feat(frontends/lean): add tactic_hint command

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

library/standard/tactic.lean

@@ -44,6 +44,8 @@ infixl ; := and_then
 notation `!` t:max     := repeat t
 -- [ t_1 | ... | t_n ] notation
 notation `[` h:100 `|` r:(foldl 100 `|` (e r, or_else r e) h) `]` := r
+-- [ t_1 || ... || t_n ] notation
+notation `[` h:100 `||` r:(foldl 100 `||` (e r, par r e) h) `]` := r
 definition try         (t : tactic) : tactic := [ t | id ]
 notation `?` t:max     := try t
 definition repeat1     (t : tactic) : tactic := t ; !t

src/emacs/lean-mode.el

@@ -24,7 +24,7 @@
 (define-generic-mode
     'lean-mode     ;; name of the mode to create
   '("--")          ;; comments start with
-  '("import" "abbreviation" "opaque_hint" "definition" "hiding" "exposing" "parameter" "parameters" "proof" "qed" "conjecture" "hypothesis" "lemma" "corollary" "variable" "variables" "print" "theorem" "axiom" "inductive" "with" "universe" "alias" "help" "environment" "options" "precedence" "postfix" "prefix" "calc_trans" "calc_subst" "calc_refl" "infix" "infixl" "infixr" "notation" "eval" "check" "exit" "coercion" "end" "using" "namespace" "builtin" "including" "class" "instance" "section" "set_option" "add_rewrite") ;; some keywords
+  '("import" "abbreviation" "opaque_hint" "tactic_hint" "definition" "hiding" "exposing" "parameter" "parameters" "proof" "qed" "conjecture" "hypothesis" "lemma" "corollary" "variable" "variables" "print" "theorem" "axiom" "inductive" "with" "universe" "alias" "help" "environment" "options" "precedence" "postfix" "prefix" "calc_trans" "calc_subst" "calc_refl" "infix" "infixl" "infixr" "notation" "eval" "check" "exit" "coercion" "end" "using" "namespace" "builtin" "including" "class" "instance" "section" "set_option" "add_rewrite") ;; some keywords
   '(("\\_<\\(Bool\\|Int\\|Nat\\|Real\\|Type\\|TypeU\\|ℕ\\|ℤ\\)\\_>" . 'font-lock-type-face)
     ("\\_<\\(calc\\|have\\|show\\|by\\|in\\|let\\|forall\\|fun\\|exists\\|if\\|then\\|else\\|assume\\|take\\|obtain\\|from\\)\\_>" . font-lock-keyword-face)
     ("\"[^\"]*\"" . 'font-lock-string-face)

src/frontends/lean/CMakeLists.txt

@@ -4,6 +4,6 @@ parser_pos_provider.cpp builtin_cmds.cpp builtin_exprs.cpp
 interactive.cpp notation_cmd.cpp calc.cpp
 decl_cmds.cpp util.cpp inductive_cmd.cpp elaborator.cpp
 dependencies.cpp parser_bindings.cpp proof_qed_ext.cpp
-class.cpp pp_options.cpp)
+class.cpp pp_options.cpp tactic_hint.cpp)
 
 target_link_libraries(lean_frontend ${LEAN_LIBS})

src/frontends/lean/builtin_cmds.cpp

@@ -23,6 +23,7 @@ Author: Leonardo de Moura
 #include "frontends/lean/proof_qed_ext.h"
 #include "frontends/lean/decl_cmds.h"
 #include "frontends/lean/class.h"
+#include "frontends/lean/tactic_hint.h"
 
 namespace lean {
 static name g_raw("raw");
@@ -306,6 +307,7 @@ cmd_table init_cmd_table() {
     register_calc_cmds(r);
     register_proof_qed_cmds(r);
     register_class_cmds(r);
+    register_tactic_hint_cmd(r);
     return r;
 }
 

src/frontends/lean/class.cpp

@@ -57,17 +57,22 @@ struct class_config {
 template class scoped_ext<class_config>;
 typedef scoped_ext<class_config> class_ext;
 
