<expr> no_confusion_type_args;
for (unsigned i = 0; i < nparams + nindices; i++)
no_confusion_type_args.push_back(args[i]);
no_confusion_type_args.push_back(P);
no_confusion_type_args.push_back(v1);
no_confusion_type_args.push_back(v1);
expr no_confusion_type_app = mk_app(mk_constant(no_confusion_type_decl.get_name(), ls), no_confusion_type_args);
expr type_former = Fun(type_former_args, no_confusion_type_app);
// create cases_on
levels clvls = ls;
expr cases_on = mk_app(mk_app(mk_constant(cases_decl.get_name(), clvls), nparams, args.data()), type_former);
cases_on = mk_app(mk_app(cases_on, nindices, args.data() + nparams), v1);
expr cot = tc.infer(cases_on).first;
while (is_pi(cot)) {
buffer<expr> minor_args;
expr minor = to_telescope(tc, binding_domain(cot), minor_args);
lean_assert(!minor_args.empty());
expr H = minor_args.back();
expr Ht = mlocal_type(H);
buffer<expr> refl_args;
while (is_pi(Ht)) {
buffer<expr> eq_args;
expr eq_fn = get_app_args(binding_domain(Ht), eq_args);
if (const_name(eq_fn) == eq_name) {
refl_args.push_back(mk_app(mk_constant(eq_refl_name, const_levels(eq_fn)), eq_args[0], eq_args[1]));
} else {
refl_args.push_back(mk_app(mk_constant(heq_refl_name, const_levels(eq_fn)), eq_args[0], eq_args[1]));
}
Ht = binding_body(Ht);
}
expr pr = mk_app(H, refl_args);
cases_on = mk_app(cases_on, Fun(minor_args, pr));
cot = binding_body(cot);
}
expr gen = Fun(H11, cases_on);
// Now, we use gen to build the final proof using eq.rec
//
// eq.rec InductiveType v1 (fun (a : InductiveType), v1 = a -> no_confusion_type Params Indices v1 a) gen v2 H12 H12
//
name eq_rec_name{"eq", "rec"};
expr eq_rec = mk_app(mk_constant(eq_rec_name, {head(ls), v_lvl}), v_type, v1);
// create eq_rec type_former
// (fun (a : InductiveType), v1 = a -> no_confusion_type Params Indices v1 a)
expr a = mk_local(ngen.next(), "a", v_type, binder_info());
expr H1a = mk_local(ngen.next(), "H1a", mk_app(eq_v, v1, a), binder_info());
// reusing no_confusion_type_args... we just replace the last argument with a
no_confusion_type_args.pop_back();
no_confusion_type_args.push_back(a);
expr no_confusion_type_app_1a = mk_app(mk_constant(no_confusion_type_decl.get_name(), ls), no_confusion_type_args);
expr rec_type_former = Fun(a, Pi(H1a, no_confusion_type_app_1a));
// finalize eq_rec
eq_rec = mk_app(mk_app(eq_rec, rec_type_former, gen, v2, H12), H12);
//
expr no_confusion_val = Fun(args, eq_rec);
bool opaque = false;
bool use_conv_opt = true;
declaration new_d = mk_definition(new_env, no_confusion_name, lps, no_confusion_ty, no_confusion_val,
opaque, no_confusion_type_decl.get_module_idx(), use_conv_opt);
new_env = module::add(new_env, check(new_env, new_d));
return add_protected(new_env, no_confusion_name);
}
static expr mk_cnstr(unsigned cidx) {
return mk_constant(name(*g_cnstr, cidx));
}
static environment mk_brec_on(environment const & env, name const & n, bool ind) {
if (!is_recursive_datatype(env, n))
return env;
if (is_inductive_predicate(env, n))
return env;
inductive::inductive_decls decls = *inductive::is_inductive_decl(env, n);
type_checker tc(env);
name_generator ngen;
unsigned nparams = std::get<1>(decls);
declaration ind_decl = env.get(n);
declaration rec_decl = env.get(inductive::get_elim_name(n));
// declaration below_decl = env.get(name(n, ind ? "ibelow" : "below"));
unsigned nindices = *inductive::get_num_indices(env, n);
unsigned nminors = *inductive::get_num_minor_premises(env, n);
unsigned ntypeformers = length(std::get<2>(decls));
level_param_names lps = rec_decl.get_univ_params();
bool is_reflexive = is_reflexive_datatype(tc, n);
level lvl = mk_param_univ(head(lps));
levels lvls = param_names_to_levels(tail(lps));
level rlvl;
level_param_names blps;
levels blvls; // universe level parameters of brec_on/binduction_on
// The arguments of brec_on (binduction_on) are the ones in the recursor - minor premises.
