645 lines
27 KiB
C++
645 lines
27 KiB
C++
/*
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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*/
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#include "util/interrupt.h"
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#include "util/lbool.h"
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#include "kernel/converter.h"
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#include "kernel/expr_maps.h"
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#include "kernel/instantiate.h"
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#include "kernel/free_vars.h"
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#include "kernel/type_checker.h"
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namespace lean {
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/**
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\brief Predicate for deciding whether \c d is an opaque definition or not.
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Here is the basic idea:
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1) Each definition has an opaque flag. This flag cannot be modified after a definition is added to the environment.
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The opaque flag affects the convertability check. The idea is to minimize the number of delta-reduction steps.
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We also believe it increases the modularity of Lean developments by minimizing the dependency on how things are defined.
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We should view non-opaque definitions as "inline definitions" used in programming languages such as C++.
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2) Whenever type checking an expression, the user can provide an additional set of definition names (m_extra_opaque) that
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should be considered opaque. Note that, if \c t type checks when using an extra_opaque set S, then t also type checks
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(modulo resource constraints) with the empty set. Again, the purpose of extra_opaque is to mimimize the number
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of delta-reduction steps.
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3) To be able to prove theorems about an opaque definition, we treat an opaque definition D in a module M as
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transparent when we are type checking another definition/theorem D' also in M. This rule only applies if
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D is not a theorem, nor D is in the set m_extra_opaque. To implement this feature, this class has a field
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m_module_idx that is not none when this rule should be applied.
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*/
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bool is_opaque(declaration const & d, name_set const & extra_opaque, optional<module_idx> const & mod_idx) {
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lean_assert(d.is_definition());
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if (d.is_theorem()) return true; // theorems are always opaque
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if (extra_opaque.contains(d.get_name())) return true; // extra_opaque set overrides opaque flag
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if (!d.is_opaque()) return false; // d is a transparent definition
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if (mod_idx && d.get_module_idx() == *mod_idx) return false; // the opaque definitions in mod_idx are considered transparent
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return true; // d is opaque
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}
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/** \brief Auxiliary method for \c is_delta */
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static optional<declaration> is_delta_core(environment const & env, expr const & e, name_set const & extra_opaque,
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optional<module_idx> const & mod_idx) {
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if (is_constant(e)) {
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if (auto d = env.find(const_name(e)))
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if (d->is_definition() && !is_opaque(*d, extra_opaque, mod_idx))
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return d;
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}
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return none_declaration();
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}
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/**
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\brief Return some definition \c d iff \c e is a target for delta-reduction, and the given definition is the one
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to be expanded.
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*/
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optional<declaration> is_delta(environment const & env, expr const & e, name_set const & extra_opaque, optional<module_idx> const & mod_idx) {
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return is_delta_core(env, get_app_fn(e), extra_opaque, mod_idx);
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}
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static optional<module_idx> g_opt_main_module_idx(g_main_module_idx);
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optional<declaration> is_delta(environment const & env, expr const & e, name_set const & extra_opaque) {
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return is_delta(env, e, extra_opaque, g_opt_main_module_idx);
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}
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static no_delayed_justification g_no_delayed_jst;
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pair<bool, constraint_seq> converter::is_def_eq(expr const & t, expr const & s, type_checker & c) {
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return is_def_eq(t, s, c, g_no_delayed_jst);
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}
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/** \brief Do nothing converter */
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struct dummy_converter : public converter {
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virtual pair<expr, constraint_seq> whnf(expr const & e, type_checker &) {
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return mk_pair(e, constraint_seq());
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}
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virtual pair<bool, constraint_seq> is_def_eq(expr const &, expr const &, type_checker &, delayed_justification &) {
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return mk_pair(true, constraint_seq());
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}
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virtual optional<module_idx> get_module_idx() const { return optional<module_idx>(); }
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};
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std::unique_ptr<converter> mk_dummy_converter() {
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return std::unique_ptr<converter>(new dummy_converter());
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}
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name converter::mk_fresh_name(type_checker & tc) { return tc.mk_fresh_name(); }
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pair<expr, constraint_seq> converter::infer_type(type_checker & tc, expr const & e) { return tc.infer_type(e); }
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extension_context & converter::get_extension(type_checker & tc) { return tc.get_extension(); }
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static expr g_dont_care(Const("dontcare"));
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struct default_converter : public converter {
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environment m_env;
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optional<module_idx> m_module_idx;
