Proof using iboi.
intros.
find_apply_lem_hyp in_import_in_trace_O.
find_eapply_lem_hyp output_implies_input_before_output; eauto.
eauto using input_before_output_import.
Qed.

Lemma get_IR_input_keys_log_to_IR :
forall l env_o,
get_IR_input_keys key (log_to_IR env_o l) =
map (fun e => (eClient e, eId e)) l.