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

(* Specifications: *)
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.
Lemma deduplicate_log'_ks :
forall l ks e id,
In e (deduplicate_log' l ks) ->
assoc clientId_eq_dec ks (eClient e) = Some id ->
id < (eId e).