1103 lines
40 KiB
C++
1103 lines
40 KiB
C++
/*
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Copyright (c) 2015 Daniel Selsam. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Daniel Selsam
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*/
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#include "kernel/abstract.h"
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#include "kernel/expr_maps.h"
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#include "kernel/instantiate.h"
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#include "library/constants.h"
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#include "library/expr_lt.h"
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#include "library/class_instance_resolution.h"
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#include "library/relation_manager.h"
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#include "library/blast/expr.h"
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#include "library/blast/blast_exception.h"
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#include "library/blast/blast.h"
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#include "library/blast/simplifier.h"
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#include "library/simplifier/simp_rule_set.h"
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#include "library/simplifier/ceqv.h"
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#include "library/app_builder.h"
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#include "library/num.h"
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#include "library/norm_num.h"
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#include "util/flet.h"
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#include "util/pair.h"
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#include "util/sexpr/option_declarations.h"
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#include <functional>
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#include <iostream>
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#ifndef LEAN_DEFAULT_SIMPLIFY_MAX_STEPS
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#define LEAN_DEFAULT_SIMPLIFY_MAX_STEPS 1000
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_TOP_DOWN
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#define LEAN_DEFAULT_SIMPLIFY_TOP_DOWN false
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_EXHAUSTIVE
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#define LEAN_DEFAULT_SIMPLIFY_EXHAUSTIVE true
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_MEMOIZE
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#define LEAN_DEFAULT_SIMPLIFY_MEMOIZE true
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_CONTEXTUAL
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#define LEAN_DEFAULT_SIMPLIFY_CONTEXTUAL true
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_EXPAND_MACROS
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#define LEAN_DEFAULT_SIMPLIFY_EXPAND_MACROS false
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_TRACE
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#define LEAN_DEFAULT_SIMPLIFY_TRACE false
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_FUSE
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#define LEAN_DEFAULT_SIMPLIFY_FUSE false
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#endif
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#ifndef LEAN_DEFAULT_SIMPLIFY_NUMERALS
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#define LEAN_DEFAULT_SIMPLIFY_NUMERALS false
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#endif
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namespace lean {
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namespace blast {
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using simp::result;
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/* Names */
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static name * g_simplify_unit_namespace = nullptr;
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static name * g_simplify_ac_namespace = nullptr;
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static name * g_simplify_som_namespace = nullptr;
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static name * g_simplify_numeral_namespace = nullptr;
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/* Options */
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static name * g_simplify_max_steps = nullptr;
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static name * g_simplify_top_down = nullptr;
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static name * g_simplify_exhaustive = nullptr;
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static name * g_simplify_memoize = nullptr;
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static name * g_simplify_contextual = nullptr;
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static name * g_simplify_expand_macros = nullptr;
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static name * g_simplify_trace = nullptr;
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static name * g_simplify_fuse = nullptr;
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static name * g_simplify_numerals = nullptr;
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unsigned get_simplify_max_steps() {
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return ios().get_options().get_unsigned(*g_simplify_max_steps, LEAN_DEFAULT_SIMPLIFY_MAX_STEPS);
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}
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bool get_simplify_top_down() {
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return ios().get_options().get_bool(*g_simplify_top_down, LEAN_DEFAULT_SIMPLIFY_TOP_DOWN);
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}
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bool get_simplify_exhaustive() {
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return ios().get_options().get_bool(*g_simplify_exhaustive, LEAN_DEFAULT_SIMPLIFY_EXHAUSTIVE);
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}
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bool get_simplify_memoize() {
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return ios().get_options().get_bool(*g_simplify_memoize, LEAN_DEFAULT_SIMPLIFY_MEMOIZE);
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}
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bool get_simplify_contextual() {
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return ios().get_options().get_bool(*g_simplify_contextual, LEAN_DEFAULT_SIMPLIFY_CONTEXTUAL);
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}
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bool get_simplify_expand_macros() {
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return ios().get_options().get_bool(*g_simplify_expand_macros, LEAN_DEFAULT_SIMPLIFY_EXPAND_MACROS);
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}
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bool get_simplify_trace() {
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return ios().get_options().get_bool(*g_simplify_trace, LEAN_DEFAULT_SIMPLIFY_TRACE);
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}
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bool get_simplify_fuse() {
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return ios().get_options().get_bool(*g_simplify_fuse, LEAN_DEFAULT_SIMPLIFY_FUSE);
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}
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bool get_simplify_numerals() {
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return ios().get_options().get_bool(*g_simplify_numerals, LEAN_DEFAULT_SIMPLIFY_NUMERALS);
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}
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/* Miscellaneous helpers */
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static bool is_add_app(expr const & e) {
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return is_const_app(e, get_add_name(), 4);
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}
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static bool is_mul_app(expr const & e) {
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return is_const_app(e, get_mul_name(), 4);
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}
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static bool is_neg_app(expr const & e) {
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return is_const_app(e, get_neg_name(), 3);
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}
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/* Main simplifier class */
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class simplifier {
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blast_tmp_type_context m_tmp_tctx;
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app_builder m_app_builder;
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name m_rel;
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simp_rule_sets m_srss;
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simp_rule_sets m_ctx_srss;
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/* Logging */
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unsigned m_num_steps{0};
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unsigned m_depth{0};
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/* Options */
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unsigned m_max_steps{get_simplify_max_steps()};
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bool m_top_down{get_simplify_top_down()};
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bool m_exhaustive{get_simplify_exhaustive()};
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bool m_memoize{get_simplify_memoize()};
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bool m_contextual{get_simplify_contextual()};
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bool m_expand_macros{get_simplify_expand_macros()};
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bool m_trace{get_simplify_trace()};
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bool m_fuse{get_simplify_fuse()};
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bool m_numerals{get_simplify_numerals()};
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/* Cache */
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struct key {
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name m_rel;
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expr m_e;
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unsigned m_hash;
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key(name const & rel, expr const & e):
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m_rel(rel), m_e(e), m_hash(hash(rel.hash(), abstract_hash(e))) { }
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};
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struct key_hash_fn {
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unsigned operator()(key const & k) const { return k.m_hash; }
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};
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struct key_eq_fn {
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bool operator()(key const & k1, key const & k2) const {
