Proof using.
induction log; simpl; intuition.
repeat break_match; subst; simpl in *;
rewrite get_IR_input_keys_defn; auto using f_equal.
Qed.

Lemma get_IR_output_of_log_to_IR :
forall env log,
get_IR_output_keys _ (log_to_IR env log) =
map (fun e => (eClient e, eId e)) log.