1082 lines
45 KiB
C++
1082 lines
45 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 <utility>
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#include <vector>
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#include "util/flet.h"
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#include "util/freset.h"
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#include "util/interrupt.h"
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#include "kernel/type_checker.h"
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#include "kernel/free_vars.h"
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#include "kernel/instantiate.h"
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#include "kernel/abstract.h"
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#include "kernel/normalizer.h"
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#include "kernel/kernel.h"
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#include "kernel/max_sharing.h"
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#include "library/heq_decls.h"
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#include "library/cast_decls.h"
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#include "library/kernel_bindings.h"
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#include "library/expr_pair.h"
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#include "library/hop_match.h"
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#include "library/expr_lt.h"
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#include "library/simplifier/rewrite_rule_set.h"
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#ifndef LEAN_SIMPLIFIER_PROOFS
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#define LEAN_SIMPLIFIER_PROOFS true
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#endif
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#ifndef LEAN_SIMPLIFIER_CONTEXTUAL
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#define LEAN_SIMPLIFIER_CONTEXTUAL true
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#endif
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#ifndef LEAN_SIMPLIFIER_SINGLE_PASS
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#define LEAN_SIMPLIFIER_SINGLE_PASS false
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#endif
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#ifndef LEAN_SIMPLIFIER_BETA
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#define LEAN_SIMPLIFIER_BETA true
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#endif
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#ifndef LEAN_SIMPLIFIER_ETA
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#define LEAN_SIMPLIFIER_ETA true
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#endif
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#ifndef LEAN_SIMPLIFIER_EVAL
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#define LEAN_SIMPLIFIER_EVAL true
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#endif
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#ifndef LEAN_SIMPLIFIER_UNFOLD
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#define LEAN_SIMPLIFIER_UNFOLD false
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#endif
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#ifndef LEAN_SIMPLIFIER_CONDITIONAL
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#define LEAN_SIMPLIFIER_CONDITIONAL true
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#endif
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#ifndef LEAN_SIMPLIFIER_MEMOIZE
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#define LEAN_SIMPLIFIER_MEMOIZE true
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#endif
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#ifndef LEAN_SIMPLIFIER_MAX_STEPS
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#define LEAN_SIMPLIFIER_MAX_STEPS std::numeric_limits<unsigned>::max()
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#endif
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namespace lean {
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static name g_simplifier_proofs {"simplifier", "proofs"};
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static name g_simplifier_contextual {"simplifier", "contextual"};
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static name g_simplifier_single_pass {"simplifier", "single_pass"};
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static name g_simplifier_beta {"simplifier", "beta"};
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static name g_simplifier_eta {"simplifier", "eta"};
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static name g_simplifier_eval {"simplifier", "eval"};
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static name g_simplifier_unfold {"simplifier", "unfold"};
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static name g_simplifier_conditional {"simplifier", "conditional"};
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static name g_simplifier_memoize {"simplifier", "memoize"};
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static name g_simplifier_max_steps {"simplifier", "max_steps"};
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RegisterBoolOption(g_simplifier_proofs, LEAN_SIMPLIFIER_PROOFS, "(simplifier) generate proofs");
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RegisterBoolOption(g_simplifier_contextual, LEAN_SIMPLIFIER_CONTEXTUAL, "(simplifier) contextual simplification");
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RegisterBoolOption(g_simplifier_single_pass, LEAN_SIMPLIFIER_SINGLE_PASS, "(simplifier) if false then the simplifier keeps applying simplifications as long as possible");
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RegisterBoolOption(g_simplifier_beta, LEAN_SIMPLIFIER_BETA, "(simplifier) use beta-reduction");
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RegisterBoolOption(g_simplifier_eta, LEAN_SIMPLIFIER_ETA, "(simplifier) use eta-reduction");
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RegisterBoolOption(g_simplifier_eval, LEAN_SIMPLIFIER_EVAL, "(simplifier) apply reductions based on computation");
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RegisterBoolOption(g_simplifier_unfold, LEAN_SIMPLIFIER_UNFOLD, "(simplifier) unfolds non-opaque definitions");
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RegisterBoolOption(g_simplifier_conditional, LEAN_SIMPLIFIER_CONDITIONAL, "(simplifier) conditional rewriting");
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RegisterBoolOption(g_simplifier_memoize, LEAN_SIMPLIFIER_MEMOIZE, "(simplifier) memoize/cache intermediate results");
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RegisterUnsignedOption(g_simplifier_max_steps, LEAN_SIMPLIFIER_MAX_STEPS, "(simplifier) maximum number of steps");
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bool get_simplifier_proofs(options const & opts) { return opts.get_bool(g_simplifier_proofs, LEAN_SIMPLIFIER_PROOFS); }
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bool get_simplifier_contextual(options const & opts) { return opts.get_bool(g_simplifier_contextual, LEAN_SIMPLIFIER_CONTEXTUAL); }
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bool get_simplifier_single_pass(options const & opts) { return opts.get_bool(g_simplifier_single_pass, LEAN_SIMPLIFIER_SINGLE_PASS); }
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bool get_simplifier_beta(options const & opts) { return opts.get_bool(g_simplifier_beta, LEAN_SIMPLIFIER_BETA); }
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bool get_simplifier_eta(options const & opts) { return opts.get_bool(g_simplifier_eta, LEAN_SIMPLIFIER_ETA); }
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bool get_simplifier_eval(options const & opts) { return opts.get_bool(g_simplifier_eval, LEAN_SIMPLIFIER_EVAL); }
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bool get_simplifier_unfold(options const & opts) { return opts.get_bool(g_simplifier_unfold, LEAN_SIMPLIFIER_UNFOLD); }
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bool get_simplifier_conditional(options const & opts) { return opts.get_bool(g_simplifier_conditional, LEAN_SIMPLIFIER_CONDITIONAL); }
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bool get_simplifier_memoize(options const & opts) { return opts.get_bool(g_simplifier_memoize, LEAN_SIMPLIFIER_MEMOIZE); }
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unsigned get_simplifier_max_steps(options const & opts) { return opts.get_unsigned(g_simplifier_max_steps, LEAN_SIMPLIFIER_MAX_STEPS); }
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static name g_local("local");
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static name g_C("C");
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static name g_x("x");
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static name g_unique = name::mk_internal_unique_name();
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class simplifier_fn {
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struct result {
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expr m_out; // the result of a simplification step
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optional<expr> m_proof; // a proof that the result is equal to the input (when m_proofs_enabled)
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bool m_heq_proof; // true if the proof is for heterogeneous equality
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explicit result(expr const & out, bool heq_proof = false):
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m_out(out), m_heq_proof(heq_proof) {}
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result(expr const & out, expr const & pr, bool heq_proof = false):
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m_out(out), m_proof(pr), m_heq_proof(heq_proof) {}
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result(expr const & out, optional<expr> const & pr, bool heq_proof = false):
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m_out(out), m_proof(pr), m_heq_proof(heq_proof) {}
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};
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typedef std::vector<rewrite_rule_set> rule_sets;
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typedef expr_map<result> cache;
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typedef std::vector<congr_theorem_info const *> congr_thms;
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ro_environment m_env;
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type_checker m_tc;
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bool m_has_heq;
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bool m_has_cast;
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context m_ctx;
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rule_sets m_rule_sets;