+name get_class_name(environment const & env, expr const & e) {
+    if (!is_constant(e))
+        throw exception("class expected, expression is not a constant");
+    name const & c_name = const_name(e);
+    declaration c_d = env.get(c_name);
+    if (c_d.is_definition() && !c_d.is_opaque())
+        throw exception(sstream() << "invalid class, '" << c_name << "' is a transparent definition");
+    return c_name;
+}
+
 environment add_instance(environment const & env, name const & n) {
     declaration d = env.get(n);
     expr type = d.get_type();
     while (is_pi(type))
         type = binding_body(type);
-    if (!is_constant(get_app_fn(type)))
+    name c = get_class_name(env, get_app_fn(type));
-        throw exception(sstream() << "invalid class instance declaration '" << n << "' resultant type must be a class");
-    name const & c = const_name(get_app_fn(type));
-    declaration c_d = env.get(c);
-    if (c_d.is_definition() && !c_d.is_opaque())
-        throw exception(sstream() << "invalid class instance declaration, '" << c << "' is not a valid class, it is a transparent definition");
     return class_ext::add_entry(env, get_dummy_ios(), class_entry(c, n));
 }
 

src/frontends/lean/class.h

@@ -17,6 +17,6 @@ environment add_instance(environment const & env, name const & n);
 bool is_class(environment const & env, name const & c);
 /** \brief Return the instances of the given class. */
 list<name> get_class_instances(environment const & env, name const & c);
-
+name get_class_name(environment const & env, expr const & e);
 void register_class_cmds(cmd_table & r);
 }

src/frontends/lean/elaborator.cpp

@@ -31,6 +31,7 @@ Author: Leonardo de Moura
 #include "library/error_handling/error_handling.h"
 #include "frontends/lean/local_decls.h"
 #include "frontends/lean/class.h"
+#include "frontends/lean/tactic_hint.h"
 
 #ifndef LEAN_DEFAULT_ELABORATOR_LOCAL_INSTANCES
 #define LEAN_DEFAULT_ELABORATOR_LOCAL_INSTANCES true
@@ -122,7 +123,7 @@ public:
 class elaborator {
     typedef list<expr> context;
     typedef std::vector<constraint> constraint_vect;
-    typedef name_map<expr> tactic_hints;
+    typedef name_map<expr> local_tactic_hints;
     typedef name_map<expr> mvar2meta;
     typedef std::unique_ptr<type_checker> type_checker_ptr;
 
@@ -136,7 +137,7 @@ class elaborator {
     pos_info_provider * m_pos_provider; // optional expression position information used when reporting errors.
     justification       m_accumulated; // accumulate justification of eagerly used substitutions
     constraint_vect     m_constraints; // constraints that must be solved for the elaborated term to be type correct.
-    tactic_hints        m_tactic_hints; // mapping from metavariable name ?m to tactic expression that should be used to solve it.
+    local_tactic_hints  m_local_tactic_hints; // mapping from metavariable name ?m to tactic expression that should be used to solve it.
                                               // this mapping is populated by the 'by tactic-expr' expression.
     mvar2meta           m_mvar2meta; // mapping from metavariable ?m to the (?m l_1 ... l_n) where [l_1 ... l_n] are the local constants
                                      // representing the context where ?m was created.
@@ -552,7 +553,7 @@ public:
         lean_assert(is_by(e));
         expr tac = visit(get_by_arg(e));
         expr m   = mk_meta(t, e.get_tag());
-        m_tactic_hints.insert(mlocal_name(get_app_fn(m)), tac);
+        m_local_tactic_hints.insert(mlocal_name(get_app_fn(m)), tac);
         return m;
     }
 
@@ -876,7 +877,7 @@ public:
     }
 
     optional<expr> get_pre_tactic_for(substitution & subst, expr const & mvar, name_set & visited) {
-        if (auto it = m_tactic_hints.find(mlocal_name(mvar))) {
+        if (auto it = m_local_tactic_hints.find(mlocal_name(mvar))) {
             expr pre_tac = subst.instantiate(*it);
             pre_tac = solve_unassigned_mvars(subst, pre_tac, visited);
             check_exact_tacs(pre_tac, subst);
@@ -899,6 +900,58 @@ public:
         }
     }
 