// The universe we map to is also different (l+1 for below of reflexive types) and (0 fo ibelow).
expr ref_type;
if (ind) {
// we are eliminating to Prop
blps = tail(lps);
blvls = lvls;
rlvl = mk_level_zero();
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_level_zero());
} else if (is_reflexive) {
blps = lps;
blvls = cons(lvl, lvls);
rlvl = get_datatype_level(ind_decl.get_type());
// if rlvl is of the form (max 1 l), then rlvl <- l
if (is_max(rlvl) && is_one(max_lhs(rlvl)))
rlvl = max_rhs(rlvl);
rlvl = mk_max(mk_succ(lvl), rlvl);
// inner_prod, inner_prod_intro, pr1, pr2 do not use the same universe levels for
// reflective datatypes.
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_succ(lvl));
} else {
// we can simplify the universe levels for non-reflexive datatypes
blps = lps;
blvls = cons(lvl, lvls);
rlvl = mk_max(mk_level_one(), lvl);
ref_type = rec_decl.get_type();
}
buffer<expr> ref_args;
to_telescope(ngen, ref_type, ref_args);
if (ref_args.size() != nparams + ntypeformers + nminors + nindices + 1)
throw_corrupted(n);
// args contains the brec_on/binduction_on arguments
buffer<expr> args;
buffer<name> typeformer_names;
// add parameters and typeformers
for (unsigned i = 0; i < nparams; i++)
args.push_back(ref_args[i]);
for (unsigned i = nparams; i < nparams + ntypeformers; i++) {
args.push_back(ref_args[i]);
typeformer_names.push_back(mlocal_name(ref_args[i]));
}
// add indices and major premise
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
args.push_back(ref_args[i]);
// create below terms (one per datatype)
// (below.{lvls} params type-formers)
// Remark: it also creates the result type
buffer<expr> belows;
expr result_type;
unsigned k = 0;
for (auto const & decl : std::get<2>(decls)) {
name const & n1 = inductive::inductive_decl_name(decl);
if (n1 == n) {
result_type = ref_args[nparams + k];
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
result_type = mk_app(result_type, ref_args[i]);
}
k++;
name bname = name(n1, ind ? "ibelow" : "below");
expr below = mk_constant(bname, blvls);
for (unsigned i = 0; i < nparams; i++)
below = mk_app(below, ref_args[i]);
for (unsigned i = nparams; i < nparams + ntypeformers; i++)
below = mk_app(below, ref_args[i]);
belows.push_back(below);
}
// create functionals (one for each type former)
// Pi idxs t, below idxs t -> C idxs t
buffer<expr> Fs;
name F_name("F");
for (unsigned i = nparams, j = 0; i < nparams + ntypeformers; i++, j++) {
expr const & C = ref_args[i];
buffer<expr> F_args;
to_telescope(ngen, mlocal_type(C), F_args);
expr F_result = mk_app(C, F_args);
expr F_below = mk_app(belows[j], F_args);
F_args.push_back(mk_local(ngen.next(), "f", F_below, binder_info()));
expr F_type = Pi(F_args, F_result);
expr F = mk_local(ngen.next(), F_name.append_after(j+1), F_type, binder_info());
Fs.push_back(F);
args.push_back(F);
}
// We define brec_on/binduction_on using the recursor for this type
levels rec_lvls = cons(rlvl, lvls);
expr rec = mk_constant(rec_decl.get_name(), rec_lvls);
// add parameters to rec
for (unsigned i = 0; i < nparams; i++)
rec = mk_app(rec, ref_args[i]);
// add type formers to rec
// Pi indices t, prod (C ... t) (below ... t)
for (unsigned i = nparams, j = 0; i < nparams + ntypeformers; i++, j++) {
expr const & C = ref_args[i];
buffer<expr> C_args;
to_telescope(ngen, mlocal_type(C), C_args);
expr C_t = mk_app(C, C_args);
expr below_t = mk_app(belows[j], C_args);
expr prod = mk_prod(tc, C_t, below_t, ind);
rec = mk_app(rec, Fun(C_args, prod));
}
// add minor premises to rec
for (unsigned i = nparams + ntypeformers, j = 0; i < nparams + ntypeformers + nminors; i++, j++) {
