Definition var_env_or (ve1 ve2 : var_env) (x : ad) := ve1 x || ve2 x.
Definition var_env'_or (ve1 ve2 : var_env') (x : nat) := ve1 x || ve2 x.

Lemma renamefS :
forall (be : bool_expr) (n : nat),
n < N ->
renamef (renfnad (S n)) be =
subst (ap n) (Var (ap' N n)) (renamef (renfnad n) be).