+    optional<tactic> get_local_tactic_hint(substitution & subst, expr const & mvar, name_set & visited) {
+        if (auto pre_tac = get_pre_tactic_for(subst, mvar, visited)) {
+            return pre_tactic_to_tactic(*pre_tac, mvar);
+        } else {
+            return optional<tactic>();
+        }
+    }
+
+    /** \brief Try to instantiate meta-variable \c mvar (modulo its state ps) using the given tactic.
+        If it succeeds, then update subst with the solution.
+        If \c report_failure is true, then display error messages. to the user.
+        Return true iff the metavariable \c mvar has been assigned.
+    */
+    bool try_using(substitution & subst, expr const & mvar, proof_state const & ps, tactic const & tac, bool report_failure) {
+        lean_assert(length(ps.get_goals()) == 1);
+        lean_assert(get_app_fn(head(ps.get_goals()).get_meta()) == mvar); // make sure ps is a really a proof state for mvar.
+        try {
+            proof_state_seq seq = tac(m_env, m_ios, ps);
+            auto r = seq.pull();
+            if (!r) {
+                // tactic failed to produce any result
+                if (report_failure)
+                    display_unsolved_proof_state(mvar, ps, "tactic failed");
+                return false;
+            } else if (!empty(r->first.get_goals())) {
+                // tactic contains unsolved subgoals
+                if (report_failure)
+                    display_unsolved_proof_state(mvar, r->first, "unsolved subgoals");
+                return false;
+            } else {
+                subst = r->first.get_subst();
+                expr v = subst.instantiate(mvar);
+                subst = subst.assign(mlocal_name(mvar), v);
+                return true;
+            }
+        } catch (tactic_exception & ex) {
+            if (report_failure) {
+                regular out(m_env, m_ios);
+                display_error_pos(out, m_pos_provider, ex.get_expr());
+                out << " tactic failed: " << ex.what() << "\n";
+            }
+            return false;
+        }
+    }
+
+    bool try_using(substitution & subst, expr const & mvar, proof_state const & ps, list<tactic_hint_entry> const & tacs) {
+        return std::any_of(tacs.begin(), tacs.end(),
+                           [&](tactic_hint_entry const & e) {
+                               return try_using(subst, mvar, ps, e.get_tactic(), false);
+                           });
+    }
+
     void solve_unassigned_mvar(substitution & subst, expr mvar, name_set & visited) {
         if (visited.contains(mlocal_name(mvar)))
             return;
@@ -916,30 +969,15 @@ public:
         // first solve unassigned metavariables in type
         type = solve_unassigned_mvars(subst, type, visited);
         proof_state ps(goals(goal(*meta, type)), subst, m_ngen.mk_child());
-        optional<expr> pre_tac = get_pre_tactic_for(subst, mvar, visited);
+        if (auto local_hint = get_local_tactic_hint(subst, mvar, visited)) {
-        if (!pre_tac)
+            try_using(subst, mvar, ps, *local_hint, true);
-            return;
-        optional<tactic> tac   = pre_tactic_to_tactic(*pre_tac, mvar);
-        if (!tac)
-            return;
-        try {
-            proof_state_seq seq = (*tac)(m_env, m_ios, ps);
-            auto r = seq.pull();
-            if (!r) {
-                // tactic failed to produce any result
-                display_unsolved_proof_state(mvar, ps, "tactic failed");
-            } else if (!empty(r->first.get_goals())) {
-                // tactic contains unsolved subgoals
-                display_unsolved_proof_state(mvar, r->first, "unsolved subgoals");
         } else {
-                subst = r->first.get_subst();
+            if (is_constant(get_app_fn(type))) {
-                expr v = subst.instantiate(mvar);
+                name cls = const_name(get_app_fn(type));
-                subst = subst.assign(mlocal_name(mvar), v);
+                if (try_using(subst, mvar, ps, get_tactic_hints(m_env, cls)))
+                    return;
             }
-        } catch (tactic_exception & ex) {
+            try_using(subst, mvar, ps, get_tactic_hints(m_env));
-            regular out(m_env, m_ios);
-            display_error_pos(out, m_pos_provider, ex.get_expr());
-            out << " tactic failed: " << ex.what() << "\n";
         }
     }
 

src/frontends/lean/parser.cpp

@@ -469,22 +469,22 @@ expr parser::mk_Type() {
 }
 
 std::tuple<expr, level_param_names> parser::elaborate_relaxed(expr const & e) {
-    parser_pos_provider pp(m_pos_table, get_stream_name(), m_last_cmd_pos);
+    parser_pos_provider pp = get_pos_provider();
     return ::lean::elaborate(m_env, m_local_level_decls, m_ios, e, &pp, false);
 }
 
