End Subsume.
Hint Resolve osub_reflexive osub_Some_Some subsume_reflexive
subsume_transitive subsume_empty subsume_insert subsume_map : subsume.
(* ---------------------------------------------------------------------------- *)
(* Extending an environment with a list of bindings found in a pattern. *)
(* Note that we cannot define the concatenation of two environments, because we view environments as total functions, so we do not have precise control over their domain. Only a list has finite domain. *)
(* Concatenation is just an iterated version of [insert 0]. *)
Fixpoint concat (A : Type) (e1 : env A) (e2 : list A) : env A :=
match e2 with
| nil =>
e1
| cons a e2 =>
concat (insert 0 a e1) e2
end.
(* Concatenation acts upon the length of the environment in an obvious manner. *)

(* Specifications: *)
Lemma omega_hint_1:
forall n,
n <= (n + 1) - 1.
Lemma length_concat:
forall A (e2 : list A) (e1 : env A) n1 n,
length e1 <= n1 ->
n1 + length e2 = n ->
length (concat e1 e2) <= n.