418 lines
18 KiB
C++
418 lines
18 KiB
C++
/*
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Copyright (c) 2015 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 "kernel/instantiate.h"
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#include "kernel/abstract.h"
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#include "library/util.h"
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#include "library/locals.h"
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#include "library/constants.h"
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#include "library/replace_visitor.h"
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#include "library/congr_lemma_manager.h"
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namespace lean {
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class congr_lemma_manager::imp {
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app_builder & m_builder;
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fun_info_manager & m_fmanager;
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type_context & m_ctx;
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bool m_ignore_inst_implicit;
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struct key {
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expr m_fn;
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unsigned m_nargs;
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unsigned m_hash;
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key(expr const & fn, unsigned nargs):
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m_fn(fn), m_nargs(nargs), m_hash(hash(m_fn.hash(), m_nargs)) {}
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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_fn == k2.m_fn && k1.m_nargs == k2.m_nargs;
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}
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};
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std::unordered_map<key, result, key_hash_fn, key_eq_fn> m_simp_cache;
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std::unordered_map<key, result, key_hash_fn, key_eq_fn> m_cache;
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expr infer(expr const & e) { return m_ctx.infer(e); }
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expr whnf(expr const & e) { return m_ctx.whnf(e); }
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/** \brief (Try to) cast expression \c e to the given type using the equations \c eqs.
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\c deps contains the indices of the relevant equalities.
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\remark deps is sorted. */
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expr cast(expr const & e, expr const & type, list<unsigned> const & deps, buffer<optional<expr>> const & eqs) {
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if (!deps)
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return e;
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unsigned d = head(deps);
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auto major = eqs[d];
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if (!major) {
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return cast(e, type, tail(deps), eqs);
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} else {
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expr lhs, rhs;
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lean_verify(is_eq(mlocal_type(*major), lhs, rhs));
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lean_assert(is_local(lhs));
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lean_assert(is_local(rhs));
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// We compute the new type by replacing rhs with lhs, and major with (refl lhs).
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expr motive, new_type;
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bool use_drec;
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if (depends_on(type, *major)) {
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// Since the type depends on the major, we must use dependent elimination.
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// and the motive just abstract rhs and *major
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use_drec = true;
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motive = Fun(rhs, Fun(*major, type));
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// We compute new_type by replacing rhs with lhs and *major with eq.refl(lhs) in type.
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new_type = instantiate(abstract_local(type, rhs), lhs);
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expr refl = m_builder.mk_eq_refl(lhs);
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new_type = instantiate(abstract_local(new_type, *major), refl);
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} else {
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// type does not depend on the *major.
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// So, the motive is just (fun rhs, type), and
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// new_type just replaces rhs with lhs.
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use_drec = false;
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motive = Fun(rhs, type);
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new_type = instantiate(abstract_local(type, rhs), lhs);
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}
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expr minor = cast(e, new_type, tail(deps), eqs);
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if (use_drec) {
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return m_builder.mk_eq_drec(motive, minor, *major);
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} else {
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return m_builder.mk_eq_rec(motive, minor, *major);
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}
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}
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}
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bool has_cast(buffer<congr_arg_kind> const & kinds) {
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return std::find(kinds.begin(), kinds.end(), congr_arg_kind::Cast) != kinds.end();
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}
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expr mk_simple_congr_proof(expr const & fn, buffer<expr> const & lhss,
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buffer<optional<expr>> const & eqs, buffer<congr_arg_kind> const & kinds) {
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unsigned i = 0;
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for (; i < kinds.size(); i++) {
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if (kinds[i] != congr_arg_kind::Fixed)
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break;
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}
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expr g = mk_app(fn, i, lhss.data());
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if (i == kinds.size())
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return m_builder.mk_eq_refl(g);
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lean_assert(kinds[i] == congr_arg_kind::Eq);
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lean_assert(eqs[i]);
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expr pr = m_builder.mk_congr_arg(g, *eqs[i]);
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i++;
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for (; i < kinds.size(); i++) {
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if (kinds[i] == congr_arg_kind::Eq) {
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pr = m_builder.mk_congr(pr, *eqs[i]);
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} else {
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lean_assert(kinds[i] == congr_arg_kind::Fixed);
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pr = m_builder.mk_congr_fun(pr, lhss[i]);
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}
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}
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return pr;
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}
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expr mk_congr_proof(unsigned i, expr const & lhs, expr const & rhs, buffer<optional<expr>> const & eqs) {
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if (i == eqs.size()) {
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return m_builder.mk_eq_refl(rhs);
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} else if (!eqs[i]) {
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return mk_congr_proof(i+1, lhs, rhs, eqs);
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} else {
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expr major = *eqs[i];
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expr x_1, x_2;
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lean_verify(is_eq(mlocal_type(major), x_1, x_2));
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lean_assert(is_local(x_1));
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lean_assert(is_local(x_2));
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expr motive_eq = m_builder.mk_eq(lhs, rhs);
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expr motive = Fun(x_2, Fun(major, motive_eq));
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// We compute the new_rhs by replacing x_2 with x_1 and major with (eq.refl x_1) in rhs.