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return k1.m_rel == k2.m_rel && abstract_is_equal(k1.m_e, k2.m_e);
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}
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};
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typedef std::unordered_map<key, result, key_hash_fn, key_eq_fn> simplify_cache;
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simplify_cache m_cache;
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optional<result> cache_lookup(expr const & e);
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void cache_save(expr const & e, result const & r);
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/* Basic helpers */
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bool using_eq() { return m_rel == get_eq_name(); }
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bool is_dependent_fn(expr const & f) {
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expr f_type = m_tmp_tctx->whnf(m_tmp_tctx->infer(f));
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lean_assert(is_pi(f_type));
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return has_free_vars(binding_body(f_type));
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}
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simp_rule_sets add_to_srss(simp_rule_sets const & _srss, buffer<expr> & ls) {
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simp_rule_sets srss = _srss;
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for (unsigned i = 0; i < ls.size(); i++) {
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expr & l = ls[i];
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if (m_trace) {
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ios().get_diagnostic_channel() << "Local: " << l << " : " << mlocal_type(l) << "\n";
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}
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tmp_type_context tctx(env(), ios());
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srss = add(tctx, srss, mlocal_name(l), tctx.infer(l), l);
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}
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return srss;
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}
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expr unfold_reducible_instances(expr const & e);
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expr remove_unnecessary_casts(expr const & e);
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bool instantiate_emetas(blast_tmp_type_context & tmp_tctx, unsigned num_emeta,
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list<expr> const & emetas, list<bool> const & instances);
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/* Results */
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result lift_from_eq(expr const & e_old, result const & r_eq);
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result join(result const & r1, result const & r2);
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result funext(result const & r, expr const & l);
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result finalize(result const & r);
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/* Simplification */
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result simplify(expr const & e, simp_rule_sets const & srss);
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result simplify(expr const & e, bool is_root);
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result simplify_lambda(expr const & e);
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result simplify_pi(expr const & e);
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result simplify_app(expr const & e);
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result simplify_fun(expr const & e);
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optional<result> simplify_numeral(expr const & e);
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/* Proving */
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optional<expr> prove(expr const & thm);
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optional<expr> prove(expr const & thm, simp_rule_sets const & srss);
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/* Rewriting */
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result rewrite(expr const & e);
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result rewrite(expr const & e, simp_rule_sets const & srss);
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result rewrite(expr const & e, simp_rule const & sr);
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/* Congruence */
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result congr_fun_arg(result const & r_f, result const & r_arg);
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result congr(result const & r_f, result const & r_arg);
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result congr_fun(result const & r_f, expr const & arg);
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result congr_arg(expr const & f, result const & r_arg);
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result congr_funs(result const & r_f, buffer<expr> const & args);
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result try_congrs(expr const & e);
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result try_congr(expr const & e, congr_rule const & cr);
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template<typename F>
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optional<result> synth_congr(expr const & e, F && simp);
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/* Fusion */
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result maybe_fuse(expr const & e, bool is_root);
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result fuse(expr const & e);
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expr_pair split_summand(expr const & e, expr const & f_mul, expr const & one);
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public:
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simplifier(name const & rel): m_app_builder(*m_tmp_tctx), m_rel(rel) { }
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result operator()(expr const & e, simp_rule_sets const & srss) { return simplify(e, srss); }
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};
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/* Cache */
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optional<result> simplifier::cache_lookup(expr const & e) {
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auto it = m_cache.find(key(m_rel, e));
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if (it == m_cache.end()) return optional<result>();
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/* The cache ignores subsingletons, so we may need to
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synthesize a congruence lemma. */
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expr e_old = it->first.m_e;
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result r_old = it->second;
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lean_assert(abstract_is_equal(e, e_old));
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if (e == e_old) return optional<result>(r_old);
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lean_assert(is_app(e_old));
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buffer<expr> new_args, old_args;
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expr const & f_new = get_app_args(e, new_args);
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DEBUG_CODE(expr const & f_old =) get_app_args(e_old, old_args);
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lean_assert(f_new == f_old);
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auto congr_lemma = mk_congr_lemma(f_new, new_args.size());
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if (!congr_lemma) return optional<result>();
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expr proof = congr_lemma->get_proof();
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expr type = congr_lemma->get_type();
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unsigned i = 0;
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for_each(congr_lemma->get_arg_kinds(), [&](congr_arg_kind const & ckind) {
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if (ckind != congr_arg_kind::Cast) {
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lean_assert(new_args[i] == old_args[i]);
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}
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proof = mk_app(proof, new_args[i]);
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type = instantiate(binding_body(type), new_args[i]);
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expr rfl;
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switch (ckind) {
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case congr_arg_kind::Fixed:
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break;
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case congr_arg_kind::Eq:
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rfl = m_app_builder.mk_eq_refl(old_args[i]);
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proof = mk_app(proof, old_args[i], rfl);
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type = instantiate(binding_body(type), old_args[i]);
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type = instantiate(binding_body(type), rfl);
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break;
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case congr_arg_kind::Cast:
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proof = mk_app(proof, old_args[i]);
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type = instantiate(binding_body(type), old_args[i]);
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break;
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}
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i++;
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});
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lean_assert(is_eq(type));
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result r_congr = result(e_old, proof);
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return optional<result>(join(r_congr, r_old));
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}
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void simplifier::cache_save(expr const & e, result const & r) {
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m_cache.insert(mk_pair(key(m_rel, e), r));
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}
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/* Results */
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result simplifier::lift_from_eq(expr const & e_old, result const & r_eq) {
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if (!r_eq.has_proof()) return r_eq;
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lean_assert(r_eq.has_proof());
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/* r_eq.get_proof() : e_old = r_eq.get_new() */
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/* goal : e_old <m_rel> r_eq.get_new() */
