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).