Section PartialMapSimulations.
Context {base_fst : BaseParams}.
Context {base_snd : BaseParams}.
Context {multi_fst : MultiParams base_fst}.
Context {multi_snd : MultiParams base_snd}.
Context {base_map : BaseParamsPartialMap base_fst base_snd}.
Context {name_map : MultiParamsNameTotalMap multi_fst multi_snd}.
Context {msg_map : MultiParamsMsgPartialMap multi_fst multi_snd}.
Context {name_map_bijective : MultiParamsNameTotalMapBijective name_map}.
Context {multi_map_congr : MultiParamsPartialMapCongruency base_map name_map msg_map}.
Definition pt_map_trace_occ (e : @name _ multi_fst * (@input base_fst + list (@output base_fst))) :
option (@name _ multi_snd * (@input base_snd + list (@output base_snd))) :=
match e with
| (n, inl io) =>
match pt_map_input io with
| Some io' => Some (tot_map_name n, inl io')
| None => None
end
| (n, inr out) => Some (tot_map_name n, inr (filterMap pt_map_output out))
end.
Definition pt_map_packet (p : @packet _ multi_fst)  :=
match p with
| mkPacket src dst m =>
match pt_map_msg m with
| Some m' => Some (mkPacket (tot_map_name src) (tot_map_name dst) m')
| None => None
end
end.
Definition pt_map_net (net : @network _ multi_fst) : @network _ multi_snd :=
{| nwPackets := filterMap pt_map_packet net.(nwPackets) ;
nwState := fun n => pt_map_data (net.(nwState) (tot_map_name_inv n)) |}.
Proof using name_map_bijective multi_map_congr msg_map.
apply functional_extensionality => n.
have H_eq := pt_init_handlers_eq.
rewrite H_eq {H_eq}.
by rewrite tot_map_name_inverse_inv.
Qed.
Proof using.
elim => //=.
case => /= n m l IH dst.
rewrite /pt_map_name_msg /=.
case pt_map_msg => [m'|] H_eq //=.
by rewrite IH.
Qed.
Proof using.
move => l h.
elim: l => //=.
case => n m l IH.
rewrite /pt_map_name_msg /=.
case pt_map_msg => [m'|] //.
by rewrite IH.
Qed.
Proof using.
move => l; case => /= src dst m p.
case H_m: (pt_map_msg m) => [m'|] // H_eq.
injection H_eq => H_eq_p.
rewrite -H_eq_p /=.
exact: filterMap_pt_map_packet_map_eq.
Qed.
Proof using name_map_bijective.
move => f h d.
apply functional_extensionality => n.
rewrite /update /=.
case name_eq_dec => H_dec; case name_eq_dec => H_dec' //.
rewrite -H_dec in H_dec'.
by rewrite tot_map_name_inverse_inv in H_dec'.
rewrite H_dec' in H_dec.
by rewrite tot_map_name_inv_inverse in H_dec.
Qed.
Proof using name_map_bijective.
move => net d p p'.
case: p => src dst m.
case: p' => src' dst' m' /=.
case H_eq: (pt_map_msg _) => [m0|] // H_eq'.
inversion H_eq'; subst.
move {H_eq H_eq'}.
exact: pt_map_update_eq.
Qed.

Theorem step_async_pt_mapped_simulation_1 :
forall net net' tr,
@step_async _ multi_fst net net' tr ->
@step_async _ multi_snd (pt_map_net net) (pt_map_net net') (filterMap pt_map_trace_occ tr) \/
(pt_map_net net' = pt_map_net net /\ filterMap trace_non_empty_out (filterMap pt_map_trace_occ tr) = []).