(* Two lemmas about equations of the form [insert x a1 e1 = insert x a2 e2]. Note that we have [a1 = a2], but not [e1 = e2], due to padding. *)

(* Specifications: *)
Lemma insert_eq_insert_1:
forall A x a1 a2 (e1 e2 : env A),
insert x a1 e1 = insert x a2 e2 ->
a1 = a2.
Lemma insert_eq_insert_2:
forall A x a1 a2 (e1 e2 : env A),
insert x a1 e1 = insert x a2 e2 ->
forall b,
insert x b e1 = insert x b e2.