417 lines
15 KiB
C++
417 lines
15 KiB
C++
/*
|
|
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Author: Leonardo de Moura
|
|
*/
|
|
#include "util/list_fn.h"
|
|
#include "kernel/kernel.h"
|
|
#include "kernel/free_vars.h"
|
|
#include "kernel/for_each_fn.h"
|
|
#include "kernel/type_checker.h"
|
|
#include "kernel/instantiate.h"
|
|
#include "kernel/occurs.h"
|
|
#include "library/expr_pair.h"
|
|
#include "library/kernel_bindings.h"
|
|
#include "library/equality.h"
|
|
|
|
namespace lean {
|
|
static name g_Hc("Hc"); // auxiliary name for if-then-else
|
|
|
|
bool is_ceq(ro_environment const & env, optional<ro_metavar_env> const & menv, expr e);
|
|
|
|
/**
|
|
\brief Auxiliary functional object for creating "conditional equations"
|
|
*/
|
|
class to_ceqs_fn {
|
|
ro_environment const & m_env;
|
|
optional<ro_metavar_env> const & m_menv;
|
|
unsigned m_idx;
|
|
|
|
static list<expr_pair> mk_singleton(expr const & e, expr const & H) {
|
|
return to_list(mk_pair(e, H));
|
|
}
|
|
|
|
expr lift_free_vars(expr const & e, unsigned d) {
|
|
return ::lean::lift_free_vars(e, d, m_menv);
|
|
}
|
|
|
|
name mk_aux_name() {
|
|
if (m_idx == 0) {
|
|
m_idx = 1;
|
|
return g_Hc;
|
|
} else {
|
|
name r = name(g_Hc, m_idx);
|
|
m_idx++;
|
|
return r;
|
|
}
|
|
}
|
|
|
|
list<expr_pair> apply(expr const & e, expr const & H) {
|
|
if (is_equality(e)) {
|
|
return mk_singleton(e, H);
|
|
} else if (is_neq(e)) {
|
|
expr T = arg(e, 1);
|
|
expr lhs = arg(e, 2);
|
|
expr rhs = arg(e, 3);
|
|
expr new_e = mk_iff(mk_eq(T, lhs, rhs), False);
|
|
expr new_H = mk_neq_elim_th(T, lhs, rhs, H);
|
|
return mk_singleton(new_e, new_H);
|
|
} else if (is_not(e)) {
|
|
expr a = arg(e, 1);
|
|
if (is_not(a)) {
|
|
return apply(arg(a, 1), mk_not_not_elim_th(arg(a, 1), H));
|
|
} else if (is_neq(a)) {
|
|
return mk_singleton(update_app(a, 0, mk_eq_fn()),
|
|
mk_not_neq_elim_th(arg(a, 1), arg(a, 2), arg(a, 3), H));
|
|
} else if (is_or(a)) {
|
|
return apply(mk_and(mk_not(arg(a, 1)), mk_not(arg(a, 2))),
|
|
mk_not_or_elim_th(arg(a, 1), arg(a, 2), H));
|
|
} else {
|
|
expr new_e = mk_iff(a, False);
|
|
expr new_H = mk_eqf_intro_th(a, H);
|
|
return mk_singleton(new_e, new_H);
|
|
}
|
|
} else if (is_and(e)) {
|
|
expr a1 = arg(e, 1);
|
|
expr a2 = arg(e, 2);
|
|
expr new_H1 = mk_and_eliml_th(a1, a2, H);
|
|
expr new_H2 = mk_and_elimr_th(a1, a2, H);
|
|
return append(apply(a1, new_H1), apply(a2, new_H2));
|
|
} else if (is_pi(e)) {
|
|
expr new_e = abst_body(e);
|
|
expr new_H = mk_app(lift_free_vars(H, 1), mk_var(0));
|
|
auto ceqs = apply(new_e, new_H);
|
|
if (length(ceqs) == 1 && new_e == car(ceqs).first) {
|
|
return mk_singleton(e, H);
|
|
} else {
|
|
return map(ceqs, [&](expr_pair const & e_H) -> expr_pair {
|
|
