1145 lines
50 KiB
C++
1145 lines
50 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 "util/sstream.h"
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#include "kernel/abstract.h"
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#include "kernel/instantiate.h"
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#include "kernel/inductive/inductive.h"
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#include "kernel/error_msgs.h"
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#include "library/io_state_stream.h"
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#include "library/locals.h"
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#include "library/util.h"
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#include "library/constants.h"
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#include "library/reducible.h"
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#include "library/class_instance_synth.h"
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#include "library/tactic/tactic.h"
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#include "library/tactic/expr_to_tactic.h"
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#include "library/tactic/inversion_tactic.h"
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#include "library/tactic/generalize_tactic.h"
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#include "library/tactic/clear_tactic.h"
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namespace lean {
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namespace inversion {
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result::result(list<goal> const & gs, list<list<expr>> const & args, list<implementation_list> const & imps,
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list<rename_map> const & rs, name_generator const & ngen, substitution const & subst):
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m_goals(gs), m_args(args), m_implementation_lists(imps),
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m_renames(rs), m_ngen(ngen), m_subst(subst) {
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lean_assert_eq(length(m_goals), length(m_args));
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lean_assert_eq(length(m_goals), length(m_implementation_lists));
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lean_assert_eq(length(m_goals), length(m_renames));
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}
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}
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/**
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\brief Given eq_rec of the form <tt>@eq.rec.{l₂ l₁} A a (λ (a' : A) (h : a = a'), B a') b a p</tt>,
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apply the eq_rec_eq definition to produce the equality
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b = @eq.rec.{l₂ l₁} A a (λ (a' : A) (h : a = a'), B a') b a p
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The eq_rec_eq definition is of the form
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definition eq_rec_eq.{l₁ l₂} {A : Type.{l₁}} {B : A → Type.{l₂}} [h : is_hset A] {a : A} (b : B a) (p : a = a) :
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b = @eq.rec.{l₂ l₁} A a (λ (a' : A) (h : a = a'), B a') b a p :=
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...
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*/
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optional<expr> apply_eq_rec_eq(type_checker & tc, io_state const & ios, list<expr> const & ctx, expr const & eq_rec) {
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buffer<expr> args;
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expr const & eq_rec_fn = get_app_args(eq_rec, args);
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if (args.size() != 6)
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return none_expr();
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expr const & p = args[5];
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if (!is_local(p) || !is_eq_a_a(tc, mlocal_type(p)))
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return none_expr();
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expr const & A = args[0];
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auto is_hset_A = mk_hset_instance(tc, ios, ctx, A);
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if (!is_hset_A)
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return none_expr();
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levels ls = const_levels(eq_rec_fn);
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level l2 = head(ls);
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level l1 = head(tail(ls));
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expr C = tc.whnf(args[2]).first;
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if (!is_lambda(C))
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return none_expr();
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expr a1 = mk_local(tc.mk_fresh_name(), binding_domain(C));
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C = tc.whnf(instantiate(binding_body(C), a1)).first;
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if (!is_lambda(C))
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return none_expr();
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C = binding_body(C);
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if (!closed(C))
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return none_expr();
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expr B = Fun(a1, C);
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expr a = args[1];
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expr b = args[3];
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expr r = mk_constant(get_eq_rec_eq_name(), {l1, l2});
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return some_expr(mk_app({r, A, B, *is_hset_A, a, b, p}));
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}
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typedef inversion::implementation_ptr implementation_ptr;
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typedef inversion::implementation_list implementation_list;
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static void replace(implementation_list const & imps, unsigned sz, expr const * from, expr const * to) {
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lean_assert(std::all_of(from, from+sz, is_local));
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for (implementation_ptr const & imp : imps) {
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imp->update_exprs([&](expr const & e) {
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return instantiate_rev(abstract_locals(e, sz, from), sz, to);
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});
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}
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}
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static void replace(implementation_list const & imps, buffer<expr> const & from, buffer<expr> const & to) {
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lean_assert(from.size() == to.size());
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return replace(imps, from.size(), from.data(), to.data());
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}
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static void replace(implementation_list const & imps, expr const & from, expr const & to) {
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return replace(imps, 1, &from, &to);
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}
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class inversion_tac {
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environment const & m_env;
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io_state const & m_ios;
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type_checker & m_tc;
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list<name> m_ids;
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name_generator m_ngen;
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substitution m_subst;
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bool m_dep_elim;
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bool m_proof_irrel;
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unsigned m_nparams;
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unsigned m_nindices;
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unsigned m_nminors;
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declaration m_I_decl;
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declaration m_cases_on_decl;
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bool m_throw_tactic_exception; // if true, then throw tactic_exception in case of failure, instead of returning none
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bool m_clear_elim; // if true, then clear eliminated variables
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expr whnf(expr const & e) { return m_tc.whnf(e).first; }
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expr infer_type(expr const & e) { return m_tc.infer(e).first; }
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void init_inductive_info(name const & n) {
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m_dep_elim = inductive::has_dep_elim(m_env, n);
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m_nindices = *inductive::get_num_indices(m_env, n);
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m_nparams = *inductive::get_num_params(m_env, n);
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// This tactic is bases on cases_on construction which only has
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// minor premises for the introduction rules of this datatype.
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// For non-mutually recursive datatypes inductive::get_num_intro_rules == inductive::get_num_minor_premises
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m_nminors = *inductive::get_num_intro_rules(m_env, n);
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m_I_decl = m_env.get(n);
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m_cases_on_decl = m_env.get({n, "cases_on"});
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}
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bool is_inversion_applicable(expr const & t) {
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buffer<expr> args;
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expr const & fn = get_app_args(t, args);
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if (!is_constant(fn))
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return false;
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if (!inductive::is_inductive_decl(m_env, const_name(fn)))
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return false;
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if (!m_env.find(name{const_name(fn), "cases_on"}) || !m_env.find(get_eq_name()))
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return false;
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if (m_proof_irrel && !m_env.find(get_heq_name()))
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return false;
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init_inductive_info(const_name(fn));
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if (args.size() != m_nindices + m_nparams)
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return false;
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return true;
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}
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pair<expr, expr> mk_eq(expr const & lhs, expr const & rhs) {
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expr lhs_type = infer_type(lhs);
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expr rhs_type = infer_type(rhs);
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level l = sort_level(m_tc.ensure_type(lhs_type).first);
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constraint_seq cs;
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if (m_tc.is_def_eq(lhs_type, rhs_type, justification(), cs) && !cs) {
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return mk_pair(mk_app(mk_constant(get_eq_name(), to_list(l)), lhs_type, lhs, rhs),
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mk_app(mk_constant(get_eq_refl_name(), to_list(l)), lhs_type, lhs));
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} else {
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return mk_pair(mk_app(mk_constant(get_heq_name(), to_list(l)), lhs_type, lhs, rhs_type, rhs),
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mk_app(mk_constant(get_heq_refl_name(), to_list(l)), lhs_type, lhs));
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}
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}
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void assign(goal const & g, expr const & val) {
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::lean::assign(m_subst, g, val);
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}
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/** \brief We say h has independent indices IF
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1- it is *not* an indexed inductive family, OR
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2- it is an indexed inductive family, but all indices are distinct local constants,
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and all hypotheses of g different from h and indices, do not depend on the indices.