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expr l = m_tmp_tctx->mk_tmp_local(m_tmp_tctx->infer(e_old));
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expr motive_local = m_app_builder.mk_app(m_rel, e_old, l);
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/* motive = fun y, e_old <m_rel> y */
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expr motive = Fun(l, motive_local);
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/* Rxx = x <m_rel> x */
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expr Rxx = m_app_builder.mk_refl(m_rel, e_old);
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expr pf = m_app_builder.mk_eq_rec(motive, Rxx, r_eq.get_proof());
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return result(r_eq.get_new(), pf);
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}
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result simplifier::join(result const & r1, result const & r2) {
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/* Assumes that both results are with respect to the same relation */
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if (!r1.has_proof()) {
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return r2;
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} else if (!r2.has_proof()) {
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lean_assert(r1.has_proof());
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return result(r2.get_new(), r1.get_proof());
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} else {
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/* If they both have proofs, we need to glue them together with transitivity. */
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lean_assert(r1.has_proof() && r2.has_proof());
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expr trans = m_app_builder.mk_trans(m_rel, r1.get_proof(), r2.get_proof());
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return result(r2.get_new(), trans);
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}
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}
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result simplifier::funext(result const & r, expr const & l) {
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// theorem funext {f₁ f₂ : Πx : A, B x} : (∀x, f₁ x = f₂ x) → f₁ = f₂ :=
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expr e = Fun(l, r.get_new());
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if (!r.has_proof()) return result(e);
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expr pf = m_app_builder.mk_app(get_funext_name(), Fun(l, r.get_proof()));
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return result(e, pf);
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}
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result simplifier::finalize(result const & r) {
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if (r.has_proof()) return r;
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expr pf = m_app_builder.mk_refl(m_rel, r.get_new());
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return result(r.get_new(), pf);
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}
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/* Simplification */
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result simplifier::simplify(expr const & e, simp_rule_sets const & srss) {
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flet<simp_rule_sets> set_srss(m_srss, srss);
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simplify_cache fresh_cache;
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flet<simplify_cache> set_simplify_cache(m_cache, fresh_cache);
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return simplify(e, true);
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}
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result simplifier::simplify(expr const & e, bool is_root) {
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m_num_steps++;
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flet<unsigned> inc_depth(m_depth, m_depth+1);
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if (m_trace) {
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ios().get_diagnostic_channel() << m_depth << "." << m_rel << ": " << e << "\n";
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}
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if (m_num_steps > m_max_steps)
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throw blast_exception("simplifier failed, maximum number of steps exceeded", e);
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if (m_memoize) {
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if (auto it = cache_lookup(e)) return *it;
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}
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if (m_numerals) {
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// TODO(dhs): cache these?
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if (auto r = simplify_numeral(e)) return *r;
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}
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result r(e);
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if (m_top_down) r = join(r, rewrite(whnf(r.get_new())));
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r.update(whnf(r.get_new()));
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switch (r.get_new().kind()) {
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case expr_kind::Local:
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case expr_kind::Meta:
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case expr_kind::Sort:
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case expr_kind::Constant:
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// no-op
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break;
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case expr_kind::Var:
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lean_unreachable();
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case expr_kind::Macro:
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if (m_expand_macros) {
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if (auto m = m_tmp_tctx->expand_macro(e)) r = join(r, simplify(whnf(*m), is_root));
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}
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break;
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case expr_kind::Lambda:
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if (using_eq()) r = join(r, simplify_lambda(r.get_new()));
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break;
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case expr_kind::Pi:
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r = join(r, simplify_pi(r.get_new()));
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break;
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case expr_kind::App:
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r = join(r, simplify_app(r.get_new()));
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break;
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}
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if (!m_top_down) r = join(r, rewrite(whnf(r.get_new())));
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if (r.get_new() == e && !using_eq()) {
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result r_eq;
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{
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flet<name> use_eq(m_rel, get_eq_name());
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r_eq = simplify(r.get_new(), is_root);
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}
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r = join(r, lift_from_eq(r.get_new(), r_eq));
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}
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if (m_exhaustive && r.has_proof()) r = join(r, simplify(r.get_new(), is_root));
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if (m_memoize) cache_save(e, r);
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if (m_fuse && using_eq()) r = join(r, maybe_fuse(r.get_new(), is_root));
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return r;
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}
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result simplifier::simplify_lambda(expr const & e) {
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lean_assert(is_lambda(e));
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expr t = e;
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buffer<expr> ls;
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while (is_lambda(t)) {
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expr d = instantiate_rev(binding_domain(t), ls.size(), ls.data());
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expr l = m_tmp_tctx->mk_tmp_local(d, binding_info(t));
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ls.push_back(l);
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t = instantiate(binding_body(t), l);
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}
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result r = simplify(t, false);
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for (int i = ls.size() - 1; i >= 0; --i) r = funext(r, ls[i]);
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return r;
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}
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result simplifier::simplify_pi(expr const & e) {
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lean_assert(is_pi(e));
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return try_congrs(e);
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}
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expr simplifier::unfold_reducible_instances(expr const & e) {
|
|
buffer<expr> args;
|
|
expr f = get_app_args(e, args);
|
|
fun_info f_info = get_fun_info(f, args.size());
|
|
int i = -1;
|
|
for_each(f_info.get_params_info(), [&](param_info const & p_info) {
|
|
i++;
|
|
if (p_info.is_inst_implicit() && !p_info.is_subsingleton()) {
|
|
// ios().get_diagnostic_channel() << "normalize: "
|
|
// << args[i] << "\n==>\n" << blast::normalize(args[i]) << "\n";
|
|
args[i] = blast::normalize(args[i]);
|
|
}
|
|
});
|
|
return mk_app(f, args);
|
|
}
|
|
|
|
result simplifier::simplify_app(expr const & _e) {
|
|
lean_assert(is_app(_e));
|
|
expr e = unfold_reducible_instances(_e);
|
|
|
|
/* (1) Try user-defined congruences */
|
|
result r_user = try_congrs(e);
|
|
if (r_user.has_proof()) {
|
|
if (using_eq()) return join(r_user, simplify_fun(r_user.get_new()));
|
|
else return r_user;
|
|
}
|
|
|
|
/* (2) Synthesize congruence lemma */
|
|
if (using_eq()) {
|
|
optional<result> r_args = synth_congr(e, [&](expr const & e) {
|
|
return simplify(e, false);
|
|
});
|
|
if (r_args) return join(*r_args, simplify_fun(r_args->get_new()));
|
|
}
|
|
|
|
/* (3) Fall back on generic binary congruence */
|
|
if (using_eq()) {
|
|
expr const & f = app_fn(e);
|
|
expr const & arg = app_arg(e);
|
|
|
|
// TODO(dhs): it is not clear if this recursive call should be considered
|
|
// a root or not, though does not matter since if + were being applied,
|
|
// we would have synthesized a congruence rule in step (2).
