level mk_succ(level l, unsigned k) { while (k > 0) { --k; l = mk_succ(l); } return l; }
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); }
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); }
vm_obj level_succ(vm_obj const & o) { return to_obj(mk_succ(to_level(o))); }
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)); }
level normalize(level const & l) { auto p = to_offset(l); level const & r = p.first; switch (kind(r)) { case level_kind::Succ: lean_unreachable(); // LCOV_EXCL_LINE case level_kind::Zero: case level_kind::Param: case level_kind::Global: case level_kind::Meta: return l; case level_kind::IMax: { auto l1 = normalize(imax_lhs(r)); auto l2 = normalize(imax_rhs(r)); if (!is_eqp(l1, imax_lhs(r)) || !is_eqp(l2, imax_rhs(r))) return mk_succ(mk_imax(l1, l2), p.second); else return l; } case level_kind::Max: { buffer<level> todo; buffer<level> args; push_max_args(r, todo); for (level const & a : todo) push_max_args(normalize(a), args); std::sort(args.begin(), args.end(), is_norm_lt); buffer<level> & rargs = todo; rargs.clear(); unsigned i = 0; if (is_explicit(args[i])) { // find max explicit univierse while (i+1 < args.size() && is_explicit(args[i+1])) i++; lean_assert(is_explicit(args[i])); unsigned k = to_offset(args[i]).second; // an explicit universe k is subsumed by succ^k(l) unsigned j = i+1; for (; j < args.size(); j++) { if (to_offset(args[j]).second >= k) break; } if (j < args.size()) { // explicit universe was subsumed by succ^k'(l) where k' >= k i++; } } rargs.push_back(args[i]); auto p_prev = to_offset(args[i]); i++; for (; i < args.size(); i++) { auto p_curr = to_offset(args[i]); if (p_prev.first == p_curr.first) { if (p_prev.second < p_curr.second) { p_prev = p_curr; rargs.pop_back(); rargs.push_back(args[i]); } } else { p_prev = p_curr; rargs.push_back(args[i]); } } for (level & a : rargs) a = mk_succ(a, p.second); return mk_max(rargs); }} lean_unreachable(); // LCOV_EXCL_LINE }
level update_succ(level const & l, level const & new_arg) { if (is_eqp(succ_of(l), new_arg)) return l; else return mk_succ(new_arg); }