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cache m_cache;
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max_sharing_fn m_max_sharing;
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congr_thms m_congr_thms;
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unsigned m_contextual_depth; // number of contextual simplification steps in the current branch
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unsigned m_num_steps; // number of steps performed
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// Configuration
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bool m_proofs_enabled;
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bool m_contextual;
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bool m_single_pass;
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bool m_beta;
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bool m_eta;
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bool m_eval;
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bool m_unfold;
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bool m_conditional;
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bool m_memoize;
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unsigned m_max_steps;
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struct set_context {
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flet<context> m_set;
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freset<cache> m_reset_cache;
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set_context(simplifier_fn & s, context const & new_ctx):m_set(s.m_ctx, new_ctx), m_reset_cache(s.m_cache) {}
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};
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struct updt_rule_set {
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rewrite_rule_set & m_rs;
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rewrite_rule_set m_old;
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updt_rule_set(rewrite_rule_set & rs, expr const & fact, expr const & proof):m_rs(rs), m_old(m_rs) {
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m_rs.insert(g_local, fact, proof);
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}
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~updt_rule_set() { m_rs = m_old; }
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};
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static expr mk_lambda(name const & n, expr const & d, expr const & b) {
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return ::lean::mk_lambda(n, d, b);
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}
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/**
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\brief Return a lambda with body \c new_body, and name and domain from abst.
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*/
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static expr mk_lambda(expr const & abst, expr const & new_body) {
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return mk_lambda(abst_name(abst), abst_domain(abst), new_body);
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}
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bool is_proposition(expr const & e) {
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return m_tc.is_proposition(e, m_ctx);
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}
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bool is_definitionally_equal(expr const & t1, expr const & t2) {
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return m_tc.is_definitionally_equal(t1, t2, m_ctx);
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}
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expr infer_type(expr const & e) {
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return m_tc.infer_type(e, m_ctx);
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}
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expr ensure_pi(expr const & e) {
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return m_tc.ensure_pi(e, m_ctx);
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}
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expr normalize(expr const & e) {
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normalizer & proc = m_tc.get_normalizer();
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return proc(e, m_ctx, true);
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}
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/**
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\brief Auxiliary method for converting a proof H of (@eq A a b) into (@eq B a b) when
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type A is convertible to B, but not definitionally equal.
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*/
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expr translate_eq_proof(expr const & A, expr const & a, expr const & b, expr const & H, expr const & B) {
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return mk_subst_th(A, a, b, mk_lambda(g_x, A, mk_eq(B, a, mk_var(0))), mk_refl_th(B, a), H);
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}
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expr mk_congr1_th(expr const & f_type, expr const & f, expr const & new_f, expr const & a, expr const & Heq_f) {
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expr const & A = abst_domain(f_type);
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expr B = lower_free_vars(abst_body(f_type), 1, 1);
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return ::lean::mk_congr1_th(A, B, f, new_f, a, Heq_f);
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}
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expr mk_congr2_th(expr const & f_type, expr const & a, expr const & new_a, expr const & f, expr Heq_a) {
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expr const & A = abst_domain(f_type);
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expr B = lower_free_vars(abst_body(f_type), 1, 1);
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expr a_type = infer_type(a);
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if (!is_definitionally_equal(A, a_type))
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Heq_a = translate_eq_proof(a_type, a, new_a, Heq_a, A);
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return ::lean::mk_congr2_th(A, B, a, new_a, f, Heq_a);
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}
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expr mk_congr_th(expr const & f_type, expr const & f, expr const & new_f, expr const & a, expr const & new_a,
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expr const & Heq_f, expr Heq_a) {
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expr const & A = abst_domain(f_type);
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expr B = lower_free_vars(abst_body(f_type), 1, 1);
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expr a_type = infer_type(a);
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if (!is_definitionally_equal(A, a_type))
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Heq_a = translate_eq_proof(a_type, a, new_a, Heq_a, A);
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return ::lean::mk_congr_th(A, B, f, new_f, a, new_a, Heq_f, Heq_a);
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}
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optional<expr> mk_hcongr_th(expr const & f_type, expr const & new_f_type, expr const & f, expr const & new_f,
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expr const & a, expr const & new_a, expr const & Heq_f, expr Heq_a, bool Heq_a_is_heq) {
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expr const & A = abst_domain(f_type);
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expr const & new_A = abst_domain(new_f_type);
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expr a_type = infer_type(a);
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expr new_a_type = infer_type(new_a);
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if (!is_definitionally_equal(A, a_type) || !is_definitionally_equal(new_A, new_a_type)) {
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if (Heq_a_is_heq) {
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if (is_definitionally_equal(a_type, new_a_type) && is_definitionally_equal(A, new_A)) {
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Heq_a = mk_to_eq_th(a_type, a, new_a, Heq_a);
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Heq_a_is_heq = false;
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} else {
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return none_expr(); // we don't know how to handle this case
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}
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}
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Heq_a = translate_eq_proof(a_type, a, new_a, Heq_a, A);
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}
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if (!Heq_a_is_heq)
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Heq_a = mk_to_heq_th(A, a, new_a, Heq_a);
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return some_expr(::lean::mk_hcongr_th(A,
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new_A,
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mk_lambda(f_type, abst_body(f_type)),
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mk_lambda(new_f_type, abst_body(new_f_type)),
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f, new_f, a, new_a, Heq_f, Heq_a));
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}
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/**
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\brief Given
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a = b_res.m_out with proof b_res.m_proof
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b_res.m_out = c with proof H_bc
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This method returns a new result r s.t. r.m_out == c and a proof of a = c
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*/
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result mk_trans_result(expr const & a, result const & b_res, expr const & c, expr const & H_bc) {
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if (m_proofs_enabled) {
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if (!b_res.m_proof) {
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// The proof of a = b is reflexivity
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return result(c, H_bc);
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} else {
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expr const & b = b_res.m_out;
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expr new_proof;