|
|
result r_f = simplify(f, false);
|
|
|
|
if (is_dependent_fn(f)) {
|
|
if (r_f.has_proof()) return congr_fun(r_f, arg);
|
|
else return mk_app(r_f.get_new(), arg);
|
|
} else {
|
|
result r_arg = simplify(arg, false);
|
|
return congr_fun_arg(r_f, r_arg);
|
|
}
|
|
}
|
|
return result(e);
|
|
}
|
|
|
|
result simplifier::simplify_fun(expr const & e) {
|
|
lean_assert(is_app(e));
|
|
buffer<expr> args;
|
|
expr const & f = get_app_args(e, args);
|
|
result r_f = simplify(f, true);
|
|
return congr_funs(r_f, args);
|
|
}
|
|
|
|
optional<result> simplifier::simplify_numeral(expr const & e) {
|
|
if (is_num(e)) { return optional<result>(result(e)); }
|
|
|
|
try {
|
|
expr_pair r = mk_norm_num(*m_tmp_tctx, e);
|
|
return optional<result>(result(r.first, r.second));
|
|
} catch (exception e) {
|
|
return optional<result>();
|
|
}
|
|
}
|
|
|
|
/* Proving */
|
|
|
|
optional<expr> simplifier::prove(expr const & thm) {
|
|
flet<name> set_name(m_rel, get_iff_name());
|
|
result r_cond = simplify(thm, true);
|
|
if (is_constant(r_cond.get_new()) && const_name(r_cond.get_new()) == get_true_name()) {
|
|
expr pf = m_app_builder.mk_app(get_iff_elim_right_name(),
|
|
finalize(r_cond).get_proof(),
|
|
mk_constant(get_true_intro_name()));
|
|
return some_expr(pf);
|
|
}
|
|
return none_expr();
|
|
}
|
|
|
|
optional<expr> simplifier::prove(expr const & thm, simp_rule_sets const & srss) {
|
|
flet<name> set_name(m_rel, get_iff_name());
|
|
result r_cond = simplify(thm, srss);
|
|
if (is_constant(r_cond.get_new()) && const_name(r_cond.get_new()) == get_true_name()) {
|
|
expr pf = m_app_builder.mk_app(get_iff_elim_right_name(),
|
|
finalize(r_cond).get_proof(),
|
|
mk_constant(get_true_intro_name()));
|
|
return some_expr(pf);
|
|
}
|
|
return none_expr();
|
|
}
|
|
|
|
/* Rewriting */
|
|
|
|
result simplifier::rewrite(expr const & e) {
|
|
result r(e);
|
|
while (true) {
|
|
result r_ctx = rewrite(r.get_new(), m_ctx_srss);
|
|
result r_new = rewrite(r_ctx.get_new(), m_srss);
|
|
if (!r_ctx.has_proof() && !r_new.has_proof()) break;
|
|
r = join(join(r, r_ctx), r_new);
|
|
}
|
|
return r;
|
|
}
|
|
|
|
result simplifier::rewrite(expr const & e, simp_rule_sets const & srss) {
|
|
result r(e);
|
|
|
|
simp_rule_set const * sr = srss.find(m_rel);
|
|
if (!sr) return r;
|
|
|
|
list<simp_rule> const * srs = sr->find_simp(e);
|
|
if (!srs) return r;
|
|
|
|
for_each(*srs, [&](simp_rule const & sr) {
|
|
result r_new = rewrite(r.get_new(), sr);
|
|
if (!r_new.has_proof()) return;
|
|
r = join(r, r_new);
|
|
});
|
|
return r;
|
|
}
|
|
|
|
result simplifier::rewrite(expr const & e, simp_rule const & sr) {
|
|
blast_tmp_type_context tmp_tctx(sr.get_num_umeta(), sr.get_num_emeta());
|
|
|
|
if (m_trace) {
|
|
expr new_lhs = tmp_tctx->instantiate_uvars_mvars(sr.get_lhs());
|
|
expr new_rhs = tmp_tctx->instantiate_uvars_mvars(sr.get_rhs());
|
|
ios().get_diagnostic_channel()
|
|
<< "TRYREW(" << sr.get_id() << ") "
|
|
<< "[" << new_lhs << " =?= " << new_rhs << "]\n";
|
|
}
|
|
|
|
if (!tmp_tctx->is_def_eq(e, sr.get_lhs())) return result(e);
|
|
|
|
if (m_trace) {
|
|
expr new_lhs = tmp_tctx->instantiate_uvars_mvars(sr.get_lhs());
|
|
expr new_rhs = tmp_tctx->instantiate_uvars_mvars(sr.get_rhs());
|
|
ios().get_diagnostic_channel()
|
|
<< "REW(" << sr.get_id() << ") "
|
|
<< "[" << new_lhs << " =?= " << new_rhs << "]\n";
|
|
}
|
|
|
|
if (!instantiate_emetas(tmp_tctx, sr.get_num_emeta(), sr.get_emetas(), sr.get_instances())) return result(e);
|
|
|
|
for (unsigned i = 0; i < sr.get_num_umeta(); i++) {
|
|
if (!tmp_tctx->is_uvar_assigned(i)) return result(e);
|
|
}
|
|
|
|
expr new_lhs = tmp_tctx->instantiate_uvars_mvars(sr.get_lhs());
|
|
expr new_rhs = tmp_tctx->instantiate_uvars_mvars(sr.get_rhs());
|
|
|
|
if (sr.is_perm()) {
|
|
if (!is_lt(new_rhs, new_lhs, false))
|
|
return result(e);
|
|
}
|
|
|
|
expr pf = tmp_tctx->instantiate_uvars_mvars(sr.get_proof());
|
|
return result(new_rhs, pf);
|
|
}
|
|
|
|
/* Congruence */
|
|
|
|
result simplifier::congr_fun_arg(result const & r_f, result const & r_arg) {
|
|
if (!r_f.has_proof() && !r_arg.has_proof()) return result(mk_app(r_f.get_new(), r_arg.get_new()));
|
|
else if (!r_f.has_proof()) return congr_arg(r_f.get_new(), r_arg);
|
|
else if (!r_arg.has_proof()) return congr_fun(r_f, r_arg.get_new());
|
|
else return congr(r_f, r_arg);
|
|
}
|
|
|
|
result simplifier::congr(result const & r_f, result const & r_arg) {
|
|
lean_assert(r_f.has_proof() && r_arg.has_proof());
|
|
// theorem congr {A B : Type} {f₁ f₂ : A → B} {a₁ a₂ : A} (H₁ : f₁ = f₂) (H₂ : a₁ = a₂) : f₁ a₁ = f₂ a₂
|
|
expr e = mk_app(r_f.get_new(), r_arg.get_new());
|
|
expr pf = m_app_builder.mk_congr(r_f.get_proof(), r_arg.get_proof());
|
|
return result(e, pf);
|
|
}
|
|
|
|
result simplifier::congr_fun(result const & r_f, expr const & arg) {
|
|
lean_assert(r_f.has_proof());
|
|
// theorem congr_fun {A : Type} {B : A → Type} {f g : Π x, B x} (H : f = g) (a : A) : f a = g a
|
|
expr e = mk_app(r_f.get_new(), arg);
|
|
expr pf = m_app_builder.mk_congr_fun(r_f.get_proof(), arg);
|
|
return result(e, pf);
|
|
}
|
|
|
|
result simplifier::congr_arg(expr const & f, result const & r_arg) {
|
|
lean_assert(r_arg.has_proof());
|
|
// theorem congr_arg {A B : Type} {a₁ a₂ : A} (f : A → B) : a₁ = a₂ → f a₁ = f a₂
|
|
expr e = mk_app(f, r_arg.get_new());
|
|
expr pf = m_app_builder.mk_congr_arg(f, r_arg.get_proof());
|
|
return result(e, pf);
|
|
}
|
|
|
|
/* Note: handles reflexivity */
|
|
result simplifier::congr_funs(result const & r_f, buffer<expr> const & args) {
|
|
