feat(builtin): use namespaces when defining nat, int and real.

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

src/builtin/int.lean

@@ -2,63 +2,67 @@ Import nat.
 
 Variable Int : Type.
 Alias ℤ : Int.
-
-Builtin Int::numeral.
-
-Builtin Int::add : Int → Int → Int.
-Infixl 65 + : Int::add.
-
-Builtin Int::mul : Int → Int → Int.
-Infixl 70 * : Int::mul.
-
-Builtin Int::div : Int → Int → Int.
-Infixl 70 div : Int::div.
-
-Builtin Int::le : Int → Int → Bool.
-Infix  50 <= : Int::le.
-Infix  50 ≤  : Int::le.
-
-Definition Int::ge (a b : Int) : Bool := b ≤ a.
-Infix 50 >= : Int::ge.
-Infix 50 ≥  : Int::ge.
-
-Definition Int::lt (a b : Int) : Bool := ¬ (a ≥ b).
-Infix 50 <  : Int::lt.
-
-Definition Int::gt (a b : Int) : Bool := ¬ (a ≤ b).
-Infix 50 >  : Int::gt.
-
-Definition Int::sub (a b : Int) : Int := a + -1 * b.
-Infixl 65 - : Int::sub.
-
-Definition Int::neg (a : Int) : Int := -1 * a.
-Notation 75 - _ : Int::neg.
-
-Definition Int::mod (a b : Int) : Int := a - b * (a div b).
-Infixl 70 mod : Int::mod.
-
 Builtin nat_to_int : Nat → Int.
 Coercion nat_to_int.
 
-Definition Int::divides (a b : Int) : Bool := (b mod a) = 0.
+Namespace Int.
-Infix 50 | : Int::divides.
+Builtin numeral.
 
-Definition Int::abs (a : Int) : Int := if (0 ≤ a) a (- a).
+Builtin add : Int → Int → Int.
-Notation 55 | _ | : Int::abs.
+Infixl 65 + : add.
 
-Definition Nat::sub (a b : Nat) : Int := nat_to_int a - nat_to_int b.
+Builtin mul : Int → Int → Int.
-Infixl 65 - : Nat::sub.
+Infixl 70 * : mul.
 
-Definition Nat::neg (a : Nat) : Int := - (nat_to_int a).
+Builtin div : Int → Int → Int.
-Notation 75 - _ : Nat::neg.
+Infixl 70 div : div.
 
-SetOpaque Int::sub true.
+Builtin le : Int → Int → Bool.
-SetOpaque Int::neg true.
+Infix  50 <= : le.
-SetOpaque Int::mod true.
+Infix  50 ≤  : le.
-SetOpaque Int::divides true.
+
-SetOpaque Int::abs true.
+Definition ge (a b : Int) : Bool := b ≤ a.
-SetOpaque Int::ge true.
+Infix 50 >= : ge.
-SetOpaque Int::lt true.
+Infix 50 ≥  : ge.
-SetOpaque Int::gt true.
+
-SetOpaque Nat::sub true.
+Definition lt (a b : Int) : Bool := ¬ (a ≥ b).
-SetOpaque Nat::neg true.
+Infix 50 <  : lt.
+
+Definition gt (a b : Int) : Bool := ¬ (a ≤ b).
+Infix 50 >  : gt.
+
+Definition sub (a b : Int) : Int := a + -1 * b.
+Infixl 65 - : sub.
+
+Definition neg (a : Int) : Int := -1 * a.
+Notation 75 - _ : neg.
+
+Definition mod (a b : Int) : Int := a - b * (a div b).
+Infixl 70 mod : mod.
+
+Definition divides (a b : Int) : Bool := (b mod a) = 0.
+Infix 50 | : divides.
+
+Definition abs (a : Int) : Int := if (0 ≤ a) a (- a).
+Notation 55 | _ | : abs.
+
+SetOpaque sub true.
+SetOpaque neg true.
+SetOpaque mod true.
+SetOpaque divides true.
+SetOpaque abs true.
+SetOpaque ge true.
+SetOpaque lt true.
+SetOpaque gt true.
+EndNamespace.
+
+Namespace Nat.
+Definition sub (a b : Nat) : Int := nat_to_int a - nat_to_int b.
+Infixl 65 - : sub.
+
+Definition neg (a : Nat) : Int := - (nat_to_int a).
+Notation 75 - _ : neg.
+
+SetOpaque sub true.
+SetOpaque neg true.
+EndNamespace.

src/builtin/nat.lean

@@ -3,32 +3,36 @@ Import kernel.
 Variable Nat : Type.
 Alias ℕ : Nat.
 