expr new_e = mk_pi(abst_name(e), abst_domain(e), e_H.first);
|
|
expr new_H = mk_lambda(abst_name(e), abst_domain(e), e_H.second);
|
|
return mk_pair(new_e, new_H);
|
|
});
|
|
}
|
|
} else if (is_ite(e)) {
|
|
expr c = arg(e, 2);
|
|
expr not_c = mk_not(c);
|
|
expr c1 = lift_free_vars(c, 1);
|
|
expr a1 = lift_free_vars(arg(e, 3), 1);
|
|
expr b1 = lift_free_vars(arg(e, 4), 1);
|
|
expr H1 = lift_free_vars(H, 1);
|
|
auto then_ceqs = apply(a1, mk_if_imp_then_th(c1, a1, b1, H1, mk_var(0)));
|
|
auto else_ceqs = apply(b1, mk_if_imp_else_th(c1, a1, b1, H1, mk_var(0)));
|
|
name Hc = mk_aux_name();
|
|
auto new_then_ceqs = map(then_ceqs, [&](expr_pair const & e_H) {
|
|
expr new_e = mk_pi(Hc, c, e_H.first);
|
|
expr new_H = mk_lambda(Hc, c, e_H.second);
|
|
return mk_pair(new_e, new_H);
|
|
});
|
|
auto new_else_ceqs = map(else_ceqs, [&](expr_pair const & e_H) {
|
|
expr new_e = mk_pi(Hc, not_c, e_H.first);
|
|
expr new_H = mk_lambda(Hc, not_c, e_H.second);
|
|
return mk_pair(new_e, new_H);
|
|
});
|
|
return append(new_then_ceqs, new_else_ceqs);
|
|
} else {
|
|
return mk_singleton(mk_iff(e, True), mk_eqt_intro_th(e, H));
|
|
}
|
|
}
|
|
public:
|
|
to_ceqs_fn(ro_environment const & env, optional<ro_metavar_env> const & menv):m_env(env), m_menv(menv), m_idx(0) {}
|
|
|
|
list<expr_pair> operator()(expr const & e, expr const & H) {
|
|
return filter(apply(e, H), [&](expr_pair const & p) { return is_ceq(m_env, m_menv, p.first); });
|
|
}
|
|
};
|
|
|
|
list<expr_pair> to_ceqs(ro_environment const & env, optional<ro_metavar_env> const & menv, expr const & e, expr const & H) {
|
|
return to_ceqs_fn(env, menv)(e, H);
|
|
}
|
|
|
|
static name g_unique = name::mk_internal_unique_name();
|
|
|
|
bool is_ceq(ro_environment const & env, optional<ro_metavar_env> const & menv, expr e) {
|
|
buffer<bool> found_args;
|
|
// Define a procedure for marking arguments found.
|
|
auto visitor_fn = [&](expr const & e, unsigned offset) {
|
|
if (is_var(e)) {
|
|
unsigned vidx = var_idx(e);
|
|
if (vidx >= offset) {
|
|
vidx -= offset;
|
|
if (vidx >= found_args.size()) {
|
|
// it is a free variable
|
|
} else {
|
|
found_args[found_args.size() - vidx - 1] = true;
|
|
}
|
|
}
|
|
}
|
|
return true;
|
|
};
|
|
type_checker tc(env);
|
|
context ctx;
|
|
buffer<expr> hypothesis; // arguments that are propositions
|
|
expr e_prime = e; // in e_prime we replace local variables with fresh constants
|
|
unsigned next_idx = 0;
|
|
while (is_pi(e)) {
|
|
if (!found_args.empty()) {
|
|
// Support for dependent types.
|
|
// We may find the instantiation for the previous variables
|
|
// by matching the type.
|
|
for_each(abst_domain(e), visitor_fn);
|
|
}
|
|
// If a variable is a proposition, than if doesn't need to occurr in the lhs.
