feat(frontends/lean): add 'universe variable' command

We can declare variables anywhere. So, we must also be able do declare
"universe" variables anywhere. Here is a minimal example that requires
this feature

```
-- We want A and B to be in the same universe
universe variable l
variable A : Type.{l}
variable B : Type.{l}
definition tst := A = B
```

The following doesn't work because A and B are in different universes
```
variable A : Type
variable B : Type
definition tst := A = B
```

The following works, but tst is not universe polymorphic, since l is
one *fixed* global universe
```
universe l
variable A : Type.{l}
variable B : Type.{l}
definition tst := A = B
```

src/frontends/lean/decl_cmds.cpp

@@ -24,8 +24,8 @@ Author: Leonardo de Moura
 #include "frontends/lean/tokens.h"
 
 namespace lean {
-static environment declare_universe(parser & p, environment env, name const & n) {
-    if (in_section_or_context(env)) {
+static environment declare_universe(parser & p, environment env, name const & n, bool local) {
+    if (in_section_or_context(env) || local) {
         p.add_local_level(n, mk_param_univ(n));
     } else {
         name const & ns = get_namespace(env);
@@ -37,23 +37,37 @@ static environment declare_universe(parser & p, environment env, name const & n)
     return env;
 }
 
-environment universe_cmd(parser & p) {
-    name n = p.check_id_next("invalid 'universe' command, identifier expected");
-    return declare_universe(p, p.env(), n);
-}
-
-environment universes_cmd(parser & p) {
+static environment universes_cmd_core(parser & p, bool local) {
     if (!p.curr_is_identifier())
         throw parser_error("invalid 'universes' command, identifier expected", p.pos());
     environment env = p.env();
     while (p.curr_is_identifier()) {
         name n = p.get_name_val();
         p.next();
-        env = declare_universe(p, env, n);
+        env = declare_universe(p, env, n, local);
     }
     return env;
 }
 
+static environment universe_cmd(parser & p) {
+    if (p.curr_is_token(get_variables_tk())) {
+        p.next();
+        return universes_cmd_core(p, true);
+    } else {
+        bool local = false;
+        if (p.curr_is_token(get_variable_tk())) {
+            p.next();
+            local = true;
+        }
+        name n = p.check_id_next("invalid 'universe' command, identifier expected");
+        return declare_universe(p, p.env(), n, local);
+    }
+}
+
+static environment universes_cmd(parser & p) {
+    return universes_cmd_core(p, false);
+}
+
 bool parse_univ_params(parser & p, buffer<name> & ps) {
     if (p.curr_is_token(get_llevel_curly_tk())) {
         p.next();
@@ -383,7 +397,7 @@ environment definition_cmd_core(parser & p, bool is_theorem, bool is_opaque, boo
         value = Fun_as_is(section_value_ps, value, p);
         update_univ_parameters(ls_buffer, collect_univ_params(value, collect_univ_params(type)), p);
         ls = to_list(ls_buffer.begin(), ls_buffer.end());
-        levels section_ls = collect_section_nonvar_levels(p, ls);
+        levels section_ls = collect_local_nonvar_levels(p, ls);
         remove_section_variables(p, section_ps);
         if (!section_ps.empty()) {
             expr ref = mk_section_local_ref(real_n, section_ls, section_ps);

src/frontends/lean/inductive_cmd.cpp

@@ -608,7 +608,7 @@ struct inductive_cmd_fn {
         buffer<expr> section_params_only(section_params);
         remove_section_variables(m_p, section_params_only);
         // Create aliases/local refs
-        levels section_levels = collect_section_nonvar_levels(m_p, ls);
+        levels section_levels = collect_local_nonvar_levels(m_p, ls);
         for (auto & d : decls) {
             name d_name = inductive_decl_name(d);
             name d_short_name(d_name.get_string());

src/frontends/lean/token_table.cpp

@@ -75,7 +75,7 @@ void init_token_table(token_table & t) {
          {"using", 0}, {"|", 0}, {"!", g_max_prec}, {"with", 0}, {"...", 0}, {",", 0},
          {".", 0}, {":", 0}, {"::", 0}, {"calc", 0}, {":=", 0}, {"--", 0}, {"#", 0},
          {"(*", 0}, {"/-", 0}, {"begin", g_max_prec}, {"proof", g_max_prec}, {"qed", 0}, {"@", g_max_prec},
-         {"sorry", g_max_prec}, {"+", g_plus_prec}, {g_cup, g_cup_prec}, {"->", g_arrow_prec},
+         {"sorry", g_max_prec}, {"+", g_plus_prec}, {g_cup, g_cup_prec}, {"->", g_arrow_prec}, {"local", 0},
          {nullptr, 0}};
 
