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 renamef_id : forall be : bool_expr, renamef (fun x => x) be = be.