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bool m_memoize;
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name_set m_extra_opaque;
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expr_struct_map<expr> m_whnf_core_cache;
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expr_struct_map<pair<expr, constraint_seq>> m_whnf_cache;
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default_converter(environment const & env, optional<module_idx> mod_idx, bool memoize, name_set const & extra_opaque):
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m_env(env), m_module_idx(mod_idx), m_memoize(memoize), m_extra_opaque(extra_opaque) {
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}
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constraint mk_eq_cnstr(expr const & lhs, expr const & rhs, justification const & j) {
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return ::lean::mk_eq_cnstr(lhs, rhs, j, static_cast<bool>(m_module_idx));
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}
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optional<expr> expand_macro(expr const & m, type_checker & c) {
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lean_assert(is_macro(m));
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return macro_def(m).expand(m, get_extension(c));
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}
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/** \brief Apply normalizer extensions to \c e. */
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optional<pair<expr, constraint_seq>> norm_ext(expr const & e, type_checker & c) {
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return m_env.norm_ext()(e, get_extension(c));
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}
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optional<expr> d_norm_ext(expr const & e, type_checker & c, constraint_seq & cs) {
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if (auto r = norm_ext(e, c)) {
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cs = cs + r->second;
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return some_expr(r->first);
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} else {
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return none_expr();
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}
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}
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/** \brief Return true if \c e may be reduced later after metavariables are instantiated. */
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bool may_reduce_later(expr const & e, type_checker & c) {
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return m_env.norm_ext().may_reduce_later(e, get_extension(c));
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}
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/** \brief Try to apply eta-reduction to \c e. */
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expr try_eta(expr const & e) {
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lean_assert(is_lambda(e));
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expr const & b = binding_body(e);
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if (is_lambda(b)) {
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expr new_b = try_eta(b);
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if (is_eqp(b, new_b)) {
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return e;
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} else if (is_app(new_b) && is_var(app_arg(new_b), 0) && !has_free_var(app_fn(new_b), 0)) {
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return lower_free_vars(app_fn(new_b), 1);
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} else {
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return update_binding(e, binding_domain(e), new_b);
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}
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} else if (is_app(b) && is_var(app_arg(b), 0) && !has_free_var(app_fn(b), 0)) {
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return lower_free_vars(app_fn(b), 1);
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} else {
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return e;
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}
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}
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/** \brief Weak head normal form core procedure. It does not perform delta reduction nor normalization extensions. */
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expr whnf_core(expr const & e, type_checker & c) {
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check_system("whnf");
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// handle easy cases
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switch (e.kind()) {
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case expr_kind::Var: case expr_kind::Sort: case expr_kind::Meta: case expr_kind::Local:
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case expr_kind::Pi: case expr_kind::Constant:
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return e;
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case expr_kind::Lambda: case expr_kind::Macro: case expr_kind::App:
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break;
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}
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// check cache
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if (m_memoize) {
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auto it = m_whnf_core_cache.find(e);
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if (it != m_whnf_core_cache.end())
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return it->second;
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}
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// do the actual work
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expr r;
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switch (e.kind()) {
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case expr_kind::Var: case expr_kind::Sort: case expr_kind::Meta: case expr_kind::Local:
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case expr_kind::Pi: case expr_kind::Constant:
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lean_unreachable(); // LCOV_EXCL_LINE
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case expr_kind::Lambda:
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r = (m_env.eta()) ? try_eta(e) : e;
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break;
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case expr_kind::Macro:
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if (auto m = expand_macro(e, c))
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r = whnf_core(*m, c);
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else
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r = e;
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break;
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case expr_kind::App: {
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buffer<expr> args;
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expr f0 = get_app_rev_args(e, args);
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expr f = whnf_core(f0, c);
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if (is_lambda(f)) {
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unsigned m = 1;
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unsigned num_args = args.size();
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while (is_lambda(binding_body(f)) && m < num_args) {
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f = binding_body(f);
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m++;
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}
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lean_assert(m <= num_args);
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r = whnf_core(mk_rev_app(instantiate(binding_body(f), m, args.data() + (num_args - m)), num_args - m, args.data()), c);
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} else {
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r = f == f0 ? e : whnf_core(mk_rev_app(f, args.size(), args.data()), c);
