Section SerializedMsgCorrect.
Context {orig_base_params : BaseParams}.
Context {orig_multi_params : MultiParams orig_base_params}.
Context {orig_failure_params : FailureParams orig_multi_params}.
Context {orig_name_overlay_params : NameOverlayParams orig_multi_params}.
Context {orig_fail_msg_params : FailMsgParams orig_multi_params}.
Context {orig_new_msg_params : NewMsgParams orig_multi_params}.
Context {orig_msg_serializer : Serializer msg}.
Definition serialize_packet p :=
@mkPacket _ serialized_multi_params
(@pSrc _ orig_multi_params p)
(pDst p)
(serialize_top (serialize (pBody p))).
Definition serialize_net (net : @network _ orig_multi_params) : (@network _ serialized_multi_params) :=
@mkNetwork _ serialized_multi_params (map serialize_packet (nwPackets net)) net.(nwState).
Definition serialize_onet (net : @ordered_network _ orig_multi_params) : (@ordered_network _ serialized_multi_params) :=
@mkONetwork _ serialized_multi_params (fun src dst => map (fun v => serialize_top (serialize v)) (net.(onwPackets) src dst)) net.(onwState).
Definition serialize_odnet (net : @ordered_dynamic_network _ orig_multi_params) : (@ordered_dynamic_network _ serialized_multi_params) :=
@mkODNetwork _ serialized_multi_params net.(odnwNodes) (fun src dst => map (fun v => serialize_top (serialize v)) (net.(odnwPackets) src dst)) net.(odnwState).
Instance orig_multi_params_name_tot_map :
MultiParamsNameTotalMap orig_multi_params serialized_multi_params :=
{
tot_map_name := id ;
tot_map_name_inv := id
}.
Instance orig_multi_params_name_tot_map_bijective :
MultiParamsNameTotalMapBijective orig_multi_params_name_tot_map :=
{
tot_map_name_inv_inverse := fun _ => eq_refl ;
tot_map_name_inverse_inv := fun _ => eq_refl
}.
Instance orig_base_params_tot_map :
BaseParamsTotalMap orig_base_params serialized_base_params :=
{
tot_map_data := id;
tot_map_input := id;
tot_map_output := id
}.
Instance orig_multi_params_tot_msg_map :
MultiParamsMsgTotalMap orig_multi_params serialized_multi_params :=
{
tot_map_msg := fun v => serialize_top (serialize v)
}.
Instance orig_failure_params_tot_map_congruency : FailureParamsTotalMapCongruency orig_failure_params serialized_failure_params orig_base_params_tot_map :=
{
tot_reboot_eq := fun _ => eq_refl
}.
Instance orig_multi_name_overlay_params_tot_map_congruency : NameOverlayParamsTotalMapCongruency orig_name_overlay_params serialized_name_overlay_params orig_multi_params_name_tot_map :=
{
tot_adjacent_to_fst_snd := fun _ _ => conj (fun H => H) (fun H => H)
}.
Instance orig_multi_fail_msg_params_tot_map_congruency : FailMsgParamsTotalMapCongruency orig_fail_msg_params serialized_fail_msg_params orig_multi_params_tot_msg_map :=
{
tot_fail_msg_fst_snd := eq_refl
}.
Instance orig_multi_new_msg_params_tot_map_congruency : NewMsgParamsTotalMapCongruency orig_new_msg_params serialized_new_msg_params orig_multi_params_tot_msg_map :=
{
tot_new_msg_fst_snd := eq_refl
}.
Program Instance orig_multi_params_map_congruency :
MultiParamsTotalMapCongruency orig_base_params_tot_map
orig_multi_params_name_tot_map orig_multi_params_tot_msg_map :=
{
tot_init_handlers_eq := fun _ => eq_refl ;
tot_net_handlers_eq := _ ;
tot_input_handlers_eq := _
}.
