Definition var_env''le (ve1 ve2 : var_env'') :=
forall x : ad, in_dom _ x ve1 = true -> in_dom _ x ve2 = true.

Lemma increasing_seq :
forall (n : nat) (f : nat -> nat),
(forall k : nat, f k <= n) ->
(forall k : nat, f k <= f (S k)) ->
exists m : nat, m <= n - f 0 /\ f m = f (S m).