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}
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break;
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}}
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if (m_memoize)
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m_whnf_core_cache.insert(mk_pair(e, r));
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return r;
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}
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bool is_opaque(declaration const & d) const {
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return ::lean::is_opaque(d, m_extra_opaque, m_module_idx);
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}
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/** \brief Expand \c e if it is non-opaque constant with weight >= w */
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expr unfold_name_core(expr e, unsigned w) {
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if (is_constant(e)) {
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if (auto d = m_env.find(const_name(e))) {
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if (d->is_definition() && !is_opaque(*d) && d->get_weight() >= w)
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return unfold_name_core(instantiate_value_univ_params(*d, const_levels(e)), w);
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}
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}
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return e;
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}
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/**
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\brief Expand constants and application where the function is a constant.
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The unfolding is only performend if the constant corresponds to
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a non-opaque definition with weight >= w.
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*/
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expr unfold_names(expr const & e, unsigned w) {
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if (is_app(e)) {
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expr f0 = get_app_fn(e);
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expr f = unfold_name_core(f0, w);
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if (is_eqp(f, f0)) {
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return e;
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} else {
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buffer<expr> args;
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get_app_rev_args(e, args);
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return mk_rev_app(f, args);
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}
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} else {
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return unfold_name_core(e, w);
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}
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}
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/**
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\brief Return some definition \c d iff \c e is a target for delta-reduction, and the given definition is the one
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to be expanded.
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*/
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optional<declaration> is_delta(expr const & e) {
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return ::lean::is_delta(m_env, get_app_fn(e), m_extra_opaque, m_module_idx);
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}
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/**
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\brief Weak head normal form core procedure that perform delta reduction for non-opaque constants with
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weight greater than or equal to \c w.
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This method is based on <tt>whnf_core(expr const &)</tt> and \c unfold_names.
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\remark This method does not use normalization extensions attached in the environment.
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*/
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expr whnf_core(expr e, unsigned w, type_checker & c) {
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while (true) {
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expr new_e = unfold_names(whnf_core(e, c), w);
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if (is_eqp(e, new_e))
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return e;
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e = new_e;
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}
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}
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/** \brief Put expression \c t in weak head normal form */
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virtual pair<expr, constraint_seq> whnf(expr const & e_prime, type_checker & c) {
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// Do not cache easy cases
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switch (e_prime.kind()) {
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case expr_kind::Var: case expr_kind::Sort: case expr_kind::Meta: case expr_kind::Local: case expr_kind::Pi:
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return to_ecs(e_prime);
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case expr_kind::Lambda: case expr_kind::Macro: case expr_kind::App: case expr_kind::Constant:
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break;
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}
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expr e = e_prime;
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// check cache
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if (m_memoize) {
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auto it = m_whnf_cache.find(e);
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if (it != m_whnf_cache.end())
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return it->second;
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}
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expr t = e;
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constraint_seq cs;
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while (true) {
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expr t1 = whnf_core(t, 0, c);
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if (auto new_t = d_norm_ext(t1, c, cs)) {
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t = *new_t;
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} else {
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auto r = mk_pair(t1, cs);
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if (m_memoize)
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m_whnf_cache.insert(mk_pair(e, r));
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return r;
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}
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}
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}
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expr whnf(expr const & e_prime, type_checker & c, constraint_seq & cs) {
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auto r = whnf(e_prime, c);
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cs = cs + r.second;
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return r.first;
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}
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pair<bool, constraint_seq> to_bcs(bool b) { return mk_pair(b, constraint_seq()); }
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pair<bool, constraint_seq> to_bcs(bool b, constraint const & c) { return mk_pair(b, constraint_seq(c)); }
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pair<bool, constraint_seq> to_bcs(bool b, constraint_seq const & cs) { return mk_pair(b, cs); }
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/**
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\brief Given lambda/Pi expressions \c t and \c s, return true iff \c t is def eq to \c s.
