Proof using name_map_bijective multi_map_congr msg_map.
apply functional_extensionality => n.
have H_eq := pt_ext_init_handlers_eq.
rewrite H_eq {H_eq}.
by rewrite tot_map_name_inverse_inv.
Qed.
Definition pt_ext_map_net (net : @network  _ multi_fst) : @network _ multi_snd :=
{| nwPackets := filterMap pt_map_packet net.(nwPackets) ;
nwState := fun n => pt_ext_map_data (net.(nwState) (tot_map_name_inv n)) (tot_map_name_inv n) |}.

Lemma pt_ext_map_update_eq :
forall f h d,
(fun n : name => pt_ext_map_data (update name_eq_dec f h d (tot_map_name_inv n)) (tot_map_name_inv n)) =
update name_eq_dec (fun n : name => pt_ext_map_data (f (tot_map_name_inv n)) (tot_map_name_inv n)) (tot_map_name h) (pt_ext_map_data d h).