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)) |}.
Proof using name_map_bijective.
move => net d h.
apply functional_extensionality => n.
rewrite /update /=.
repeat break_match => //.
- by find_reverse_rewrite; find_rewrite_lem tot_map_name_inverse_inv.
- by find_rewrite; find_rewrite_lem tot_map_name_inv_inverse.
Qed.
Proof using name_map_bijective.
move => f.
case => src dst m d.
exact: tot_map_update_eq.
Qed.
Proof using.
elim => //=.
case => /= n m' l IH h.
by rewrite IH.
Qed.
Proof using name_map_bijective multi_map_congr msg_map.
apply functional_extensionality => n.
by rewrite tot_init_handlers_eq tot_map_name_inverse_inv.
Qed.
Proof using.
case => //=.
case.
move => n ol tr n' lo H_eq.
case: tr H_eq => //=.
case: ol => //=.
move => out H_eq.
exists n; exists out.
split => //.
by inversion H_eq.
Qed.

Lemma tot_map_trace_occ_in_inv :
forall tr h inp out,
map tot_map_trace_occ tr = [(h, inl inp); (h, inr out)] ->
exists h', exists inp', exists out', tr = [(h', inl inp'); (h', inr out')] /\
tot_map_name h' = h /\ map tot_map_output out' = out /\ tot_map_input inp' = inp.