Proof using.
intros.
rewrite get_IR_input_keys_log_to_IR.
eauto using NoDup_deduplicate_log.
Qed.

(* Specifications: *)
Lemma NoDup_output_log :
forall l env_o,
NoDup (get_IR_output_keys key (log_to_IR env_o (deduplicate_log l))).