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t and s are definitionally equal
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iff
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domain(t) is definitionally equal to domain(s)
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and
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body(t) is definitionally equal to body(s)
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*/
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bool is_def_eq_binding(expr t, expr s, type_checker & c, delayed_justification & jst, constraint_seq & cs) {
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lean_assert(t.kind() == s.kind());
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lean_assert(is_binding(t));
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expr_kind k = t.kind();
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buffer<expr> subst;
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do {
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optional<expr> var_s_type;
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if (binding_domain(t) != binding_domain(s)) {
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var_s_type = instantiate_rev(binding_domain(s), subst.size(), subst.data());
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expr var_t_type = instantiate_rev(binding_domain(t), subst.size(), subst.data());
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if (!is_def_eq(var_t_type, *var_s_type, c, jst, cs))
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return false;
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}
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if (!closed(binding_body(t)) || !closed(binding_body(s))) {
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// local is used inside t or s
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if (!var_s_type)
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var_s_type = instantiate_rev(binding_domain(s), subst.size(), subst.data());
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subst.push_back(mk_local(mk_fresh_name(c), binding_name(s), *var_s_type, binding_info(s)));
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} else {
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subst.push_back(g_dont_care); // don't care
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}
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t = binding_body(t);
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s = binding_body(s);
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} while (t.kind() == k && s.kind() == k);
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return is_def_eq(instantiate_rev(t, subst.size(), subst.data()),
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instantiate_rev(s, subst.size(), subst.data()), c, jst, cs);
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}
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bool is_def_eq(level const & l1, level const & l2, delayed_justification & jst, constraint_seq & cs) {
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if (is_equivalent(l1, l2)) {
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return true;
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} else if (has_meta(l1) || has_meta(l2)) {
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cs = cs + constraint_seq(mk_level_eq_cnstr(l1, l2, jst.get()));
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return true;
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} else {
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return false;
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}
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}
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bool is_def_eq(levels const & ls1, levels const & ls2, type_checker & c, delayed_justification & jst, constraint_seq & cs) {
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if (is_nil(ls1) && is_nil(ls2)) {
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return true;
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} else if (!is_nil(ls1) && !is_nil(ls2)) {
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return
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is_def_eq(head(ls1), head(ls2), jst, cs) &&
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is_def_eq(tail(ls1), tail(ls2), c, jst, cs);
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} else {
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return false;
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}
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}
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static pair<lbool, constraint_seq> to_lbcs(lbool l) { return mk_pair(l, constraint_seq()); }
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static pair<lbool, constraint_seq> to_lbcs(lbool l, constraint const & c) { return mk_pair(l, constraint_seq(c)); }
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static pair<lbool, constraint_seq> to_lbcs(pair<bool, constraint_seq> const & bcs) {
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return mk_pair(to_lbool(bcs.first), bcs.second);
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}
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/** \brief This is an auxiliary method for is_def_eq. It handles the "easy cases". */
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lbool quick_is_def_eq(expr const & t, expr const & s, type_checker & c, delayed_justification & jst, constraint_seq & cs) {
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if (t == s)
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return l_true; // t and s are structurally equal
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if (is_meta(t) || is_meta(s)) {
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// if t or s is a metavariable (or the application of a metavariable), then add constraint
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cs = cs + constraint_seq(mk_eq_cnstr(t, s, jst.get()));
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return l_true;
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}
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if (t.kind() == s.kind()) {
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switch (t.kind()) {
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case expr_kind::Lambda: case expr_kind::Pi:
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return to_lbool(is_def_eq_binding(t, s, c, jst, cs));
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case expr_kind::Sort:
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return to_lbool(is_def_eq(sort_level(t), sort_level(s), c, jst, cs));
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case expr_kind::Meta:
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lean_unreachable(); // LCOV_EXCL_LINE
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case expr_kind::Var: case expr_kind::Local: case expr_kind::App:
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case expr_kind::Constant: case expr_kind::Macro:
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// We do not handle these cases in this method.
