Definition cons_inj {A} {a1 a2} {n} {v1 v2 : t A n}
 (eq : a1 :: v1 = a2 :: v2) : a1 = a2 /\ v1 = v2 :=
   match eq in _ = x return caseS _ (fun a2' _ v2' => fun v1' => a1 = a2' /\ v1' = v2') x v1
   with | eq_refl => conj eq_refl eq_refl
   end.

Lemma take_app : forall {A} {n} (v:t A n) {m} (w:t A m) le, take n le (append v w) = v.