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3- if not m_dep_elim, then the conclusion does not depend on the indices.
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*/
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bool has_indep_indices(goal const & g, expr const & h, expr const & h_type) {
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if (m_nindices == 0)
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return true;
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buffer<expr> args;
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get_app_args(h_type, args);
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lean_assert(m_nindices <= args.size());
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unsigned fidx = args.size() - m_nindices;
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for (unsigned i = fidx; i < args.size(); i++) {
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if (!is_local(args[i]))
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return false; // the indices must be local constants
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for (unsigned j = 0; j < i; j++) {
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if (is_local(args[j]) && mlocal_name(args[j]) == mlocal_name(args[i]))
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return false; // the indices must be distinct local constants
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}
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}
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if (!m_dep_elim) {
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expr const & g_type = g.get_type();
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if (depends_on(g_type, h))
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return false;
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}
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buffer<expr> hyps;
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g.get_hyps(hyps);
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for (expr const & h1 : hyps) {
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if (mlocal_name(h1) == mlocal_name(h))
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continue;
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if (std::any_of(args.end() - m_nindices, args.end(),
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[&](expr const & arg) { return mlocal_name(arg) == mlocal_name(h1); }))
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continue;
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// h1 is not h nor any of the indices
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// Thus, it must not depend on the indices
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if (depends_on_any(h1, m_nindices, args.end() - m_nindices))
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return false;
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}
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return true;
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}
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/** \brief Split hyps into two "telescopes".
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- non_deps : hypotheses that do not depend on H
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- deps : hypotheses that depend on H (directly or indirectly)
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*/
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void split_deps(buffer<expr> const & hyps, expr const & H, buffer<expr> & non_deps, buffer<expr> & deps,
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bool clear_H = false) {
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for (expr const & hyp : hyps) {
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expr const & hyp_type = mlocal_type(hyp);
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if (depends_on(hyp_type, H) || depends_on_any(hyp_type, deps)) {
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deps.push_back(hyp);
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} else if (hyp != H || !clear_H) {
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non_deps.push_back(hyp);
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}
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}
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}
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/** \brief Given a goal of the form
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Ctx, h : I A j, D |- T
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where the type of h is the inductive datatype (I A j) where A are parameters,
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and j the indices. Generate the goal
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Ctx, h : I A j, D, j' : J, h' : I A j' |- j == j' -> h == h' -> T
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Remark: (j == j' -> h == h') is a "telescopic" equality. If the environment
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is proof irrelevant, it is built using homogenous and heterogeneous equalities.
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If the environment is proof relevant, it is built using homogeneous equality
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and the eq.rec (this construction is much more complex than the proof irrelevant
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one).
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Remark: j is sequence of terms, and j' a sequence of local constants.
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The original goal is solved if we can solve the produced goal.
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\remark h_type is mlocal_type(h) after normalization
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*/
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goal generalize_indices(goal const & g, expr const & h, expr const & h_type) {
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buffer<expr> hyps;
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g.get_hyps(hyps);
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expr m = g.get_meta();
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expr m_type = g.get_type();
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name h_new_name = get_unused_name(local_pp_name(h), hyps);
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buffer<expr> I_args;
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expr const & I = get_app_args(h_type, I_args);
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expr h_new_type = mk_app(I, I_args.size() - m_nindices, I_args.data());
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expr d = whnf(infer_type(h_new_type));
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name t_prefix("t");
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unsigned nidx = 1;
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if (m_proof_irrel) {
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unsigned eq_idx = 1;
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name eq_prefix("H");
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buffer<expr> ts;
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buffer<expr> eqs;
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buffer<expr> refls;
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auto add_eq = [&](expr const & lhs, expr const & rhs) {
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pair<expr, expr> p = mk_eq(lhs, rhs);
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expr new_eq = p.first;
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expr new_refl = p.second;
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eqs.push_back(mk_local(m_ngen.next(), g.get_unused_name(eq_prefix, eq_idx), new_eq, binder_info()));
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refls.push_back(new_refl);
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};
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for (unsigned i = I_args.size() - m_nindices; i < I_args.size(); i++) {
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expr t_type = binding_domain(d);
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expr t = mk_local(m_ngen.next(), g.get_unused_name(t_prefix, nidx), t_type, binder_info());
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expr const & index = I_args[i];
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add_eq(index, t);
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h_new_type = mk_app(h_new_type, t);
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hyps.push_back(t);
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ts.push_back(t);
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d = instantiate(binding_body(d), t);
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}
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expr h_new = mk_local(m_ngen.next(), h_new_name, h_new_type, local_info(h));
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if (m_dep_elim)
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add_eq(h, h_new);
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hyps.push_back(h_new);
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expr new_type = Pi(eqs, g.get_type());
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expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(hyps, new_type)), hyps);
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goal new_g(new_meta, new_type);
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expr val = mk_app(mk_app(mk_app(Fun(ts, Fun(h_new, new_meta)), m_nindices, I_args.end() - m_nindices), h),
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refls);
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assign(g, val);
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return new_g;
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} else {
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// proof relevant version
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buffer<expr> ss;
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buffer<expr> ts;
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buffer<expr> refls;
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for (unsigned i = I_args.size() - m_nindices; i < I_args.size(); i++) {
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expr t_type = binding_domain(d);
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expr t = mk_local(m_ngen.next(), g.get_unused_name(t_prefix, nidx), t_type, binder_info());
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h_new_type = mk_app(h_new_type, t);
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ss.push_back(I_args[i]);
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refls.push_back(mk_refl(m_tc, I_args[i]));
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hyps.push_back(t);
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ts.push_back(t);
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d = instantiate(binding_body(d), t);
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}
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expr h_new = mk_local(m_ngen.next(), h_new_name, h_new_type, local_info(h));
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ts.push_back(h_new);
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ss.push_back(h);
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refls.push_back(mk_refl(m_tc, h));
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hyps.push_back(h_new);
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buffer<expr> eqs;
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mk_telescopic_eq(m_tc, ss, ts, eqs);
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ts.pop_back();
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expr new_type = Pi(eqs, g.get_type());
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expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(hyps, new_type)), hyps);
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goal new_g(new_meta, new_type);
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expr val = mk_app(mk_app(mk_app(Fun(ts, Fun(h_new, new_meta)), m_nindices, I_args.end() - m_nindices), h),
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refls);
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assign(g, val);
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return new_g;
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}
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}
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/** \brief Generalize h dependencies.
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This tactic uses this method only if has_indep_indices(hyps, h, h_type) returns true.