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break;
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}
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}
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return l_undef; // This is not an "easy case"
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}
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/**
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\brief Return true if arguments of \c t are definitionally equal to arguments of \c s.
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This method is used to implement an optimization in the method \c is_def_eq.
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*/
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bool is_def_eq_args(expr t, expr s, type_checker & c, delayed_justification & jst, constraint_seq & cs) {
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while (is_app(t) && is_app(s)) {
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if (!is_def_eq(app_arg(t), app_arg(s), c, jst, cs))
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return false;
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t = app_fn(t);
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s = app_fn(s);
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}
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return !is_app(t) && !is_app(s);
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}
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/** \brief Return true iff t is a constant named f_name or an application of the form (f_name a_1 ... a_k) */
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bool is_app_of(expr t, name const & f_name) {
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t = get_app_fn(t);
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return is_constant(t) && const_name(t) == f_name;
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}
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/** \brief Try to solve (fun (x : A), B) =?= s by trying eta-expansion on s */
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bool try_eta_expansion(expr const & t, expr const & s, type_checker & c, delayed_justification & jst, constraint_seq & cs) {
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if (is_lambda(t) && !is_lambda(s)) {
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auto tcs = infer_type(c, s);
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auto wcs = whnf(tcs.first, c);
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expr s_type = wcs.first;
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if (!is_pi(s_type))
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return false;
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expr new_s = mk_lambda(binding_name(s_type), binding_domain(s_type), mk_app(s, Var(0)), binding_info(s_type));
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auto dcs = is_def_eq(t, new_s, c, jst);
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if (!dcs.first)
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return false;
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cs = cs + dcs.second + wcs.second + tcs.second;
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return true;
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} else {
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return false;
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}
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}
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bool is_def_eq(expr const & t, expr const & s, type_checker & c, delayed_justification & jst, constraint_seq & cs) {
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auto bcs = is_def_eq(t, s, c, jst);
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if (bcs.first) {
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cs = cs + bcs.second;
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return true;
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} else {
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return false;
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}
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}
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/** Return true iff t is definitionally equal to s. */
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virtual pair<bool, constraint_seq> is_def_eq(expr const & t, expr const & s, type_checker & c, delayed_justification & jst) {
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check_system("is_definitionally_equal");
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constraint_seq cs;
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lbool r = quick_is_def_eq(t, s, c, jst, cs);
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if (r != l_undef) return to_bcs(r == l_true, cs);
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// apply whnf (without using delta-reduction or normalizer extensions)
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expr t_n = whnf_core(t, c);
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expr s_n = whnf_core(s, c);
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if (!is_eqp(t_n, t) || !is_eqp(s_n, s)) {
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r = quick_is_def_eq(t_n, s_n, c, jst, cs);
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if (r != l_undef) return to_bcs(r == l_true, cs);
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}
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// lazy delta-reduction and then normalizer extensions
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while (true) {
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// first, keep applying lazy delta-reduction while applicable
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while (true) {
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auto d_t = is_delta(t_n);
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auto d_s = is_delta(s_n);
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if (!d_t && !d_s) {
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break;
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} else if (d_t && !d_s) {
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t_n = whnf_core(unfold_names(t_n, 0), c);
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} else if (!d_t && d_s) {
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s_n = whnf_core(unfold_names(s_n, 0), c);
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} else if (d_t->get_weight() > d_s->get_weight()) {
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t_n = whnf_core(unfold_names(t_n, d_s->get_weight() + 1), c);
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} else if (d_t->get_weight() < d_s->get_weight()) {
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s_n = whnf_core(unfold_names(s_n, d_t->get_weight() + 1), c);
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} else {
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lean_assert(d_t && d_s && d_t->get_weight() == d_s->get_weight());
