lean2/src/library/tactic/inversion_tactic.cpp
Leonardo de Moura 1f13bfa4f7 feat(library/tactic/inversion_tactic): add inversion::apply procedure
The new procedure is essentially a "customized" version of the
inversion (aka cases) tactic for the equations package
2015-01-01 19:33:14 -08:00

758 lines
34 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 <utility>
#include "util/sstream.h"
#include "kernel/abstract.h"
#include "kernel/instantiate.h"
#include "kernel/inductive/inductive.h"
#include "library/io_state_stream.h"
#include "library/locals.h"
#include "library/util.h"
#include "library/reducible.h"
#include "library/tactic/tactic.h"
#include "library/tactic/expr_to_tactic.h"
#include "library/tactic/class_instance_synth.h"
#include "library/tactic/inversion_tactic.h"
namespace lean {
namespace inversion {
result::result(list<goal> const & gs, list<unsigned> const & num_args, list<implementation_list> const & imps,
name_generator const & ngen, substitution const & subst):
m_goals(gs), m_num_args(num_args), m_implementation_lists(imps),
m_ngen(ngen), m_subst(subst) {
lean_assert_eq(length(m_goals), length(m_num_args));
lean_assert_eq(length(m_goals), length(m_implementation_lists));
}
}
/**
\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>,
apply the eq_rec_eq definition to produce the equality
b = @eq.rec.{l₂ l₁} A a (λ (a' : A) (h : a = a'), B a') b a p
The eq_rec_eq definition is of the form
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) :
b = @eq.rec.{l₂ l₁} A a (λ (a' : A) (h : a = a'), B a') b a p :=
...
*/
optional<expr> apply_eq_rec_eq(type_checker & tc, io_state const & ios, list<expr> const & ctx, expr const & eq_rec) {
buffer<expr> args;
expr const & eq_rec_fn = get_app_args(eq_rec, args);
if (args.size() != 6)
return none_expr();
expr const & p = args[5];
if (!is_local(p) || !is_eq_a_a(tc, mlocal_type(p)))
return none_expr();
expr const & A = args[0];
auto is_hset_A = mk_hset_instance(tc, ios, ctx, A);
if (!is_hset_A)
return none_expr();
levels ls = const_levels(eq_rec_fn);
level l2 = head(ls);
level l1 = head(tail(ls));
expr C = tc.whnf(args[2]).first;
if (!is_lambda(C))
return none_expr();
expr a1 = mk_local(tc.mk_fresh_name(), binding_domain(C));
C = tc.whnf(instantiate(binding_body(C), a1)).first;
if (!is_lambda(C))
return none_expr();
C = binding_body(C);
if (!closed(C))
return none_expr();
expr B = Fun(a1, C);
expr a = args[1];
expr b = args[3];
expr r = mk_constant("eq_rec_eq", {l1, l2});
return some_expr(mk_app({r, A, B, *is_hset_A, a, b, p}));
}
typedef inversion::implementation_ptr implementation_ptr;
typedef inversion::implementation_list implementation_list;
static void abstract_locals(implementation_list const & imps, unsigned nlocals, expr const * locals) {
for (implementation_ptr const & imp : imps) {
imp->update_exprs([&](expr const & e) { return abstract_locals(e, nlocals, locals); });
}
}
static void instantiate(implementation_list const & imps, expr const & local) {
for (implementation_ptr const & imp : imps) {
imp->update_exprs([&](expr const & e) { return instantiate(e, local); });
}
}
static void abstract_locals(implementation_list const & imps, buffer<expr> const & locals) {
abstract_locals(imps, locals.size(), locals.data());
}
static void abstract_local(implementation_list const & imps, expr const & local) {
abstract_locals(imps, 1, &local);
}
class inversion_tac {
environment const & m_env;
io_state const & m_ios;
type_checker & m_tc;
list<name> m_ids;
name_generator m_ngen;
substitution m_subst;
bool m_dep_elim;
bool m_proof_irrel;
unsigned m_nparams;
unsigned m_nindices;
unsigned m_nminors;
declaration m_I_decl;
declaration m_cases_on_decl;
void init_inductive_info(name const & n) {
m_dep_elim = inductive::has_dep_elim(m_env, n);
m_nindices = *inductive::get_num_indices(m_env, n);
m_nparams = *inductive::get_num_params(m_env, n);
// This tactic is bases on cases_on construction which only has
// minor premises for the introduction rules of this datatype.
