feat(library/tactic): add first step of 'inversion' tactic

library/tools/tactic.lean

@@ -57,6 +57,7 @@ opaque definition revert     (e : expr)   : tactic := builtin
 opaque definition unfold     (e : expr)   : tactic := builtin
 opaque definition exact      (e : expr)   : tactic := builtin
 opaque definition trace      (s : string) : tactic := builtin
+opaque definition inversion  (e : expr)   : tactic := builtin
 
 notation a `↦` b := rename a b
 

src/library/tactic/CMakeLists.txt

@@ -1,7 +1,7 @@
 add_library(tactic goal.cpp proof_state.cpp tactic.cpp elaborate.cpp
 apply_tactic.cpp intros_tactic.cpp rename_tactic.cpp trace_tactic.cpp
 exact_tactic.cpp unfold_tactic.cpp generalize_tactic.cpp
-whnf_tactic.cpp revert_tactic.cpp clear_tactic.cpp expr_to_tactic.cpp
-util.cpp init_module.cpp)
+inversion_tactic.cpp whnf_tactic.cpp revert_tactic.cpp
+clear_tactic.cpp expr_to_tactic.cpp util.cpp init_module.cpp)
 
 target_link_libraries(tactic ${LEAN_LIBS})

src/library/tactic/init_module.cpp

@@ -17,6 +17,7 @@ Author: Leonardo de Moura
 #include "library/tactic/whnf_tactic.h"
 #include "library/tactic/clear_tactic.h"
 #include "library/tactic/revert_tactic.h"
+#include "library/tactic/inversion_tactic.h"
 
 namespace lean {
 void initialize_tactic_module() {
@@ -33,9 +34,11 @@ void initialize_tactic_module() {
     initialize_whnf_tactic();
     initialize_clear_tactic();
     initialize_revert_tactic();
+    initialize_inversion_tactic();
 }
 
 void finalize_tactic_module() {
+    finalize_inversion_tactic();
     finalize_revert_tactic();
     finalize_clear_tactic();
     finalize_whnf_tactic();