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The hypotheses that depend on h are stored in deps.
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*/
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goal generalize_dependecies(goal const & g, expr const & h, buffer<expr> & deps) {
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buffer<expr> hyps;
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g.get_hyps(hyps);
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hyps.erase_elem(h);
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buffer<expr> non_deps;
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split_deps(hyps, h, non_deps, deps);
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buffer<expr> & new_hyps = non_deps;
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new_hyps.push_back(h);
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expr new_type = Pi(deps, g.get_type());
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expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(new_hyps, new_type)), new_hyps);
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goal new_g(new_meta, new_type);
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expr val = mk_app(new_meta, deps);
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assign(g, val);
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return new_g;
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}
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/**
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\brief Given a goal of the form
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Ctx, h : I A j |- T[h]
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produce the subgoals
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Ctx, h : I A j |- b_1 : B_1 -> T [C_1 A b_1]
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...
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Ctx, h : I A j |- b_n : B_n -> T [C_n A b_n]
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where the C_i's are the constructors (aka introduction rules) of the inductive datatype h.
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The hypothesis h must be the last hypothesis in the input goal. Consequently, no other
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hypothesis depends on it.
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The implementation list is external auxiliary data associated with constructors.
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For example, we use it to compile recursive equations. Each implementation corresponds to one
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equation. Each equation is tagged with a constructor C_i. The constructor is retrieved using
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the virtual method get_constructor_name. We split the list imps in n lists. One for each
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subgoal. An implementation associated with the constructor i is in the i-th resulting list.
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\remark This method assumes the indices j are local constants, and only h and j may depend on j.
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*/
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std::pair<list<goal>, list<implementation_list>> apply_cases_on(goal const & g, implementation_list const & imps, bool has_indep_indices) {
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buffer<expr> hyps;
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g.get_hyps(hyps);
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expr const & h = hyps.back();
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expr const & h_type = whnf(mlocal_type(h));
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buffer<expr> I_args;
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expr const & I = get_app_args(h_type, I_args);
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name const & I_name = const_name(I);
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expr g_type = g.get_type();
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expr cases_on;
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level g_lvl = sort_level(m_tc.ensure_type(g_type).first);
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if (m_cases_on_decl.get_num_univ_params() != m_I_decl.get_num_univ_params()) {
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cases_on = mk_constant({I_name, "cases_on"}, cons(g_lvl, const_levels(I)));
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} else {
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if (!is_zero(g_lvl)) {
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if (m_throw_tactic_exception)
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throw tactic_exception(sstream() << "invalid 'cases' tactic, '" << const_name(I) << "' can only eliminate to Prop");
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else
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throw inversion_exception();
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}
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cases_on = mk_constant({I_name, "cases_on"}, const_levels(I));
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}
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// add params
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cases_on = mk_app(cases_on, m_nparams, I_args.data());
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// add type former
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expr type_former = g_type;
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if (m_dep_elim)
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type_former = Fun(h, type_former);
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type_former = Fun(m_nindices, I_args.end() - m_nindices, type_former);
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cases_on = mk_app(cases_on, type_former);
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// add indices
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cases_on = mk_app(cases_on, m_nindices, I_args.end() - m_nindices);
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// add h
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cases_on = mk_app(cases_on, h);
|
|
buffer<name> intro_names;
|
|
get_intro_rule_names(m_env, I_name, intro_names);
|
|
lean_assert(m_nminors == intro_names.size());
|
|
buffer<expr> new_hyps;
|
|
if (has_indep_indices)
|
|
new_hyps.append(hyps.size() - 1, hyps.data());
|
|
else
|
|
new_hyps.append(hyps.size() - m_nindices - 1, hyps.data());
|
|
|
|
// add a subgoal for each minor premise of cases_on
|
|
expr cases_on_type = whnf(infer_type(cases_on));
|
|
buffer<goal> new_goals;
|
|
buffer<implementation_list> new_imps;
|
|
for (unsigned i = 0; i < m_nminors; i++) {
|
|
expr new_type = binding_domain(cases_on_type);
|
|
expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(new_hyps, new_type)), new_hyps);
|
|
goal new_g(new_meta, new_type);
|
|
new_goals.push_back(new_g);
|
|
cases_on = mk_app(cases_on, new_meta);
|
|
cases_on_type = whnf(binding_body(cases_on_type)); // the minor premises do not depend on each other
|
|
name const & intro_name = intro_names[i];
|
|
new_imps.push_back(filter(imps, [&](implementation_ptr const & imp) { return imp->get_constructor_name() == intro_name; }));
|
|
}
|
|
assign(g, cases_on);
|
|
return mk_pair(to_list(new_goals), to_list(new_imps));
|
|
}
|
|
|
|
// Store in \c r the number of arguments for each cases_on minor.
|
|
void get_minors_nargs(buffer<unsigned> & r) {
|
|
expr cases_on_type = m_cases_on_decl.get_type();
|
|
for (unsigned i = 0; i < m_nparams + 1 + m_nindices + 1; i++)
|
|
cases_on_type = binding_body(cases_on_type);
|
|
for (unsigned i = 0; i < m_nminors; i++) {
|
|
expr minor_type = binding_domain(cases_on_type);
|
|
unsigned nargs = 0;
|
|
while (is_pi(minor_type)) {
|
|
nargs++;
|
|
minor_type = binding_body(minor_type);
|
|
}
|
|
r.push_back(nargs);
|
|
cases_on_type = binding_body(cases_on_type);
|
|
}
|
|
}
|
|
|
|
/**
|
|
The method apply_cases_on produces subgoals of the form
|
|
|
|
Ctx, h : I A j |- b_i : B_i -> T [C_i A b_i]
|
|
|
|
The b_i are the arguments of the constructor C_i.
|
|
This method moves the b_i's to the hypotheses list.