expr minor = ref_args[i];
expr minor_type = mlocal_type(minor);
buffer<expr> minor_args;
minor_type = to_telescope(ngen, minor_type, minor_args);
buffer<expr> pairs;
for (expr & minor_arg : minor_args) {
buffer<expr> minor_arg_args;
expr minor_arg_type = to_telescope(tc, mlocal_type(minor_arg), minor_arg_args);
if (auto k = is_typeformer_app(typeformer_names, minor_arg_type)) {
buffer<expr> C_args;
get_app_args(minor_arg_type, C_args);
expr new_minor_arg_type = mk_prod(tc, minor_arg_type, mk_app(belows[*k], C_args), ind);
minor_arg = update_mlocal(minor_arg, Pi(minor_arg_args, new_minor_arg_type));
if (minor_arg_args.empty()) {
pairs.push_back(minor_arg);
} else {
expr r = mk_app(minor_arg, minor_arg_args);
expr r_1 = Fun(minor_arg_args, mk_pr1(tc, r, ind));
expr r_2 = Fun(minor_arg_args, mk_pr2(tc, r, ind));
pairs.push_back(mk_pair(tc, r_1, r_2, ind));
}
}
}
expr b = foldr([&](expr const & a, expr const & b) { return mk_pair(tc, a, b, ind); },
[&]() { return mk_unit_mk(rlvl, ind); },
pairs.size(), pairs.data());
unsigned F_idx = *is_typeformer_app(typeformer_names, minor_type);
expr F = Fs[F_idx];
buffer<expr> F_args;
get_app_args(minor_type, F_args);
F_args.push_back(b);
expr new_arg = mk_pair(tc, mk_app(F, F_args), b, ind);
rec = mk_app(rec, Fun(minor_args, new_arg));
}
// add indices and major to rec
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
rec = mk_app(rec, ref_args[i]);
name brec_on_name = name(n, ind ? "binduction_on" : "brec_on");
expr brec_on_type = Pi(args, result_type);
expr brec_on_value = Fun(args, mk_pr1(tc, rec, ind));
bool use_conv_opt = true;
declaration new_d = mk_definition(env, brec_on_name, blps, brec_on_type, brec_on_value,
use_conv_opt);
environment new_env = module::add(env, check(env, new_d));
new_env = set_reducible(new_env, brec_on_name, reducible_status::Reducible);
if (!ind)
new_env = add_unfold_hint(new_env, brec_on_name, nparams + nindices + ntypeformers);
return add_protected(new_env, brec_on_name);
}
optional<environment> mk_no_confusion_type(environment const & env, name const & n) {
optional<inductive::inductive_decls> decls = inductive::is_inductive_decl(env, n);
if (!decls)
throw exception(sstream() << "error in 'no_confusion' generation, '" << n << "' is not an inductive datatype");
if (is_inductive_predicate(env, n))
return optional<environment>(); // type is a proposition
name_generator ngen;
unsigned nparams = std::get<1>(*decls);
declaration ind_decl = env.get(n);
declaration cases_decl = env.get(name(n, "cases_on"));
level_param_names lps = cases_decl.get_univ_params();
level rlvl = mk_param_univ(head(lps));
levels ilvls = param_names_to_levels(tail(lps));
if (length(ilvls) != length(ind_decl.get_univ_params()))
return optional<environment>(); // type does not have only a restricted eliminator
expr ind_type = instantiate_type_univ_params(ind_decl, ilvls);
name eq_name("eq");
name heq_name("heq");
// All inductive datatype parameters and indices are arguments
buffer<expr> args;
ind_type = to_telescope(ngen, ind_type, args, some(mk_implicit_binder_info()));
if (!is_sort(ind_type) || args.size() < nparams)
throw_corrupted(n);
lean_assert(!(env.impredicative() && is_zero(sort_level(ind_type))));
unsigned nindices = args.size() - nparams;
// Create inductive datatype
expr I = mk_app(mk_constant(n, ilvls), args);
// Add (P : Type)
expr P = mk_local(ngen.next(), "P", mk_sort(rlvl), binder_info());
args.push_back(P);
// add v1 and v2 elements of the inductive type
expr v1 = mk_local(ngen.next(), "v1", I, binder_info());
expr v2 = mk_local(ngen.next(), "v2", I, binder_info());
