(* Pain, pain. *)

(* Specifications: *)
Lemma length_insert_reverse_1:
forall A (e : env A) k x a,
length (insert x a e) <= k ->
x < k.
Lemma length_insert_reverse_2:
forall A (e : env A) k x a,
length (insert x a e) <= k + 1 ->
length e <= k.
Lemma length_insert_independent:
forall A (e : env A) k x a,
length (insert x a e) <= k ->
forall y o,
y < k ->
length (raw_insert y o e) <= k.