|
|
*/
|
|
std::pair<list<goal>, list<list<expr>>> intros_minors_args(list<goal> gs) {
|
|
buffer<unsigned> minors_nargs;
|
|
get_minors_nargs(minors_nargs);
|
|
lean_assert(length(gs) == minors_nargs.size());
|
|
buffer<goal> new_gs;
|
|
buffer<list<expr>> new_args;
|
|
for (unsigned i = 0; i < minors_nargs.size(); i++) {
|
|
goal const & g = head(gs);
|
|
unsigned nargs = minors_nargs[i];
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
buffer<expr> new_hyps;
|
|
new_hyps.append(hyps);
|
|
expr g_type = g.get_type();
|
|
buffer<expr> curr_new_args;
|
|
for (unsigned i = 0; i < nargs; i++) {
|
|
expr type = binding_domain(g_type);
|
|
name new_h_name;
|
|
if (m_ids) {
|
|
new_h_name = head(m_ids);
|
|
m_ids = tail(m_ids);
|
|
} else {
|
|
new_h_name = binding_name(g_type);
|
|
}
|
|
expr new_h = mk_local(m_ngen.next(), get_unused_name(new_h_name, new_hyps), type, binder_info());
|
|
curr_new_args.push_back(new_h);
|
|
new_hyps.push_back(new_h);
|
|
g_type = instantiate(binding_body(g_type), new_h);
|
|
}
|
|
new_args.push_back(to_list(curr_new_args));
|
|
g_type = head_beta_reduce(g_type);
|
|
expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(new_hyps, g_type)), new_hyps);
|
|
goal new_g(new_meta, g_type);
|
|
new_gs.push_back(new_g);
|
|
expr val = Fun(nargs, new_hyps.end() - nargs, new_meta);
|
|
assign(g, val);
|
|
gs = tail(gs);
|
|
}
|
|
return mk_pair(to_list(new_gs), to_list(new_args));
|
|
}
|
|
|
|
// auxiliary exception used to interrupt execution
|
|
struct inversion_exception {};
|
|
|
|
[[ noreturn ]] void throw_ill_formed_goal() {
|
|
if (m_throw_tactic_exception)
|
|
throw tactic_exception("invalid 'cases' tactic, ill-formed goal");
|
|
else
|
|
throw inversion_exception();
|
|
}
|
|
|
|
[[ noreturn ]] void throw_ill_typed_goal() {
|
|
if (m_throw_tactic_exception)
|
|
throw tactic_exception("invalid 'cases' tactic, ill-typed goal");
|
|
else
|
|
throw inversion_exception();
|
|
}
|
|
|
|
[[ noreturn ]] void throw_lift_down_failure() {
|
|
if (m_throw_tactic_exception)
|
|
throw tactic_exception("invalid 'cases' tactic, lift.down failed");
|
|
else
|
|
throw inversion_exception();
|
|
}
|
|
|
|
void throw_unification_eq_rec_failure(goal const & g, expr const & eq) {
|
|
if (m_throw_tactic_exception) {
|
|
throw tactic_exception([=](formatter const & fmt) {
|
|
format r("invalid 'cases' tactic, unification failed to eliminate eq.rec in the homogeneous equality");
|
|
r += pp_indent_expr(fmt, eq);
|
|
r += compose(line(), format("auxiliary goal at time of failure"));
|
|
r += nest(get_pp_indent(fmt.get_options()), compose(line(), g.pp(fmt)));
|
|
return r;
|
|
});
|
|
} else {
|
|
throw inversion_exception();
|
|
}
|
|
}
|
|
|
|
/** \brief Process goal of the form: Pi (H : eq.rec A s C a s p = b), R
|
|
The idea is to reduce it to
|
|
Pi (H : a = b), R
|
|
when A is a hset
|
|
and then invoke intro_next_eq recursively.
|
|
|
|
\remark \c type is the type of \c g after some normalization
|
|
*/
|
|
goal intro_next_eq_rec(goal const & g, expr const & type) {
|
|
lean_assert(is_pi(type));
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
expr const & eq = binding_domain(type);
|
|
expr const & lhs = app_arg(app_fn(eq));
|
|
expr const & rhs = app_arg(eq);
|
|
lean_assert(is_eq_rec(lhs));
|
|
// lhs is of the form (eq.rec A s C a s p)
|
|
// aux_eq is a term of type ((eq.rec A s C a s p) = a)
|
|
auto aux_eq = apply_eq_rec_eq(m_tc, m_ios, to_list(hyps), lhs);
|
|
if (!aux_eq || has_expr_metavar_relaxed(*aux_eq)) {
|
|
throw_unification_eq_rec_failure(g, eq);
|
|
}
|
|
buffer<expr> lhs_args;
|
|
get_app_args(lhs, lhs_args);
|
|
expr const & reduced_lhs = lhs_args[3];
|
|
expr new_eq = ::lean::mk_eq(m_tc, reduced_lhs, rhs);
|
|
expr new_type = update_binding(type, new_eq, binding_body(type));
|
|
expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(hyps, new_type)), hyps);
|
|
goal new_g(new_meta, new_type);
|
|
// create assignment for g
|
|
expr A = infer_type(lhs);
|
|
level lvl = sort_level(m_tc.ensure_type(A).first);
|
|
// old_eq : eq.rec A s C a s p = b
|
|
expr old_eq = mk_local(m_ngen.next(), binding_name(type), eq, binder_info());
|
|
// aux_eq : a = eq.rec A s C a s p
|
|
expr trans_eq = mk_app({mk_constant(get_eq_trans_name(), {lvl}), A, reduced_lhs, lhs, rhs, *aux_eq, old_eq});
|
|
// trans_eq : a = b
|
|
expr val = Fun(old_eq, mk_app(new_meta, trans_eq));
|
|
assign(g, val);
|
|
return intro_next_eq(new_g);
|
|
}
|
|
|
|
/**
|
|
\brief Process goal of the form: Ctx |- Pi (H : a == b), R when a and b have the same type
|
|
The idea is to reduce it to
|
|
Ctx, H : a = b |- R
|
|
This method is only used if the environment has a proof irrelevant Prop.