// congr_fun : ∀ {A : Type} {B : A → Type} {f g : Π (x : A), B x}, f = g → (∀ (a : A), f a = g a)
|
|
expr e = r_f.get_new();
|
|
for (unsigned i = 0; i < args.size(); ++i) {
|
|
e = mk_app(e, args[i]);
|
|
}
|
|
if (!r_f.has_proof()) return result(e);
|
|
expr pf = r_f.get_proof();
|
|
for (unsigned i = 0; i < args.size(); ++i) {
|
|
pf = m_app_builder.mk_app(get_congr_fun_name(), pf, args[i]);
|
|
}
|
|
return result(e, pf);
|
|
}
|
|
|
|
result simplifier::try_congrs(expr const & e) {
|
|
simp_rule_set const * sr = get_simp_rule_sets(env()).find(m_rel);
|
|
if (!sr) return result(e);
|
|
|
|
list<congr_rule> const * crs = sr->find_congr(e);
|
|
if (!crs) return result(e);
|
|
|
|
result r(e);
|
|
for_each(*crs, [&](congr_rule const & cr) {
|
|
if (r.has_proof()) return;
|
|
r = try_congr(e, cr);
|
|
});
|
|
return r;
|
|
}
|
|
|
|
result simplifier::try_congr(expr const & e, congr_rule const & cr) {
|
|
blast_tmp_type_context tmp_tctx(cr.get_num_umeta(), cr.get_num_emeta());
|
|
|
|
if (!tmp_tctx->is_def_eq(e, cr.get_lhs())) return result(e);
|
|
|
|
if (m_trace) {
|
|
ios().get_diagnostic_channel() << "<" << cr.get_id() << "> "
|
|
<< e << " === " << cr.get_lhs() << "\n";
|
|
}
|
|
|
|
/* First, iterate over the congruence hypotheses */
|
|
bool failed = false;
|
|
bool simplified = false;
|
|
list<expr> const & congr_hyps = cr.get_congr_hyps();
|
|
for_each(congr_hyps, [&](expr const & m) {
|
|
if (failed) return;
|
|
buffer<expr> ls;
|
|
expr m_type = tmp_tctx->instantiate_uvars_mvars(tmp_tctx->infer(m));
|
|
|
|
while (is_pi(m_type)) {
|
|
expr d = instantiate_rev(binding_domain(m_type), ls.size(), ls.data());
|
|
expr l = tmp_tctx->mk_tmp_local(d, binding_info(m_type));
|
|
lean_assert(!has_metavar(l));
|
|
ls.push_back(l);
|
|
m_type = instantiate(binding_body(m_type), l);
|
|
}
|
|
|
|
expr h_rel, h_lhs, h_rhs;
|
|
lean_verify(is_simp_relation(env(), m_type, h_rel, h_lhs, h_rhs) && is_constant(h_rel));
|
|
{
|
|
flet<name> set_name(m_rel, const_name(h_rel));
|
|
flet<simp_rule_sets> set_ctx_srss(m_ctx_srss, add_to_srss(m_ctx_srss, ls));
|
|
|
|
h_lhs = tmp_tctx->instantiate_uvars_mvars(h_lhs);
|
|
lean_assert(!has_metavar(h_lhs));
|
|
|
|
result r_congr_hyp = simplify(h_lhs, m_srss);
|
|
if (r_congr_hyp.has_proof()) simplified = true;
|
|
r_congr_hyp = finalize(r_congr_hyp);
|
|
expr hyp = finalize(r_congr_hyp).get_proof();
|
|
|
|
if (!tmp_tctx->is_def_eq(m, Fun(ls, hyp))) failed = true;
|
|
}
|
|
});
|
|
|
|
if (failed || !simplified) return result(e);
|
|
|
|
if (!instantiate_emetas(tmp_tctx, cr.get_num_emeta(), cr.get_emetas(), cr.get_instances())) return result(e);
|
|
|
|
for (unsigned i = 0; i < cr.get_num_umeta(); i++) {
|
|
if (!tmp_tctx->is_uvar_assigned(i)) return result(e);
|
|
}
|
|
|
|
expr e_s = tmp_tctx->instantiate_uvars_mvars(cr.get_rhs());
|
|
expr pf = tmp_tctx->instantiate_uvars_mvars(cr.get_proof());
|
|
return result(e_s, pf);
|
|
}
|
|
|
|
bool simplifier::instantiate_emetas(blast_tmp_type_context & tmp_tctx, unsigned num_emeta, list<expr> const & emetas,
|
|
list<bool> const & instances) {
|
|
bool failed = false;
|
|
unsigned i = num_emeta;
|
|
for_each2(emetas, instances, [&](expr const & m, bool const & is_instance) {
|
|
i--;
|
|
if (failed) return;
|
|
expr m_type = tmp_tctx->instantiate_uvars_mvars(tmp_tctx->infer(m));
|
|
lean_assert(!has_metavar(m_type));
|
|
|
|
if (is_instance) {
|
|
if (auto v = tmp_tctx->mk_class_instance(m_type)) {
|
|
if (!tmp_tctx->force_assign(m, *v)) {
|
|
if (m_trace) {
|
|
ios().get_diagnostic_channel() << "unable to assign instance for: " << m_type << "\n";
|
|
}
|
|
failed = true;
|
|
return;
|
|
}
|
|
} else {
|
|
if (m_trace) {
|
|
ios().get_diagnostic_channel() << "unable to synthesize instance for: " << m_type << "\n";
|
|
}
|
|
failed = true;
|
|
return;
|
|
}
|
|
}
|
|
|
|
if (tmp_tctx->is_mvar_assigned(i)) return;
|
|
|
|
if (tmp_tctx->is_prop(m_type)) {
|
|
if (auto pf = prove(m_type)) {
|
|
lean_verify(tmp_tctx->is_def_eq(m, *pf));
|
|
return;
|
|
}
|
|
}
|
|
|
|
if (m_trace) {
|
|
ios().get_diagnostic_channel() << "failed to assign: " << m << " : " << m_type << "\n";
|
|
}
|
|
|
|
failed = true;
|
|
return;
|
|
});
|
|
|
|
return !failed;
|
|
}
|
|
|
|
template<typename F>
|
|
optional<result> simplifier::synth_congr(expr const & e, F && simp) {
|
|
static_assert(std::is_same<typename std::result_of<F(expr const & e)>::type, result>::value,
|
|
"synth_congr: simp must take expressions to results");
|
|
lean_assert(is_app(e));
|
|
buffer<expr> args;
|
|
expr f = get_app_args(e, args);
|
|
auto congr_lemma = mk_congr_lemma_for_simp(f, args.size());
|
|
if (!congr_lemma) return optional<result>();
|
|
expr proof = congr_lemma->get_proof();
|
|
expr type = congr_lemma->get_type();
|
|
unsigned i = 0;
|
|
bool has_proof = false;
|
|
bool has_cast = false;
|
|
buffer<expr> locals;
|
|
for_each(congr_lemma->get_arg_kinds(), [&](congr_arg_kind const & ckind) {
|
|
proof = mk_app(proof, args[i]);
|
|
type = instantiate(binding_body(type), args[i]);
|
|
if (ckind == congr_arg_kind::Eq) {
|
|
result r_arg = simp(args[i]);
|
|
if (r_arg.has_proof()) has_proof = true;
|
|
r_arg = finalize(r_arg);
|
|
proof = mk_app(proof, r_arg.get_new(), r_arg.get_proof());
|
|
type = instantiate(binding_body(type), r_arg.get_new());
|
|
type = instantiate(binding_body(type), r_arg.get_proof());
|
|
} else if (ckind == congr_arg_kind::Cast) {
|
|
has_cast = true;
|
|
}
|
|
i++;
|
|
});
|
|
lean_assert(is_eq(type));
|
|
buffer<expr> type_args;
|
|
get_app_args(type, type_args);
|
|
expr e_new = remove_unnecessary_casts(type_args[2]);