|
|
// So, we mark it as true.
|
|
if (tc.is_proposition(abst_domain(e), ctx, menv)) {
|
|
found_args.push_back(true);
|
|
hypothesis.push_back(abst_domain(e_prime));
|
|
} else {
|
|
found_args.push_back(false);
|
|
}
|
|
ctx = extend(ctx, abst_name(e), abst_domain(e));
|
|
next_idx++;
|
|
e_prime = instantiate(abst_body(e_prime), mk_constant(name(g_unique, next_idx)), menv);
|
|
e = abst_body(e);
|
|
}
|
|
expr lhs, rhs;
|
|
if (is_equality(e, lhs, rhs)) {
|
|
// traverse lhs, and mark all variables that occur there in is_lhs.
|
|
for_each(lhs, visitor_fn);
|
|
if (std::find(found_args.begin(), found_args.end(), false) != found_args.end())
|
|
return false;
|
|
// basic looping ceq detection: the left-hand-side should not occur in the right-hand-side,
|
|
// nor it should occur in any of the hypothesis
|
|
expr lhs_prime, rhs_prime;
|
|
lean_verify(is_equality(e_prime, lhs_prime, rhs_prime));
|
|
return
|
|
!occurs(lhs_prime, rhs_prime) &&
|
|
std::all_of(hypothesis.begin(), hypothesis.end(), [&](expr const & h) { return !occurs(lhs_prime, h); });
|
|
} else {
|
|
return false;
|
|
}
|
|
}
|
|
|
|
static bool is_permutation(expr const & lhs, expr const & rhs, unsigned offset, buffer<optional<unsigned>> & p) {
|
|
if (lhs.kind() != rhs.kind())
|
|
return false;
|
|
switch (lhs.kind()) {
|
|
case expr_kind::Constant: case expr_kind::Type: case expr_kind::Value: case expr_kind::MetaVar:
|
|
return lhs == rhs;
|
|
case expr_kind::Var:
|
|
if (var_idx(lhs) < offset) {
|
|
return lhs == rhs; // locally bound variable
|
|
} else if (var_idx(lhs) - offset < p.size()) {
|
|
if (p[var_idx(lhs) - offset]) {
|
|
return *(p[var_idx(lhs) - offset]) == var_idx(rhs);
|
|
} else {
|
|
p[var_idx(lhs) - offset] = var_idx(rhs);
|
|
return true;
|
|
}
|
|
} else {
|
|
return lhs == rhs; // free variable
|
|
}
|
|
case expr_kind::HEq:
|
|
return
|
|
is_permutation(heq_lhs(lhs), heq_lhs(rhs), offset, p) &&
|
|
is_permutation(heq_rhs(lhs), heq_rhs(rhs), offset, p);
|
|
case expr_kind::Proj:
|
|
return proj_first(lhs) == proj_first(rhs) && is_permutation(proj_arg(lhs), proj_arg(rhs), offset, p);
|
|
case expr_kind::Pair:
|
|
return
|
|
is_permutation(pair_first(lhs), pair_first(rhs), offset, p) &&
|
|
is_permutation(pair_second(lhs), pair_second(rhs), offset, p) &&
|
|
is_permutation(pair_type(lhs), pair_type(rhs), offset, p);
|
|
case expr_kind::Lambda: case expr_kind::Pi: case expr_kind::Sigma:
|
|
return
|
|
is_permutation(abst_domain(lhs), abst_domain(rhs), offset, p) &&
|
|
is_permutation(abst_body(lhs), abst_body(rhs), offset+1, p);
|
|
case expr_kind::App:
|
|
if (num_args(lhs) == num_args(rhs)) {
|
|
for (unsigned i = 0; i < num_args(lhs); i++) {
|
|
if (!is_permutation(arg(lhs, i), arg(rhs, i), offset, p))
|
|
return false;
|
|
}
|
|
return true;
|
|
} else {
|
|
return false;
|
|
}
|
|
case expr_kind::Let:
|
|
if (static_cast<bool>(let_type(lhs)) != static_cast<bool>(let_type(rhs)))
|
|
return false;
|
|
if (static_cast<bool>(let_type(lhs)) && !is_permutation(*let_type(lhs), *let_type(rhs), offset, p))
|
|
return false;
|
|
return
|
|
is_permutation(let_value(lhs), let_value(rhs), offset, p) &&
|
|
is_permutation(let_body(lhs), let_value(rhs), offset+1, p);
|
|
}
|
|
lean_unreachable();
|
|
}
|
|
|
|
bool is_permutation_ceq(expr e) {
|
|
unsigned num_args = 0;
|
|
while (is_pi(e)) {
|
|
e = abst_body(e);
|
|
num_args++;
|
|
}
|
|
expr lhs, rhs;
|
|
if (is_equality(e, lhs, rhs)) {
|
|
buffer<optional<unsigned>> permutation;
|
|
permutation.resize(num_args);
|
|
return is_permutation(lhs, rhs, 0, permutation);
|
|
} else {
|
|
return false;
|
|
}
|
|
}
|
|
|
|
// Quick approximate test for e == (Type U).
