286 lines
13 KiB
C++
286 lines
13 KiB
C++
/*
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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*/
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#include "util/sstream.h"
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#include "kernel/environment.h"
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#include "kernel/instantiate.h"
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#include "kernel/abstract.h"
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#include "kernel/inductive/inductive.h"
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#include "kernel/type_checker.h"
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#include "library/protected.h"
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#include "library/module.h"
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#include "library/util.h"
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#include "library/reducible.h"
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namespace lean {
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static void throw_corrupted(name const & n) {
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throw exception(sstream() << "error in 'no_confusion' generation, '" << n << "' inductive datatype declaration is corrupted");
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}
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optional<environment> mk_no_confusion_type(environment const & env, name const & n) {
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optional<inductive::inductive_decls> decls = inductive::is_inductive_decl(env, n);
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if (!decls)
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throw exception(sstream() << "error in 'no_confusion' generation, '" << n << "' is not an inductive datatype");
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if (is_inductive_predicate(env, n))
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return optional<environment>(); // type is a proposition
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name_generator ngen;
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bool impredicative = env.impredicative();
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unsigned nparams = std::get<1>(*decls);
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declaration ind_decl = env.get(n);
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declaration cases_decl = env.get(name(n, "cases_on"));
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level_param_names lps = cases_decl.get_univ_params();
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level plvl = mk_param_univ(head(lps));
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levels ilvls = param_names_to_levels(tail(lps));
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level rlvl;
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expr ind_type = instantiate_type_univ_params(ind_decl, ilvls);
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level ind_lvl = get_datatype_level(ind_type);
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if (impredicative)
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rlvl = plvl;
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else
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rlvl = mk_max(plvl, ind_lvl);
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if (length(ilvls) != length(ind_decl.get_univ_params()))
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return optional<environment>(); // type does not have only a restricted eliminator
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name eq_name("eq");
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name heq_name("heq");
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// All inductive datatype parameters and indices are arguments
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buffer<expr> args;
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ind_type = to_telescope(ngen, ind_type, args, some(mk_implicit_binder_info()));
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if (!is_sort(ind_type) || args.size() < nparams)
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throw_corrupted(n);
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lean_assert(!(env.impredicative() && is_zero(sort_level(ind_type))));
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unsigned nindices = args.size() - nparams;
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// Create inductive datatype
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expr I = mk_app(mk_constant(n, ilvls), args);
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// Add (P : Type)
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expr P = mk_local(ngen.next(), "P", mk_sort(plvl), binder_info());
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args.push_back(P);
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// add v1 and v2 elements of the inductive type
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expr v1 = mk_local(ngen.next(), "v1", I, binder_info());
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expr v2 = mk_local(ngen.next(), "v2", I, binder_info());
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args.push_back(v1);
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args.push_back(v2);
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expr R = mk_sort(rlvl);
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expr Pres;
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if (impredicative)
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Pres = P;
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else
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Pres = mk_app(mk_constant("lift", {plvl, ind_lvl}), P);
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name no_confusion_type_name{n, "no_confusion_type"};
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expr no_confusion_type_type = Pi(args, R);
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// Create type former
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buffer<expr> type_former_args;
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for (unsigned i = nparams; i < nparams + nindices; i++)
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type_former_args.push_back(args[i]);
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type_former_args.push_back(v1);
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expr type_former = Fun(type_former_args, R);
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// Create cases_on
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levels clvls = levels(mk_succ(rlvl), ilvls);
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expr cases_on = mk_app(mk_app(mk_constant(cases_decl.get_name(), clvls), nparams, args.data()), type_former);
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cases_on = mk_app(cases_on, nindices, args.data() + nparams);
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expr cases_on1 = mk_app(cases_on, v1);
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expr cases_on2 = mk_app(cases_on, v2);
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type_checker tc(env);
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expr t1 = tc.infer(cases_on1).first;
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expr t2 = tc.infer(cases_on2).first;
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buffer<expr> outer_cases_on_args;
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unsigned idx1 = 0;
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while (is_pi(t1)) {
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buffer<expr> minor1_args;
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expr minor1 = to_telescope(tc, binding_domain(t1), minor1_args);
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expr curr_t2 = t2;
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buffer<expr> inner_cases_on_args;
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unsigned idx2 = 0;
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while (is_pi(curr_t2)) {
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buffer<expr> minor2_args;
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expr minor2 = to_telescope(tc, binding_domain(curr_t2), minor2_args);
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if (idx1 != idx2) {
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// infeasible case, constructors do not match