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bool heq_proof = false;
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if (b_res.m_heq_proof) {
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expr b_type = infer_type(b);
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new_proof = ::lean::mk_htrans_th(infer_type(a), b_type, b_type, /* b and c must have the same type */
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a, b, c, *b_res.m_proof, mk_to_heq_th(b_type, b, c, H_bc));
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heq_proof = true;
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} else {
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new_proof = ::lean::mk_trans_th(infer_type(a), a, b, c, *b_res.m_proof, H_bc);
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}
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return result(c, new_proof, heq_proof);
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}
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} else {
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return result(c);
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}
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}
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/**
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\brief Given
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a = b_res.m_out with proof b_res.m_proof
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b_res.m_out = c_res.m_out with proof c_res.m_proof
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This method returns a new result r s.t. r.m_out == c and a proof of a = c_res.m_out
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*/
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result mk_trans_result(expr const & a, result const & b_res, result const & c_res) {
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if (m_proofs_enabled) {
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if (!b_res.m_proof) {
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// the proof of a == b is reflexivity
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return c_res;
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} else if (!c_res.m_proof) {
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// the proof of b == c is reflexivity
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return result(c_res.m_out, *b_res.m_proof, b_res.m_heq_proof);
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} else {
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bool heq_proof = b_res.m_heq_proof || c_res.m_heq_proof;
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expr new_proof;
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expr const & b = b_res.m_out;
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expr const & c = c_res.m_out;
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if (heq_proof) {
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expr a_type = infer_type(a);
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expr b_type = infer_type(b);
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expr c_type = infer_type(c);
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expr H_ab = *b_res.m_proof;
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if (!b_res.m_heq_proof)
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H_ab = mk_to_heq_th(a_type, a, b, H_ab);
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expr H_bc = *c_res.m_proof;
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if (!c_res.m_heq_proof)
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H_bc = mk_to_heq_th(b_type, b, c, H_bc);
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new_proof = ::lean::mk_htrans_th(a_type, b_type, c_type, a, b, c, H_ab, H_bc);
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} else {
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new_proof = ::lean::mk_trans_th(infer_type(a), a, b, c, *b_res.m_proof, *c_res.m_proof);
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}
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return result(c, new_proof, heq_proof);
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}
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} else {
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// proof generation is disabled
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return c_res;
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}
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}
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expr mk_app_prefix(unsigned i, expr const & a) {
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lean_assert(i > 0);
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if (i == 1)
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return arg(a, 0);
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else
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return mk_app(i, &arg(a, 0));
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}
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expr mk_app_prefix(unsigned i, buffer<expr> const & args) {
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lean_assert(i > 0);
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if (i == 1)
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return args[0];
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else
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return mk_app(i, args.data());
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}
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result simplify_app(expr const & e) {
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if (m_has_cast && is_cast(e)) {
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// e is of the form (cast A B H a)
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expr A = arg(e, 1);
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expr B = arg(e, 2);
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expr H = arg(e, 3);
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expr a = arg(e, 4);
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if (m_proofs_enabled) {
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result res_a = simplify(a);
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expr c = res_a.m_out;
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if (res_a.m_proof) {
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expr Hec;
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expr Hac = *res_a.m_proof;
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if (!res_a.m_heq_proof) {
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Hec = ::lean::mk_htrans_th(A, B, B, e, a, c,
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update_app(e, 0, mk_cast_heq_fn()), // cast A B H a == a
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mk_to_heq_th(B, a, c, Hac)); // a == c
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} else {
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Hec = ::lean::mk_htrans_th(A, B, infer_type(c), e, a, c,
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update_app(e, 0, mk_cast_heq_fn()), // cast A B H a == a
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Hac); // a == c
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}
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return result(c, Hec, true);
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} else {
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// c is definitionally equal to a
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// So, we use cast_heq theorem cast_heq : cast A B H a == a
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return result(c, update_app(e, 0, mk_cast_heq_fn()), true);
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}
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} else {
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return simplify(arg(e, 4));
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}
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}
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if (m_contextual) {
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expr const & f = arg(e, 0);
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for (auto congr_th : m_congr_thms) {
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if (congr_th->get_fun() == f)
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return simplify_app_congr(e, *congr_th);
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}
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}
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return simplify_app_default(e);
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}
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/**
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\brief Make sure the proof in rhs is using homogeneous equality, and return true.
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If it is not possible to transform it in a homogeneous equality proof, then return false.
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*/
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bool ensure_homogeneous(expr const & lhs, result & rhs) {
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if (rhs.m_heq_proof) {
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// try to convert back to homogeneous
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lean_assert(rhs.m_proof);
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expr lhs_type = infer_type(lhs);
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expr rhs_type = infer_type(rhs.m_out);
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if (is_definitionally_equal(lhs_type, rhs_type)) {
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// move back to homogeneous equality using to_eq
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rhs.m_proof = mk_to_eq_th(lhs_type, lhs, rhs.m_out, *rhs.m_proof);
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return true;
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} else {
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return false;
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}
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} else {
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return true;
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}
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}
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expr get_proof(result const & rhs) {
|
|
if (rhs.m_proof) {
|
|
return *rhs.m_proof;
|
|
} else {
|
|
// lhs and rhs are definitionally equal
|
|
return mk_refl_th(infer_type(rhs.m_out), rhs.m_out);
|
|
}
|
|
}
|
|
|
|
/**
|
|
\brief Simplify \c e using the given congruence theorem.