|
|
*/
|
|
goal intro_next_heq(goal const & g) {
|
|
lean_assert(m_proof_irrel);
|
|
expr const & type = g.get_type();
|
|
expr eq = binding_domain(type);
|
|
lean_assert(const_name(get_app_fn(eq)) == get_heq_name());
|
|
buffer<expr> args;
|
|
expr const & heq_fn = get_app_args(eq, args);
|
|
constraint_seq cs;
|
|
if (m_tc.is_def_eq(args[0], args[2], justification(), cs) && !cs) {
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
expr new_eq = mk_app(mk_constant(get_eq_name(), const_levels(heq_fn)), args[0], args[1], args[3]);
|
|
expr new_hyp = mk_local(m_ngen.next(), g.get_unused_name(binding_name(type)), new_eq, binder_info());
|
|
expr new_type = instantiate(binding_body(type), new_hyp);
|
|
hyps.push_back(new_hyp);
|
|
expr new_mvar = mk_metavar(m_ngen.next(), Pi(hyps, new_type));
|
|
expr new_meta = mk_app(new_mvar, hyps);
|
|
goal new_g(new_meta, new_type);
|
|
hyps.pop_back();
|
|
expr H = mk_local(m_ngen.next(), g.get_unused_name(binding_name(type)), binding_domain(type), binder_info());
|
|
expr to_eq = mk_app(mk_constant(get_heq_to_eq_name(), const_levels(heq_fn)), args[0], args[1], args[3], H);
|
|
expr val = Fun(H, mk_app(mk_app(new_mvar, hyps), to_eq));
|
|
assign(g, val);
|
|
return new_g;
|
|
} else {
|
|
if (m_throw_tactic_exception) {
|
|
throw tactic_exception("invalid 'cases' tactic, unification failed to reduce heterogeneous equality into homogeneous one");
|
|
} else {
|
|
throw inversion_exception();
|
|
}
|
|
}
|
|
}
|
|
|
|
/** \brief Process goal of the form: Ctx |- Pi (H : a = b), R
|
|
The idea is to reduce it to
|
|
Ctx, H : a = b |- R
|
|
|
|
\remark \c type is the type of \c g after some normalization
|
|
*/
|
|
goal intro_next_eq_simple(goal const & g, expr const & type) {
|
|
expr eq = binding_domain(type);
|
|
lean_assert(const_name(get_app_fn(eq)) == get_eq_name());
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
expr new_hyp = mk_local(m_ngen.next(), g.get_unused_name(binding_name(type)), binding_domain(type), binder_info());
|
|
expr new_type = instantiate(binding_body(type), new_hyp);
|
|
hyps.push_back(new_hyp);
|
|
expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(hyps, new_type)), hyps);
|
|
goal new_g(new_meta, new_type);
|
|
expr val = Fun(new_hyp, new_meta);
|
|
assign(g, val);
|
|
return new_g;
|
|
}
|
|
|
|
goal intro_next_eq(goal const & g) {
|
|
expr type = g.get_type();
|
|
if (!is_pi(type))
|
|
throw_ill_formed_goal();
|
|
expr eq = binding_domain(type);
|
|
expr const & eq_fn = get_app_fn(eq);
|
|
if (!is_constant(eq_fn))
|
|
throw_ill_formed_goal();
|
|
if (const_name(eq_fn) == get_eq_name()) {
|
|
expr const & lhs = app_arg(app_fn(eq));
|
|
expr const & rhs = app_arg(eq);
|
|
expr new_lhs = whnf(lhs);
|
|
expr new_rhs = whnf(rhs);
|
|
if (lhs != new_lhs || rhs != new_rhs) {
|
|
eq = mk_app(app_fn(app_fn(eq)), new_lhs, new_rhs);
|
|
type = update_binding(type, eq, binding_body(type));
|
|
}
|
|
if (!m_proof_irrel && is_eq_rec(new_lhs)) {
|
|
return intro_next_eq_rec(g, type);
|
|
} else {
|
|
return intro_next_eq_simple(g, type);
|
|
}
|
|
} else if (m_proof_irrel && const_name(eq_fn) == get_heq_name()) {
|
|
return intro_next_heq(g);
|
|
} else {
|
|
throw_ill_formed_goal();
|
|
}
|
|
}
|
|
|
|
/**
|
|
\brief The no_confusion constructions uses lifts in the proof relevant version.
|
|
We must apply lift.down to eliminate the auxiliary lift.
|
|
*/
|
|
expr lift_down(expr const & v) {
|
|
if (auto r = lift_down_if_hott(m_tc, v))
|
|
return *r;
|
|
else
|
|
throw_lift_down_failure();
|
|
}
|
|
|
|
buffer<expr> m_c_args; // current intro/constructor args that may be renamed by unify_eqs
|
|
rename_map m_renames;
|
|
implementation_list m_imps;
|
|
|
|
void store_rename(expr const & old_hyp, expr const & new_hyp) {
|
|
for (expr & c_arg : m_c_args) {
|
|
if (is_local(c_arg) && mlocal_name(old_hyp) == mlocal_name(c_arg))
|
|
c_arg = new_hyp;
|
|
}
|
|
if (is_local(new_hyp))
|
|
m_renames.insert(mlocal_name(old_hyp), mlocal_name(new_hyp));
|
|
}
|
|
|
|
/** \brief Update m_renames with old_hyps --> new_hyps. */
|
|
void store_renames(buffer<expr> const & old_hyps, buffer<expr> const & new_hyps) {
|
|
lean_assert(old_hyps.size() == new_hyps.size());
|
|
for (unsigned i = 0; i < old_hyps.size(); i++) {
|
|
store_rename(old_hyps[i], new_hyps[i]);
|
|
}
|
|
}
|
|
|
|
[[ noreturn ]] void throw_unify_eqs_failure(goal const & g) {
|
|
if (m_throw_tactic_exception) {
|
|
throw tactic_exception([=](formatter const & fmt) {
|
|
format r("invalid 'cases' tactic, unification failed");
|
|
r += compose(line(), format("auxiliary goal at time of failure"));
|
|
r += nest(get_pp_indent(fmt.get_options()), compose(line(), g.pp(fmt)));
|
|
return r;
|
|
});
|
|
} else {
|
|
throw inversion_exception();
|
|
}
|
|
}
|
|
|
|
// Remark: it also updates m_renames and m_imps
|
|
optional<goal> unify_eqs(goal g, unsigned neqs) {
|
|
if (neqs == 0)
|
|
return optional<goal>(g); // done
|
|
g = intro_next_eq(g);
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
lean_assert(!hyps.empty());
|
|
expr Heq = hyps.back();
|
|
buffer<expr> Heq_args;
|
|
get_app_args(mlocal_type(Heq), Heq_args);
|
|
expr const & A = whnf(Heq_args[0]);
|
|
expr lhs = whnf(Heq_args[1]);
|
|
expr rhs = whnf(Heq_args[2]);
|
|
constraint_seq cs;
|
|
if (m_proof_irrel && m_tc.is_def_eq(lhs, rhs, justification(), cs) && !cs) {
|
|
// deletion transition: t == t
|
|
hyps.pop_back(); // remove t == t equality
|
|
expr new_type = g.get_type();
|
|
expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(hyps, new_type)), hyps);
|
|
goal new_g(new_meta, new_type);
|
|
assign(g, new_meta);
|
|
return unify_eqs(new_g, neqs-1);
|
|
}
|
|
buffer<expr> lhs_args, rhs_args;
|
|
expr const & lhs_fn = get_app_args(lhs, lhs_args);
|
|
expr const & rhs_fn = get_app_args(rhs, rhs_args);
|
|
expr const & g_type = g.get_type();
|
|
level const & g_lvl = sort_level(m_tc.ensure_type(g_type).first);
|
|
if (is_constant(lhs_fn) &&
|
|
is_constant(rhs_fn) &&
|
|
inductive::is_intro_rule(m_env, const_name(lhs_fn)) &&
|
|
inductive::is_intro_rule(m_env, const_name(rhs_fn))) {
|
|
buffer<expr> A_args;
|
|
expr const & A_fn = get_app_args(A, A_args);
|
|
if (!is_constant(A_fn) || !inductive::is_inductive_decl(m_env, const_name(A_fn)))
|
|
throw_ill_typed_goal();
|
|
name no_confusion_name(const_name(A_fn), "no_confusion");
|
|
if (!m_env.find(no_confusion_name)) {
|
|
if (m_throw_tactic_exception)
|
|
throw tactic_exception(sstream() << "invalid 'cases' tactic, construction '" << no_confusion_name << "' is not available in the environment");
|
|
else
|
|
throw inversion_exception();
|
|
}
|
|
expr no_confusion = mk_app(mk_app(mk_constant(no_confusion_name, cons(g_lvl, const_levels(A_fn))), A_args), g_type, lhs, rhs, Heq);
|
|
if (const_name(lhs_fn) == const_name(rhs_fn)) {
|
|
// injectivity transition
|
|
expr new_type = binding_domain(whnf(infer_type(no_confusion)));
|
|
if (m_proof_irrel || !depends_on(g_type, hyps.back()))
|
|
hyps.pop_back(); // remove processed equality
|
|
expr new_mvar = mk_metavar(m_ngen.next(), Pi(hyps, new_type));
|
|
expr new_meta = mk_app(new_mvar, hyps);
|
|
goal new_g(new_meta, new_type);
|
|
expr val = lift_down(mk_app(no_confusion, new_meta));
|
|
assign(g, val);
|
|
unsigned A_nparams = *inductive::get_num_params(m_env, const_name(A_fn));
|
|
lean_assert(lhs_args.size() >= A_nparams);
|
|
return unify_eqs(new_g, neqs - 1 + lhs_args.size() - A_nparams);
|
|
} else {
|
|
// conflict transition, Heq is of the form c_1 ... = c_2 ..., where c_1 and c_2 are different constructors/intro rules.