|
|
if (has_proof) return optional<result>(result(e_new, proof));
|
|
else return optional<result>(result(e_new));
|
|
}
|
|
|
|
expr simplifier::remove_unnecessary_casts(expr const & e) {
|
|
buffer<expr> args;
|
|
expr f = get_app_args(e, args);
|
|
fun_info f_info = get_fun_info(f, args.size());
|
|
int i = -1;
|
|
for_each(f_info.get_params_info(), [&](param_info const & p_info) {
|
|
i++;
|
|
if (p_info.is_subsingleton()) {
|
|
while (is_constant(get_app_fn(args[i]))) {
|
|
buffer<expr> cast_args;
|
|
expr f_cast = get_app_args(args[i], cast_args);
|
|
name n_f = const_name(f_cast);
|
|
if (n_f == get_eq_rec_name() || n_f == get_eq_drec_name() || n_f == get_eq_nrec_name()) {
|
|
lean_assert(cast_args.size() == 6);
|
|
expr major_premise = cast_args[5];
|
|
expr f_major_premise = get_app_fn(major_premise);
|
|
if (is_constant(f_major_premise) && const_name(f_major_premise) == get_eq_refl_name())
|
|
args[i] = cast_args[3];
|
|
else
|
|
return;
|
|
} else {
|
|
return;
|
|
}
|
|
}
|
|
}
|
|
});
|
|
return mk_app(f, args);
|
|
}
|
|
|
|
/* Fusion */
|
|
|
|
result simplifier::maybe_fuse(expr const & e, bool is_root) {
|
|
// ios().get_diagnostic_channel() << "maybe_fuse(" << e << ", " << is_root << ")\n";
|
|
if (!is_app(e)) return result(e);
|
|
if (is_root && is_add_app(e)) return fuse(e);
|
|
if (!is_root && is_add_app(e)) return result(e);
|
|
lean_assert(!is_add_app(e));
|
|
/* At this point we know we are an application of something other than + */
|
|
optional<result> r = synth_congr(e, [&](expr const & arg) {
|
|
// ios().get_diagnostic_channel() << "synth_fuse: " << arg << "\n";
|
|
if (is_add_app(arg)) return fuse(arg);
|
|
else return result(arg);
|
|
});
|
|
if (r) return *r;
|
|
else return result(e);
|
|
}
|
|
|
|
result simplifier::fuse(expr const & e) {
|
|
lean_assert(is_add_app(e));
|
|
// ios().get_diagnostic_channel() << "FUSE: " << e << "\n\n";
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flet<bool> no_recursive_fusion(m_fuse, false);
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/* Note: we assume all negations are on the outside of the summands */
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/* Example
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1. Input (e): n1 * x1 * n2 * x2 + x3 + - (x1 * x2 * n3) + (- x3) + x4
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2. Split summands: (n1 * n2) * (x1 * x2) + 1 * x3 + (- n3) * (x1 * x2) + (- 1) * x3 + 1 * x4
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3. Group by variable (e_grp): ((n1 * n2) * (x1 * x2) + (- n3) * (x1 * x2)) + (1 * x3 + (-1) * x3) + x4
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4. Introduce locals (e_grp_ls): ((n1 * n2) * l1 + (- n3) * l1) + (1 * l2 + (-1) * l2) + l3
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5. Fuse (e_fused_ls): (n1 * n2 + (- n3)) * l1 + (1 + (- 1)) * l2 + 1 * l3
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6. Simplify coefficients (e_simp_ls): (n1 * n2 + (- n3)) * l1 + l3
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7. Instantiate locals (e_simp): (n1 * n2 + (- n3)) * (x1 * x2) + x4
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Proofs
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1. Prove (1) == (3) using simplify with [ac]
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2. Prove (4) == (5) using simplify with [som]
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3. Prove (5) == (6) using simplify with [numeral]
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4. Prove (4) == (6) by transitivity of proofs (2) and (3)
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5. Prove (3) == (7) by instantiating proof (4)
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6. Prove (1) == (7) by transitivity of proofs (1) and (5)
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*/
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/* Construct useful expressions */
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buffer<expr> args;
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get_app_args(e, args);
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expr T = args[0];
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expr f_add, f_mul, zero, one;
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try {
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f_add = m_app_builder.mk_partial_add(T);
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f_mul = m_app_builder.mk_partial_mul(T);
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zero = m_app_builder.mk_zero(T);
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one = m_app_builder.mk_one(T);
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expr left_distrib = m_app_builder.mk_partial_left_distrib(T);
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} catch (app_builder_exception & ex) {
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ios().get_diagnostic_channel() << "Cannot synthesize necessary instances\n";
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return result(e);
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}
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/* (1 ==> 2) Split each summand into (a) numerals and (b) variables */
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buffer<expr> numerals;
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buffer<expr> variables;
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expr s = e;
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while (is_add_app(s)) {