|
|
See congr.h for a description of congr_theorem_info.
|
|
*/
|
|
result simplify_app_congr(expr const & e, congr_theorem_info const & cg_thm) {
|
|
lean_assert(is_app(e));
|
|
lean_assert(arg(e, 0) == cg_thm.get_fun());
|
|
buffer<expr> new_args;
|
|
bool changed = false;
|
|
new_args.resize(num_args(e));
|
|
new_args[0] = arg(e, 0);
|
|
buffer<expr> proof_args_buf;
|
|
expr * proof_args;
|
|
if (m_proofs_enabled) {
|
|
proof_args_buf.resize(cg_thm.get_num_proof_args() + 1);
|
|
proof_args_buf[0] = cg_thm.get_proof();
|
|
proof_args = proof_args_buf.data()+1;
|
|
}
|
|
for (auto const & info : cg_thm.get_arg_info()) {
|
|
unsigned pos = info.get_arg_pos();
|
|
expr const & a = arg(e, pos);
|
|
if (info.should_simplify()) {
|
|
optional<congr_theorem_info::context> const & ctx = info.get_context();
|
|
if (!ctx) {
|
|
// argument does not have a context
|
|
result res_a = simplify(a);
|
|
new_args[pos] = res_a.m_out;
|
|
if (m_proofs_enabled) {
|
|
if (!ensure_homogeneous(a, res_a))
|
|
return simplify_app_default(e); // fallback to default congruence
|
|
proof_args[info.get_pos_at_proof()] = a;
|
|
proof_args[*info.get_new_pos_at_proof()] = new_args[pos];
|
|
proof_args[*info.get_proof_pos_at_proof()] = get_proof(res_a);
|
|
}
|
|
} else {
|
|
unsigned dep_pos = ctx->get_arg_pos();
|
|
expr H = ctx->use_new_val() ? new_args[dep_pos] : arg(e, dep_pos);
|
|
if (!ctx->is_pos_dep())
|
|
H = mk_not(H);
|
|
// We will simplify the \c a under the hypothesis H
|
|
if (!m_proofs_enabled) {
|
|
// Contextual reasoning without proofs.
|
|
expr dummy_proof; // we don't need a proof
|
|
updt_rule_set update(m_rule_sets[0], H, dummy_proof);
|
|
result res_a = simplify(a);
|
|
new_args[pos] = res_a.m_out;
|
|
} else {
|
|
// We have to introduce H in the context, so first we lift the free variables in \c a
|
|
flet<unsigned> set_depth(m_contextual_depth, m_contextual_depth+1);
|
|
expr H_proof = mk_constant(name(g_unique, m_contextual_depth));
|
|
updt_rule_set update(m_rule_sets[0], H, H_proof);
|
|
freset<cache> m_reset_cache(m_cache); // must reset cache for the recursive call because we updated the rule_sets
|
|
result res_a = simplify(a);
|
|
if (!ensure_homogeneous(a, res_a))
|
|
return simplify_app_default(e); // fallback to default congruence
|
|
new_args[pos] = res_a.m_out;
|
|
proof_args[info.get_pos_at_proof()] = a;
|
|
proof_args[*info.get_new_pos_at_proof()] = new_args[pos];
|
|
name C_name(g_C, m_contextual_depth); // H_name is a cryptic unique name
|
|
proof_args[*info.get_proof_pos_at_proof()] = mk_lambda(C_name, H, abstract(get_proof(res_a), H_proof));
|
|
}
|
|
}
|
|
if (new_args[pos] != a)
|
|
changed = true;
|
|
} else {
|
|
// argument should not be simplified
|
|
new_args[pos] = arg(e, pos);
|
|
if (m_proofs_enabled)
|
|
proof_args[info.get_pos_at_proof()] = arg(e, pos);
|
|
}
|
|
}
|
|
|
|
if (!changed) {
|
|
return rewrite_app(e, result(e));
|
|
} else if (!m_proofs_enabled) {
|
|
return rewrite_app(e, result(mk_app(new_args)));
|
|
} else {
|
|
return rewrite_app(e, result(mk_app(new_args), mk_app(proof_args_buf)));
|
|
}
|
|
}
|
|
|
|
result simplify_app_default(expr const & e) {
|
|
lean_assert(is_app(e));
|
|
buffer<expr> new_args;
|
|
buffer<optional<expr>> proofs; // used only if m_proofs_enabled
|
|
buffer<expr> f_types, new_f_types; // used only if m_proofs_enabled
|
|
buffer<bool> heq_proofs; // used only if m_has_heq && m_proofs_enabled
|
|
bool changed = false;
|
|
expr f = arg(e, 0);
|
|
expr f_type = infer_type(f);
|
|
result res_f = simplify(f);
|
|
expr new_f = res_f.m_out;
|
|
expr new_f_type;
|
|
if (new_f != f)
|
|
changed = true;
|
|
new_args.push_back(new_f);
|
|
if (m_proofs_enabled) {
|
|
proofs.push_back(res_f.m_proof);
|
|
f_types.push_back(f_type);
|
|
new_f_type = res_f.m_heq_proof ? infer_type(new_f) : f_type;
|
|
new_f_types.push_back(new_f_type);
|
|
if (m_has_heq)
|
|
heq_proofs.push_back(res_f.m_heq_proof);
|
|
}
|
|
unsigned num = num_args(e);
|
|
for (unsigned i = 1; i < num; i++) {
|
|
f_type = ensure_pi(f_type);
|
|
bool f_arrow = is_arrow(f_type);
|
|
expr const & a = arg(e, i);
|
|
result res_a(a);
|
|
if (m_has_heq || f_arrow) {
|
|
res_a = simplify(a);
|
|
if (res_a.m_out != a)
|
|
changed = true;
|
|
}
|
|
expr new_a = res_a.m_out;
|
|
new_args.push_back(new_a);
|
|
if (m_proofs_enabled) {
|
|
proofs.push_back(res_a.m_proof);
|
|
if (m_has_heq)
|
|
heq_proofs.push_back(res_a.m_heq_proof);
|
|
bool changed_f_type = !is_eqp(f_type, new_f_type);
|
|
if (f_arrow) {
|
|
f_type = lower_free_vars(abst_body(f_type), 1, 1);
|
|
new_f_type = changed_f_type ? lower_free_vars(abst_body(new_f_type), 1, 1) : f_type;