// For non-mutually recursive datatypes inductive::get_num_intro_rules == inductive::get_num_minor_premises
m_nminors = *inductive::get_num_intro_rules(m_env, n);
m_I_decl = m_env.get(n);
m_cases_on_decl = m_env.get({n, "cases_on"});
}
bool is_inversion_applicable(expr const & t) {
buffer<expr> args;
expr const & fn = get_app_args(t, args);
if (!is_constant(fn))
return false;
if (!inductive::is_inductive_decl(m_env, const_name(fn)))
return false;
if (!m_env.find(name{const_name(fn), "cases_on"}) || !m_env.find(name("eq")))
return false;
if (m_proof_irrel && !m_env.find(name("heq")))
return false;
init_inductive_info(const_name(fn));
if (args.size() != m_nindices + m_nparams)
return false;
return true;
}
pair<expr, expr> mk_eq(expr const & lhs, expr const & rhs) {
expr lhs_type = m_tc.infer(lhs).first;
expr rhs_type = m_tc.infer(rhs).first;
level l = sort_level(m_tc.ensure_type(lhs_type).first);
constraint_seq cs;
if (m_tc.is_def_eq(lhs_type, rhs_type, justification(), cs) && !cs) {
return mk_pair(mk_app(mk_constant("eq", to_list(l)), lhs_type, lhs, rhs),
mk_app(mk_constant({"eq", "refl"}, to_list(l)), rhs_type, rhs));
} else {
return mk_pair(mk_app(mk_constant("heq", to_list(l)), lhs_type, lhs, rhs_type, rhs),
mk_app(mk_constant({"heq", "refl"}, to_list(l)), rhs_type, rhs));
}
}
void assign(name const & n, expr const & val) {
m_subst.assign(n, val);
}
goal generalize_indices(goal const & g, expr const & h, expr const & h_type) {
buffer<expr> hyps;
g.get_hyps(hyps);
expr m = g.get_meta();
expr m_type = g.get_type();
name h_new_name = get_unused_name(local_pp_name(h), hyps);
buffer<expr> I_args;
expr const & I = get_app_args(h_type, I_args);
expr h_new_type = mk_app(I, I_args.size() - m_nindices, I_args.data());
expr d = m_tc.whnf(m_tc.infer(h_new_type).first).first;
name t_prefix("t");
unsigned nidx = 1;
if (m_proof_irrel) {
unsigned eq_idx = 1;
name eq_prefix("H");
buffer<expr> ts;
buffer<expr> eqs;
buffer<expr> refls;
auto add_eq = [&](expr const & lhs, expr const & rhs) {
pair<expr, expr> p = mk_eq(lhs, rhs);
expr new_eq = p.first;
expr new_refl = p.second;
eqs.push_back(mk_local(m_ngen.next(), g.get_unused_name(eq_prefix, eq_idx), new_eq, binder_info()));
refls.push_back(new_refl);
};
for (unsigned i = I_args.size() - m_nindices; i < I_args.size(); i++) {
expr t_type = binding_domain(d);
expr t = mk_local(m_ngen.next(), g.get_unused_name(t_prefix, nidx), t_type, binder_info());
expr const & index = I_args[i];
add_eq(t, index);
h_new_type = mk_app(h_new_type, t);
hyps.push_back(t);
ts.push_back(t);
d = instantiate(binding_body(d), t);
}
expr h_new = mk_local(m_ngen.next(), h_new_name, h_new_type, local_info(h));
if (m_dep_elim)
add_eq(h_new, h);
hyps.push_back(h_new);
expr new_type = Pi(eqs, 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);
expr val = g.abstract(mk_app(mk_app(mk_app(Fun(ts, Fun(h_new, new_meta)), m_nindices, I_args.end() - m_nindices), h),
refls));
assign(g.get_name(), val);
return new_g;
} else {
// proof relevant version
buffer<expr> ss;
buffer<expr> ts;
buffer<expr> refls;
for (unsigned i = I_args.size() - m_nindices; i < I_args.size(); i++) {
expr t_type = binding_domain(d);
expr t = mk_local(m_ngen.next(), g.get_unused_name(t_prefix, nidx), t_type, binder_info());
h_new_type = mk_app(h_new_type, t);
ss.push_back(I_args[i]);
refls.push_back(mk_refl(m_tc, I_args[i]));
hyps.push_back(t);
ts.push_back(t);
d = instantiate(binding_body(d), t);
}
expr h_new = mk_local(m_ngen.next(), h_new_name, h_new_type, local_info(h));
ts.push_back(h_new);
ss.push_back(h);
refls.push_back(mk_refl(m_tc, h));
hyps.push_back(h_new);
buffer<expr> eqs;
mk_telescopic_eq(m_tc, ss, ts, eqs);
ts.pop_back();
expr new_type = Pi(eqs, 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);
expr val = g.abstract(mk_app(mk_app(mk_app(Fun(ts, Fun(h_new, new_meta)), m_nindices, I_args.end() - m_nindices), h),
refls));
assign(g.get_name(), val);
return new_g;
}
}
