Proof using overlay_map_congr name_map_bijective fail_msg_map_congr.
move => h failed.
rewrite tot_fail_msg_fst_snd.
exact: map_map_pair_eq.
Qed.
Definition tot_map_onet (onet : @ordered_network _ multi_fst) : @ordered_network _ multi_snd :=
{| onwPackets := fun src dst => map tot_map_msg (onet.(onwPackets) (tot_map_name_inv src) (tot_map_name_inv dst)) ;
onwState := fun n => tot_map_data (onet.(onwState) (tot_map_name_inv n)) |}.

(* Specifications: *)
Lemma map_msg_update2 :
forall f ms to from,
(fun src dst => map tot_map_msg (update2 name_eq_dec f from to ms (tot_map_name_inv src) (tot_map_name_inv dst))) =
update2 name_eq_dec (fun src0 dst0 : name => map tot_map_msg (f (tot_map_name_inv src0) (tot_map_name_inv dst0)))
(tot_map_name from) (tot_map_name to) (map tot_map_msg ms).