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

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