args.push_back(v1);
args.push_back(v2);
expr R = mk_sort(rlvl);
name no_confusion_type_name{n, "no_confusion_type"};
expr no_confusion_type_type = Pi(args, R);
// Create type former
buffer<expr> type_former_args;
for (unsigned i = nparams; i < nparams + nindices; i++)
type_former_args.push_back(args[i]);
type_former_args.push_back(v1);
expr type_former = Fun(type_former_args, R);
// Create cases_on
levels clvls = levels(mk_succ(rlvl), ilvls);
expr cases_on = mk_app(mk_app(mk_constant(cases_decl.get_name(), clvls), nparams, args.data()), type_former);
cases_on = mk_app(cases_on, nindices, args.data() + nparams);
expr cases_on1 = mk_app(cases_on, v1);
expr cases_on2 = mk_app(cases_on, v2);
type_checker tc(env);
expr t1 = tc.infer(cases_on1).first;
expr t2 = tc.infer(cases_on2).first;
buffer<expr> outer_cases_on_args;
unsigned idx1 = 0;
while (is_pi(t1)) {
buffer<expr> minor1_args;
expr minor1 = to_telescope(tc, binding_domain(t1), minor1_args);
expr curr_t2 = t2;
buffer<expr> inner_cases_on_args;
unsigned idx2 = 0;
while (is_pi(curr_t2)) {
buffer<expr> minor2_args;
expr minor2 = to_telescope(tc, binding_domain(curr_t2), minor2_args);
if (idx1 != idx2) {
// infeasible case, constructors do not match
inner_cases_on_args.push_back(Fun(minor2_args, P));
} else {
if (minor1_args.size() != minor2_args.size())
throw_corrupted(n);
buffer<expr> rtype_hyp;
// add equalities
for (unsigned i = 0; i < minor1_args.size(); i++) {
expr lhs = minor1_args[i];
expr rhs = minor2_args[i];
expr lhs_type = mlocal_type(lhs);
expr rhs_type = mlocal_type(rhs);
level l = sort_level(tc.ensure_type(lhs_type).first);
expr h_type;
if (tc.is_def_eq(lhs_type, rhs_type).first) {
h_type = mk_app(mk_constant(eq_name, to_list(l)), lhs_type, lhs, rhs);
} else {
h_type = mk_app(mk_constant(heq_name, to_list(l)), lhs_type, lhs, rhs_type, rhs);
}
rtype_hyp.push_back(mk_local(ngen.next(), local_pp_name(lhs).append_after("_eq"), h_type, binder_info()));
}
inner_cases_on_args.push_back(Fun(minor2_args, mk_arrow(Pi(rtype_hyp, P), P)));
}
idx2++;
curr_t2 = binding_body(curr_t2);
}
outer_cases_on_args.push_back(Fun(minor1_args, mk_app(cases_on2, inner_cases_on_args)));
idx1++;
t1 = binding_body(t1);
}
expr no_confusion_type_value = Fun(args, mk_app(cases_on1, outer_cases_on_args));
bool opaque = false;
bool use_conv_opt = true;
declaration new_d = mk_definition(env, no_confusion_type_name, lps, no_confusion_type_type, no_confusion_type_value,
opaque, ind_decl.get_module_idx(), use_conv_opt);
environment new_env = module::add(env, check(env, new_d));
return some(add_protected(new_env, no_confusion_type_name));
}
static environment mk_below(environment const & env, name const & n, bool ibelow) {
if (!is_recursive_datatype(env, n))
return env;
if (is_inductive_predicate(env, n))
return env;
inductive::inductive_decls decls = *inductive::is_inductive_decl(env, n);
type_checker tc(env);
name_generator ngen;
unsigned nparams = std::get<1>(decls);
declaration ind_decl = env.get(n);
declaration rec_decl = env.get(inductive::get_elim_name(n));
unsigned nindices = *inductive::get_num_indices(env, n);
unsigned nminors = *inductive::get_num_minor_premises(env, n);
unsigned ntypeformers = length(std::get<2>(decls));
level_param_names lps = rec_decl.get_univ_params();
bool is_reflexive = is_reflexive_datatype(tc, n);
level lvl = mk_param_univ(head(lps));
levels lvls = param_names_to_levels(tail(lps));
level_param_names blvls; // universe level parameters of ibelow/below
level rlvl; // universe level of the resultant type
// The arguments of below (ibelow) are the ones in the recursor - minor premises.