std::pair<list<goal>, list<implementation_list>> apply_cases_on(goal const & g, implementation_list const & imps) {
buffer<expr> hyps;
g.get_hyps(hyps);
expr const & h = hyps.back();
expr const & h_type = mlocal_type(h);
buffer<expr> I_args;
expr const & I = get_app_args(h_type, I_args);
name const & I_name = const_name(I);
expr g_type = g.get_type();
expr cases_on;
if (length(m_cases_on_decl.get_univ_params()) != length(m_I_decl.get_univ_params())) {
level g_lvl = sort_level(m_tc.ensure_type(g_type).first);
cases_on = mk_constant({I_name, "cases_on"}, cons(g_lvl, const_levels(I)));
} else {
cases_on = mk_constant({I_name, "cases_on"}, const_levels(I));
}
// add params
cases_on = mk_app(cases_on, m_nparams, I_args.data());
// add type former
expr type_former = g_type;
if (m_dep_elim)
type_former = Fun(h, type_former);
type_former = Fun(m_nindices, I_args.end() - m_nindices, type_former);
cases_on = mk_app(cases_on, type_former);
// add indices
cases_on = mk_app(cases_on, m_nindices, I_args.end() - m_nindices);
// add h
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;
new_hyps.append(hyps.size() - m_nindices - 1, hyps.data());
// add a subgoal for each minor premise of cases_on
expr cases_on_type = m_tc.whnf(m_tc.infer(cases_on).first).first;
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 = m_tc.whnf(binding_body(cases_on_type)).first; // 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; }));
}
expr val = g.abstract(cases_on);
assign(g.get_name(), val);
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);
}
}
std::pair<list<goal>, list<unsigned>> 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;
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();
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());
new_hyps.push_back(new_h);
g_type = instantiate(binding_body(g_type), new_h);
}
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 = g.abstract(Fun(nargs, new_hyps.end() - nargs, new_meta));
assign(g.get_name(), val);
gs = tail(gs);
}
return mk_pair(to_list(new_gs), to_list(minors_nargs));
}
struct inversion_exception : public exception {
inversion_exception(char const * msg):exception(msg) {}
inversion_exception(sstream const & strm):exception(strm) {}
};
[[ noreturn ]] void throw_ill_formed_goal() {
throw inversion_exception("ill-formed goal");
}
[[ noreturn ]] void throw_ill_typed_goal() {
throw inversion_exception("ill-typed goal");
}
void throw_unification_eq_rec_failure() {
throw inversion_exception("unification failed to eliminate eq.rec in homogeneous equality");
}
// 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)
throw_unification_eq_rec_failure();
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 = m_tc.infer(lhs).first;
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(name{"eq", "trans"}, {lvl}), A, reduced_lhs, lhs, rhs, *aux_eq, old_eq});
// trans_eq : a = b
expr val = g.abstract(Fun(old_eq, mk_app(new_meta, trans_eq)));
assign(g.get_name(), val);
return intro_next_eq(new_g);
}
// 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)) == "heq");
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("eq", 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({"heq", "to_eq"}, const_levels(heq_fn)), args[0], args[1], args[3], H);
expr val = g.abstract(Fun(H, mk_app(mk_app(new_mvar, hyps), to_eq)));
assign(g.get_name(), val);
return new_g;
} else {
throw inversion_exception("unification failed to reduce heterogeneous equality into homogeneous one");
}
}
// 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)) == "eq");
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 = g.abstract(Fun(new_hyp, new_meta));
assign(g.get_name(), 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) == "eq") {
expr const & lhs = app_arg(app_fn(eq));
expr const & rhs = app_arg(eq);
expr new_lhs = m_tc.whnf(lhs).first;
expr new_rhs = m_tc.whnf(rhs).first;
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) == "heq") {
return intro_next_heq(g);
} else {
throw_ill_formed_goal();
}
}
// Split hyps into two "telescopes".