|
|
} else if (is_eqp(a, new_a)) {
|
|
f_type = pi_body_at(f_type, a);
|
|
new_f_type = changed_f_type ? pi_body_at(new_f_type, a) : f_type;
|
|
} else {
|
|
f_type = pi_body_at(f_type, a);
|
|
new_f_type = pi_body_at(new_f_type, new_a);
|
|
}
|
|
f_types.push_back(f_type);
|
|
new_f_types.push_back(new_f_type);
|
|
}
|
|
}
|
|
|
|
if (!changed) {
|
|
return rewrite_app(e, result(e));
|
|
} else if (!m_proofs_enabled) {
|
|
return rewrite_app(e, result(mk_app(new_args)));
|
|
} else {
|
|
expr out = mk_app(new_args);
|
|
unsigned i = 0;
|
|
while (i < num && !proofs[i]) {
|
|
// skip "reflexive" proofs
|
|
i++;
|
|
}
|
|
if (i == num)
|
|
return rewrite_app(e, result(out));
|
|
expr pr;
|
|
bool heq_proof = false;
|
|
if (i == 0) {
|
|
pr = *(proofs[0]);
|
|
heq_proof = m_has_heq && heq_proofs[0];
|
|
} else if (m_has_heq && (heq_proofs[i] || !is_arrow(f_types[i-1]))) {
|
|
expr f = mk_app_prefix(i, new_args);
|
|
expr pr_i = *proofs[i];
|
|
auto new_pr = mk_hcongr_th(f_types[i-1], f_types[i-1], f, f, arg(e, i), new_args[i],
|
|
mk_hrefl_th(f_types[i-1], f), pr_i, heq_proofs[i]);
|
|
if (!new_pr)
|
|
return rewrite_app(e, result(e)); // failed to create congruence proof
|
|
pr = *new_pr;
|
|
heq_proof = true;
|
|
} else {
|
|
expr f = mk_app_prefix(i, new_args);
|
|
pr = mk_congr2_th(f_types[i-1], arg(e, i), new_args[i], f, *(proofs[i]));
|
|
}
|
|
i++;
|
|
for (; i < num; i++) {
|
|
expr f = mk_app_prefix(i, e);
|
|
expr new_f = mk_app_prefix(i, new_args);
|
|
if (proofs[i]) {
|
|
expr pr_i = *proofs[i];
|
|
if (m_has_heq && heq_proofs[i]) {
|
|
if (!heq_proof)
|
|
pr = mk_to_heq_th(f_types[i], f, new_f, pr);
|
|
auto new_pr = mk_hcongr_th(f_types[i-1], new_f_types[i-1], f, new_f, arg(e, i), new_args[i], pr, pr_i, true);
|
|
if (!new_pr)
|
|
return rewrite_app(e, result(e)); // failed to create congruence proof
|
|
pr = *new_pr;
|
|
heq_proof = true;
|
|
} else if (heq_proof) {
|
|
auto new_pr = mk_hcongr_th(f_types[i-1], new_f_types[i-1], f, new_f, arg(e, i), new_args[i], pr, pr_i, heq_proofs[i]);
|
|
if (!new_pr)
|
|
return rewrite_app(e, result(e)); // failed to create congruence proof
|
|
pr = *new_pr;
|
|
} else {
|
|
pr = mk_congr_th(f_types[i-1], f, new_f, arg(e, i), new_args[i], pr, pr_i);
|
|
}
|
|
} else if (heq_proof) {
|
|
auto new_pr = mk_hcongr_th(f_types[i-1], new_f_types[i-1], f, new_f, arg(e, i), arg(e, i),
|
|
pr, mk_refl_th(infer_type(arg(e, i)), arg(e, i)), false);
|
|
if (!new_pr)
|
|
return rewrite_app(e, result(e)); // failed to create congruence proof
|
|
pr = *new_pr;
|
|
} else {
|
|
lean_assert(!heq_proof);
|
|
pr = mk_congr1_th(f_types[i-1], f, new_f, arg(e, i), pr);
|
|
}
|
|
}
|
|
return rewrite_app(e, result(out, pr, heq_proof));
|
|
}
|
|
}
|
|
|
|
/** \brief Return true when \c e is a value from the point of view of the simplifier */
|
|
static bool is_value(expr const & e) {
|
|
// Currently only semantic attachments are treated as value.
|
|
// We may relax that in the future.
|
|
return ::lean::is_value(e);
|
|
}
|
|
|
|
/**
|
|
\brief Return true iff the simplifier should use the evaluator/normalizer to reduce application
|
|
*/
|
|
bool evaluate_app(expr const & e) const {
|
|
lean_assert(is_app(e));
|
|
// only evaluate if it is enabled
|
|
if (!m_eval)
|
|
return false;
|
|
// if all arguments are values, we should evaluate
|
|
if (std::all_of(args(e).begin()+1, args(e).end(), [](expr const & a) { return is_value(a); }))
|
|
return true;
|
|
// The previous test fails for equality/disequality because the first arguments are types.
|
|
// Should we have something more general for cases like that?
|
|
// Some possibilities:
|
|
// - We have a table mapping constants to argument positions. The positions tell the simplifier
|
|
// which arguments must be value to trigger evaluation.
|
|
// - We have an external predicate that is invoked by the simplifier to decide whether to normalize/evaluate an
|
|
// expression.
|
|
unsigned num = num_args(e);
|
|
return
|
|
(is_eq(e) || is_neq(e) || is_heq(e)) &&
|
|
is_value(arg(e, num-2)) &&
|
|
is_value(arg(e, num-1));
|
|
}
|
|
|
|
/**
|
|
\brief Given (applications) lhs and rhs s.t. lhs = rhs.m_out
|
|
with proof rhs.m_proof, this method applies rewrite rules, beta
|
|
and evaluation to \c rhs.m_out, and return a new result object
|
|
new_rhs s.t. lhs = new_rhs.m_out with proof new_rhs.m_proof
|
|
|
|
\pre is_app(lhs)
|
|
\pre is_app(rhs.m_out)
|
|
*/
|
|
result rewrite_app(expr const & lhs, result const & rhs) {
|
|
lean_assert(is_app(rhs.m_out));
|
|
lean_assert(is_app(lhs));
|
|
if (evaluate_app(rhs.m_out)) {
|
|
// try to evaluate if all arguments are values.