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if (is_app(t_n) && is_app(s_n) && is_eqp(*d_t, *d_s)) {
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// If t_n and s_n are both applications of the same (non-opaque) definition,
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if (has_expr_metavar(t_n) || has_expr_metavar(s_n)) {
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// We let the unifier deal with cases such as
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// (f ...) =?= (f ...)
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// when t_n or s_n contains metavariables
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break;
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} else {
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// Optimization:
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// We try to check if their arguments are definitionally equal.
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// If they are, then t_n and s_n must be definitionally equal, and we can
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// skip the delta-reduction step.
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// If the flag use_conv_opt() is not true, then we skip this optimization
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if (!is_opaque(*d_t) && d_t->use_conv_opt() &&
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is_def_eq_args(t_n, s_n, c, jst, cs))
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return to_bcs(true, cs);
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}
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}
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t_n = whnf_core(unfold_names(t_n, d_t->get_weight() - 1), c);
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s_n = whnf_core(unfold_names(s_n, d_s->get_weight() - 1), c);
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}
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r = quick_is_def_eq(t_n, s_n, c, jst, cs);
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if (r != l_undef) return to_bcs(r == l_true, cs);
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}
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// try normalizer extensions
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auto new_t_n = d_norm_ext(t_n, c, cs);
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auto new_s_n = d_norm_ext(s_n, c, cs);
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if (!new_t_n && !new_s_n)
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break; // t_n and s_n are in weak head normal form
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if (new_t_n)
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t_n = whnf_core(*new_t_n, c);
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if (new_s_n)
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s_n = whnf_core(*new_s_n, c);
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r = quick_is_def_eq(t_n, s_n, c, jst, cs);
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if (r != l_undef) return to_bcs(r == l_true, cs);
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}
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if (is_constant(t_n) && is_constant(s_n) && const_name(t_n) == const_name(s_n) &&
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is_def_eq(const_levels(t_n), const_levels(s_n), c, jst, cs))
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return to_bcs(true, cs);
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if (is_local(t_n) && is_local(s_n) && mlocal_name(t_n) == mlocal_name(s_n) &&
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is_def_eq(mlocal_type(t_n), mlocal_type(s_n), c, jst, cs))
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return to_bcs(true, cs);
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optional<declaration> d_t, d_s;
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bool delay_check = false;
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if (has_expr_metavar(t_n) || has_expr_metavar(s_n)) {
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d_t = is_delta(t_n);
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d_s = is_delta(s_n);
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if (d_t && d_s && is_eqp(*d_t, *d_s))
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delay_check = true;
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else if (may_reduce_later(t_n, c) && may_reduce_later(s_n, c))
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delay_check = true;
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}
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// At this point, t_n and s_n are in weak head normal form (modulo meta-variables and proof irrelevance)
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if (!delay_check && is_app(t_n) && is_app(s_n)) {
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buffer<expr> t_args;
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buffer<expr> s_args;
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expr t_fn = get_app_args(t_n, t_args);
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expr s_fn = get_app_args(s_n, s_args);
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constraint_seq cs_prime = cs;
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if (is_def_eq(t_fn, s_fn, c, jst, cs_prime) && t_args.size() == s_args.size()) {
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unsigned i = 0;
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for (; i < t_args.size(); i++) {
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if (!is_def_eq(t_args[i], s_args[i], c, jst, cs_prime))
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break;
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}
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if (i == t_args.size()) {
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return to_bcs(true, cs_prime);
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}
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}
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}
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if (try_eta_expansion(t_n, s_n, c, jst, cs) ||
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try_eta_expansion(s_n, t_n, c, jst, cs))
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return to_bcs(true, cs);
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if (m_env.prop_proof_irrel()) {
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// Proof irrelevance support for Prop (aka Type.{0})
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auto tcs = infer_type(c, t);
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auto scs = infer_type(c, s);
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expr t_type = tcs.first;
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expr s_type = scs.first;
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// remark: is_prop returns true only if t_type reducible to Prop.