|
|
expr val = lift_down(no_confusion);
|
|
assign(g, val);
|
|
return optional<goal>(); // goal has been solved
|
|
}
|
|
}
|
|
if (is_local(rhs)) {
|
|
// solution transition, Heq is of the form t = y, where y is a local constant
|
|
|
|
// assume the current goal is of the form
|
|
//
|
|
// non_deps, deps[rhs], H : lhs = rhs |- C[rhs]
|
|
//
|
|
// We use non_deps to denote hypotheses that do not depend on rhs,
|
|
// and deps[rhs] to denote hypotheses that depend on it.
|
|
//
|
|
// The resultant goal is of the form
|
|
//
|
|
// non_deps, deps[lhs] |- C[lhs]
|
|
//
|
|
// Now, assume ?m1 is a solution for the resultant goal.
|
|
// Then,
|
|
//
|
|
// @eq.rec A lhs (fun rhs, Pi(deps[rhs], C[rhs])) (?m1 non_deps) rhs H deps[rhs]
|
|
//
|
|
// is a solution for the original goal.
|
|
// Remark: A is the type of lhs and rhs
|
|
hyps.pop_back(); // remove processed equality
|
|
buffer<expr> non_deps, deps;
|
|
bool clear_rhs = m_clear_elim;
|
|
split_deps(hyps, rhs, non_deps, deps, clear_rhs);
|
|
if (deps.empty() && !depends_on(g_type, rhs)) {
|
|
// eq.rec is not necessary
|
|
buffer<expr> & new_hyps = non_deps;
|
|
expr new_type = g_type;
|
|
expr new_mvar = mk_metavar(m_ngen.next(), Pi(new_hyps, new_type));
|
|
expr new_meta = mk_app(new_mvar, new_hyps);
|
|
goal new_g(new_meta, new_type);
|
|
assign(g, new_meta);
|
|
return unify_eqs(new_g, neqs-1);
|
|
} else {
|
|
expr deps_g_type = Pi(deps, g_type);
|
|
level eq_rec_lvl1 = sort_level(m_tc.ensure_type(deps_g_type).first);
|
|
level eq_rec_lvl2 = sort_level(m_tc.ensure_type(A).first);
|
|
expr tformer;
|
|
if (m_proof_irrel) {
|
|
tformer = Fun(rhs, deps_g_type);
|
|
} else {
|
|
if (depends_on(lhs, rhs)) {
|
|
// tformer must be of the form
|
|
// fun x, fun (Heq : lhs = x, ...)
|
|
// If the lhs contains occurrences of rhs, then
|
|
// we would produce the following ill-typed tformer
|
|
// fun x, fun (Heq : lhs[x] = x, ...)
|
|
throw_unify_eqs_failure(g);
|
|
}
|
|
tformer = Fun(rhs, Fun(Heq, deps_g_type));
|
|
}
|
|
expr eq_rec = mk_constant(get_eq_rec_name(), {eq_rec_lvl1, eq_rec_lvl2});
|
|
eq_rec = mk_app(eq_rec, A, lhs, tformer);
|
|
buffer<expr> new_hyps;
|
|
new_hyps.append(non_deps);
|
|
expr new_type = instantiate(abstract_local(deps_g_type, rhs), lhs);
|
|
store_rename(rhs, lhs);
|
|
replace(m_imps, rhs, lhs);
|
|
if (!m_proof_irrel) {
|
|
new_type = abstract_local(new_type, Heq);
|
|
new_type = instantiate(new_type, mk_refl(m_tc, lhs));
|
|
}
|
|
buffer<expr> new_deps;
|
|
for (unsigned i = 0; i < deps.size(); i++) {
|
|
expr new_hyp = mk_local(m_ngen.next(), binding_name(new_type), binding_domain(new_type),
|
|
binding_info(new_type));
|
|
new_hyps.push_back(new_hyp);
|
|
new_deps.push_back(new_hyp);
|
|
new_type = instantiate(binding_body(new_type), new_hyp);
|
|
}
|
|
replace(m_imps, deps, new_deps);
|
|
lean_assert(deps.size() == new_deps.size());
|
|
store_renames(deps, new_deps);
|
|
expr new_mvar = mk_metavar(m_ngen.next(), Pi(new_hyps, new_type));
|
|
expr new_meta = mk_app(new_mvar, new_hyps);
|
|
goal new_g(new_meta, new_type);
|
|
expr eq_rec_minor = mk_app(new_mvar, non_deps);
|
|
eq_rec = mk_app(eq_rec, eq_rec_minor, rhs, Heq);
|
|
expr val = mk_app(eq_rec, deps);
|
|
assign(g, val);
|
|
return unify_eqs(new_g, neqs-1);
|
|
}
|
|
} else if (is_local(lhs)) {
|
|
// flip equation and reduce to previous case
|
|
expr symm_eq = mk_eq(rhs, lhs).first;
|
|
hyps.pop_back(); // remove processed equality
|
|
if (!depends_on(g_type, Heq)) {
|
|
expr new_type = mk_arrow(symm_eq, g_type);
|
|
expr new_mvar = mk_metavar(m_ngen.next(), Pi(hyps, new_type));
|
|
expr new_meta = mk_app(new_mvar, hyps);
|
|
goal new_g(new_meta, new_type);
|
|
expr Heq_inv = mk_symm(m_tc, Heq);
|
|
expr val = mk_app(new_meta, Heq_inv);
|
|
assign(g, val);
|
|
return unify_eqs(new_g, neqs);
|
|
} else {
|
|
// Let C[Heq] be the original conclusion which depends on the equality eq being processed.