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inner_cases_on_args.push_back(Fun(minor2_args, Pres));
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} else {
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if (minor1_args.size() != minor2_args.size())
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throw_corrupted(n);
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buffer<expr> rtype_hyp;
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// add equalities
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if (env.prop_proof_irrel()) {
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// proof irrelevance version using heterogeneous equality
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for (unsigned i = 0; i < minor1_args.size(); i++) {
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expr lhs = minor1_args[i];
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expr rhs = minor2_args[i];
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expr lhs_type = mlocal_type(lhs);
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expr rhs_type = mlocal_type(rhs);
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level l = sort_level(tc.ensure_type(lhs_type).first);
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expr h_type;
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if (tc.is_def_eq(lhs_type, rhs_type).first) {
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h_type = mk_app(mk_constant(eq_name, to_list(l)), lhs_type, lhs, rhs);
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} else {
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h_type = mk_app(mk_constant(heq_name, to_list(l)), lhs_type, lhs, rhs_type, rhs);
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}
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rtype_hyp.push_back(mk_local(ngen.next(), local_pp_name(lhs).append_after("_eq"), h_type, binder_info()));
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}
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} else {
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// we use telescope equality (with casts) when proof irrelevance is not available
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mk_telescopic_eq(tc, minor1_args, minor2_args, rtype_hyp);
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}
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inner_cases_on_args.push_back(Fun(minor2_args, mk_arrow(Pi(rtype_hyp, P), Pres)));
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}
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idx2++;
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curr_t2 = binding_body(curr_t2);
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}
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outer_cases_on_args.push_back(Fun(minor1_args, mk_app(cases_on2, inner_cases_on_args)));
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idx1++;
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t1 = binding_body(t1);
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}
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expr no_confusion_type_value = Fun(args, mk_app(cases_on1, outer_cases_on_args));
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bool opaque = false;
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bool use_conv_opt = true;
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declaration new_d = mk_definition(env, no_confusion_type_name, lps, no_confusion_type_type, no_confusion_type_value,
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opaque, ind_decl.get_module_idx(), use_conv_opt);
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environment new_env = module::add(env, check(env, new_d));
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new_env = set_reducible(new_env, no_confusion_type_name, reducible_status::On);
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return some(add_protected(new_env, no_confusion_type_name));
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}
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environment mk_no_confusion(environment const & env, name const & n) {
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optional<environment> env1 = mk_no_confusion_type(env, n);
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if (!env1)
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return env;
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environment new_env = *env1;
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type_checker tc(new_env);
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bool impredicative = env.impredicative();
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inductive::inductive_decls decls = *inductive::is_inductive_decl(new_env, n);
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unsigned nparams = std::get<1>(decls);
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name_generator ngen;
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declaration ind_decl = env.get(n);
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declaration no_confusion_type_decl = new_env.get(name{n, "no_confusion_type"});
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declaration cases_decl = new_env.get(name(n, "cases_on"));
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level_param_names lps = no_confusion_type_decl.get_univ_params();
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levels ls = param_names_to_levels(lps);
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expr ind_type = instantiate_type_univ_params(ind_decl, tail(ls));
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level ind_lvl = get_datatype_level(ind_type);
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expr no_confusion_type_type = instantiate_type_univ_params(no_confusion_type_decl, ls);
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name eq_name("eq");
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name heq_name("heq");
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name eq_refl_name{"eq", "refl"};
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name heq_refl_name{"heq", "refl"};
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buffer<expr> args;
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expr type = no_confusion_type_type;
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type = to_telescope(ngen, type, args, some(mk_implicit_binder_info()));
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lean_assert(args.size() >= nparams + 3);
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unsigned nindices = args.size() - nparams - 3; // 3 is for P v1 v2
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expr range = mk_app(mk_constant(no_confusion_type_decl.get_name(), ls), args);
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expr P = args[args.size()-3];
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expr v1 = args[args.size()-2];
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expr v2 = args[args.size()-1];
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expr v_type = mlocal_type(v1);
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expr lift_up;
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if (!impredicative) {
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lift_up = mk_app(mk_constant(name{"lift", "up"}, {head(ls), ind_lvl}), P);
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}
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level v_lvl = sort_level(tc.ensure_type(v_type).first);
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expr eq_v = mk_app(mk_constant(eq_name, to_list(v_lvl)), v_type);
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expr H12 = mk_local(ngen.next(), "H12", mk_app(eq_v, v1, v2), binder_info());
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lean_assert(impredicative != inductive::has_dep_elim(env, eq_name));
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args.push_back(H12);
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name no_confusion_name{n, "no_confusion"};
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expr no_confusion_ty = Pi(args, range);
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// The gen proof is of the form
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// (fun H11 : v1 = v1, cases_on Params (fun Indices v1, no_confusion_type Params Indices P v1 v1) Indices v1
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// <for-each case>
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// (fun H : (equations -> P), H (refl) ... (refl))
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// ...