// - non_deps : hypotheses that do not depend on rhs
// - deps : hypotheses that depend on rhs (directly or indirectly)
void split_deps(buffer<expr> const & hyps, expr const & rhs, buffer<expr> & non_deps, buffer<expr> & deps) {
for (expr const & hyp : hyps) {
expr const & hyp_type = mlocal_type(hyp);
if (depends_on(hyp_type, rhs) || std::any_of(deps.begin(), deps.end(), [&](expr const & dep) { return depends_on(hyp_type, dep); })) {
deps.push_back(hyp);
} else {
non_deps.push_back(hyp);
}
}
}
// 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 (!m_proof_irrel) {
expr v_type = m_tc.whnf(m_tc.infer(v).first).first;
if (!is_app(v_type))
throw_unification_eq_rec_failure();
expr const & lift = app_fn(v_type);
if (!is_constant(lift) || const_name(lift) != "lift")
throw_unification_eq_rec_failure();
return mk_app(mk_constant(name{"lift", "down"}, const_levels(lift)), app_arg(v_type), v);
} else {
return v;
}
}
typedef optional<std::pair<goal, implementation_list>> unify_result;
unify_result unify_eqs(goal g, implementation_list imps, unsigned neqs) {
if (neqs == 0)
return unify_result(g, imps); // done
g = intro_next_eq(g);
buffer<expr> hyps;
g.get_hyps(hyps);
lean_assert(!hyps.empty());
expr eq = hyps.back();
buffer<expr> eq_args;
get_app_args(mlocal_type(eq), eq_args);
expr const & A = m_tc.whnf(eq_args[0]).first;
expr lhs = m_tc.whnf(eq_args[1]).first;
expr rhs = m_tc.whnf(eq_args[2]).first;
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);
expr val = g.abstract(new_meta);
assign(g.get_name(), val);
return unify_eqs(new_g, imps, 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))
throw inversion_exception(sstream() << "construction '" << no_confusion_name << "' is not available in the environment");
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, eq);
if (const_name(lhs_fn) == const_name(rhs_fn)) {
// injectivity transition
expr new_type = binding_domain(m_tc.whnf(m_tc.infer(no_confusion).first).first);
if (m_proof_irrel)
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 = g.abstract(lift_down(mk_app(no_confusion, new_meta)));
assign(g.get_name(), 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, imps, neqs - 1 + lhs_args.size() - A_nparams);
} else {
// conflict transition, eq is of the form c_1 ... = c_2 ..., where c_1 and c_2 are different constructors/intro rules.
