diff --git a/library/logic/core/prop.lean b/library/logic/core/prop.lean
index f582d5189b..0201454329 100644
--- a/library/logic/core/prop.lean
+++ b/library/logic/core/prop.lean
@@ -2,10 +2,7 @@
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Authors: Leonardo de Moura, Jeremy Avigad
 
-import general_notation
-
-definition Prop [reducible] := Type.{0}
-
+import general_notation type
 
 -- implication
 -- -----------
diff --git a/library/type.lean b/library/type.lean
index bdbc8fc664..6762fe490a 100644
--- a/library/type.lean
+++ b/library/type.lean
@@ -2,6 +2,9 @@
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Authors: Leonardo de Moura
 
-definition Type₁ [reducible] := Type.{1}
-definition Type₂ [reducible] := Type.{2}
-definition Type₃ [reducible] := Type.{3}
+notation `Prop`:max  := Type.{0}
+notation `Type₁`:max := Type.{1}
+notation `Type₂`:max := Type.{2}
+notation `Type₃`:max := Type.{3}
+notation `Type'`:max := Type.{_+1}
+notation `Type₊`:max := Type.{_+1}
diff --git a/src/emacs/lean-input.el b/src/emacs/lean-input.el
index d855237b58..d56ece0b56 100644
--- a/src/emacs/lean-input.el
+++ b/src/emacs/lean-input.el
@@ -252,6 +252,12 @@ order for the change to take effect."
   ("inn" . ("∉"))
   ("nin" . ("∌"))
 
+  ;; Types
+
+  ("T1" . ("Type₁"))
+  ("T2" . ("Type₂"))
+  ("T+" . ("Type₊"))
+
   ;; Intersections, unions etc.
 
   ("intersection" . ,(lean-input-to-string-list "∩⋂∧⋀⋏⨇⊓⨅⋒∏ ⊼      ⨉"))
@@ -421,7 +427,7 @@ order for the change to take effect."
   ;; Big/small, black/white.
 
   ("t" . ,(lean-input-to-string-list "▸▹►▻◂◃◄◅▴▵▾▿◢◿◣◺◤◸◥◹"))
-  ("T" . ,(lean-input-to-string-list "◀◁▶▷▲△▼▽◬◭◮"))
+  ("Tr" . ,(lean-input-to-string-list "◀◁▶▷▲△▼▽◬◭◮"))
 
   ("tb" . ,(lean-input-to-string-list "◂▸▴▾◄►◢◣◤◥"))
   ("tw" . ,(lean-input-to-string-list "◃▹▵▿◅▻◿◺◸◹"))
diff --git a/src/frontends/lean/builtin_exprs.cpp b/src/frontends/lean/builtin_exprs.cpp
index b3c2bbc3eb..3e26994a5b 100644
--- a/src/frontends/lean/builtin_exprs.cpp
+++ b/src/frontends/lean/builtin_exprs.cpp
@@ -36,10 +36,6 @@ static expr parse_Type(parser & p, unsigned, expr const *, pos_info const & pos)
     }
 }
 
-static expr parse_Type_prime(parser & p, unsigned, expr const *, pos_info const & pos) {
-    return p.save_pos(mk_sort(mk_succ(mk_level_placeholder())), pos);
-}
-
 static expr parse_let(parser & p, pos_info const & pos);
 static expr parse_let_body(parser & p, pos_info const & pos) {
     if (p.curr_is_token(get_comma_tk())) {
@@ -407,7 +403,6 @@ parse_table init_nud_table() {
     r = r.add({transition("fun", Binders), transition(",", mk_scoped_expr_action(x0))}, x0);
     r = r.add({transition("Pi", Binders), transition(",", mk_scoped_expr_action(x0, 0, false))}, x0);
     r = r.add({transition("Type", mk_ext_action(parse_Type))}, x0);
-    r = r.add({transition("Type'", mk_ext_action(parse_Type_prime))}, x0);
     r = r.add({transition("let", mk_ext_action(parse_let_expr))}, x0);
     r = r.add({transition("calc", mk_ext_action(parse_calc_expr))}, x0);
     r = r.add({transition("#", mk_ext_action(parse_overwrite_notation))}, x0);
diff --git a/src/frontends/lean/token_table.cpp b/src/frontends/lean/token_table.cpp
index 008c66f78a..289051addb 100644
--- a/src/frontends/lean/token_table.cpp
+++ b/src/frontends/lean/token_table.cpp
@@ -72,7 +72,7 @@ void init_token_table(token_table & t) {
         {{"fun", 0}, {"Pi", 0}, {"let", 0}, {"in", 0}, {"have", 0}, {"show", 0}, {"obtain", 0}, {"by", 0}, {"then", 0},
          {"from", 0}, {"(", g_max_prec}, {")", 0}, {"{", g_max_prec}, {"}", 0}, {"_", g_max_prec},
          {"[", g_max_prec}, {"]", 0}, {"⦃", g_max_prec}, {"⦄", 0}, {".{", 0}, {"Type", g_max_prec},
-         {"Type'", g_max_prec}, {"using", 0}, {"|", 0}, {"!", g_max_prec}, {"with", 0}, {"...", 0}, {",", 0},
+         {"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},
          {"including", 0}, {"sorry", g_max_prec}, {"+", g_plus_prec}, {g_cup, g_cup_prec}, {"->", g_arrow_prec},
diff --git a/tests/lean/prodtst.lean b/tests/lean/prodtst.lean
new file mode 100644
index 0000000000..2da0c56af9
--- /dev/null
+++ b/tests/lean/prodtst.lean
@@ -0,0 +1,7 @@
+import type
+
+inductive prod (A B : Type₊) :=
+mk : A → B → prod A B
+
+set_option pp.universes true
+check @prod
diff --git a/tests/lean/prodtst.lean.expected.out b/tests/lean/prodtst.lean.expected.out
new file mode 100644
index 0000000000..f563850f2b
--- /dev/null
+++ b/tests/lean/prodtst.lean.expected.out
@@ -0,0 +1 @@
+prod.{l_1 l_2} : Type.{succ l_1} → Type.{succ l_2} → Type.{max (succ l_1) (succ l_2)}