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expr new_rhs = instantiate(abstract_local(rhs, x_2), x_1);
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expr x1_refl = m_builder.mk_eq_refl(x_1);
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new_rhs = instantiate(abstract_local(new_rhs, major), x1_refl);
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expr minor = mk_congr_proof(i+1, lhs, new_rhs, eqs);
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return m_builder.mk_eq_drec(motive, minor, major);
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}
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}
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optional<result> mk_congr_simp(expr const & fn, buffer<param_info> const & pinfos, buffer<congr_arg_kind> const & kinds) {
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try {
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expr fn_type = whnf(infer(fn));
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name e_name("e");
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buffer<expr> lhss;
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buffer<expr> rhss; // it contains the right-hand-side argument
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buffer<optional<expr>> eqs; // for Eq args, it contains the equality
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buffer<expr> hyps; // contains lhss + rhss + eqs
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for (unsigned i = 0; i < pinfos.size(); i++) {
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if (!is_pi(fn_type)) {
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return optional<result>();
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}
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expr lhs = m_ctx.mk_tmp_local(binding_name(fn_type), binding_domain(fn_type));
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lhss.push_back(lhs);
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hyps.push_back(lhs);
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switch (kinds[i]) {
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case congr_arg_kind::Eq: {
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expr rhs = m_ctx.mk_tmp_local(binding_name(fn_type), binding_domain(fn_type));
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expr eq_type = m_builder.mk_eq(lhs, rhs);
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rhss.push_back(rhs);
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expr eq = m_ctx.mk_tmp_local(e_name.append_after(eqs.size()+1), eq_type);
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eqs.push_back(some_expr(eq));
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hyps.push_back(rhs);
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hyps.push_back(eq);
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break;
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}
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case congr_arg_kind::Fixed:
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rhss.push_back(lhs);
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eqs.push_back(none_expr());
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break;
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case congr_arg_kind::Cast: {
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expr rhs_type = mlocal_type(lhs);
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rhs_type = instantiate_rev(abstract_locals(rhs_type, lhss.size()-1, lhss.data()), rhss.size(), rhss.data());
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expr rhs = cast(lhs, rhs_type, pinfos[i].get_dependencies(), eqs);
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rhss.push_back(rhs);
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eqs.push_back(none_expr());
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break;
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}}
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fn_type = whnf(instantiate(binding_body(fn_type), lhs));
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}
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expr lhs = mk_app(fn, lhss);
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expr rhs = mk_app(fn, rhss);
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expr eq = m_builder.mk_eq(lhs, rhs);
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expr congr_type = Pi(hyps, eq);
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expr congr_proof;
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if (has_cast(kinds)) {
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congr_proof = mk_congr_proof(0, lhs, rhs, eqs);
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} else {
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congr_proof = mk_simple_congr_proof(fn, lhss, eqs, kinds);
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}
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congr_proof = Fun(hyps, congr_proof);
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return optional<result>(congr_type, congr_proof, to_list(kinds));
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} catch (app_builder_exception &) {
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return optional<result>();
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}
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}
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optional<result> mk_congr(expr const & fn, optional<result> const & simp_lemma,
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buffer<param_info> const & pinfos, buffer<congr_arg_kind> const & kinds) {
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try {
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expr fn_type1 = whnf(infer(fn));
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expr fn_type2 = fn_type1;
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name e_name("e");
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buffer<expr> lhss;
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buffer<expr> rhss; // it contains the right-hand-side argument
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buffer<optional<expr>> eqs; // for Eq args, it contains the equality
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buffer<expr> hyps; // contains lhss + rhss + eqs
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buffer<expr> simp_lemma_args;
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for (unsigned i = 0; i < pinfos.size(); i++) {
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if (!is_pi(fn_type1)) {
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return optional<result>();
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}
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expr lhs = m_ctx.mk_tmp_local(binding_name(fn_type1), binding_domain(fn_type1));
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expr rhs;
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lhss.push_back(lhs);
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hyps.push_back(lhs);
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simp_lemma_args.push_back(lhs);
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switch (kinds[i]) {
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case congr_arg_kind::Eq: {
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lean_assert(m_ctx.is_def_eq(binding_domain(fn_type1), binding_domain(fn_type2)));
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rhs = m_ctx.mk_tmp_local(binding_name(fn_type2), binding_domain(fn_type2));
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expr eq_type = m_builder.mk_eq(lhs, rhs);
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rhss.push_back(rhs);
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expr eq = m_ctx.mk_tmp_local(e_name.append_after(eqs.size()+1), eq_type);
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eqs.push_back(some_expr(eq));
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hyps.push_back(rhs);
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hyps.push_back(eq);
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simp_lemma_args.push_back(rhs);
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simp_lemma_args.push_back(eq);
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break;
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}