Next Obligation.
rewrite /tot_mapped_net_handlers /= /tot_map_name_msgs /= /id /=.
repeat break_let.
rewrite /serialized_net_handlers.
rewrite serialize_deserialize_top_id.
rewrite /serialize_handler_result.
repeat break_let.
find_injection.
rewrite map_id.
set l1 := map _ l.
set l2 := map _ l.
suff H_suff: l1 = l2 by rewrite H_suff.
rewrite /l1 /l2.
clear.
elim: l => //=.
move => p l IH.
rewrite IH /= /serialize_name_msg_tuple /=.
by break_let.
Qed.
Next Obligation.
rewrite /tot_mapped_input_handlers /=.
repeat break_let.
unfold serialized_input_handlers in *.
rewrite /serialize_handler_result /id /tot_map_name_msgs /tot_map_name /= /id /=.
repeat break_let.
find_injection.
rewrite map_id.
set l1 := map _ l.
set l2 := map _ l.
suff H_suff: l1 = l2 by rewrite H_suff.
rewrite /l1 /l2.
clear.
elim: l => //=.
move => p l IH.
rewrite IH /= /serialize_name_msg_tuple /=.
by break_let.
Qed.
Proof using.
move => net.
rewrite /tot_map_odnet.
rewrite /tot_map_name /= /id /= map_id.
rewrite /serialize_odnet.
set f := fun _ => match _ with _ => _ end.
by have ->: odnwState net = f by rewrite /f; apply functional_extensionality => n; case: odnwState.
Qed.
Proof using.
move => net net' out H_step.
apply step_async_tot_mapped_simulation_1 in H_step.
rewrite /tot_map_name /tot_map_net /= /id /= in H_step.
rewrite /tot_map_trace_occ /= /id /= in H_step.
rewrite /tot_map_packet /= /id /= in H_step.
rewrite /serialize_net.
move: H_step.
set fp := fun p => _.
have H_eq: fp = serialize_packet by rewrite /fp ; apply functional_extensionality; case => /= src dst m.
rewrite H_eq {H_eq fp}.
set mp := fun e : name * _ => _.
suff H_eq': map mp out = out by rewrite H_eq'.
elim: out => //=.
case => a.
case => /= [i|o] l IH; rewrite IH //.
by rewrite map_id.
Qed.
Proof using.
move => net tr H_step.
apply step_async_tot_mapped_simulation_star_1 in H_step.
rewrite /tot_map_name /tot_map_net /= /id /= in H_step.
rewrite /tot_map_trace_occ /= /id /= in H_step.
rewrite /tot_map_packet /= /id /= in H_step.
rewrite /serialize_net.
move: H_step.
set fp := fun p : packet => _.
have H_eq: fp = serialize_packet by rewrite /fp ; apply functional_extensionality; case => /= src dst m.
rewrite H_eq {H_eq fp}.
set mp := fun e : name * _ => _.
suff H_eq': map mp tr = tr by rewrite H_eq'.
elim: tr => //=.
case => a.
case => /= [i|o] l IH; rewrite IH //.
by rewrite map_id.
Qed.
Proof using.
move => net net' failed failed' tr H_step.
apply step_failure_tot_mapped_simulation_1 in H_step.
rewrite /tot_map_name /tot_map_net /= /id /= in H_step.
rewrite /tot_map_trace_occ /= /id /= in H_step.
rewrite /tot_map_packet /= /id /= in H_step.
rewrite /serialize_net.
move: H_step.
rewrite 2!map_id.
set fp := fun p : packet => _.
have H_eq: fp = serialize_packet by rewrite /fp ; apply functional_extensionality; case => /= src dst m.
rewrite H_eq {H_eq fp}.
set mp := fun e : name * _ => _.
suff H_eq': map mp tr = tr by rewrite H_eq'.
elim: tr => //=.
case => a.
case => /= [i|o] l IH; rewrite IH //.
by rewrite map_id.
Qed.