-Builtin Nat::numeral.
+Namespace Nat.
 
-Builtin Nat::add : Nat → Nat → Nat.
+Builtin numeral.
-Infixl 65 +  : Nat::add.
 
-Builtin Nat::mul : Nat → Nat → Nat.
+Builtin add : Nat → Nat → Nat.
-Infixl 70 *  : Nat::mul.
+Infixl 65 +  : add.
 
-Builtin Nat::le  : Nat → Nat → Bool.
+Builtin mul : Nat → Nat → Nat.
-Infix  50 <= : Nat::le.
+Infixl 70 *  : mul.
-Infix  50 ≤  : Nat::le.
 
-Definition Nat::ge (a b : Nat) := b ≤ a.
+Builtin le  : Nat → Nat → Bool.
-Infix 50 >= : Nat::ge.
+Infix  50 <= : le.
-Infix 50 ≥  : Nat::ge.
+Infix  50 ≤  : le.
 
-Definition Nat::lt (a b : Nat) := ¬ (a ≥ b).
+Definition ge (a b : Nat) := b ≤ a.
-Infix 50 <  : Nat::lt.
+Infix 50 >= : ge.
+Infix 50 ≥  : ge.
 
-Definition Nat::gt (a b : Nat) := ¬ (a ≤ b).
+Definition lt (a b : Nat) := ¬ (a ≥ b).
-Infix 50 >  : Nat::gt.
+Infix 50 <  : lt.
 
-Definition Nat::id (a : Nat) := a.
+Definition gt (a b : Nat) := ¬ (a ≤ b).
-Notation 55 | _ | : Nat::id.
+Infix 50 >  : gt.
 
-SetOpaque Nat::ge true.
+Definition id (a : Nat) := a.
-SetOpaque Nat::lt true.
+Notation 55 | _ | : id.
-SetOpaque Nat::gt true.
+
-SetOpaque Nat::id true.
+SetOpaque ge true.
+SetOpaque lt true.
+SetOpaque gt true.
+SetOpaque id true.
+
+EndNamespace.

src/builtin/real.lean

@@ -2,43 +2,43 @@ Import int.
 
 Variable Real : Type.
 Alias ℝ : Real.
-
-Builtin Real::numeral.
-
-Builtin Real::add : Real → Real → Real.
-Infixl 65 + : Real::add.
-
-Builtin Real::mul : Real → Real → Real.
-Infixl 70 * : Real::mul.
-
-Builtin Real::div : Real → Real → Real.
-Infixl 70 / : Real::div.
-
-Builtin Real::le : Real → Real → Bool.
-Infix  50 <= : Real::le.
-Infix  50 ≤  : Real::le.
-
-Definition Real::ge (a b : Real) : Bool := b ≤ a.
-Infix 50 >= : Real::ge.
-Infix 50 ≥  : Real::ge.
-
-Definition Real::lt (a b : Real) : Bool := ¬ (a ≥ b).
-Infix 50 <  : Real::lt.
-
-Definition Real::gt (a b : Real) : Bool := ¬ (a ≤ b).
-Infix 50 >  : Real::gt.
-
-Definition Real::sub (a b : Real) : Real := a + -1.0 * b.
-Infixl 65 - : Real::sub.
-
-Definition Real::neg (a : Real) : Real := -1.0 * a.
-Notation 75 - _ : Real::neg.
-
-Definition Real::abs (a : Real) : Real := if (0.0 ≤ a) a (- a).
-Notation 55 | _ | : Real::abs.
-
 Builtin int_to_real : Int → Real.
 Coercion int_to_real.
-
 Definition nat_to_real (a : Nat) : Real := int_to_real (nat_to_int a).
 Coercion nat_to_real.
+
+Namespace Real.
+Builtin numeral.
+
+Builtin add : Real → Real → Real.
+Infixl 65 + : add.
+
+Builtin mul : Real → Real → Real.
+Infixl 70 * : mul.
+
+Builtin div : Real → Real → Real.
+Infixl 70 / : div.
+
+Builtin le : Real → Real → Bool.
+Infix  50 <= : le.
+Infix  50 ≤  : le.
+
+Definition ge (a b : Real) : Bool := b ≤ a.
+Infix 50 >= : ge.
+Infix 50 ≥  : ge.
+
+Definition lt (a b : Real) : Bool := ¬ (a ≥ b).
+Infix 50 <  : lt.
+
+Definition gt (a b : Real) : Bool := ¬ (a ≤ b).
+Infix 50 >  : gt.
+
+Definition sub (a b : Real) : Real := a + -1.0 * b.
+Infixl 65 - : sub.
+
+Definition neg (a : Real) : Real := -1.0 * a.
+Notation 75 - _ : neg.
+
+Definition abs (a : Real) : Real := if (0.0 ≤ a) a (- a).
+Notation 55 | _ | : abs.
+EndNamespace.

