Section PartialExtendedMapSimulations.
Context {base_fst : BaseParams}.
Context {base_snd : BaseParams}.
Context {multi_fst : MultiParams base_fst}.
Context {multi_snd : MultiParams base_snd}.
Context {name_map : MultiParamsNameTotalMap multi_fst multi_snd}.
Context {msg_map : MultiParamsMsgPartialMap multi_fst multi_snd}.
Context {multi_map : MultiParamsPartialExtendedMap multi_fst multi_snd}.
Context {name_map_bijective : MultiParamsNameTotalMapBijective name_map}.
Context {multi_map_congr : MultiParamsPartialExtendedMapCongruency name_map msg_map multi_map}.
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) |}.
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 H_dec.
- case: H_dec'.
rewrite -H_dec.
by rewrite tot_map_name_inverse_inv.
- 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_ext_map_update_eq.
Qed.
Proof using name_map_bijective multi_map_congr.
move => net net' tr.
case => {net net' tr}.
- move => net net' p ms ms' out d l H_eq H_hnd H_eq'.
destruct (pt_map_packet p) eqn:?.
left.
rewrite H_eq' /= /pt_ext_map_net /=.
have H_eq_dst: tot_map_name (pDst p) = pDst p0.
case: p H_eq H_hnd H_eq' Heqo => /= src dst m H_eq H_hnd H_eq'.
case (pt_map_msg m) => //= m' H_m.
by inversion H_m.
destruct (net_handlers (pDst p0) (pSrc p0) (pBody p0) (pt_ext_map_data (nwState net (pDst p)) (pDst p))) eqn:?.
destruct p1 as [out' d'].
exists [(pDst p0, inr out')].
apply @StepAsync_deliver with (xs := filterMap pt_map_packet ms) (ys := filterMap pt_map_packet ms') (d := pt_ext_map_data d (pDst p)) (l := filterMap pt_map_name_msg l).
* rewrite /= H_eq filterMap_app /=.
case H_p: (pt_map_packet _) => [p1|]; last by rewrite H_p in Heqo.
by rewrite H_p in Heqo; injection Heqo => H_eq_p; rewrite H_eq_p.
* rewrite /=.
rewrite -{2}H_eq_dst tot_map_name_inv_inverse.
case: p H_eq H_hnd H_eq' Heqo H_eq_dst Heqp1 => /= src dst mg H_eq H_hnd H_eq'.
case H_m: (pt_map_msg mg) => [mg'|] //.
case: p0 H_eq' => src' dst' m0 H_eq' H_eq_p.
inversion H_eq_p; subst.
move => H_eq_dst H_eq_n {H_eq_p H_eq_dst}.
simpl in *.
have H_q := @pt_ext_net_handlers_some _ _ _ _ _ _ _ multi_map_congr dst src mg (nwState net dst) _ _ _ _ H_m H_eq_n.
rewrite /pt_ext_mapped_net_handlers in H_q.
rewrite H_hnd in H_q.
find_inversion.
by rewrite tot_map_name_inv_inverse.
* rewrite /= /pt_ext_map_net /= 2!filterMap_app.
rewrite (filterMap_pt_map_packet_map_eq_some _ _ Heqo).
by rewrite (pt_ext_map_update_eq_some _ _ _ Heqo).
right.
rewrite H_eq' /= {H_eq'}.
rewrite /pt_ext_map_net /=.
case: p H_eq H_hnd Heqo => /= src dst m H_eq H_hnd.
case H_m: (pt_map_msg _) => [m'|] // H_eq' {H_eq'}.
rewrite 2!filterMap_app H_eq filterMap_app /=.
case H_m': (pt_map_msg _) => [m'|]; first by rewrite H_m' in H_m.
have [H_d H_l] := pt_ext_net_handlers_none _ _ _ _ H_m H_hnd.
rewrite (filterMap_pt_map_name_msg_empty_eq _ dst H_l) /=.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2.
rewrite /nwS1 /nwS2 /=.
apply functional_extensionality => n.
rewrite /update /=.
case name_eq_dec => H_dec //.
by rewrite H_dec H_d.
by rewrite H_eq_s.
- move => h net net' out inp d l H_hnd H_eq.
destruct (pt_ext_map_input inp h (nwState net h)) eqn:?.
left.
destruct (input_handlers (tot_map_name h) i (pt_ext_map_data (nwState net h) h)) eqn:?.
destruct p  as [out' d'].
exists [(tot_map_name h, inl i); (tot_map_name h, inr out')].
apply (@StepAsync_input _ _ _ _ _ _ _ (pt_ext_map_data d h) (filterMap pt_map_name_msg l)).
rewrite /=.
have H_q := @pt_ext_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h inp (nwState net h) _ _ _ _ Heqo Heqp.
rewrite /pt_ext_mapped_input_handlers /= in H_q.
rewrite H_hnd in H_q.
find_inversion.
by rewrite tot_map_name_inv_inverse.
rewrite /= H_eq /= /pt_ext_map_net /= filterMap_app filterMap_pt_map_packet_map_eq.
by rewrite -pt_ext_map_update_eq.
right.
rewrite H_eq /pt_ext_map_net /=.
have [H_d H_l] := pt_ext_input_handlers_none _ _ _ Heqo H_hnd.
rewrite filterMap_app.
rewrite (filterMap_pt_map_name_msg_empty_eq _ h H_l) /=.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2.
rewrite /nwS1 /nwS2 /=.
apply functional_extensionality => n.
rewrite /update /=.
case name_eq_dec => H_dec //.
by rewrite H_dec H_d.
by rewrite H_eq_s.
Qed.

Corollary step_async_pt_ext_mapped_simulation_star_1 :
forall net tr,
@step_async_star _ multi_fst step_async_init net tr ->
exists tr', @step_async_star _ multi_snd step_async_init (pt_ext_map_net net) tr'.