diff --git a/src/ciclib/reduce.ml b/src/ciclib/reduce.ml
new file mode 100644
index 00000000..d55a83dd
--- /dev/null
+++ b/src/ciclib/reduce.ml
@@ -0,0 +1,28 @@
+open Types_
+module T = Term
+
+type machine = { hd: T.t; args: T.t list; env: T.t list }
+
+let t_of_machine (m : machine) : T.t =
+  T.app_l (T.DB.subst_db_l m.hd m.env) m.args
+
+let rec reduce (m : machine) : machine =
+  match T.view m.hd, m.args with
+  | E_bound_var v, _ when v.bv_idx < List.length m.env ->
+    (* lookup in env *)
+    let t' = List.nth m.env v.bv_idx in
+    reduce { m with hd = t'; env = [] }
+  | E_lam (_, _, _ty_arg, body), arg :: tl_args ->
+    (* beta-reduce*)
+    let m' = { hd = body; env = arg :: m.env; args = tl_args } in
+    reduce m'
+  | E_app (f, a), _ ->
+    (* push onto m.args *)
+    let a = T.DB.subst_db_l a m.env in
+    reduce { m with hd = f; args = a :: m.args }
+  | (E_bound_var _ | E_type _ | E_const (_, _) | E_lam _ | E_pi _), _ -> m
+
+let beta_reduce (t : T.t) : T.t =
+  let hd, args = T.unfold_app t in
+  let m = { hd; args; env = [] } in
+  reduce m |> t_of_machine
diff --git a/src/ciclib/reduce.mli b/src/ciclib/reduce.mli
new file mode 100644
index 00000000..3debe500
--- /dev/null
+++ b/src/ciclib/reduce.mli
@@ -0,0 +1,5 @@
+(** Normalization *)
+
+module T = Term
+
+val beta_reduce : T.t -> T.t
diff --git a/src/ciclib/sidekick_cic_lib.ml b/src/ciclib/sidekick_cic_lib.ml
index fae77242..1ab9629c 100644
--- a/src/ciclib/sidekick_cic_lib.ml
+++ b/src/ciclib/sidekick_cic_lib.ml
@@ -2,3 +2,4 @@ module Term = Term
 module Bvar = Bvar
 module Const = Const
 module Level = Level
+module Reduce = Reduce
diff --git a/src/ciclib/term.ml b/src/ciclib/term.ml
index 6508179e..2d6996f4 100644
--- a/src/ciclib/term.ml
+++ b/src/ciclib/term.ml
@@ -174,9 +174,9 @@ let map_shallow ~f (e : term) : term =
       make_ (E_pi (b, name, tyv', bod'))
 
 (* shift open bound variables of [e] by [n] *)
-let db_shift_ ~make (e : term) (n : int) =
+let db_shift_ (e : term) (n : int) =
   let rec loop e k : term =
-    if is_closed e then
+    if is_closed e || db_depth e < k then
       e
     else if is_type e then
       e
@@ -184,7 +184,7 @@ let db_shift_ ~make (e : term) (n : int) =
       match view e with
       | E_bound_var bv ->
         if bv.bv_idx >= k then
-          make (E_bound_var (Bvar.make (bv.bv_idx + n)))
+          make_ (E_bound_var (Bvar.make (bv.bv_idx + n)))
         else
           e
       | _ ->
@@ -203,21 +203,25 @@ let db_shift_ ~make (e : term) (n : int) =
     loop e 0
 
 (* replace DB0 in [e] with [u] *)
-let db_0_replace_ ~make e ~by:u : term =
+let db_replace_ e (env : t list) : term =
+  let len_env = List.length env 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_type e then
       e
     else if db_depth e < k then
+      (* no open variable *)
       e
     else (
       match view e with
       | E_const _ -> e
-      | E_bound_var bv when bv.bv_idx = k ->
-        (* replace [bv] with [u], and shift [u] to account for the
+      | E_bound_var v when v.bv_idx >= k && v.bv_idx < k + len_env ->
+        (* replace [v] with [env[v-k]], and shift it to account for the
            [k] intermediate binders we traversed to get to [bv] *)
-        db_shift_ ~make u k
+        let u = List.nth env (v.bv_idx - k) in
+        db_shift_ u k
       | _ ->
         map_shallow e ~f:(fun inb u ->
             aux u
@@ -229,24 +233,27 @@ let db_0_replace_ ~make e ~by:u : term =
   in
   if is_closed e then
     e
+  else if len_env = 0 then
+    e
   else
     aux e 0
 
-let type_of_univ lvl : term = make_ (E_type lvl)
-let type_of_univ_int i : term = type_of_univ (Level.of_int i)
+let[@inline] type_of_univ lvl : term = make_ (E_type lvl)
+let[@inline] type_of_univ_int i : term = type_of_univ (Level.of_int i)
 let type_ : term = type_of_univ Level.one
-let bvar v : term = make_ (E_bound_var v)
-let bvar_i i : term = make_ (E_bound_var (Bvar.make i))
-let const c args : term = make_ (E_const (c, args))
-let app f a = make_ (E_app (f, a))
-let app_l f l = List.fold_left app f l
-let lam b str ~var_ty bod : term = make_ (E_lam (b, str, var_ty, bod))
-let pi b str ~var_ty bod : term = make_ (E_pi (b, str, var_ty, bod))
+let[@inline] bvar v : term = make_ (E_bound_var v)
+let[@inline] bvar_i i : term = make_ (E_bound_var (Bvar.make i))
+let[@inline] const c args : term = make_ (E_const (c, args))
+let[@inline] app f a = make_ (E_app (f, a))
+let[@inline] app_l f l = List.fold_left app f l
+let[@inline] lam b str ~var_ty bod : term = make_ (E_lam (b, str, var_ty, bod))
+let[@inline] pi b str ~var_ty bod : term = make_ (E_pi (b, str, var_ty, bod))
 
 module DB = struct
-  let subst_db0 e ~by : t = db_0_replace_ ~make:make_ e ~by
+  let[@inline] subst_db0 e ~by : t = db_replace_ e [ by ]
+  let[@inline] subst_db_l e env : t = db_replace_ e env
 
-  let shift t ~by : t =
+  let[@inline] shift t ~by : t =
     assert (by >= 0);
-    db_shift_ ~make:make_ t by
+    db_shift_ t by
 end
diff --git a/src/ciclib/term.mli b/src/ciclib/term.mli
index 7565c597..f87338ed 100644
--- a/src/ciclib/term.mli
+++ b/src/ciclib/term.mli
@@ -76,8 +76,9 @@ module DB : sig
       the term [by]. This is useful, for example, to implement beta-reduction.
 
       For example, with [t] being [_[0] = (\x. _[2] _[1] x[0])],
-      [subst_db0 store t ~by:"hello"] is ["hello" = (\x. _[2] "hello" x[0])].
-  *)
+      [subst_db0 store t ~by:"hello"] is ["hello" = (\x. _[2] "hello" x[0])]. *)
+
+  val subst_db_l : t -> t list -> t
 
   val shift : t -> by:int -> t
   (** [shift t ~by] shifts all bound variables in [t] that are not