Proof using.
intros. induction l; simpl in *; intuition.
repeat break_match; subst; compute; simpl; f_equal; auto.
Qed.

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