|
|
expr new_rhs = normalize(rhs.m_out);
|
|
if (is_value(new_rhs)) {
|
|
// We don't need to create a new proof term since rhs.m_out and new_rhs are
|
|
// definitionally equal.
|
|
return rewrite(lhs, result(new_rhs, rhs.m_proof, rhs.m_heq_proof));
|
|
}
|
|
}
|
|
|
|
expr f = arg(rhs.m_out, 0);
|
|
if (m_beta && is_lambda(f)) {
|
|
expr new_rhs = head_beta_reduce(rhs.m_out);
|
|
// rhs.m_out and new_rhs are also definitionally equal
|
|
return rewrite(lhs, result(new_rhs, rhs.m_proof, rhs.m_heq_proof));
|
|
}
|
|
|
|
return rewrite(lhs, rhs);
|
|
}
|
|
|
|
|
|
bool found_all_args(unsigned num, buffer<optional<expr>> const & subst, buffer<expr> & new_args) {
|
|
for (unsigned i = 0; i < num; i++) {
|
|
if (!subst[i])
|
|
return false;
|
|
new_args[i+1] = *subst[i];
|
|
}
|
|
return true;
|
|
}
|
|
|
|
/**
|
|
\brief Given lhs and rhs s.t. lhs = rhs.m_out with proof rhs.m_proof,
|
|
this method applies rewrite rules, beta and evaluation to \c rhs.m_out,
|
|
and return a new result object new_rhs s.t.
|
|
lhs = new_rhs.m_out
|
|
with proof new_rhs.m_proof
|
|
*/
|
|
result rewrite(expr const & lhs, result const & rhs) {
|
|
expr target = rhs.m_out;
|
|
buffer<optional<expr>> subst;
|
|
buffer<expr> new_args;
|
|
expr new_rhs;
|
|
expr new_proof;
|
|
auto check_rule_fn = [&](rewrite_rule const & rule) -> bool {
|
|
unsigned num = rule.get_num_args();
|
|
subst.clear();
|
|
subst.resize(num);
|
|
if (hop_match(rule.get_lhs(), target, subst, optional<ro_environment>(m_env))) {
|
|
new_args.clear();
|
|
new_args.resize(num+1);
|
|
if (found_all_args(num, subst, new_args)) {
|
|
// easy case: all arguments found
|
|
new_rhs = instantiate(rule.get_rhs(), num, new_args.data() + 1);
|
|
if (rule.is_permutation() && !is_lt(new_rhs, target, false))
|
|
return false;
|
|
if (m_proofs_enabled) {
|
|
if (num > 0) {
|
|
new_args[0] = rule.get_proof();
|
|
new_proof = mk_app(new_args);
|
|
} else {
|
|
new_proof = rule.get_proof();
|
|
}
|
|
}
|
|
return true;
|
|
} else {
|
|
// Conditional rewriting: we try to fill the missing
|
|
// arguments by trying to find a proof for ones that are
|
|
// propositions.
|
|
expr ceq = rule.get_ceq();
|
|
buffer<expr> & proof_args = new_args;
|
|
proof_args.clear();
|
|
if (m_proofs_enabled)
|
|
proof_args.push_back(rule.get_proof());
|
|
for (unsigned i = 0; i < num; i++) {
|
|
lean_assert(is_pi(ceq));
|
|
if (subst[i]) {
|
|
ceq = instantiate(abst_body(ceq), *subst[i]);
|
|
if (m_proofs_enabled)
|
|
proof_args.push_back(*subst[i]);
|
|
} else {
|
|
expr d = abst_domain(ceq);
|
|
if (is_proposition(d)) {
|
|
result d_res = simplify(d);
|
|
if (d_res.m_out == True) {
|
|
if (m_proofs_enabled) {
|
|
expr d_proof;
|
|
if (!d_res.m_proof) {
|
|
// No proof available. So d should be definitionally equal to True
|
|
d_proof = mk_trivial();
|
|
} else {
|
|
d_proof = mk_eqt_elim_th(d, *d_res.m_proof);
|
|
}
|
|
ceq = instantiate(abst_body(ceq), d_proof);
|
|
proof_args.push_back(d_proof);
|
|
} else if (is_arrow(ceq)) {
|
|
ceq = lower_free_vars(abst_body(ceq), 1, 1);
|
|
} else {
|
|
// The body of ceq depends on this argument,
|
|
// but proof generation is not enabled.
|
|
// So, we should fail
|
|
return false;
|
|
}
|
|
} else {
|
|
// failed to prove proposition
|
|
return false;
|
|
}
|
|
} else {
|
|
// failed, the argument is not a proposition
|
|
return false;
|
|
}
|
|
}
|
|
}
|
|
new_proof = mk_app(proof_args);
|
|
new_rhs = arg(ceq, num_args(ceq) - 1);
|
|
if (rule.is_permutation() && !is_lt(new_rhs, target, false))
|
|
return false;
|
|
return true;
|
|
}
|
|
}
|
|
return false;
|
|
};
|
|
// Traverse all rule sets
|
|
for (rewrite_rule_set const & rs : m_rule_sets) {
|
|
if (rs.find_match(target, check_rule_fn)) {
|
|
// the result is in new_rhs and proof at new_proof
|
|
result new_r1 = mk_trans_result(lhs, rhs, new_rhs, new_proof);
|
|
if (m_single_pass) {
|
|
return new_r1;
|
|
} else {
|
|
result new_r2 = simplify(new_r1.m_out);
|
|
return mk_trans_result(lhs, new_r1, new_r2);
|
|
}
|
|
}
|
|
}
|
|
if (!m_single_pass && lhs != rhs.m_out) {
|
|
result new_rhs = simplify(rhs.m_out);
|
|
return mk_trans_result(lhs, rhs, new_rhs);
|
|
} else {
|
|
return rhs;
|
|
}
|
|
}
|
|
|
|
result simplify_var(expr const & e) {
|
|
if (m_has_heq) {
|
|
// TODO(Leo)
|
|
return result(e);
|
|
} else {
|
|
return result(e);
|
|
}
|
|
}
|
|
|
|
result simplify_constant(expr const & e) {
|
|
lean_assert(is_constant(e));
|
|
if (m_unfold || m_eval) {
|
|
auto obj = m_env->find_object(const_name(e));
|
|
if (m_unfold && should_unfold(obj)) {
|
|
expr e = obj->get_value();
|
|
if (m_single_pass) {
|
|
return result(e);
|
|
} else {
|
|
return simplify(e);
|
|
}
|
|
}
|
|
if (m_eval && obj->is_builtin()) {
|
|
return result(obj->get_value());
|
|
}
|
|
}
|
|
return rewrite(e, result(e));
|
|
}
|
|
|
|
/**
|
|
\brief Return true iff Eta-reduction can be applied to \c e.
