diff --git a/src/core-logic/term.ml b/src/core-logic/term.ml
index fe4bd9ca..18e542e7 100644
--- a/src/core-logic/term.ml
+++ b/src/core-logic/term.ml
@@ -26,10 +26,13 @@ let[@inline] has_fvars e = (e.flags lsr store_id_bits) land 1 == 1
 let[@inline] store_uid e : int = e.flags land store_id_mask
 let[@inline] is_closed e : bool = db_depth e == 0
 
-let[@inline] ty_exn e : term =
+let[@inline] ty e : term =
   match e.ty with
-  | Some x -> x
-  | None -> assert false
+  | T_ty t -> t
+  | T_ty_delayed f ->
+    let ty = f () in
+    e.ty <- T_ty ty;
+    ty
 
 (* open an application *)
 let unfold_app (e : term) : term * term list =
@@ -45,19 +48,15 @@ let expr_pp_with_ ~pp_ids ~max_depth out (e : term) : unit =
   let rec loop k ~depth names out e =
     let pp' = loop' k ~depth:(depth + 1) names in
     (match e.view with
-    | E_type 0 -> Fmt.string out "type"
-    | E_type i -> Fmt.fprintf out "type_%d" i
+    | E_type 0 -> Fmt.string out "Type"
+    | E_type i -> Fmt.fprintf out "Type(%d)" i
     | E_var v -> Fmt.string out v.v_name
     (* | E_var v -> Fmt.fprintf out "(@[%s : %a@])" v.v_name pp v.v_ty *)
     | E_bound_var v ->
       let idx = v.bv_idx in
       (match CCList.nth_opt names idx with
-      | Some n when n <> "" -> Fmt.string out n
-      | _ ->
-        if idx < k then
-          Fmt.fprintf out "x_%d" (k - idx - 1)
-        else
-          Fmt.fprintf out "%%db_%d" (idx - k))
+      | Some n when n <> "" -> Fmt.fprintf out "%s[%d]" n idx
+      | _ -> Fmt.fprintf out "_[%d]" idx)
     | E_const c -> Const.pp out c
     | (E_app _ | E_lam _) when depth > max_depth ->
       Fmt.fprintf out "@<1>…/%d" e.id
@@ -65,7 +64,7 @@ let expr_pp_with_ ~pp_ids ~max_depth out (e : term) : unit =
       let f, args = unfold_app e in
       Fmt.fprintf out "%a@ %a" pp' f (Util.pp_list pp') args
     | E_lam ("", _ty, bod) ->
-      Fmt.fprintf out "(@[\\x_%d:@[%a@].@ %a@])" k pp' _ty
+      Fmt.fprintf out "(@[\\_:@[%a@].@ %a@])" pp' _ty
         (loop (k + 1) ~depth:(depth + 1) ("" :: names))
         bod
     | E_lam (n, _ty, bod) ->
@@ -80,7 +79,7 @@ let expr_pp_with_ ~pp_ids ~max_depth out (e : term) : unit =
         (loop (k + 1) ~depth:(depth + 1) ("" :: names))
         bod
     | E_pi ("", _ty, bod) ->
-      Fmt.fprintf out "(@[Pi x_%d:@[%a@].@ %a@])" k pp' _ty
+      Fmt.fprintf out "(@[Pi _:@[%a@].@ %a@])" pp' _ty
         (loop (k + 1) ~depth:(depth + 1) ("" :: names))
         bod
     | E_pi (n, _ty, bod) ->
@@ -125,6 +124,8 @@ module Hcons = Hashcons.Make (struct
     | E_app (f1, a1), E_app (f2, a2) -> equal f1 f2 && equal a1 a2
     | E_lam (_, ty1, bod1), E_lam (_, ty2, bod2) ->
       equal ty1 ty2 && equal bod1 bod2
+    | E_pi (_, ty1, bod1), E_pi (_, ty2, bod2) ->
+      equal ty1 ty2 && equal bod1 bod2
     | ( ( E_type _ | E_const _ | E_var _ | E_bound_var _ | E_app _ | E_lam _
         | E_pi _ ),
         _ ) ->
@@ -168,9 +169,7 @@ let iter_shallow ~f (e : term) : unit =
   match e.view with
   | E_type _ -> ()
   | _ ->
-    (match e.ty with
-    | None -> (* should be computed at build time *) assert false
-    | Some ty -> f false ty);
+    f false (ty e);
     (match e.view with
     | E_const _ -> ()
     | E_type _ -> assert false
@@ -284,13 +283,11 @@ let[@inline] is_type_ e =
   | E_type _ -> true
   | _ -> false
 