src/frontends/lean/parser.cpp

@@ -914,8 +914,11 @@ class parser::imp {
        \brief Try to find an object (Definition or Postulate) named \c
        id in the frontend/environment. If there isn't one, then tries
        to check if \c id is a builtin symbol. If it is not throws an error.
+
+       If \c implicit_args is true, then we also parse explicit arguments until
+       we have a placeholder forall implicit ones.
     */
-    expr get_name_ref(name const & id, pos_info const & p) {
+    expr get_name_ref(name const & id, pos_info const & p, bool implicit_args = true) {
         lean_assert(!m_namespace_prefixes.empty());
         auto it    = m_namespace_prefixes.end();
         auto begin = m_namespace_prefixes.begin();
@@ -926,7 +929,7 @@ class parser::imp {
             if (obj) {
                 object_kind k      = obj->kind();
                 if (k == object_kind::Definition || k == object_kind::Postulate || k == object_kind::Builtin) {
-                    if (has_implicit_arguments(m_env, obj->get_name())) {
+                    if (implicit_args && has_implicit_arguments(m_env, obj->get_name())) {
                         std::vector<bool> const & imp_args = get_implicit_arguments(m_env, obj->get_name());
                         buffer<expr> args;
                         pos_info p = pos();
@@ -2278,8 +2281,9 @@ class parser::imp {
         unsigned prec = parse_precedence();
         name op_id = parse_op_id();
         check_colon_next("invalid operator definition, ':' expected");
+        auto name_pos = pos();
         name name_id = check_identifier_next("invalid operator definition, identifier expected");
-        expr d     = mk_constant(name_id);
+        expr d       = get_name_ref(name_id, name_pos, false);
         switch (fx) {
         case fixity::Infix:   add_infix(m_env, m_io_state, op_id, prec, d); break;
         case fixity::Infixl:  add_infixl(m_env, m_io_state, op_id, prec, d); break;
@@ -2319,8 +2323,9 @@ class parser::imp {
                     if (parts.size() == 0) {
                         throw parser_error("invalid notation declaration, it must have at least one identifier", p);
                     }
+                    auto id_pos = pos();
                     name name_id = check_identifier_next("invalid notation declaration, identifier expected");
-                    expr d     = mk_constant(name_id);
+                    expr d       = get_name_ref(name_id, id_pos, false);
                     if (parts.size() == 1) {
                         if (first_placeholder && prev_placeholder) {
                             // infix: _ ID _
@@ -2593,9 +2598,10 @@ class parser::imp {
         next();
         auto id_pos = pos();
         name id = check_identifier_next("invalid builtin declaration, identifier expected");
-        auto d  = get_builtin(id);
+        name full_id = mk_full_name(id);
+        auto d  = get_builtin(full_id);
         if (!d)
-            throw parser_error(sstream() << "unknown builtin '" << id << "'", id_pos);
+            throw parser_error(sstream() << "unknown builtin '" << full_id << "'", id_pos);
         expr b = d->first;
         if (d->second) {
             m_env->add_builtin_set(b);
@@ -2623,8 +2629,8 @@ class parser::imp {
         }
         m_env->add_builtin(d->first);
         if (m_verbose)
-            regular(m_io_state) << "  Added: " << id << endl;
+            regular(m_io_state) << "  Added: " << full_id << endl;
-        register_implicit_arguments(id, bindings);
+        register_implicit_arguments(full_id, bindings);
     }
 
     void parse_namespace() {