Proof using.
intros.
rewrite get_IR_output_keys_log_to_IR.
eauto using NoDup_deduplicate_log.
Qed.
Hint Constructors exported.

Lemma exported_snoc_IO :
forall env_i env_o ir tr i o k,
exported env_i env_o ir tr ->
env_i k = Some i ->
env_o k = Some o ->
exported env_i env_o (ir ++ [IRI k; IRO k]) (tr ++ [(i, o)]).