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buffer<expr> args;
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expr f = get_app_args(s, args);
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auto n_v = split_summand(args[2], f_mul, one);
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numerals.push_back(n_v.first);
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variables.push_back(n_v.second);
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s = args[3];
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}
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auto n_v = split_summand(s, f_mul, one);
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numerals.push_back(n_v.first);
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variables.push_back(n_v.second);
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lean_assert(numerals.size() == variables.size());
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/* (2 ==> 3, 4, 5) Group the summands by variable */
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expr_bi_struct_map<list<expr> > variable_to_numerals;
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for (unsigned i = 0; i < numerals.size(); i++) {
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auto it = variable_to_numerals.find(variables[i]);
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if (it != variable_to_numerals.end()) it->second = cons(numerals[i], it->second);
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else variable_to_numerals.insert({variables[i], list<expr>(numerals[i])});
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}
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expr e_grp = zero;
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expr e_grp_ls = zero;
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expr e_fused_ls = zero;
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buffer<expr> locals;
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variables.clear();
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for (auto v_ns : variable_to_numerals) {
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expr local = m_tmp_tctx->mk_tmp_local(T);
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locals.push_back(local);
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variables.push_back(v_ns.first);
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expr term = zero;
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expr term_ls = zero;
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expr term_fused_ls = zero;
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for_each(v_ns.second, [&](expr const & n) {
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term = mk_app(f_add, mk_app(f_mul, v_ns.first, n), term);
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term_ls = mk_app(f_add, mk_app(f_mul, local, n), term_ls);
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term_fused_ls = mk_app(f_add, n, term_fused_ls);
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});
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e_grp = mk_app(f_add, term, e_grp);
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e_grp_ls = mk_app(f_add, term_ls, e_grp_ls);
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e_fused_ls = mk_app(f_add, mk_app(f_mul, term_fused_ls, local), e_fused_ls);
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}
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/* Prove (1) == (3) using simplify with [ac] */
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flet<bool> no_simplify_numerals(m_numerals, false);
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auto pf_1_3 = prove(m_app_builder.mk_eq(e, e_grp),
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get_simp_rule_sets(env(), ios(), *g_simplify_ac_namespace));
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if (!pf_1_3) {
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diagnostic(env(), ios()) << e << "\n\n =?=\n\n" << e_grp << "\n";
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throw blast_exception("Failed to prove (1) == (3) during fusion", e);
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}
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/* Prove (4) == (5) using simplify with [som] */
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auto pf_4_5 = prove(m_app_builder.mk_eq(e_grp_ls, e_fused_ls),
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get_simp_rule_sets(env(), ios(), *g_simplify_som_namespace));
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if (!pf_4_5) {
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diagnostic(env(), ios()) << e_grp_ls << "\n\n =?=\n\n" << e_fused_ls << "\n";
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throw blast_exception("Failed to prove (4) == (5) during fusion", e);
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}
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/* Prove (5) == (6) using simplify with [numeral] */
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flet<bool> simplify_numerals(m_numerals, true);
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result r_simp_ls = simplify(e_fused_ls, get_simp_rule_sets(env(), ios(), *g_simplify_unit_namespace));
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/* Prove (4) == (6) by transitivity of proofs (2) and (3) */
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|
expr pf_4_6;
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if (!r_simp_ls.has_proof()) pf_4_6 = *pf_4_5;