-let[@inline] is_a_type e = is_type_ e || is_type_ (ty_exn e)
-
 let iter_dag ?(seen = Tbl.create 8) ~iter_ty ~f e : unit =
   let rec loop e =
     if not (Tbl.mem seen e) then (
       Tbl.add seen e ();
-      if iter_ty && not (is_type_ e) then loop (ty_exn e);
+      if iter_ty && not (is_type_ e) then loop (ty e);
       f e;
       iter_shallow e ~f:(fun _ u -> loop u)
     )
@@ -335,7 +332,7 @@ module Make_ = struct
     if is_type_ e then
       0
     else (
-      let d1 = db_depth @@ ty_exn e in
+      let d1 = db_depth @@ ty e in
       let d2 =
         match view e with
         | E_type _ | E_const _ | E_var _ -> 0
@@ -351,7 +348,7 @@ module Make_ = struct
     if is_type_ e then
       false
     else
-      has_fvars (ty_exn e)
+      has_fvars (ty e)
       ||
       match view e with
       | E_var _ -> true
@@ -367,7 +364,7 @@ module Make_ = struct
   let[@inline] universe_of_ty_ (e : term) : int =
     match e.view with
     | E_type i -> i + 1
-    | _ -> universe_ (ty_exn e)
+    | _ -> universe_ (ty e)
 
   module T_int_tbl = CCHashtbl.Make (struct
     type t = term * int
@@ -376,11 +373,12 @@ module Make_ = struct
     let hash (t, k) = H.combine3 27 (hash t) (H.int k)
   end)
 
+  (* shift open bound variables of [e] by [n] *)
   let db_shift_ ~make (e : term) (n : int) =
     let rec loop e k : term =
       if is_closed e then
         e
-      else if is_a_type e then
+      else if is_type_ e then
         e
       else (
         match view e with
@@ -408,8 +406,10 @@ module Make_ = struct
   let db_0_replace_ ~make e ~by:u : term =
     let cache_ = T_int_tbl.create 8 in
 
+    (* recurse in subterm [e], under [k] intermediate binders (so any
+       bound variable under k is bound by them) *)
     let rec aux e k : term =
-      if is_a_type e then
+      if is_type_ e then
         e
       else if db_depth e < k then
         e
@@ -417,7 +417,8 @@ module Make_ = struct
         match view e with
         | E_const _ -> e
         | E_bound_var bv when bv.bv_idx = k ->
-          (* replace here *)
+          (* replace [bv] with [u], and shift [u] to account for the
+             [k] intermediate binders we traversed to get to [bv] *)
           db_shift_ ~make u k
         | _ ->
           (* use the cache *)
@@ -485,24 +486,30 @@ module Make_ = struct
     | E_var v -> Var.ty v
     | E_bound_var v -> Bvar.ty v
     | E_type i -> make (E_type (i + 1))
-    | E_const c -> Const.ty c
-    | E_lam (name, ty, bod) ->
+    | E_const c ->
+      let ty = Const.ty c in
+      if not (is_closed ty) then
+        Error.errorf "const %a@ cannot have a non-closed type like %a" Const.pp
+          c pp_debug ty;
+      ty
+    | E_lam (name, ty_v, bod) ->
       (* type of [\x:tau. bod] is [pi x:tau. typeof(bod)] *)
-      let ty_bod = ty_exn bod in
-      make (E_pi (name, ty, ty_bod))
+      let ty_bod = ty bod in
+      make (E_pi (name, ty_v, ty_bod))
     | E_app (f, a) ->
       (* type of [f a], where [a:tau] and [f: Pi x:tau. ty_bod_f],
          is [ty_bod_f[x := a]] *)
-      let ty_f = ty_exn f in
-      let ty_a = ty_exn a in
+      let ty_f = ty f in
+      let ty_a = ty a in
       (match ty_f.view with
       | E_pi (_, ty_arg_f, ty_bod_f) ->
         (* check that the expected type matches *)
         if not (equal ty_arg_f ty_a) then
           Error.errorf
-            "@[<2>cannot apply %a to %a,@ expected argument type: %a@ actual: \
-             %a@]"
-            pp_debug f pp_debug a pp_debug ty_arg_f pp_debug ty_a;
+            "@[<2>cannot @[apply `%a`@]@ @[to `%a`@],@ expected argument type: \
+             `%a`@ @[actual: `%a`@]@]"
+            pp_debug f pp_debug a pp_debug_with_ids ty_arg_f pp_debug_with_ids
+            ty_a;
         db_0_replace_ ~make ty_bod_f ~by:a
       | _ ->
         Error.errorf
@@ -510,23 +517,30 @@ module Make_ = struct
           pp_debug f pp_debug ty_f)
     | E_pi (_, ty, bod) ->
       (* TODO: check the actual triplets for COC *)
-      Fmt.printf "pi %a %a@." pp_debug ty pp_debug bod;
-      let u = max (universe_of_ty_ ty) (universe_of_ty_ bod) + 1 in
+      (*Fmt.printf "pi %a %a@." pp_debug ty pp_debug bod;*)
+      let u = max (universe_of_ty_ ty) (universe_of_ty_ bod) in
       make (E_type u)
 