 std::tuple<expr, level_param_names> parser::elaborate_type(expr const & e) {
-    parser_pos_provider pp(m_pos_table, get_stream_name(), m_last_cmd_pos);
+    parser_pos_provider pp = get_pos_provider();
     return ::lean::elaborate(m_env, m_local_level_decls, m_ios, e, &pp, true, true);
 }
 
 std::tuple<expr, level_param_names> parser::elaborate_at(environment const & env, expr const & e) {
-    parser_pos_provider pp(m_pos_table, get_stream_name(), m_last_cmd_pos);
+    parser_pos_provider pp = get_pos_provider();
     return ::lean::elaborate(env, m_local_level_decls, m_ios, e, &pp);
 }
 
 std::tuple<expr, expr, level_param_names> parser::elaborate_definition(name const & n, expr const & t, expr const & v) {
-    parser_pos_provider pp(m_pos_table, get_stream_name(), m_last_cmd_pos);
+    parser_pos_provider pp = get_pos_provider();
     return ::lean::elaborate(m_env, m_local_level_decls, m_ios, n, t, v, &pp);
 }
 

src/frontends/lean/parser.h

@@ -257,9 +257,13 @@ public:
     std::tuple<expr, level_param_names> elaborate_type(expr const & e);
     /** \brief Elaborate \c e in the given environment. */
     std::tuple<expr, level_param_names> elaborate_at(environment const & env, expr const & e);
+    /** \brief Elaborate \c e (making sure the result does not have metavariables). */
+    std::tuple<expr, level_param_names> elaborate(expr const & e) { return elaborate_at(m_env, e); }
     /** \brief Elaborate the definition n : t := v */
     std::tuple<expr, expr, level_param_names> elaborate_definition(name const & n, expr const & t, expr const & v);
 
+    parser_pos_provider get_pos_provider() const { return parser_pos_provider(m_pos_table, get_stream_name(), m_last_cmd_pos); }
+
     /** parse all commands in the input stream */
     bool operator()() { return parse_commands(); }
 };

src/frontends/lean/proof_qed_ext.cpp

@@ -10,6 +10,7 @@ Author: Leonardo de Moura
 #include "library/kernel_serializer.h"
 #include "library/tactic/expr_to_tactic.h"
 #include "frontends/lean/parser.h"
+#include "frontends/lean/tactic_hint.h"
 
 namespace lean {
 // This (scoped) environment extension allows us to set a tactic to be applied before every element
@@ -84,12 +85,6 @@ optional<expr> get_proof_qed_pre_tactic(environment const & env) {
     return s.m_pre_tac;
 }
 
-static expr parse_tactic_name(parser & p) {
-    auto pos = p.pos();
-    name id  = p.check_id_next("invalid proof_qed configuration command, indentifier expected");
-    return p.id_to_expr(id, pos);
-}
-
 environment add_proof_qed_cmd(parser & p) {
     return add_proof_qed_pre_tactic(p.env(), parse_tactic_name(p));
 }