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// )
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// H11 is for creating the generalization
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expr H11 = mk_local(ngen.next(), "H11", mk_app(eq_v, v1, v1), binder_info());
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// Create the type former (fun Indices v1, no_confusion_type Params Indices P v1 v1)
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buffer<expr> type_former_args;
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for (unsigned i = nparams; i < nparams + nindices; i++)
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type_former_args.push_back(args[i]);
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type_former_args.push_back(v1);
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buffer<expr> no_confusion_type_args;
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for (unsigned i = 0; i < nparams + nindices; i++)
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no_confusion_type_args.push_back(args[i]);
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no_confusion_type_args.push_back(P);
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no_confusion_type_args.push_back(v1);
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no_confusion_type_args.push_back(v1);
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expr no_confusion_type_app = mk_app(mk_constant(no_confusion_type_decl.get_name(), ls), no_confusion_type_args);
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expr type_former = Fun(type_former_args, no_confusion_type_app);
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// create cases_on
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levels clvls;
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if (impredicative)
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clvls = ls;
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else
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clvls = cons(mk_max(head(ls), ind_lvl), tail(ls));
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expr cases_on = mk_app(mk_app(mk_constant(cases_decl.get_name(), clvls), nparams, args.data()), type_former);
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cases_on = mk_app(mk_app(cases_on, nindices, args.data() + nparams), v1);
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expr cot = tc.infer(cases_on).first;
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while (is_pi(cot)) {
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buffer<expr> minor_args;
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expr minor = to_telescope(tc, binding_domain(cot), minor_args);
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lean_assert(!minor_args.empty());
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expr H = minor_args.back();
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expr Ht = mlocal_type(H);
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buffer<expr> refl_args;
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while (is_pi(Ht)) {
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buffer<expr> eq_args;
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expr eq_fn = get_app_args(binding_domain(Ht), eq_args);
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if (const_name(eq_fn) == eq_name) {
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refl_args.push_back(mk_app(mk_constant(eq_refl_name, const_levels(eq_fn)), eq_args[0], eq_args[2]));
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} else {
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refl_args.push_back(mk_app(mk_constant(heq_refl_name, const_levels(eq_fn)), eq_args[0], eq_args[1]));
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}
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Ht = binding_body(Ht);
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}
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expr pr = mk_app(H, refl_args);
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if (!impredicative)
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pr = mk_app(lift_up, pr);
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cases_on = mk_app(cases_on, Fun(minor_args, pr));
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cot = binding_body(cot);
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}
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expr gen = cases_on;
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if (impredicative)
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gen = Fun(H11, gen);
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// Now, we use gen to build the final proof using eq.rec
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//
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// eq.rec InductiveType v1 (fun (a : InductiveType), v1 = a -> no_confusion_type Params Indices v1 a) gen v2 H12 H12
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//
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name eq_rec_name{"eq", "rec"};
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level eq_rec_l1;
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if (impredicative)
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eq_rec_l1 = head(ls);
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else
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eq_rec_l1 = mk_max(head(ls), ind_lvl);
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expr eq_rec = mk_app(mk_constant(eq_rec_name, {eq_rec_l1, v_lvl}), v_type, v1);
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// create eq_rec type_former
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// (fun (a : InductiveType), v1 = a -> no_confusion_type Params Indices v1 a)
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expr a = mk_local(ngen.next(), "a", v_type, binder_info());
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expr H1a = mk_local(ngen.next(), "H1a", mk_app(eq_v, v1, a), binder_info());
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// reusing no_confusion_type_args... we just replace the last argument with a
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no_confusion_type_args.pop_back();
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no_confusion_type_args.push_back(a);
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expr no_confusion_type_app_1a = mk_app(mk_constant(no_confusion_type_decl.get_name(), ls), no_confusion_type_args);
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if (impredicative) {
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expr rec_type_former = Fun(a, Pi(H1a, no_confusion_type_app_1a));
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// finalize eq_rec
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eq_rec = mk_app(mk_app(eq_rec, rec_type_former, gen, v2, H12), H12);
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} else {
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expr rec_type_former = Fun(a, Fun(H1a, no_confusion_type_app_1a));
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// finalize eq_rec
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eq_rec = mk_app(eq_rec, rec_type_former, gen, v2, H12);
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}
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//
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expr no_confusion_val = Fun(args, eq_rec);
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bool opaque = false;
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bool use_conv_opt = true;
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declaration new_d = mk_definition(new_env, no_confusion_name, lps, no_confusion_ty, no_confusion_val,
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opaque, no_confusion_type_decl.get_module_idx(), use_conv_opt);
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new_env = module::add(new_env, check(new_env, new_d));
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new_env = set_reducible(new_env, no_confusion_name, reducible_status::On);
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return add_protected(new_env, no_confusion_name);
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}
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}
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