|
|
// If the result is true, then \c e is definitionally equal to TypeU.
|
|
// If the result is false, then it may or may not be.
|
|
static bool is_TypeU(ro_environment const & env, expr const & e) {
|
|
if (is_type(e)) {
|
|
return e == TypeU;
|
|
} else if (is_constant(e)) {
|
|
auto obj = env->find_object(const_name(e));
|
|
return obj && obj->is_definition() && is_TypeU(obj->get_value());
|
|
} else {
|
|
return false;
|
|
}
|
|
}
|
|
|
|
bool is_safe_to_skip_check_ceq_types(ro_environment const & env, optional<ro_metavar_env> const & menv, expr ceq) {
|
|
lean_assert(is_ceq(env, menv, ceq));
|
|
type_checker tc(env);
|
|
buffer<expr> args;
|
|
buffer<bool> skip;
|
|
unsigned next_idx = 0;
|
|
bool to_check = false;
|
|
while (is_pi(ceq)) {
|
|
expr d = abst_domain(ceq);
|
|
expr a = mk_constant(name(g_unique, next_idx), d);
|
|
args.push_back(a);
|
|
if (tc.is_proposition(d, context(), menv) ||
|
|
is_TypeU(env, d)) {
|
|
// See comment at ceq.h
|
|
// 1- The argument has type (Type U). In Lean, (Type U) is the maximal universe.
|
|
// 2- The argument is a proposition.
|
|
skip.push_back(true);
|
|
} else {
|
|
skip.push_back(false);
|
|
to_check = true;
|
|
}
|
|
ceq = instantiate(abst_body(ceq), a);
|
|
next_idx++;
|
|
}
|
|
if (!to_check)
|
|
return true;
|
|
|
|
expr lhs, rhs;
|
|
lean_verify(is_equality(ceq, lhs, rhs));
|
|
|
|
auto arg_idx_core_fn = [&](expr const & e) -> optional<unsigned> {
|
|
if (is_constant(e)) {
|
|
name const & n = const_name(e);
|
|
if (!n.is_atomic() && n.get_prefix() == g_unique) {
|
|
return some(n.get_numeral());
|
|
}
|
|
}
|
|
return optional<unsigned>();
|
|
};
|
|
|
|
auto arg_idx_fn = [&](expr const & e) -> optional<unsigned> {
|
|
if (is_app(e))
|
|
return arg_idx_core_fn(arg(e, 0));
|
|
else if (is_lambda(e))
|
|
return arg_idx_core_fn(abst_body(e));
|
|
else
|
|
return arg_idx_core_fn(e);
|
|
};
|
|
|
|
// Return true if the application \c e has an argument or an
|
|
// application (f ...) where f is an argument.