src/library/tactic/inversion_tactic.cpp

@@ -0,0 +1,141 @@
+/*
+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 "kernel/abstract.h"
+#include "kernel/instantiate.h"
+#include "kernel/inductive/inductive.h"
+#include "library/locals.h"
+#include "library/tactic/tactic.h"
+#include "library/reducible.h"
+#include "library/tactic/expr_to_tactic.h"
+
+namespace lean {
+static bool is_inversion_applicable(environment const & env, expr const & t) {
+    expr const & fn = get_app_fn(t);
+    if (!is_constant(fn))
+        return false;
+    if (!inductive::is_inductive_decl(env, const_name(fn)))
+        return false;
+    if (!env.find(name{const_name(fn), "cases_on"}) ||
+        !env.find(name("eq")) || !env.find(name("heq")))
+        return false;
+    return true;
+}
+
+static pair<expr, expr> mk_eq(type_checker & tc, expr const & lhs, expr const & rhs) {
+    expr lhs_type = tc.infer(lhs).first;
+    expr rhs_type = tc.infer(rhs).first;
+    constraint_seq cs;
+    if (tc.is_def_eq(lhs_type, rhs_type, justification(), cs) && !cs) {
+        level l = sort_level(tc.ensure_type(lhs_type).first);
+        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 {
+        level l = sort_level(tc.ensure_type(lhs_type).first);
+        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));
+    }
+}
+
+tactic generalize_indices_tactic(name const & n) {
+    auto fn = [=](environment const & env, io_state const &, proof_state const & s) -> optional<proof_state> {
+        goals const & gs  = s.get_goals();
+        if (empty(gs))
+            return none_proof_state();
+        goal  g           = head(gs);
+        goals tail_gs     = tail(gs);
+        auto p = g.find_hyp(n);
+        if (!p)
+            return none_proof_state();
+        expr const & h    = p->first;
+        name_generator ngen = s.get_ngen();
+        auto tc           = mk_type_checker(env, ngen.mk_child(), s.relax_main_opaque());
+        expr h_type       = tc->whnf(mlocal_type(h)).first;
+        if (!is_inversion_applicable(env, h_type))
+            return none_proof_state();
+        buffer<expr> hyps;
+        g.get_hyps(hyps);
+        expr m            = g.get_meta();
+        expr m_type       = g.get_type();
+        auto new_subst    = s.get_subst();
+        name h_new_name   = g.get_unused_name(local_pp_name(h));
+        buffer<expr> I_args;
+        expr const & I    = get_app_args(h_type, I_args);
+        // Set 1. generalize indices
+        unsigned nindices = *inductive::get_num_indices(env, const_name(I));
+        if (nindices > 0) {
+            expr h_new_type  = mk_app(I, I_args.size() - nindices, I_args.data());
+            expr d           = tc->whnf(tc->infer(h_new_type).first).first;
+            unsigned eq_idx  = 1;
+            name eq_prefix("H");
+            buffer<expr> ts;
+            buffer<expr> eqs;
+            buffer<expr> refls;
+            for (unsigned i = I_args.size() - nindices; i < I_args.size(); i++) {
+                expr t_type = binding_domain(d);
+                expr t      = mk_local(ngen.next(), g.get_unused_name(binding_name(d)), t_type, binder_info());
+                expr const & index = I_args[i];
+                pair<expr, expr> p = mk_eq(*tc, t, index);
+                expr new_eq   = p.first;
+                expr new_refl = p.second;
+                eqs.push_back(mk_local(ngen.next(), g.get_unused_name(eq_prefix, eq_idx), new_eq, binder_info()));
+                refls.push_back(new_refl);
+                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(ngen.next(), h_new_name, h_new_type, local_info(h));
+            hyps.push_back(h_new);
+            expr new_type = Pi(eqs, g.get_type());
+            expr new_meta = mk_app(mk_metavar(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)), nindices, I_args.end() - nindices), h), refls));
+            new_subst.assign(g.get_name(), val);
+            proof_state new_s(s, goals(new_g, tail_gs), new_subst, ngen);
+            return some_proof_state(new_s);
+        } else {
+            expr h_new    = mk_local(ngen.next(), h_new_name, h_type, local_info(h));
+            hyps.push_back(h_new);
+            expr new_meta = mk_app(mk_metavar(ngen.next(), Pi(hyps, g.get_type())), hyps);
+            goal new_g(new_meta, g.get_type());
+            expr val      = g.abstract(mk_app(new_meta, h));
+            new_subst.assign(g.get_name(), val);
+            proof_state new_s(s, goals(new_g, tail_gs), new_subst, ngen);
+            return some_proof_state(new_s);
+        }
+    };
+    return tactic01(fn);
+}
+
+tactic cases_on_tactic() {
+    // TODO(Leo)
+    return id_tactic();
+}
+
+tactic inversion_eqs_tactic() {
+    // TODO(Leo)
+    return id_tactic();
+}
+
+tactic inversion_clear_tactic() {
+    // TODO(Leo)
+    return id_tactic();
+}
+
+tactic inversion_tactic(name const & n) {
+    return generalize_indices_tactic(n) << cases_on_tactic() << inversion_eqs_tactic() << inversion_clear_tactic();
+}
+
+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' tactic, argument must be an identifier");
+                     return inversion_tactic(n);
+                 });
+}
+void finalize_inversion_tactic() {}
+}

src/library/tactic/inversion_tactic.h

@@ -0,0 +1,14 @@
+/*
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Author: Leonardo de Moura
+*/
+#pragma once
+#include "library/tactic/tactic.h"
+
+namespace lean {
+tactic inversion_tactic(name const & from, name const & to);
+void initialize_inversion_tactic();
+void finalize_inversion_tactic();
+}