+  let ty_assert_false_ () = assert false
+
   (* hashconsing + computing metadata + computing type (for new terms) *)
   let rec make_ (store : store) view : term =
-    let e = { view; ty = None; id = -1; flags = 0 } in
+    let e = { view; ty = T_ty_delayed ty_assert_false_; id = -1; flags = 0 } in
     let e2 = Hcons.hashcons store.s_exprs e in
     if e == e2 then (
       (* new term, compute metadata *)
       assert (store.s_uid land store_id_mask == store.s_uid);
 
       (* first, compute type *)
-      if not (is_type_ e) then (
+      (match e.view with
+      | E_type i ->
+        (* cannot force type now, as it's an infinite tower of types.
+           Instead we will produce the type on demand. *)
+        let get_ty () = make_ store (E_type (i + 1)) in
+        e.ty <- T_ty_delayed get_ty
+      | _ ->
         let ty = compute_ty_ ~make:(make_ store) view in
-        e.ty <- Some ty
-      );
+        e.ty <- T_ty ty);
       let has_fvars = compute_has_fvars_ e in
       e2.flags <-
         (compute_db_depth_ e lsl (1 + store_id_bits))
@@ -606,11 +620,6 @@ end
 
 include Make_
 
-let get_ty store e : term =
-  match e.view with
-  | E_type i -> type_of_univ store (i + 1)
-  | _ -> ty_exn e
-
 (* re-export some internal things *)
 module Internal_ = struct
   let subst_ store ~recursive t subst =
diff --git a/src/core-logic/term.mli b/src/core-logic/term.mli
index 24d4382d..1040262a 100644
--- a/src/core-logic/term.mli
+++ b/src/core-logic/term.mli
@@ -65,12 +65,8 @@ val has_fvars : t -> bool
 (** Does the term contain free variables?
   time: O(1)  *)
 
-val ty_exn : t -> t
-(** Return the type of this term. Fails if the term is a type. *)
-
-val get_ty : store -> t -> t
-(** [get_ty store t] gets the type of [t], or computes it on demand
-    in case [t] is itself a type. *)
+val ty : t -> t
+(** Return the type of this term. *)
 
 (** {2 Creation} *)
 
diff --git a/src/core-logic/types_.ml b/src/core-logic/types_.ml
index f62a5922..69d6e95d 100644
--- a/src/core-logic/types_.ml
+++ b/src/core-logic/types_.ml
@@ -26,12 +26,14 @@ and const = { c_view: const_view; c_ops: const_ops; c_ty: term }
 and term = {
   view: term_view;
   (* computed on demand *)
-  mutable ty: term option;
+  mutable ty: term_ty_;
   mutable id: int;
   (* contains: [highest DB var | 1:has free vars | 5:ctx uid] *)
   mutable flags: int;
 }
 
+and term_ty_ = T_ty of term | T_ty_delayed of (unit -> term)
+
 module Term_ = struct
   let[@inline] equal (e1 : term) e2 : bool = e1 == e2
   let[@inline] hash (e : term) = H.int e.id