|
|
auto has_target_fn = [&](expr const & e) -> bool {
|
|
lean_assert(is_app(e));
|
|
for (unsigned i = 1; i < num_args(e); i++) {
|
|
expr const & a = arg(e, i);
|
|
if (arg_idx_fn(a))
|
|
return true;
|
|
}
|
|
return false;
|
|
};
|
|
|
|
// 3- There is an application (f x) in the left-hand-side, and
|
|
// the type expected by f is definitionally equal to the argument type.
|
|
// 4- There is an application (f (x ...)) in the left-hand-side, and
|
|
// the type expected by f is definitionally equal to the type of (x ...)
|
|
// 5- There is an application (f (fun y, x)) in the left-hand-side,
|
|
// and the type expected by f is definitionally equal to the type of (fun y, x)
|
|
std::function<void(expr const &, context const & ctx)> visit_fn =
|
|
[&](expr const & e, context const & ctx) {
|
|
if (is_app(e)) {
|
|
expr const & f = arg(e, 0);
|
|
if (has_target_fn(e)) {
|
|
expr f_type = tc.infer_type(f, ctx, menv);
|
|
for (unsigned i = 1; i < num_args(e); i++) {
|
|
f_type = tc.ensure_pi(f_type, ctx, menv);
|
|
expr const & a = arg(e, i);
|
|
auto arg_idx = arg_idx_fn(a);
|
|
if (arg_idx && !skip[*arg_idx]) {
|
|
expr const & expected_type = abst_domain(f_type);
|
|
expr const & given_type = tc.infer_type(a, ctx, menv);
|
|
if (tc.is_definitionally_equal(given_type, expected_type)) {
|
|
skip[*arg_idx] = true;
|
|
}
|
|
}
|
|
f_type = instantiate(abst_body(f_type), a);
|
|
}
|
|
}
|
|
for (expr const & a : ::lean::args(e))
|
|
visit_fn(a, ctx);
|
|
} else if (is_abstraction(e)) {
|
|
visit_fn(abst_domain(e), ctx);
|
|
visit_fn(abst_body(e), extend(ctx, abst_name(e), abst_body(e)));
|
|
}
|
|
};
|
|
|
|
visit_fn(lhs, context());
|
|
return std::all_of(skip.begin(), skip.end(), [](bool b) { return b; });
|
|
}
|
|
|
|
static int to_ceqs(lua_State * L) {
|
|
ro_shared_environment env(L, 1);
|
|
optional<ro_metavar_env> menv;
|
|
if (!lua_isnil(L, 2))
|
|
menv = to_metavar_env(L, 2);
|
|
auto r = to_ceqs(env, menv, to_expr(L, 3), to_expr(L, 4));
|
|
lua_newtable(L);
|
|
int i = 1;
|
|
for (auto p : r) {
|
|
lua_newtable(L);
|
|
push_expr(L, p.first);
|
|
lua_rawseti(L, -2, 1);
|
|
push_expr(L, p.second);
|
|
lua_rawseti(L, -2, 2);
|
|
lua_rawseti(L, -2, i);
|
|
i = i + 1;
|
|
}
|
|
return 1;
|
|
}
|
|
|
|
static int is_ceq(lua_State * L) {
|
|
ro_shared_environment env(L, 1);
|
|
optional<ro_metavar_env> menv;
|
|
if (!lua_isnil(L, 2))
|
|
menv = to_metavar_env(L, 2);
|
|
lua_pushboolean(L, is_ceq(env, menv, to_expr(L, 3)));
|
|
return 1;
|
|
}
|
|
|
|
static int is_permutation_ceq(lua_State * L) {
|
|
lua_pushboolean(L, is_permutation_ceq(to_expr(L, 1)));
|
|
return 1;
|
|
}
|
|
|
|
void open_ceq(lua_State * L) {
|
|
SET_GLOBAL_FUN(to_ceqs, "to_ceqs");
|
|
SET_GLOBAL_FUN(is_ceq, "is_ceq");
|
|
SET_GLOBAL_FUN(is_permutation_ceq, "is_permutation_ceq");
|
|
}
|
|
}
|