diff --git a/src/subsequences.v b/src/subsequences.v
index e659f0b..3bf0798 100644
--- a/src/subsequences.v
+++ b/src/subsequences.v
@@ -6,12 +6,12 @@ Import ListNotations.
 Require Import Lia.
 
 
-Definition subsequence (l s : list Type) :=
-  exists (l1: list Type) (l2 : list (list Type)),
+Definition subsequence {X: Type} (l s : list X) :=
+  exists (l1: list X) (l2 : list (list X)),
     length s = length l2
     /\ l = l1 ++ flat_map (fun e => (fst e) :: (snd e)) (combine s l2).
 
-Definition subsequence2 (l s : list Type) :=
+Definition subsequence2 {X: Type} (l s : list X) :=
   exists (t: list bool),
     length t = length l /\ s = map snd (filter fst (combine t l)).
 
@@ -112,10 +112,10 @@ Proof.
   destruct x. destruct x0.
   apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
   simpl in H0. inversion H0. exists l0. exists x0.
-  inversion H. split; reflexivity.
+  inversion H. rewrite H3. split; reflexivity.
   destruct x0. simpl in H0.
   apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
-  exists (x ++ (a::l0)). exists x0. inversion H. split. reflexivity.
+  exists (x ++ (a::l0)). exists x0. inversion H. rewrite H2. split. reflexivity.
   inversion H0. rewrite <- app_assoc. apply app_inv_head_iff.
   rewrite app_comm_cons. reflexivity.
 
@@ -130,7 +130,7 @@ Theorem subsequence2_cons_eq : forall (l1 l2: list Type) (a: Type),
 Proof.
   intros l s a. split. intro H. unfold subsequence2 in H.
   destruct H. destruct H. destruct x. inversion H0.
-  destruct b. exists (x). split. inversion H. reflexivity. simpl in H0.
+  destruct b. exists (x). split. inversion H. rewrite H2. reflexivity. simpl in H0.
   inversion H0. reflexivity. simpl in H0. inversion H.
   assert (subsequence2 l (a::s)). exists x. split; assumption.
   apply subsequence2_cons_r with (a := a). assumption.
@@ -200,7 +200,7 @@ Qed.
 
 Theorem subsequence_eq_def :
   (forall x y : Type, {x = y} + {x <> y})
-  -> (forall l s, subsequence l s <-> subsequence2 l s).
+  -> (forall l s : list Type, subsequence l s <-> subsequence2 l s).
 Proof.
   intro I. intro l. induction l.
   (* first part of the induction *)