Proof using MB.
move => n n'.
case (name_eq_dec n n') => H_dec //.
move => H_eq.
case: H_dec.
by rewrite -(tot_map_name_inv_inverse n) H_eq tot_map_name_inv_inverse.
Qed.
End TotalMapBijective.
Section TotalMapSimulations.
Context {base_fst : BaseParams}.
Context {base_snd : BaseParams}.
Context {multi_fst : MultiParams base_fst}.
Context {multi_snd : MultiParams base_snd}.
Context {base_map : BaseParamsTotalMap base_fst base_snd}.
Context {name_map : MultiParamsNameTotalMap multi_fst multi_snd}.
Context {msg_map : MultiParamsMsgTotalMap multi_fst multi_snd}.
Context {name_map_bijective : MultiParamsNameTotalMapBijective name_map}.
Context {multi_map_congr : MultiParamsTotalMapCongruency base_map name_map msg_map}.
Definition tot_map_trace_occ (e : @name _ multi_fst * (@input base_fst + list (@output base_fst))) :=
match e with
| (n, inl io) => (tot_map_name n, inl (tot_map_input io))
| (n, inr lo) => (tot_map_name n, inr (map tot_map_output lo))
end.
Definition tot_map_packet (p : @packet base_fst multi_fst)  :=
match p with
| mkPacket src dst m =>
mkPacket (tot_map_name src) (tot_map_name dst) (tot_map_msg m)
end.
Definition tot_map_net (net : @network _ multi_fst) : @network _ multi_snd :=
{| nwPackets := map tot_map_packet net.(nwPackets) ;
nwState := fun n => tot_map_data (net.(nwState) (tot_map_name_inv n)) |}.

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