expr val = g.abstract(lift_down(no_confusion));
assign(g.get_name(), val);
return unify_result(); // goal has been solved
}
}
if (is_local(rhs)) {
// solution transition, eq 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;
split_deps(hyps, rhs, non_deps, deps);
expr deps_g_type = Pi(deps, g_type);
abstract_locals(imps, deps);
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
tformer = Fun(rhs, Fun(eq, deps_g_type));
expr eq_rec = mk_constant(name{"eq", "rec"}, {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);
abstract_local(imps, rhs);
instantiate(imps, lhs);
if (!m_proof_irrel) {
new_type = abstract_local(new_type, eq);
new_type = instantiate(new_type, mk_refl(m_tc, lhs));
}
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_type = instantiate(binding_body(new_type), new_hyp);
instantiate(imps, new_hyp);
}
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, eq);
expr val = g.abstract(mk_app(eq_rec, deps));
assign(g.get_name(), val);
return unify_eqs(new_g, imps, neqs-1);
} else if (is_local(lhs)) {
// flip equation and reduce to previous case
if (m_proof_irrel)
hyps.pop_back(); // remove processed equality
expr symm_eq = mk_eq(rhs, lhs).first;
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);
level eq_symm_lvl = sort_level(m_tc.ensure_type(A).first);
expr symm_pr = mk_constant(name{"eq", "symm"}, {eq_symm_lvl});
symm_pr = mk_app(symm_pr, A, lhs, rhs, eq);
expr val = g.abstract(mk_app(new_meta, symm_pr));
assign(g.get_name(), val);
return unify_eqs(new_g, imps, neqs);
}
throw inversion_exception("unification failed");
}
auto unify_eqs(list<goal> const & gs, list<unsigned> nargs, list<implementation_list> imps) ->
std::tuple<list<goal>, list<unsigned>, list<implementation_list>> {
lean_assert(length(gs) == length(imps));
unsigned neqs = m_nindices + (m_dep_elim ? 1 : 0);
buffer<goal> new_goals;
buffer<unsigned> new_nargs;
buffer<implementation_list> new_imps;
for (goal const & g : gs) {
if (auto g_imp_pair = unify_eqs(g, head(imps), neqs)) {
new_goals.push_back(g_imp_pair->first);
new_nargs.push_back(head(nargs));
new_imps.push_back(g_imp_pair->second);
}
imps = tail(imps);
nargs = tail(nargs);
}
return std::make_tuple(to_list(new_goals), to_list(new_nargs), to_list(new_imps));
}
public:
inversion_tac(environment const & env, io_state const & ios, name_generator const & ngen,
type_checker & tc, substitution const & subst, list<name> const & ids):
m_env(env), m_ios(ios), m_tc(tc), m_ids(ids),
m_ngen(ngen), m_subst(subst) {
m_proof_irrel = m_env.prop_proof_irrel();
}
inversion_tac(environment const & env, io_state const & ios, type_checker & tc):
inversion_tac(env, ios, tc.mk_ngen(), tc, substitution(), list<name>()) {}
typedef inversion::result result;
optional<result> execute(goal const & g, name const & n, implementation_list const & imps) {
try {
auto p = g.find_hyp(n);
if (!p)
return optional<result>();
expr const & h = p->first;
expr h_type = m_tc.whnf(mlocal_type(h)).first;
if (!is_inversion_applicable(h_type))
return optional<result>();
goal g1 = generalize_indices(g, h, h_type);
auto gs_imps_pair = apply_cases_on(g1, imps);
list<goal> gs2 = gs_imps_pair.first;
list<implementation_list> new_imps = gs_imps_pair.second;
auto gs_nargs_pair = intros_minors_args(gs2);
list<goal> gs3 = gs_nargs_pair.first;
list<unsigned> nargs = gs_nargs_pair.second;
list<goal> gs4;
std::tie(gs4, nargs, new_imps) = unify_eqs(gs3, nargs, new_imps);
return optional<result>(result(gs4, nargs, new_imps, m_ngen, m_subst));
} catch (inversion_exception & ex) {
return optional<result>();
}
}
};
namespace inversion {
optional<result> apply(environment const & env, io_state const & ios, type_checker & tc,
goal const & g, name const & n, implementation_list const & imps) {
return inversion_tac(env, ios, tc).execute(g, n, 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(), ps.relax_main_opaque());
inversion_tac tac(env, ios, ngen, *tc, ps.get_subst(), ids);
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);
}
void initialize_inversion_tactic() {
register_tac(name({"tactic", "inversion"}),
[](type_checker &, elaborate_fn const &, expr const & e, pos_info_provider const *) {
name n = tactic_expr_to_id(app_arg(e), "invalid 'inversion/cases' tactic, argument must be an identifier");
return inversion_tactic(n, list<name>());
});
register_tac(name({"tactic", "inversion_with"}),
[](type_checker &, elaborate_fn const &, expr const & e, pos_info_provider const *) {
name n = tactic_expr_to_id(app_arg(app_fn(e)), "invalid 'cases-with' tactic, argument must be an identifier");
buffer<name> ids;
get_tactic_id_list_elements(app_arg(e), ids, "invalid 'cases-with' tactic, list of identifiers expected");
return inversion_tactic(n, to_list(ids.begin(), ids.end()));
});
}
void finalize_inversion_tactic() {}
}