/* Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Leonardo de Moura */ #include "kernel/find_fn.h" #include "kernel/instantiate.h" #include "kernel/abstract.h" #include "kernel/type_checker.h" #include "kernel/metavar.h" #include "kernel/inductive/inductive.h" #include "library/locals.h" #include "library/util.h" #include "library/constants.h" #include "library/unfold_macros.h" namespace lean { bool is_standard(environment const & env) { return env.prop_proof_irrel() && env.impredicative(); } bool is_def_app(environment const & env, expr const & e) { if (!is_app(e)) return false; expr const & f = get_app_fn(e); if (!is_constant(f)) return false; auto decl = env.find(const_name(f)); return decl && decl->is_definition() && !decl->is_opaque(); } optional unfold_app(environment const & env, expr const & e) { if (!is_app(e)) return none_expr(); expr const & f = get_app_fn(e); if (!is_constant(f)) return none_expr(); auto decl = env.find(const_name(f)); if (!decl || !decl->is_definition() || decl->is_opaque()) return none_expr(); expr d = instantiate_value_univ_params(*decl, const_levels(f)); buffer args; get_app_rev_args(e, args); return some_expr(apply_beta(d, args.size(), args.data())); } optional dec_level(level const & l) { switch (kind(l)) { case level_kind::Zero: case level_kind::Param: case level_kind::Global: case level_kind::Meta: return none_level(); case level_kind::Succ: return some_level(succ_of(l)); case level_kind::Max: if (auto lhs = dec_level(max_lhs(l))) { if (auto rhs = dec_level(max_rhs(l))) { return some_level(mk_max(*lhs, *rhs)); }} return none_level(); case level_kind::IMax: // Remark: the following mk_max is not a typo. The following // assertion justifies it. if (auto lhs = dec_level(imax_lhs(l))) { if (auto rhs = dec_level(imax_rhs(l))) { return some_level(mk_max(*lhs, *rhs)); }} return none_level(); } lean_unreachable(); // LCOV_EXCL_LINE } /** \brief Return true if environment has a constructor named \c c that returns an element of the inductive datatype named \c I, and \c c must have \c nparams parameters. */ bool has_constructor(environment const & env, name const & c, name const & I, unsigned nparams) { auto d = env.find(c); if (!d || d->is_definition()) return false; expr type = d->get_type(); unsigned i = 0; while (is_pi(type)) { i++; type = binding_body(type); } if (i != nparams) return false; type = get_app_fn(type); return is_constant(type) && const_name(type) == I; } bool has_unit_decls(environment const & env) { return has_constructor(env, get_unit_star_name(), get_unit_name(), 0); } bool has_eq_decls(environment const & env) { return has_constructor(env, get_eq_refl_name(), get_eq_name(), 2); } bool has_heq_decls(environment const & env) { return has_constructor(env, get_heq_refl_name(), get_heq_name(), 2); } bool has_prod_decls(environment const & env) { return has_constructor(env, get_prod_mk_name(), get_prod_name(), 4); } bool has_lift_decls(environment const & env) { return has_constructor(env, get_lift_up_name(), get_lift_name(), 2); } bool is_recursive_datatype(environment const & env, name const & n) { optional decls = inductive::is_inductive_decl(env, n); if (!decls) return false; for (inductive::inductive_decl const & decl : std::get<2>(*decls)) { for (inductive::intro_rule const & intro : inductive::inductive_decl_intros(decl)) { expr type = inductive::intro_rule_type(intro); while (is_pi(type)) { if (find(binding_domain(type), [&](expr const & e, unsigned) { return is_constant(e) && const_name(e) == n; })) { return true; } type = binding_body(type); } } } return false; } bool is_reflexive_datatype(type_checker & tc, name const & n) { environment const & env = tc.env(); name_generator ngen = tc.mk_ngen(); optional decls = inductive::is_inductive_decl(env, n); if (!decls) return false; for (inductive::inductive_decl const & decl : std::get<2>(*decls)) { for (inductive::intro_rule const & intro : inductive::inductive_decl_intros(decl)) { expr type = inductive::intro_rule_type(intro); while (is_pi(type)) { expr arg = tc.whnf(binding_domain(type)).first; if (is_pi(arg) && find(arg, [&](expr const & e, unsigned) { return is_constant(e) && const_name(e) == n; })) { return true; } expr local = mk_local(ngen.next(), binding_domain(type)); type = instantiate(binding_body(type), local); } } } return false; } level get_datatype_level(expr ind_type) { while (is_pi(ind_type)) ind_type = binding_body(ind_type); return sort_level(ind_type); } bool is_inductive_predicate(environment const & env, name const & n) { if (!is_standard(env)) return false; if (!inductive::is_inductive_decl(env, n)) return false; // n is not inductive datatype return is_zero(get_datatype_level(env.get(n).get_type())); } void get_intro_rule_names(environment const & env, name const & n, buffer & result) { if (auto decls = inductive::is_inductive_decl(env, n)) { for (inductive::inductive_decl const & decl : std::get<2>(*decls)) { if (inductive::inductive_decl_name(decl) == n) { for (inductive::intro_rule const & ir : inductive::inductive_decl_intros(decl)) result.push_back(inductive::intro_rule_name(ir)); return; } } } } optional is_constructor_app(environment const & env, expr const & e) { expr const & fn = get_app_fn(e); if (is_constant(fn)) if (auto I = inductive::is_intro_rule(env, const_name(fn))) return optional(const_name(fn)); return optional(); } optional is_constructor_app_ext(environment const & env, expr const & e) { if (auto r = is_constructor_app(env, e)) return r; expr const & f = get_app_fn(e); if (!is_constant(f)) return optional(); auto decl = env.find(const_name(f)); if (!decl || !decl->is_definition() || decl->is_opaque()) return optional(); expr const * it = &decl->get_value(); while (is_lambda(*it)) it = &binding_body(*it); return is_constructor_app_ext(env, *it); } expr instantiate_univ_param (expr const & e, name const & p, level const & l) { return instantiate_univ_params(e, to_list(p), to_list(l)); } expr to_telescope(bool pi, name_generator & ngen, expr e, buffer & telescope, optional const & binfo) { while ((pi && is_pi(e)) || (!pi && is_lambda(e))) { expr local; if (binfo) local = mk_local(ngen.next(), binding_name(e), binding_domain(e), *binfo); else local = mk_local(ngen.next(), binding_name(e), binding_domain(e), binding_info(e)); telescope.push_back(local); e = instantiate(binding_body(e), local); } return e; } expr to_telescope(name_generator & ngen, expr const & type, buffer & telescope, optional const & binfo) { return to_telescope(true, ngen, type, telescope, binfo); } expr fun_to_telescope(name_generator & ngen, expr const & e, buffer & telescope, optional const & binfo) { return to_telescope(false, ngen, e, telescope, binfo); } expr to_telescope(type_checker & tc, expr type, buffer & telescope, optional const & binfo, constraint_seq & cs) { type = tc.whnf(type, cs); while (is_pi(type)) { expr local; if (binfo) local = mk_local(tc.mk_fresh_name(), binding_name(type), binding_domain(type), *binfo); else local = mk_local(tc.mk_fresh_name(), binding_name(type), binding_domain(type), binding_info(type)); telescope.push_back(local); type = tc.whnf(instantiate(binding_body(type), local), cs); } return type; } expr to_telescope(type_checker & tc, expr type, buffer & telescope, optional const & binfo) { constraint_seq cs; return to_telescope(tc, type, telescope, binfo, cs); } expr mk_false() { return mk_constant(get_false_name()); } expr mk_empty() { return mk_constant(get_empty_name()); } expr mk_false(environment const & env) { return is_standard(env) ? mk_false() : mk_empty(); } bool is_false(expr const & e) { return is_constant(e) && const_name(e) == get_false_name(); } bool is_empty(expr const & e) { return is_constant(e) && const_name(e) == get_empty_name(); } bool is_false(environment const & env, expr const & e) { return is_standard(env) ? is_false(e) : is_empty(e); } expr mk_false_rec(type_checker & tc, expr const & f, expr const & t) { level t_lvl = sort_level(tc.ensure_type(t).first); if (is_standard(tc.env())) { return mk_app(mk_constant(get_false_rec_name(), {t_lvl}), t, f); } else { expr f_type = tc.infer(f).first; return mk_app(mk_constant(get_empty_rec_name(), {t_lvl}), mk_lambda("e", f_type, t), f); } } optional lift_down_if_hott(type_checker & tc, expr const & v) { if (is_standard(tc.env())) { return some_expr(v); } else { expr v_type = tc.whnf(tc.infer(v).first).first; if (!is_app(v_type)) return none_expr(); expr const & lift = app_fn(v_type); if (!is_constant(lift) || const_name(lift) != get_lift_name()) return none_expr(); return some_expr(mk_app(mk_constant(get_lift_down_name(), const_levels(lift)), app_arg(v_type), v)); } } static expr * g_true = nullptr; static expr * g_true_intro = nullptr; static expr * g_and = nullptr; static expr * g_and_intro = nullptr; static expr * g_and_elim_left = nullptr; static expr * g_and_elim_right = nullptr; void initialize_library_util() { g_true = new expr(mk_constant(get_true_name())); g_true_intro = new expr(mk_constant(get_true_intro_name())); g_and = new expr(mk_constant(get_and_name())); g_and_intro = new expr(mk_constant(get_and_intro_name())); g_and_elim_left = new expr(mk_constant(get_and_elim_left_name())); g_and_elim_right = new expr(mk_constant(get_and_elim_right_name())); } void finalize_library_util() { delete g_true; delete g_true_intro; delete g_and; delete g_and_intro; delete g_and_elim_left; delete g_and_elim_right; } expr mk_true() { return *g_true; } expr mk_true_intro() { return *g_true_intro; } expr mk_and(expr const & a, expr const & b) { return mk_app(*g_and, a, b); } expr mk_and_intro(type_checker & tc, expr const & Ha, expr const & Hb) { return mk_app(*g_and_intro, tc.infer(Ha).first, tc.infer(Hb).first, Ha, Hb); } expr mk_and_elim_left(type_checker & tc, expr const & H) { expr a_and_b = tc.whnf(tc.infer(H).first).first; return mk_app(*g_and_elim_left, app_arg(app_fn(a_and_b)), app_arg(a_and_b), H); } expr mk_and_elim_right(type_checker & tc, expr const & H) { expr a_and_b = tc.whnf(tc.infer(H).first).first; return mk_app(*g_and_elim_right, app_arg(app_fn(a_and_b)), app_arg(a_and_b), H); } expr mk_unit(level const & l) { return mk_constant(get_unit_name(), {l}); } expr mk_unit_mk(level const & l) { return mk_constant(get_unit_star_name(), {l}); } expr mk_prod(type_checker & tc, expr const & A, expr const & B) { level l1 = sort_level(tc.ensure_type(A).first); level l2 = sort_level(tc.ensure_type(B).first); return mk_app(mk_constant(get_prod_name(), {l1, l2}), A, B); } expr mk_pair(type_checker & tc, expr const & a, expr const & b) { expr A = tc.infer(a).first; expr B = tc.infer(b).first; level l1 = sort_level(tc.ensure_type(A).first); level l2 = sort_level(tc.ensure_type(B).first); return mk_app(mk_constant(get_prod_mk_name(), {l1, l2}), A, B, a, b); } expr mk_pr1(type_checker & tc, expr const & p) { expr AxB = tc.whnf(tc.infer(p).first).first; expr const & A = app_arg(app_fn(AxB)); expr const & B = app_arg(AxB); return mk_app(mk_constant(get_prod_pr1_name(), const_levels(get_app_fn(AxB))), A, B, p); } expr mk_pr2(type_checker & tc, expr const & p) { expr AxB = tc.whnf(tc.infer(p).first).first; expr const & A = app_arg(app_fn(AxB)); expr const & B = app_arg(AxB); return mk_app(mk_constant(get_prod_pr2_name(), const_levels(get_app_fn(AxB))), A, B, p); } expr mk_unit(level const & l, bool prop) { return prop ? mk_true() : mk_unit(l); } expr mk_unit_mk(level const & l, bool prop) { return prop ? mk_true_intro() : mk_unit_mk(l); } expr mk_prod(type_checker & tc, expr const & a, expr const & b, bool prop) { return prop ? mk_and(a, b) : mk_prod(tc, a, b); } expr mk_pair(type_checker & tc, expr const & a, expr const & b, bool prop) { return prop ? mk_and_intro(tc, a, b) : mk_pair(tc, a, b); } expr mk_pr1(type_checker & tc, expr const & p, bool prop) { return prop ? mk_and_elim_left(tc, p) : mk_pr1(tc, p); } expr mk_pr2(type_checker & tc, expr const & p, bool prop) { return prop ? mk_and_elim_right(tc, p) : mk_pr2(tc, p); } bool is_iff(expr const & e) { expr const & fn = get_app_fn(e); return is_constant(fn) && const_name(fn) == get_iff_name(); } expr mk_iff(expr const & lhs, expr const & rhs) { return mk_app(mk_constant(get_iff_name()), lhs, rhs); } expr mk_iff_refl(expr const & a) { return mk_app(mk_constant(get_iff_refl_name()), a); } expr mk_eq(type_checker & tc, expr const & lhs, expr const & rhs) { expr A = tc.whnf(tc.infer(lhs).first).first; level lvl = sort_level(tc.ensure_type(A).first); return mk_app(mk_constant(get_eq_name(), {lvl}), A, lhs, rhs); } expr mk_refl(type_checker & tc, expr const & a) { expr A = tc.whnf(tc.infer(a).first).first; level lvl = sort_level(tc.ensure_type(A).first); return mk_app(mk_constant(get_eq_refl_name(), {lvl}), A, a); } expr mk_symm(type_checker & tc, expr const & H) { expr p = tc.whnf(tc.infer(H).first).first; lean_assert(is_eq(p)); expr lhs = app_arg(app_fn(p)); expr rhs = app_arg(p); expr A = tc.infer(lhs).first; level lvl = sort_level(tc.ensure_type(A).first); return mk_app(mk_constant(get_eq_symm_name(), {lvl}), A, lhs, rhs, H); } expr mk_heq(type_checker & tc, expr const & lhs, expr const & rhs) { expr A = tc.whnf(tc.infer(lhs).first).first; expr B = tc.whnf(tc.infer(rhs).first).first; level lvl = sort_level(tc.ensure_type(A).first); return mk_app(mk_constant(get_heq_name(), {lvl}), A, lhs, B, rhs); } bool is_eq_rec(expr const & e) { expr const & fn = get_app_fn(e); return is_constant(fn) && const_name(fn) == get_eq_rec_name(); } bool is_eq(expr const & e) { expr const & fn = get_app_fn(e); return is_constant(fn) && const_name(fn) == get_eq_name(); } bool is_eq(expr const & e, expr & lhs, expr & rhs) { if (!is_eq(e) || !is_app(app_fn(e))) return false; lhs = app_arg(app_fn(e)); rhs = app_arg(e); return true; } bool is_eq_a_a(expr const & e) { if (!is_eq(e)) return false; buffer args; get_app_args(e, args); return args.size() == 3 && args[1] == args[2]; } bool is_eq_a_a(type_checker & tc, expr const & e) { if (!is_eq(e)) return false; buffer args; get_app_args(e, args); if (args.size() != 3) return false; pair d = tc.is_def_eq(args[1], args[2]); return d.first && !d.second; } bool is_heq(expr const & e) { expr const & fn = get_app_fn(e); return is_constant(fn) && const_name(fn) == get_heq_name(); } bool is_heq(expr const & e, expr & A, expr & lhs, expr & B, expr & rhs) { if (is_heq(e)) { buffer args; get_app_args(e, args); if (args.size() != 4) return false; A = args[0]; lhs = args[1]; B = args[2]; rhs = args[3]; return true; } else { return false; } } void mk_telescopic_eq(type_checker & tc, buffer const & t, buffer const & s, buffer & eqs) { lean_assert(t.size() == s.size()); lean_assert(std::all_of(s.begin(), s.end(), is_local)); lean_assert(inductive::has_dep_elim(tc.env(), get_eq_name())); buffer> t_aux; name e_name("e"); for (unsigned i = 0; i < t.size(); i++) { expr s_i = s[i]; expr s_i_ty = mlocal_type(s_i); expr s_i_ty_a = abstract_locals(s_i_ty, i, s.data()); expr t_i = t[i]; t_aux.push_back(buffer()); t_aux.back().push_back(t_i); for (unsigned j = 0; j < i; j++) { if (depends_on(s_i_ty, s[j])) { // we need to "cast" buffer ty_inst_args; for (unsigned k = 0; k <= j; k++) ty_inst_args.push_back(s[k]); for (unsigned k = j + 1; k < i; k++) ty_inst_args.push_back(t_aux[k][j+1]); lean_assert(ty_inst_args.size() == i); expr s_i_ty = instantiate_rev(s_i_ty_a, i, ty_inst_args.data()); buffer rec_args; rec_args.push_back(mlocal_type(s[j])); rec_args.push_back(t_aux[j][j]); rec_args.push_back(Fun(s[j], Fun(eqs[j], s_i_ty))); // type former ("promise") rec_args.push_back(t_i); // minor premise rec_args.push_back(s[j]); rec_args.push_back(eqs[j]); level rec_l1 = sort_level(tc.ensure_type(s_i_ty).first); level rec_l2 = sort_level(tc.ensure_type(mlocal_type(s[j])).first); t_i = mk_app(mk_constant(get_eq_rec_name(), {rec_l1, rec_l2}), rec_args.size(), rec_args.data()); } t_aux.back().push_back(t_i); } expr eq = mk_local(tc.mk_fresh_name(), e_name.append_after(i+1), mk_eq(tc, t_i, s_i), binder_info()); eqs.push_back(eq); } } void mk_telescopic_eq(type_checker & tc, buffer const & t, buffer & eqs) { lean_assert(std::all_of(t.begin(), t.end(), is_local)); lean_assert(inductive::has_dep_elim(tc.env(), get_eq_name())); buffer s; for (unsigned i = 0; i < t.size(); i++) { expr ty = mlocal_type(t[i]); ty = abstract_locals(ty, i, t.data()); ty = instantiate_rev(ty, i, s.data()); expr local = mk_local(tc.mk_fresh_name(), local_pp_name(t[i]).append_after("'"), ty, local_info(t[i])); s.push_back(local); } return mk_telescopic_eq(tc, t, s, eqs); } level mk_max(levels const & ls) { if (!ls) return mk_level_zero(); else if (!tail(ls)) return head(ls); else return mk_max(head(ls), mk_max(tail(ls))); } expr telescope_to_sigma(type_checker & tc, unsigned sz, expr const * ts, constraint_seq & cs) { lean_assert(sz > 0); unsigned i = sz - 1; expr r = mlocal_type(ts[i]); while (i > 0) { --i; expr const & a = ts[i]; expr const & A = mlocal_type(a); expr const & Ba = r; level l1 = sort_level(tc.ensure_type(A, cs)); level l2 = sort_level(tc.ensure_type(Ba, cs)); r = mk_app(mk_constant(get_sigma_name(), {l1, l2}), A, Fun(a, Ba)); } return r; } expr mk_sigma_mk(type_checker & tc, unsigned sz, expr const * ts, expr const * as, constraint_seq & cs) { lean_assert(sz > 0); if (sz == 1) return as[0]; buffer new_ts; for (unsigned i = 1; i < sz; i++) new_ts.push_back(instantiate(abstract_local(ts[i], ts[0]), as[0])); expr arg1 = mlocal_type(ts[0]); expr arg2_core = telescope_to_sigma(tc, sz-1, ts+1, cs); expr arg2 = Fun(ts[0], arg2_core); expr arg3 = as[0]; expr arg4 = mk_sigma_mk(tc, sz-1, new_ts.data(), as+1, cs); level l1 = sort_level(tc.ensure_type(arg1, cs)); level l2 = sort_level(tc.ensure_type(arg2_core, cs)); return mk_app(mk_constant(get_sigma_mk_name(), {l1, l2}), arg1, arg2, arg3, arg4); } expr mk_sigma_mk(type_checker & tc, buffer const & ts, buffer const & as, constraint_seq & cs) { lean_assert(ts.size() == as.size()); return mk_sigma_mk(tc, ts.size(), ts.data(), as.data(), cs); } expr infer_implicit_params(expr const & type, unsigned nparams, implicit_infer_kind k) { switch (k) { case implicit_infer_kind::Implicit: { bool strict = true; return infer_implicit(type, nparams, strict); } case implicit_infer_kind::RelaxedImplicit: { bool strict = false; return infer_implicit(type, nparams, strict); } case implicit_infer_kind::None: return type; } lean_unreachable(); // LCOV_EXCL_LINE } bool has_expr_metavar_relaxed(expr const & e) { if (!has_expr_metavar(e)) return false; bool found = false; for_each(e, [&](expr const & e, unsigned) { if (found || !has_expr_metavar(e)) return false; if (is_metavar(e)) { found = true; return false; } if (is_local(e)) return false; // do not visit type return true; }); return found; } constraint instantiate_metavars(constraint const & c, substitution & s) { switch (c.kind()) { case constraint_kind::Eq: return mk_eq_cnstr(s.instantiate_all(cnstr_lhs_expr(c)), s.instantiate_all(cnstr_rhs_expr(c)), c.get_justification(), relax_main_opaque(c)); case constraint_kind::LevelEq: return mk_level_eq_cnstr(s.instantiate(cnstr_lhs_level(c)), s.instantiate(cnstr_rhs_level(c)), c.get_justification()); case constraint_kind::Choice: { expr m = cnstr_expr(c); lean_assert(is_meta(m)); buffer args; expr mvar = get_app_args(m, args); mvar = update_mlocal(mvar, s.instantiate_all(mlocal_type(mvar))); for (expr & arg : args) arg = s.instantiate_all(arg); return mk_choice_cnstr(mk_app(mvar, args), cnstr_choice_fn(c), cnstr_delay_factor_core(c), cnstr_is_owner(c), c.get_justification(), relax_main_opaque(c)); }} lean_unreachable(); // LCOV_EXCL_LINE } void check_term(type_checker & tc, expr const & e) { expr tmp = unfold_untrusted_macros(tc.env(), e); tc.check_ignore_undefined_universes(tmp); } void check_term(environment const & env, expr const & e) { expr tmp = unfold_untrusted_macros(env, e); type_checker(env).check_ignore_undefined_universes(tmp); } }