|
|
expr new_Heq = update_mlocal(Heq, symm_eq);
|
|
expr new_Heq_inv = mk_symm(m_tc, new_Heq);
|
|
expr new_type = Pi(new_Heq, instantiate(abstract_local(g_type, Heq), new_Heq_inv));
|
|
expr new_mvar = mk_metavar(m_ngen.next(), Pi(hyps, new_type));
|
|
expr new_meta = mk_app(new_mvar, hyps);
|
|
goal new_g(new_meta, new_type);
|
|
// Then, we have
|
|
// new_meta : Pi (new_Heq : rhs = lhs), C[symm new_Heq]
|
|
expr Heq_inv = mk_symm(m_tc, Heq);
|
|
expr val = mk_app(new_meta, Heq_inv);
|
|
// val : C[symm (symm Heq)]
|
|
// Remark: in proof irrelevant mode (symm (symm Heq)) is definitionally equal to Heq
|
|
if (!m_proof_irrel) {
|
|
expr C = Fun(Heq, g_type);
|
|
level A_lvl = sort_level(m_tc.ensure_type(A).first);
|
|
level g_lvl = sort_level(m_tc.ensure_type(g_type).first);
|
|
expr elim_inv_inv = mk_constant(get_eq_elim_inv_inv_name(), {A_lvl, g_lvl});
|
|
val = mk_app({elim_inv_inv, A, lhs, rhs, C, Heq, val});
|
|
// val : C[Heq] as we wanted
|
|
}
|
|
assign(g, val);
|
|
return unify_eqs(new_g, neqs);
|
|
}
|
|
}
|
|
throw_unify_eqs_failure(g);
|
|
}
|
|
|
|
auto unify_eqs(list<goal> const & gs, list<list<expr>> args, list<implementation_list> imps) ->
|
|
std::tuple<list<goal>, list<list<expr>>, list<implementation_list>, list<rename_map>> {
|
|
lean_assert(length(gs) == length(imps));
|
|
unsigned neqs = m_nindices + (m_dep_elim ? 1 : 0);
|
|
buffer<goal> new_goals;
|
|
buffer<list<expr>> new_args;
|
|
buffer<implementation_list> new_imps;
|
|
buffer<rename_map> new_renames;
|
|
for (goal const & g : gs) {
|
|
flet<rename_map> set1(m_renames, rename_map());
|
|
flet<implementation_list> set2(m_imps, head(imps));
|
|
m_c_args.clear();
|
|
to_buffer(head(args), m_c_args);
|
|
if (optional<goal> new_g = unify_eqs(g, neqs)) {
|
|
new_goals.push_back(*new_g);
|
|
list<expr> new_as = to_list(m_c_args);
|
|
new_args.push_back(new_as);
|
|
new_imps.push_back(m_imps);
|
|
new_renames.push_back(m_renames);
|
|
}
|
|
imps = tail(imps);
|
|
args = tail(args);
|
|
}
|
|
return std::make_tuple(to_list(new_goals), to_list(new_args), to_list(new_imps), to_list(new_renames));
|
|
}
|
|
|
|
std::pair<goal, rename_map> intro_deps(goal const & g, buffer<expr> const & deps) {
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
buffer<expr> new_hyps;
|
|
new_hyps.append(hyps);
|
|
rename_map rs;
|
|
expr g_type = g.get_type();
|
|
for (expr const & d : deps) {
|
|
expr type = binding_domain(g_type);
|
|
expr new_d = mk_local(m_ngen.next(), get_unused_name(local_pp_name(d), new_hyps), type, local_info(d));
|
|
rs.insert(mlocal_name(d), mlocal_name(new_d));
|
|
new_hyps.push_back(new_d);
|
|
g_type = instantiate(binding_body(g_type), new_d);
|
|
}
|
|
expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(new_hyps, g_type)), new_hyps);
|
|
goal new_g(new_meta, g_type);
|
|
unsigned ndeps = deps.size();
|
|
expr val = Fun(ndeps, new_hyps.end() - ndeps, new_meta);
|
|
assign(g, val);
|
|
return mk_pair(new_g, rs);
|
|
}
|
|
|
|
std::pair<list<goal>, list<rename_map>> intro_deps(list<goal> const & gs, buffer<expr> const & deps) {
|
|
buffer<goal> new_goals;
|
|
buffer<rename_map> new_rs;
|
|
for (goal const & g : gs) {
|
|
auto p = intro_deps(g, deps);
|
|
new_goals.push_back(p.first);
|
|
new_rs.push_back(p.second);
|
|
}
|
|
return mk_pair(to_list(new_goals), to_list(new_rs));
|
|
}
|
|
|
|
goal clear_hypothesis(goal const & g, name const & h) {
|
|
if (auto p = g.find_hyp_from_internal_name(h)) {
|
|
expr const & h = p->first;
|
|
unsigned i = p->second;
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
hyps.erase(hyps.size() - i - 1);
|
|
if (depends_on(g.get_type(), h) || depends_on(i, hyps.end() - i, h)) {
|
|
return g; // other hypotheses or result type depend on h
|
|
}
|
|
expr new_type = g.get_type();
|
|
expr new_meta = mk_app(mk_metavar(m_ngen.next(), Pi(hyps, new_type)), hyps);
|
|
goal new_g(new_meta, new_type);
|
|
assign(g, new_meta);
|
|
return new_g;
|
|
} else {
|
|
return g;
|
|
}
|
|
}
|
|
|
|
// Remove hypothesis of the form (H : a = a)
|
|
goal remove_eq_refl_hypotheses(goal g) {
|
|
buffer<name> to_remove;
|
|
buffer<expr> hyps;
|
|
g.get_hyps(hyps);
|
|
for (expr const & h : hyps) {
|
|
expr const & h_type = mlocal_type(h);
|
|
if (!is_eq(h_type))
|
|
continue;
|
|
expr const & lhs = app_arg(app_fn(h_type));
|
|
expr const & rhs = app_arg(h_type);
|
|
if (lhs == rhs)
|
|
to_remove.push_back(mlocal_name(h));
|
|
}
|
|
for (name const & h : to_remove) {
|
|
g = clear_hypothesis(g, h);
|
|
}
|
|
return g;
|
|
}
|
|
|
|
list<goal> clear_hypothesis(list<goal> const & gs, list<rename_map> rs, name const & h_name, expr const & h_type) {
|
|
buffer<goal> new_gs;
|
|
optional<name> lhs_name; // If h_type is of the form lhs = rhs, and lhs is also a hypothesis, then we also remove it.