|
|
|
|
\remark Actually this is a partial test. Given,
|
|
fun x : T, f x
|
|
This method does not check whether f has type
|
|
Pi x : T, B x
|
|
This check must be performed in the caller.
|
|
Otherwise the proof (eta T (fun x : T, B x) f) will not type check.
|
|
*/
|
|
bool is_eta_target(expr const & e) const {
|
|
if (is_lambda(e)) {
|
|
expr b = abst_body(e);
|
|
return
|
|
is_app(b) && is_var(arg(b, num_args(b) - 1), 0) &&
|
|
std::all_of(begin_args(b), end_args(b) - 1, [](expr const & a) { return !has_free_var(a, 0); });
|
|
} else {
|
|
return false;
|
|
}
|
|
}
|
|
|
|
/**
|
|
\brief Given (lambdas) lhs and rhs s.t. lhs = rhs.m_out
|
|
with proof rhs.m_proof, this method applies rewrite rules, and
|
|
eta reduction, and return a new result object new_rhs s.t.
|
|
lhs = new_rhs.m_out with proof new_rhs.m_proof
|
|
|
|
\pre is_lambda(lhs)
|
|
\pre is_lambda(rhs.m_out)
|
|
*/
|
|
result rewrite_lambda(expr const & lhs, result const & rhs) {
|
|
lean_assert(is_lambda(lhs));
|
|
lean_assert(is_lambda(rhs.m_out));
|
|
if (m_eta && is_eta_target(rhs.m_out)) {
|
|
expr b = abst_body(rhs.m_out);
|
|
expr new_rhs;
|
|
if (num_args(b) > 2) {
|
|
new_rhs = mk_app(num_args(b) - 1, &arg(b, 0));
|
|
} else {
|
|
new_rhs = arg(b, 0);
|
|
}
|
|
new_rhs = lower_free_vars(new_rhs, 1, 1);
|
|
expr new_rhs_type = ensure_pi(infer_type(new_rhs));
|
|
if (m_tc.is_definitionally_equal(abst_domain(new_rhs_type), abst_domain(rhs.m_out), m_ctx)) {
|
|
if (m_proofs_enabled) {
|
|
expr new_proof = mk_eta_th(abst_domain(rhs.m_out),
|
|
mk_lambda(rhs.m_out, abst_body(new_rhs_type)),
|
|
new_rhs);
|
|
return rewrite(lhs, mk_trans_result(lhs, rhs, new_rhs, new_proof));
|
|
} else {
|
|
return rewrite(lhs, result(new_rhs));
|
|
}
|
|
}
|
|
}
|
|
return rewrite(lhs, rhs);
|
|
}
|
|
|
|
result simplify_lambda(expr const & e) {
|
|
lean_assert(is_lambda(e));
|
|
if (m_has_heq) {
|
|
// TODO(Leo)
|
|
return result(e);
|
|
} else {
|
|
set_context set(*this, extend(m_ctx, abst_name(e), abst_domain(e)));
|
|
result res_body = simplify(abst_body(e));
|
|
lean_assert(!res_body.m_heq_proof);
|
|
expr new_body = res_body.m_out;
|
|
if (is_eqp(new_body, abst_body(e)))
|
|
return rewrite_lambda(e, result(e));
|
|
expr out = mk_lambda(e, new_body);
|
|
if (!m_proofs_enabled || !res_body.m_proof)
|
|
return rewrite_lambda(e, result(out));
|
|
expr body_type = infer_type(abst_body(e));
|
|
expr pr = mk_funext_th(abst_domain(e), mk_lambda(e, body_type), e, out,
|
|
mk_lambda(e, *res_body.m_proof));
|
|
return rewrite_lambda(e, result(out, pr));
|
|
}
|
|
}
|
|
|
|
result simplify_pi(expr const & e) {
|
|
lean_assert(is_pi(e));
|
|
// TODO(Leo): handle implication, i.e., e is_proposition and is_arrow
|
|
if (m_has_heq) {
|
|
// TODO(Leo)
|
|
return result(e);
|
|
} else if (is_proposition(e)) {
|
|
set_context set(*this, extend(m_ctx, abst_name(e), abst_domain(e)));
|
|
result res_body = simplify(abst_body(e));
|
|
lean_assert(!res_body.m_heq_proof);
|
|
expr new_body = res_body.m_out;
|
|
if (is_eqp(new_body, abst_body(e)))
|
|
return rewrite(e, result(e));
|
|
expr out = mk_pi(abst_name(e), abst_domain(e), new_body);
|
|
if (!m_proofs_enabled || !res_body.m_proof)
|
|
return rewrite(e, result(out));
|
|
expr pr = mk_allext_th(abst_domain(e),
|
|
mk_lambda(e, abst_body(e)),
|
|
mk_lambda(e, abst_body(out)),
|
|
mk_lambda(e, *res_body.m_proof));
|
|
return rewrite(e, result(out, pr));
|
|
} else {
|
|
// if the environment does not contain heq axioms, then we don't simplify Pi's that are not forall's
|
|
return result(e);
|
|
}
|
|
}
|
|
|
|
result save(expr const & e, result const & r) {
|
|
if (m_memoize) {
|
|
result new_r(m_max_sharing(r.m_out), r.m_proof, r.m_heq_proof);
|
|
m_cache.insert(mk_pair(e, new_r));
|
|
return new_r;
|
|
} else {
|
|
return r;
|
|
}
|
|
}
|
|
|
|
result simplify(expr e) {
|
|
check_system("simplifier");
|
|
m_num_steps++;
|
|
if (m_num_steps > m_max_steps)
|
|
throw exception("simplifier failed, maximum number of steps exceeded");
|
|
if (m_memoize) {
|
|
e = m_max_sharing(e);
|
|