src/frontends/lean/tactic_hint.cpp

@@ -0,0 +1,124 @@
+/*
+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 <string>
+#include "util/sstream.h"
+#include "kernel/type_checker.h"
+#include "library/scoped_ext.h"
+#include "library/kernel_serializer.h"
+#include "library/tactic/tactic.h"
+#include "library/tactic/expr_to_tactic.h"
+#include "frontends/lean/tactic_hint.h"
+#include "frontends/lean/cmd_table.h"
+#include "frontends/lean/parser.h"
+#include "frontends/lean/class.h"
+
+namespace lean {
+typedef list<tactic_hint_entry> tactic_hint_entries;
+
+struct tactic_hint_state {
+    typedef rb_map<name, tactic_hint_entries, name_quick_cmp> class_tactics;
+    class_tactics        m_class_tactics;
+    tactic_hint_entries  m_tactics;
+};
+
+struct tactic_hint_config {
+    typedef tactic_hint_state   state;
+    typedef tactic_hint_entry   entry;
+    typedef tactic_hint_entries entries;
+
+    static entries add(entries const & l, entry const & e) {
+        return cons(e, filter(l, [&](entry const & e1) { return e1.m_pre_tactic != e.m_pre_tactic; }));
+    }
+
+    static void add_entry_core(state & s, entry const & e) {
+        if (e.m_class.is_anonymous())
+            s.m_tactics = add(s.m_tactics, e);
+        else if (auto it = s.m_class_tactics.find(e.m_class))
+            s.m_class_tactics.insert(e.m_class, add(*it, e));
+        else
+            s.m_class_tactics.insert(e.m_class, entries(e));
+    }
+
+    static void add_entry(environment const & env, io_state const &, state & s, entry const & e) {
+        if (!e.m_compiled) {
+            entry new_e(e);
+            new_e.m_tactic   = expr_to_tactic(env, e.m_pre_tactic, nullptr);
+            new_e.m_compiled = true;
+            add_entry_core(s, new_e);
+        } else {
+            add_entry_core(s, e);
+        }
+    }
+
+    static name const & get_class_name() {
+        static name g_class_name("tactic_hint");
+        return g_class_name;
+    }
+
+    static std::string const & get_serialization_key() {
+        static std::string g_key("tachint");
+        return g_key;
+    }
+
+    static void  write_entry(serializer & s, entry const & e) {
+        s << e.m_class << e.m_pre_tactic;
+    }
+
+    static entry read_entry(deserializer & d) {
+        entry e;
+        d >> e.m_class >> e.m_pre_tactic;
+        return e;
+    }
+};
+
+template class scoped_ext<tactic_hint_config>;
+typedef scoped_ext<tactic_hint_config> tactic_hint_ext;
+
+list<tactic_hint_entry> get_tactic_hints(environment const & env, name const & c) {
+    tactic_hint_state const & s = tactic_hint_ext::get_state(env);
+    if (auto it = s.m_class_tactics.find(c))
+        return *it;
+    else
+        return list<tactic_hint_entry>();
+}
+
+list<tactic_hint_entry> get_tactic_hints(environment const & env) {
+    tactic_hint_state const & s = tactic_hint_ext::get_state(env);
+    return s.m_tactics;
+}
+
+expr parse_tactic_name(parser & p) {
+    auto pos     = p.pos();
+    name id      = p.check_id_next("invalid tactic name, identifier expected");
+    expr pre_tac = p.id_to_expr(id, pos);
+    if (!is_constant(pre_tac))
+        throw parser_error(sstream() << "invalid tactic name, '" << id << "' is not a constant", pos);
+    expr pre_tac_type = p.env().get(const_name(pre_tac)).get_type();
+    if (!is_constant(pre_tac_type) || const_name(pre_tac_type) != name({"tactic", "tactic"}))
+        throw parser_error(sstream() << "invalid tactic name, '" << id << "' is not a tactic", pos);
+    return pre_tac;
+}
+
+static name g_lbracket("[");
+static name g_rbracket("]");
+
+environment tactic_hint_cmd(parser & p) {
+    name cls;
+    if (p.curr_is_token(g_lbracket)) {
+        p.next();
+        cls      = get_class_name(p.env(), p.parse_expr());
+        p.check_token_next(g_rbracket, "invalid tactic hint, ']' expected");
+    }
+    expr pre_tac = parse_tactic_name(p);
+    tactic tac   = expr_to_tactic(p.env(), pre_tac, nullptr);
+    return tactic_hint_ext::add_entry(p.env(), get_dummy_ios(), tactic_hint_entry(cls, pre_tac, tac));
+}
+
+void register_tactic_hint_cmd(cmd_table & r) {
+    add_cmd(r, cmd_info("tactic_hint",   "add a new tactic hint", tactic_hint_cmd));
+}
+}