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// If t_type contains metavariables, then reduction can get stuck, and is_prop will return false.
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auto pcs = is_prop(t_type, c);
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if (pcs.first) {
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auto dcs = is_def_eq(t_type, s_type, c, jst);
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if (dcs.first)
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return to_bcs(true, dcs.second + scs.second + pcs.second + tcs.second);
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} else {
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// If we can't stablish whether t_type is Prop, we try s_type.
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pcs = is_prop(s_type, c);
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if (pcs.first) {
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auto dcs = is_def_eq(t_type, s_type, c, jst);
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if (dcs.first)
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return to_bcs(true, dcs.second + scs.second + pcs.second + tcs.second);
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}
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// This procedure will miss the case where s_type and t_type cannot be reduced to Prop
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// because they contain metavariables.
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}
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}
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list<name> const & cls_proof_irrel = m_env.cls_proof_irrel();
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if (!is_nil(cls_proof_irrel)) {
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// Proof irrelevance support for classes
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auto tcs = infer_type(c, t);
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auto wcs = whnf(tcs.first, c);
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expr t_type = wcs.first;
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if (std::any_of(cls_proof_irrel.begin(), cls_proof_irrel.end(), [&](name const & cls_name) { return is_app_of(t_type, cls_name); })) {
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auto ccs = infer_type(c, s);
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auto cs_prime = tcs.second + wcs.second + ccs.second;
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if (is_def_eq(t_type, ccs.first, c, jst, cs_prime))
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return to_bcs(true, cs_prime);
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}
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}
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if (may_reduce_later(t_n, c) || may_reduce_later(s_n, c) || delay_check) {
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cs = cs + constraint_seq(mk_eq_cnstr(t_n, s_n, jst.get()));
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return to_bcs(true, cs);
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}
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return to_bcs(false);
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}
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pair<bool, constraint_seq> is_prop(expr const & e, type_checker & c) {
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auto tcs = infer_type(c, e);
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auto wcs = whnf(tcs.first, c);
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if (wcs.first == Prop)
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return to_bcs(true, wcs.second + tcs.second);
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else
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return to_bcs(false);
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}
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virtual optional<module_idx> get_module_idx() const {
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return m_module_idx;
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}
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};
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std::unique_ptr<converter> mk_default_converter(environment const & env, optional<module_idx> mod_idx,
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bool memoize, name_set const & extra_opaque) {
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return std::unique_ptr<converter>(new default_converter(env, mod_idx, memoize, extra_opaque));
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}
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std::unique_ptr<converter> mk_default_converter(environment const & env, bool unfold_opaque_main, bool memoize,
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name_set const & extra_opaque) {
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if (unfold_opaque_main)
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return mk_default_converter(env, optional<module_idx>(0), memoize, extra_opaque);
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else
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return mk_default_converter(env, optional<module_idx>(), memoize, extra_opaque);
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}
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}
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