auto it = m_cache.find(e);
|
|
if (it != m_cache.end()) {
|
|
return it->second;
|
|
}
|
|
}
|
|
switch (e.kind()) {
|
|
case expr_kind::Var: return save(e, simplify_var(e));
|
|
case expr_kind::Constant: return save(e, simplify_constant(e));
|
|
case expr_kind::Type:
|
|
case expr_kind::MetaVar:
|
|
case expr_kind::Value: return save(e, result(e));
|
|
case expr_kind::App: return save(e, simplify_app(e));
|
|
case expr_kind::Lambda: return save(e, simplify_lambda(e));
|
|
case expr_kind::Pi: return save(e, simplify_pi(e));
|
|
case expr_kind::Let: return save(e, simplify(instantiate(let_body(e), let_value(e))));
|
|
}
|
|
lean_unreachable();
|
|
}
|
|
|
|
void collect_congr_thms() {
|
|
if (m_contextual) {
|
|
for (auto const & rs : m_rule_sets) {
|
|
rs.for_each_congr([&](congr_theorem_info const & info) {
|
|
if (std::all_of(m_congr_thms.begin(), m_congr_thms.end(),
|
|
[&](congr_theorem_info const * info2) {
|
|
return info2->get_fun() != info.get_fun(); })) {
|
|
m_congr_thms.push_back(&info);
|
|
}
|
|
});
|
|
}
|
|
}
|
|
}
|
|
|
|
void set_options(options const & o) {
|
|
m_proofs_enabled = get_simplifier_proofs(o);
|
|
m_contextual = get_simplifier_contextual(o);
|
|
m_single_pass = get_simplifier_single_pass(o);
|
|
m_beta = get_simplifier_beta(o);
|
|
m_eta = get_simplifier_eta(o);
|
|
m_eval = get_simplifier_eval(o);
|
|
m_unfold = get_simplifier_unfold(o);
|
|
m_conditional = get_simplifier_conditional(o);
|
|
m_memoize = get_simplifier_memoize(o);
|
|
m_max_steps = get_simplifier_max_steps(o);
|
|
}
|
|
|
|
public:
|
|
simplifier_fn(ro_environment const & env, options const & o, unsigned num_rs, rewrite_rule_set const * rs):
|
|
m_env(env), m_tc(env) {
|
|
m_has_heq = m_env->imported("heq");
|
|
m_has_cast = m_env->imported("cast");
|
|
set_options(o);
|
|
if (m_contextual) {
|
|
// add a set of rewrite rules for contextual rewriting
|
|
m_rule_sets.push_back(rewrite_rule_set(env));
|
|
}
|
|
m_rule_sets.insert(m_rule_sets.end(), rs, rs + num_rs);
|
|
collect_congr_thms();
|
|
m_contextual_depth = 0;
|
|
}
|
|
|
|
expr_pair operator()(expr const & e, context const & ctx) {
|
|
set_context set(*this, ctx);
|
|
m_num_steps = 0;
|
|
auto r = simplify(e);
|
|
return mk_pair(r.m_out, get_proof(r));
|
|
}
|
|
};
|
|
|
|
expr_pair simplify(expr const & e, ro_environment const & env, context const & ctx, options const & opts,
|
|
unsigned num_rs, rewrite_rule_set const * rs) {
|
|
return simplifier_fn(env, opts, num_rs, rs)(e, ctx);
|
|
}
|
|
|
|
expr_pair simplify(expr const & e, ro_environment const & env, context const & ctx, options const & opts,
|
|
unsigned num_ns, name const * ns) {
|
|
buffer<rewrite_rule_set> rules;
|
|
for (unsigned i = 0; i < num_ns; i++)
|
|
rules.push_back(get_rewrite_rule_set(env, ns[i]));
|
|
return simplify(e, env, ctx, opts, num_ns, rules.data());
|
|
}
|
|
|
|
static int simplify_core(lua_State * L, ro_shared_environment const & env) {
|
|
int nargs = lua_gettop(L);
|
|
expr const & e = to_expr(L, 1);
|
|
buffer<rewrite_rule_set> rules;
|
|
if (nargs == 1) {
|
|
rules.push_back(get_rewrite_rule_set(env));
|
|
} else {
|
|
if (lua_isstring(L, 2)) {
|
|
rules.push_back(get_rewrite_rule_set(env, to_name_ext(L, 2)));
|
|
} else {
|
|
luaL_checktype(L, 2, LUA_TTABLE);
|
|
name r;
|
|
int n = objlen(L, 2);
|
|
for (int i = 1; i <= n; i++) {
|
|
lua_rawgeti(L, 2, i);
|
|
rules.push_back(get_rewrite_rule_set(env, to_name_ext(L, -1)));
|
|
lua_pop(L, 1);
|
|
}
|
|
}
|
|
}
|
|
context ctx;
|
|
options opts;
|
|
if (nargs >= 4)
|
|
ctx = to_context(L, 4);
|
|
if (nargs >= 5)
|
|
opts = to_options(L, 5);
|
|
auto r = simplify(e, env, ctx, opts, rules.size(), rules.data());
|
|
push_expr(L, r.first);
|
|
push_expr(L, r.second);
|
|
return 2;
|
|
}
|
|
|
|
static int simplify(lua_State * L) {
|
|
int nargs = lua_gettop(L);
|
|
if (nargs <= 2)
|
|
return simplify_core(L, ro_shared_environment(L));
|
|
else
|
|
return simplify_core(L, ro_shared_environment(L, 3));
|
|
}
|
|
|
|
void open_simplifier(lua_State * L) {
|
|
SET_GLOBAL_FUN(simplify, "simplify");
|
|
}
|
|
}
|