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else pf_4_6 = m_app_builder.mk_eq_trans(*pf_4_5, r_simp_ls.get_proof());
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/* Prove (3) == (7) by instantiating proof (4) */
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expr pf_3_7 = mk_app(Fun(locals, pf_4_6), variables);
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/* Prove (1) == (7) by transitivity of proofs (1) and (5) */
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expr pf_1_7 = m_app_builder.mk_eq_trans(*pf_1_3, pf_3_7);
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return result(mk_app(Fun(locals, r_simp_ls.get_new()), variables), pf_1_7);
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}
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expr_pair simplifier::split_summand(expr const & e, expr const & f_mul, expr const & one) {
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|
/* [e] is a possibly negated, possibly null product, where some of the multiplicands are numerals
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|
and the rest are treated as variables. This method does the following conversion:
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- (x1 * n1 * x2 * n3) ==> (-1 * n1 * n3) * (x1 * x2) */
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|
expr s = e;
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|
expr numeral = one;
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|
expr variable = one;
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|
if (is_neg_app(s)) {
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|
buffer<expr> args;
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|
get_app_args(s, args);
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|
numeral = m_app_builder.mk_app(get_neg_name(), one);
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|
s = args[2];
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|
}
|
|
while (is_mul_app(s)) {
|
|
buffer<expr> args;
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|
get_app_args(s, args);
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|
expr const & multiplicand = args[2];
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if (is_num(multiplicand)) {
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|
numeral = mk_app(f_mul, multiplicand, numeral);
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|
} else {
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|
if (variable == one) variable = multiplicand;
|
|
else variable = mk_app(f_mul, multiplicand, variable);
|
|
}
|
|
s = args[3];
|
|
}
|
|
if (is_num(s)) {
|
|
numeral = mk_app(f_mul, s, numeral);
|
|
} else {
|
|
if (variable == one) variable = s;
|
|
else variable = mk_app(f_mul, s, variable);
|
|
}
|
|
return mk_pair(numeral, variable);
|
|
}
|
|
|
|
/* Setup and teardown */
|
|
|
|
void initialize_simplifier() {
|
|
g_simplify_unit_namespace = new name{"simplifier", "unit"};
|
|
g_simplify_ac_namespace = new name{"simplifier", "ac"};
|
|
g_simplify_som_namespace = new name{"simplifier", "som"};
|
|
g_simplify_numeral_namespace = new name{"simplifier", "numeral"};
|
|
|
|
g_simplify_max_steps = new name{"simplify", "max_steps"};
|
|
g_simplify_top_down = new name{"simplify", "top_down"};
|
|
g_simplify_exhaustive = new name{"simplify", "exhaustive"};
|
|
g_simplify_memoize = new name{"simplify", "memoize"};
|
|
g_simplify_contextual = new name{"simplify", "contextual"};
|
|
g_simplify_expand_macros = new name{"simplify", "expand_macros"};
|
|
g_simplify_trace = new name{"simplify", "trace"};
|
|
g_simplify_fuse = new name{"simplify", "fuse"};
|
|
g_simplify_numerals = new name{"simplify", "numerals"};
|
|
|
|
register_unsigned_option(*g_simplify_max_steps, LEAN_DEFAULT_SIMPLIFY_MAX_STEPS,
|
|
"(simplify) max allowed steps in simplification");
|
|
register_bool_option(*g_simplify_top_down, LEAN_DEFAULT_SIMPLIFY_TOP_DOWN,
|
|
"(simplify) use top-down rewriting instead of bottom-up");
|
|
register_bool_option(*g_simplify_exhaustive, LEAN_DEFAULT_SIMPLIFY_EXHAUSTIVE,
|
|
"(simplify) rewrite exhaustively");
|
|
register_bool_option(*g_simplify_memoize, LEAN_DEFAULT_SIMPLIFY_MEMOIZE,
|
|
"(simplify) memoize simplifications");
|
|
register_bool_option(*g_simplify_contextual, LEAN_DEFAULT_SIMPLIFY_CONTEXTUAL,
|
|
"(simplify) use contextual simplification");
|
|
register_bool_option(*g_simplify_expand_macros, LEAN_DEFAULT_SIMPLIFY_EXPAND_MACROS,
|
|
"(simplify) expand macros");
|
|
register_bool_option(*g_simplify_trace, LEAN_DEFAULT_SIMPLIFY_TRACE,
|
|
"(simplify) trace");
|
|
register_bool_option(*g_simplify_fuse, LEAN_DEFAULT_SIMPLIFY_FUSE,
|
|
"(simplify) fuse addition in distrib structures");
|
|
register_bool_option(*g_simplify_numerals, LEAN_DEFAULT_SIMPLIFY_NUMERALS,
|
|
"(simplify) simplify (+, *, -, /) over numerals");
|
|
}
|
|
|
|
void finalize_simplifier() {
|
|
delete g_simplify_numerals;
|
|
delete g_simplify_fuse;
|
|
delete g_simplify_trace;
|
|
delete g_simplify_expand_macros;
|
|
delete g_simplify_contextual;
|
|
delete g_simplify_memoize;
|
|
delete g_simplify_exhaustive;
|
|
delete g_simplify_top_down;
|
|
delete g_simplify_max_steps;
|
|
|
|
delete g_simplify_numeral_namespace;
|
|
delete g_simplify_som_namespace;
|
|
delete g_simplify_ac_namespace;
|
|
delete g_simplify_unit_namespace;
|
|
}
|
|
|
|
/* Entry point */
|
|
result simplify(name const & rel, expr const & e, simp_rule_sets const & srss) {
|
|
return simplifier(rel)(e, srss);
|
|
}
|
|
|
|
}}
|