Proof using.
induction 1; intros; simpl; auto.
Qed.

(* Specifications: *)
Lemma exported_snoc_IU :
forall env_i env_o ir tr i k o,
exported env_i env_o ir tr ->
env_i k = Some i ->
env_o k = None ->
exported env_i env_o (ir ++ [IRI k; IRU k]) (tr ++ [(i, o)]).
Lemma log_to_IR_app :
forall xs ys env,
log_to_IR env (xs ++ ys) = log_to_IR env xs ++ log_to_IR env ys.