     char const * commands[] =

src/frontends/lean/tokens.cpp

@@ -65,6 +65,7 @@ static name * g_definition   = nullptr;
 static name * g_theorem      = nullptr;
 static name * g_axiom        = nullptr;
 static name * g_variable     = nullptr;
+static name * g_variables    = nullptr;
 static name * g_opaque       = nullptr;
 static name * g_instance     = nullptr;
 static name * g_priority     = nullptr;
@@ -148,6 +149,7 @@ void initialize_tokens() {
     g_opaque       = new name("opaque");
     g_axiom        = new name("axiom");
     g_variable     = new name("variable");
+    g_variables    = new name("variables");
     g_instance     = new name("[instance]");
     g_priority     = new name("[priority");
     g_coercion     = new name("[coercion]");
@@ -193,6 +195,7 @@ void finalize_tokens() {
     delete g_theorem;
     delete g_opaque;
     delete g_axiom;
+    delete g_variables;
     delete g_variable;
     delete g_instance;
     delete g_priority;
@@ -312,6 +315,7 @@ name const & get_definition_tk() { return *g_definition; }
 name const & get_theorem_tk() { return *g_theorem; }
 name const & get_axiom_tk() { return *g_axiom; }
 name const & get_variable_tk() { return *g_variable; }
+name const & get_variables_tk() { return *g_variables; }
 name const & get_opaque_tk() { return *g_opaque; }
 name const & get_instance_tk() { return *g_instance; }
 name const & get_priority_tk() { return *g_priority; }

src/frontends/lean/tokens.h

@@ -67,6 +67,7 @@ name const & get_definition_tk();
 name const & get_theorem_tk();
 name const & get_axiom_tk();
 name const & get_variable_tk();
+name const & get_variables_tk();
 name const & get_opaque_tk();
 name const & get_instance_tk();
 name const & get_priority_tk();

src/frontends/lean/util.cpp

@@ -60,8 +60,8 @@ void sort_section_params(expr_struct_set const & locals, parser const & p, buffe
         });
 }
 
-// Return the levels in \c ls that are defined in the section
-levels collect_section_nonvar_levels(parser & p, level_param_names const & ls) {
+// Return the local levels in \c ls that are not associated with variables
+levels collect_local_nonvar_levels(parser & p, level_param_names const & ls) {
     buffer<level> section_ls_buffer;
     for (name const & l : ls) {
         if (p.get_local_level_index(l) && !p.is_local_level_variable(l))

src/frontends/lean/util.h

@@ -22,8 +22,10 @@ void check_atomic(name const & n);
 void check_in_section_or_context(parser const & p);
 bool is_root_namespace(name const & n);
 name remove_root_prefix(name const & n);
-/** \brief Return the levels in \c ls that are defined in the section, but are not tagged as variables. */
-levels collect_section_nonvar_levels(parser & p, level_param_names const & ls);
+/** \brief Return the local levels in \c ls that are not tagged as variables.
+    A local level is tagged as variable if it associated with a variable.
+*/
+levels collect_local_nonvar_levels(parser & p, level_param_names const & ls);
 /** \brief Collect local (section) constants occurring in type and value, sort them, and store in section_ps */
 void collect_section_locals(expr const & type, expr const & value, parser const & p, buffer<expr> & section_ps);
 /** \brief Copy the local parameters to \c section_ps, then sort \c section_ps (using the order in which they were declared). */

tests/lean/univ_vars.lean

@@ -0,0 +1,28 @@
+import logic
+set_option pp.universes true
+
+universe u
+variable A : Type.{u}
+
+definition id1 (a : A) : A := a
+check @id1
+
+variable B : Type
+
+definition id2 (a : B) : B := a
+check @id2
+
+universe variable k
+variable C : Type.{k}
+
+definition id3 (a : C) := a
+
+check @id3
+
+universe variables l m
+variable A₁ : Type.{l}
+variable A₂ : Type.{l}
+definition foo (a₁ : A₁) (a₂ : A₂) := a₁ == a₂
+check @foo
+
+check Type.{m}

tests/lean/univ_vars.lean.expected.out

@@ -0,0 +1,5 @@
+id1 : Π (A : Type.{u}), A → A
+id2.{l_2} : Π (B : Type.{l_2}), B → B
+id3.{k} : Π (C : Type.{k}), C → C
+foo.{l} : Π (A₁ A₂ : Type.{l}), A₁ → A₂ → Prop
+Type.{m} : Type.{succ m}