src/frontends/lean/tactic_hint.h

@@ -0,0 +1,30 @@
+/*
+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"
+#include "frontends/lean/cmd_table.h"
+
+namespace lean {
+class tactic_hint_entry {
+    friend class tactic_hint_config;
+    name   m_class;
+    expr   m_pre_tactic;
+    tactic m_tactic;
+    bool   m_compiled;
+public:
+    tactic_hint_entry():m_compiled(false) {}
+    tactic_hint_entry(name const & c, expr const & pre_tac, tactic const & tac):
+        m_class(c), m_pre_tactic(pre_tac), m_tactic(tac), m_compiled(true) {}
+    expr const & get_pre_tactic() const { return m_pre_tactic; }
+    tactic const & get_tactic() const { return m_tactic; }
+};
+list<tactic_hint_entry> get_tactic_hints(environment const & env, name const & cls_name);
+list<tactic_hint_entry> get_tactic_hints(environment const & env);
+class parser;
+expr parse_tactic_name(parser & p);
+void register_tactic_hint_cmd(cmd_table & r);
+}

src/frontends/lean/token_table.cpp

@@ -77,7 +77,7 @@ token_table init_token_table() {
                                "abbreviation", "opaque_hint", "evaluate", "check", "print", "end", "namespace", "section", "import",
                                "abbreviation", "inductive", "record", "structure", "module", "universe",
                                "precedence", "infixl", "infixr", "infix", "postfix", "prefix", "notation",
-                               "exit", "set_option", "using", "calc_subst", "calc_refl", "calc_trans",
+                               "exit", "set_option", "using", "calc_subst", "calc_refl", "calc_trans", "tactic_hint",
                                "add_proof_qed", "set_proof_qed", "#setline", "class", "instance", nullptr};
 
     std::pair<char const *, char const *> aliases[] =

src/kernel/type_checker.h

@@ -113,6 +113,7 @@ public:
     ~type_checker();
 
     environment const & env() const { return m_env; }
+    name_generator mk_ngen() { return m_gen.mk_child(); }
 
     /**
        \brief Return the type of \c t.

src/library/tactic/expr_to_tactic.cpp

@@ -84,14 +84,36 @@ tactic expr_to_tactic(type_checker & tc, expr e, pos_info_provider const * p) {
         // unfold definition
         buffer<expr> locals;
         get_app_rev_args(e, locals);
-        v = instantiate_univ_params(v, it->get_univ_params(), const_levels(f));
+        level_param_names const & ps = it->get_univ_params();
+        levels ls = const_levels(f);
+        unsigned num_ps = length(ps);
+        unsigned num_ls = length(ls);
+        if (num_ls > num_ps)
+            throw expr_to_tactic_exception(e, sstream() << "invalid number of universes");
+        if (num_ls < num_ps) {
+            buffer<level> extra_ls;
+            name_generator ngen = tc.mk_ngen();
+            for (unsigned i = num_ls; i < num_ps; i++)
+                extra_ls.push_back(mk_meta_univ(ngen.next()));
+            ls = append(ls, to_list(extra_ls.begin(), extra_ls.end()));
+        }
+        v = instantiate_univ_params(v, ps, ls);
         v = apply_beta(v, locals.size(), locals.data());
         return expr_to_tactic(tc, v, p);
     }
 }
 
+static name g_tmp_prefix = name::mk_internal_unique_name();
+MK_THREAD_LOCAL_GET(unsigned, get_expr_tac_id, 0)
+static name_generator next_name_generator() {
+    unsigned & c = get_expr_tac_id();
+    unsigned r = c;
+    c++;
+    return name_generator(name(g_tmp_prefix, r));
+}
+
 tactic expr_to_tactic(environment const & env, expr const & e, pos_info_provider const * p) {
-    type_checker tc(env);
+    type_checker tc(env, next_name_generator());
     return expr_to_tactic(tc, e, p);
 }
 

tests/lean/run/tactic24.lean

@@ -0,0 +1,23 @@
+import standard
+using tactic
+
+definition my_tac1 := apply @refl
+definition my_tac2 := repeat (apply @and_intro; assumption)
+
+tactic_hint my_tac1
+tactic_hint my_tac2
+
+theorem T1 {A : Type.{2}} (a : A) : a = a
+:= _
+
+theorem T2 {a b c : Bool} (Ha : a) (Hb : b) (Hc : c) : a ∧ b ∧ c
+:= _
+
+definition my_tac3 := fixpoint (λ f, [apply @or_intro_left; f  |
+                                      apply @or_intro_right; f |
+                                      assumption])
+
+tactic_hint [or] my_tac3
+
+theorem T3 {a b c : Bool} (Hb : b) : a ∨ b ∨ c
+:= _