// The universe we map to is also different (l+1 for below of reflexive types) and (0 fo ibelow).
expr ref_type;
expr Type_result;
if (ibelow) {
// we are eliminating to Prop
blvls = tail(lps);
rlvl = mk_level_zero();
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_level_zero());
} else if (is_reflexive) {
blvls = lps;
rlvl = get_datatype_level(ind_decl.get_type());
// if rlvl is of the form (max 1 l), then rlvl <- l
if (is_max(rlvl) && is_one(max_lhs(rlvl)))
rlvl = max_rhs(rlvl);
rlvl = mk_max(mk_succ(lvl), rlvl);
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_succ(lvl));
} else {
// we can simplify the universe levels for non-reflexive datatypes
blvls = lps;
rlvl = mk_max(mk_level_one(), lvl);
ref_type = rec_decl.get_type();
}
Type_result = mk_sort(rlvl);
buffer<expr> ref_args;
to_telescope(ngen, ref_type, ref_args);
if (ref_args.size() != nparams + ntypeformers + nminors + nindices + 1)
throw_corrupted(n);
// args contains the below/ibelow arguments
buffer<expr> args;
buffer<name> typeformer_names;
// add parameters and typeformers
for (unsigned i = 0; i < nparams; i++)
args.push_back(ref_args[i]);
for (unsigned i = nparams; i < nparams + ntypeformers; i++) {
args.push_back(ref_args[i]);
typeformer_names.push_back(mlocal_name(ref_args[i]));
}
// we ignore minor premises in below/ibelow
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
args.push_back(ref_args[i]);
// We define below/ibelow using the recursor for this type
levels rec_lvls = cons(mk_succ(rlvl), lvls);
expr rec = mk_constant(rec_decl.get_name(), rec_lvls);
for (unsigned i = 0; i < nparams; i++)
rec = mk_app(rec, args[i]);
// add type formers
for (unsigned i = nparams; i < nparams + ntypeformers; i++) {
buffer<expr> targs;
to_telescope(ngen, mlocal_type(args[i]), targs);
rec = mk_app(rec, Fun(targs, Type_result));
}
// add minor premises
for (unsigned i = nparams + ntypeformers; i < nparams + ntypeformers + nminors; i++) {
expr minor = ref_args[i];
expr minor_type = mlocal_type(minor);
buffer<expr> minor_args;
minor_type = to_telescope(ngen, minor_type, minor_args);
buffer<expr> prod_pairs;
for (expr & minor_arg : minor_args) {
buffer<expr> minor_arg_args;
expr minor_arg_type = to_telescope(tc, mlocal_type(minor_arg), minor_arg_args);
if (is_typeformer_app(typeformer_names, minor_arg_type)) {
expr fst = mlocal_type(minor_arg);
minor_arg = update_mlocal(minor_arg, Pi(minor_arg_args, Type_result));
expr snd = Pi(minor_arg_args, mk_app(minor_arg, minor_arg_args));
prod_pairs.push_back(mk_prod(tc, fst, snd, ibelow));
}
}
expr new_arg = foldr([&](expr const & a, expr const & b) { return mk_prod(tc, a, b, ibelow); },
[&]() { return mk_unit(rlvl, ibelow); },
prod_pairs.size(), prod_pairs.data());
rec = mk_app(rec, Fun(minor_args, new_arg));
}
// add indices and major premise
for (unsigned i = nparams + ntypeformers; i < args.size(); i++) {
rec = mk_app(rec, args[i]);
}
name below_name = ibelow ? name{n, "ibelow"} : name{n, "below"};
expr below_type = Pi(args, Type_result);
expr below_value = Fun(args, rec);
bool use_conv_opt = true;
declaration new_d = mk_definition(env, below_name, blvls, below_type, below_value,
use_conv_opt);
environment new_env = module::add(env, check(env, new_d));
new_env = set_reducible(new_env, below_name, reducible_status::Reducible);
if (!ibelow)
new_env = add_unfold_hint(new_env, below_name, nparams + nindices + ntypeformers);
return add_protected(new_env, below_name);
}
struct pnode *add_symbol_constant(struct pnode *parms, int value)
{
return mk_list(HEAD(parms), mk_list(mk_constant(value), NULL));
}