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case congr_arg_kind::Fixed:
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rhs = lhs;
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rhss.push_back(rhs);
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eqs.push_back(none_expr());
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break;
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case congr_arg_kind::Cast: {
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rhs = m_ctx.mk_tmp_local(binding_name(fn_type2), binding_domain(fn_type2));
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rhss.push_back(rhs);
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eqs.push_back(none_expr());
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hyps.push_back(rhs);
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break;
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}}
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fn_type1 = whnf(instantiate(binding_body(fn_type1), lhs));
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fn_type2 = whnf(instantiate(binding_body(fn_type2), rhs));
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}
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expr pr1 = mk_app(simp_lemma->get_proof(), simp_lemma_args);
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expr type1 = simp_lemma->get_type();
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while (is_pi(type1))
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type1 = binding_body(type1);
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type1 = instantiate_rev(type1, simp_lemma_args.size(), simp_lemma_args.data());
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expr lhs1, rhs1;
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lean_verify(is_eq(type1, lhs1, rhs1));
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// build proof2
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expr rhs2 = mk_app(fn, rhss);
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expr eq = m_builder.mk_eq(lhs1, rhs2);
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expr congr_type = Pi(hyps, eq);
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// build proof that rhs1 = rhs2
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unsigned i;
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for (i = 0; i < kinds.size(); i++) {
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if (kinds[i] == congr_arg_kind::Cast && !pinfos[i].is_prop())
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break;
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}
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if (i == kinds.size()) {
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// rhs1 and rhs2 are definitionally equal
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expr congr_proof = Fun(hyps, pr1);
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return optional<result>(congr_type, congr_proof, to_list(kinds));
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}
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buffer<expr> rhss1;
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get_app_args(rhs1, rhss1);
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lean_assert(rhss.size() == rhss1.size());
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expr a = mk_app(fn, i, rhss1.data());
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expr pr2 = m_builder.mk_eq_refl(a);
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for (; i < kinds.size(); i++) {
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if (kinds[i] == congr_arg_kind::Cast && !pinfos[i].is_prop()) {
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lean_assert(pinfos[i].is_subsingleton());
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expr r1 = rhss1[i];
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expr r2 = rhss[i];
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expr r1_eq_r2 = m_builder.mk_app(get_subsingleton_elim_name(), r1, r2);
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pr2 = m_builder.mk_congr(pr2, r1_eq_r2);
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} else {
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pr2 = m_builder.mk_congr_fun(pr2, rhss[i]);
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}
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}
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expr congr_proof = Fun(hyps, m_builder.mk_eq_trans(pr1, pr2));
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return optional<result>(congr_type, congr_proof, to_list(kinds));
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} catch (app_builder_exception &) {
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return optional<result>();
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}
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}
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public:
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imp(app_builder & b, fun_info_manager & fm, bool ignore_inst_implicit):
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m_builder(b), m_fmanager(fm), m_ctx(fm.ctx()), m_ignore_inst_implicit(ignore_inst_implicit) {}
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optional<result> mk_congr_simp(expr const & fn) {
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fun_info finfo = m_fmanager.get(fn);
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return mk_congr_simp(fn, finfo.get_arity());
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}
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optional<result> mk_congr_simp(expr const & fn, unsigned nargs) {
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auto r = m_simp_cache.find(key(fn, nargs));
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if (r != m_simp_cache.end())
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return optional<result>(r->second);
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fun_info finfo = m_fmanager.get(fn, nargs);
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list<unsigned> const & result_deps = finfo.get_dependencies();
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buffer<congr_arg_kind> kinds;
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buffer<param_info> pinfos;
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to_buffer(finfo.get_params_info(), pinfos);
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kinds.resize(pinfos.size(), congr_arg_kind::Eq);
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for (unsigned i = 0; i < pinfos.size(); i++) {
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if (std::find(result_deps.begin(), result_deps.end(), i) != result_deps.end()) {
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kinds[i] = congr_arg_kind::Fixed;
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} else if (pinfos[i].is_subsingleton()) {
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if (empty(pinfos[i].get_dependencies()))
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kinds[i] = congr_arg_kind::Fixed;
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else
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kinds[i] = congr_arg_kind::Cast;
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} else if (pinfos[i].is_inst_implicit()) {
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lean_assert(!pinfos[i].is_subsingleton());
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kinds[i] = congr_arg_kind::Fixed;
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}
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}
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for (unsigned i = 0; i < pinfos.size(); i++) {
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for (unsigned j = i+1; j < pinfos.size(); j++) {
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auto j_deps = pinfos[j].get_dependencies();
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if (std::find(j_deps.begin(), j_deps.end(), i) != j_deps.end() &&
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kinds[j] == congr_arg_kind::Eq) {
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// We must fix i because there is a j that depends on i,
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// and j is not fixed nor a cast-fixed.