|
|
if (is_eq(h_type) && is_local(app_arg(app_fn(h_type)))) {
|
|
lhs_name = mlocal_name(app_arg(app_fn(h_type)));
|
|
}
|
|
for (goal const & g : gs) {
|
|
rename_map const & m = head(rs);
|
|
goal new_g = clear_hypothesis(g, m.find(h_name));
|
|
if (lhs_name)
|
|
new_g = clear_hypothesis(new_g, *lhs_name);
|
|
if (!m_proof_irrel)
|
|
new_g = remove_eq_refl_hypotheses(new_g);
|
|
new_gs.push_back(new_g);
|
|
rs = tail(rs);
|
|
}
|
|
return to_list(new_gs);
|
|
}
|
|
|
|
public:
|
|
inversion_tac(environment const & env, io_state const & ios, name_generator const & ngen,
|
|
type_checker & tc, substitution const & subst, list<name> const & ids,
|
|
bool throw_tactic_ex, bool clear_elim):
|
|
m_env(env), m_ios(ios), m_tc(tc), m_ids(ids),
|
|
m_ngen(ngen), m_subst(subst), m_throw_tactic_exception(throw_tactic_ex),
|
|
m_clear_elim(clear_elim) {
|
|
m_proof_irrel = m_env.prop_proof_irrel();
|
|
}
|
|
|
|
inversion_tac(environment const & env, io_state const & ios, type_checker & tc, bool clear_elim):
|
|
inversion_tac(env, ios, tc.mk_ngen(), tc, substitution(), list<name>(), false, clear_elim) {}
|
|
|
|
typedef inversion::result result;
|
|
|
|
optional<result> execute(goal const & g, expr const & h, implementation_list const & imps) {
|
|
try {
|
|
expr h_type = whnf(mlocal_type(h));
|
|
if (!is_inversion_applicable(h_type))
|
|
return optional<result>();
|
|
if (has_indep_indices(g, h, h_type)) {
|
|
buffer<expr> deps;
|
|
goal g1 = generalize_dependecies(g, h, deps);
|
|
auto gs_imps_pair = apply_cases_on(g1, imps, true);
|
|
list<goal> gs2 = gs_imps_pair.first;
|
|
list<implementation_list> new_imps = gs_imps_pair.second;
|
|
auto gs_args_pair = intros_minors_args(gs2);
|
|
list<goal> gs3 = gs_args_pair.first;
|
|
list<list<expr>> args = gs_args_pair.second;
|
|
auto gs_rs_pair = intro_deps(gs3, deps);
|
|
list<goal> gs4 = gs_rs_pair.first;
|
|
list<rename_map> rs = gs_rs_pair.second;
|
|
return optional<result>(result(gs4, args, new_imps, rs, m_ngen, m_subst));
|
|
} else {
|
|
goal g1 = generalize_indices(g, h, h_type);
|
|
auto gs_imps_pair = apply_cases_on(g1, imps, false);
|
|
list<goal> gs2 = gs_imps_pair.first;
|
|
list<implementation_list> new_imps = gs_imps_pair.second;
|
|
auto gs_args_pair = intros_minors_args(gs2);
|
|
list<goal> gs3 = gs_args_pair.first;
|
|
list<list<expr>> args = gs_args_pair.second;
|
|
list<goal> gs4;
|
|
list<rename_map> rs;
|
|
std::tie(gs4, args, new_imps, rs) = unify_eqs(gs3, args, new_imps);
|
|
gs4 = clear_hypothesis(gs4, rs, mlocal_name(h), h_type);
|
|
return optional<result>(result(gs4, args, new_imps, rs, m_ngen, m_subst));
|
|
}
|
|
} catch (inversion_exception & ex) {
|
|
return optional<result>();
|
|
}
|
|
}
|
|
|
|
optional<result> execute(goal const & g, name const & n, implementation_list const & imps) {
|
|
auto p = g.find_hyp(n);
|
|
if (!p) {
|
|
if (m_throw_tactic_exception)
|
|
throw tactic_exception(sstream() << "invalid 'cases' tactic, unknown hypothesis '" << n << "'");
|
|
return optional<result>();
|
|
}
|
|
expr const & h = p->first;
|
|
return execute(g, h, imps);
|
|
}
|
|
};
|
|
|
|
namespace inversion {
|
|
optional<result> apply(environment const & env, io_state const & ios, type_checker & tc,
|
|
goal const & g, expr const & h, implementation_list const & imps, bool clear_elim) {
|
|
return inversion_tac(env, ios, tc, clear_elim).execute(g, h, imps);
|
|
}
|
|
}
|
|
|
|
tactic inversion_tactic(name const & n, list<name> const & ids) {
|
|
auto fn = [=](environment const & env, io_state const & ios, proof_state const & ps) -> optional<proof_state> {
|
|
goals const & gs = ps.get_goals();
|
|
if (empty(gs))
|
|
return none_proof_state();
|
|
goal g = head(gs);
|
|
goals tail_gs = tail(gs);
|
|
name_generator ngen = ps.get_ngen();
|
|
std::unique_ptr<type_checker> tc = mk_type_checker(env, ngen.mk_child());
|
|
bool clear_elim = true;
|
|
inversion_tac tac(env, ios, ngen, *tc, ps.get_subst(), ids, ps.report_failure(), clear_elim);
|
|
if (auto res = tac.execute(g, n, implementation_list())) {
|
|
proof_state new_s(ps, append(res->m_goals, tail_gs), res->m_subst, res->m_ngen);
|
|
return some_proof_state(new_s);
|
|
} else {
|
|
return none_proof_state();
|
|
}
|
|
};
|
|
return tactic01(fn);
|
|
}
|
|
|
|
tactic inversion_tactic(elaborate_fn const & elab, expr const & H, list<name> const & ids) {
|
|
auto fn = [=](environment const & env, io_state const & ios, proof_state const & ps) {
|
|
name Haux{"H", "cases"};
|
|
auto new_ps = generalize_core(env, ios, elab, H, Haux, ps, "cases", true);
|
|
if (!new_ps)
|
|
return proof_state_seq();
|
|
goal g = head(new_ps->get_goals());
|
|
expr new_H = app_arg(g.get_meta());
|
|
lean_assert(is_local(new_H));
|
|
name H_name = local_pp_name(new_H);
|
|
return inversion_tactic(H_name, ids)(env, ios, *new_ps);
|
|
};
|
|
return tactic(fn);
|
|
}
|
|
|
|
void initialize_inversion_tactic() {
|
|
register_tac(get_tactic_cases_name(),
|
|
[](type_checker &, elaborate_fn const & elab, expr const & e, pos_info_provider const *) {
|
|
check_tactic_expr(app_arg(app_fn(e)), "invalid 'cases' tactic, argument must be an expression");
|
|
expr H = get_tactic_expr_expr(app_arg(app_fn(e)));
|
|
buffer<name> ids;
|
|
get_tactic_id_list_elements(app_arg(e), ids, "invalid 'cases' tactic, list of identifiers expected");
|
|
if (is_local(H))
|
|
return inversion_tactic(local_pp_name(H), to_list(ids.begin(), ids.end()));
|
|
else
|
|
return inversion_tactic(elab, H, to_list(ids.begin(), ids.end()));
|
|
});
|
|
}
|
|
void finalize_inversion_tactic() {}
|
|
}
|