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kinds[i] = congr_arg_kind::Fixed;
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break;
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}
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}
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}
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auto new_r = mk_congr_simp(fn, pinfos, kinds);
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if (new_r) {
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m_simp_cache.insert(mk_pair(key(fn, nargs), *new_r));
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return new_r;
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} else if (has_cast(kinds)) {
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// remove casts and try again
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for (unsigned i = 0; i < kinds.size(); i++) {
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if (kinds[i] == congr_arg_kind::Cast)
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kinds[i] = congr_arg_kind::Fixed;
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}
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auto new_r = mk_congr_simp(fn, pinfos, kinds);
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if (new_r) {
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m_simp_cache.insert(mk_pair(key(fn, nargs), *new_r));
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return new_r;
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} else {
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return new_r;
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}
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} else {
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return new_r;
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}
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}
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optional<result> mk_congr(expr const & fn) {
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fun_info finfo = m_fmanager.get(fn);
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return mk_congr(fn, finfo.get_arity());
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}
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optional<result> mk_congr(expr const & fn, unsigned nargs) {
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auto r = m_cache.find(key(fn, nargs));
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if (r != m_cache.end())
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return optional<result>(r->second);
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fun_info finfo = m_fmanager.get(fn, nargs);
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optional<result> simp_lemma = mk_congr_simp(fn, nargs);
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if (!simp_lemma)
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return optional<result>();
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buffer<congr_arg_kind> kinds;
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buffer<param_info> pinfos;
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to_buffer(simp_lemma->get_arg_kinds(), kinds);
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to_buffer(finfo.get_params_info(), pinfos);
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// For congr lemmas we have the following restriction:
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// if a Cast arg is subsingleton, it is not a proposition,
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// and it is a dependent argument, then we mark it as fixed.
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// This restriction doesn't affect the standard library,
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// but it simplifies the implementation.
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lean_assert(kinds.size() == pinfos.size());
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bool has_cast = false;
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for (unsigned i = 0; i < kinds.size(); i++) {
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if (!pinfos[i].is_prop() && pinfos[i].is_subsingleton() && pinfos[i].is_dep()) {
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kinds[i] = congr_arg_kind::Fixed;
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}
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if (kinds[i] == congr_arg_kind::Cast)
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has_cast = true;
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}
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if (!has_cast) {
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m_cache.insert(mk_pair(key(fn, nargs), *simp_lemma));
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return simp_lemma; // simp_lemma will be identical to regular congr lemma
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}
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auto new_r = mk_congr(fn, simp_lemma, pinfos, kinds);
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if (new_r)
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m_cache.insert(mk_pair(key(fn, nargs), *new_r));
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return new_r;
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}
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};
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congr_lemma_manager::congr_lemma_manager(app_builder & b, fun_info_manager & fm, bool ignore_inst_implicit):
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m_ptr(new imp(b, fm, ignore_inst_implicit)) {
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}
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congr_lemma_manager::~congr_lemma_manager() {
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}
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auto congr_lemma_manager::mk_congr_simp(expr const & fn) -> optional<result> {
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return m_ptr->mk_congr_simp(fn);
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}
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auto congr_lemma_manager::mk_congr_simp(expr const & fn, unsigned nargs) -> optional<result> {
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return m_ptr->mk_congr_simp(fn, nargs);
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}
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auto congr_lemma_manager::mk_congr(expr const & fn) -> optional<result> {
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return m_ptr->mk_congr(fn);
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}
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auto congr_lemma_manager::mk_congr(expr const & fn, unsigned nargs) -> optional<result> {
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return m_ptr->mk_congr(fn, nargs);
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}
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}
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