Proof using.
move => net failed tr H_step.
apply step_failure_tot_mapped_simulation_star_1 in H_step.
rewrite /tot_map_name /tot_map_net /= /id /= in H_step.
rewrite /tot_map_packet /= /id /= in H_step.
rewrite /serialize_net.
move: H_step.
rewrite map_id.
set fp := fun p : packet => _.
have H_eq: fp = serialize_packet by rewrite /fp ; apply functional_extensionality; case => /= src dst m.
rewrite H_eq {H_eq fp}.
rewrite /tot_map_trace_occ /=.
set fp := fun e : name * _ => _.
suff H_eq': map fp tr = tr by rewrite H_eq'.
elim: tr => //=.
case => a.
case => /= [i|o] l IH; rewrite IH //.
by rewrite map_id.
Qed.
Proof using.
move => net net' failed failed' tr H_step.
have H_eq_n: failed = map (@tot_map_name _ _ _ _ orig_multi_params_name_tot_map) failed by rewrite /tot_map_name /= map_id.
rewrite {1}H_eq_n.
have H_eq_n': failed' = map (@tot_map_name _ _ _ _ orig_multi_params_name_tot_map) failed' by rewrite /tot_map_name /= map_id.
rewrite {1}H_eq_n'.
have H_eq_tr: tr = map (@tot_map_trace _ _ _ _ _ orig_multi_params_name_tot_map) tr.
rewrite /tot_map_trace /= /id /= {H_step}.
elim: tr => //=.
by case => a; case => [i|o] l IH; rewrite -IH.
rewrite H_eq_tr.
exact: step_ordered_failure_tot_mapped_simulation_1.
Qed.
Proof using.
move => net failed tr H_st.
have ->: failed = map (@tot_map_name _ _ _ _ orig_multi_params_name_tot_map) failed by rewrite /tot_map_name /= map_id.
have ->: tr = map (@tot_map_trace _ _ _ _ _ orig_multi_params_name_tot_map) tr.
rewrite /tot_map_trace /= /id /= {H_st}.
elim: tr => //=.
by case => a; case => [i|o] l IH; rewrite -IH.
exact: step_ordered_failure_tot_mapped_simulation_star_1.
Qed.
Proof using.
move => net net' failed failed' tr H_nd H_step.
have H_eq_n: failed = map (@tot_map_name _ _ _ _ orig_multi_params_name_tot_map) failed by rewrite /tot_map_name /= map_id.
rewrite {1}H_eq_n.
have H_eq_n': failed' = map (@tot_map_name _ _ _ _ orig_multi_params_name_tot_map) failed' by rewrite /tot_map_name /= map_id.
rewrite {1}H_eq_n'.
rewrite 2!serialize_odnet_tot_map_odnet.
have ->: tr = map (@tot_map_trace _ _ _ _ _ orig_multi_params_name_tot_map) tr.
rewrite /tot_map_trace /= /id /= {H_step}.
elim: tr => //=.
by case => a; case => [i|o] l IH; rewrite -IH.
exact: step_ordered_dynamic_failure_tot_mapped_simulation_1.
Qed.
Proof using.
move => net failed tr H_st.
have ->: failed = map (@tot_map_name _ _ _ _ orig_multi_params_name_tot_map) failed by rewrite /tot_map_name /= map_id.
rewrite serialize_odnet_tot_map_odnet.
have ->: tr = map (@tot_map_trace _ _ _ _ _ orig_multi_params_name_tot_map) tr.
rewrite /tot_map_trace /= /id /= {H_st}.
elim: tr => //=.
by case => a; case => [i|o] l IH; rewrite -IH.
exact: step_ordered_dynamic_failure_tot_mapped_simulation_star_1.
Qed.
Definition deserialize_packet (p : @packet _ serialized_multi_params) : option (@packet _ orig_multi_params) :=
match deserialize_top deserialize (pBody p) with
| None => None
| Some body =>
Some (@mkPacket _ orig_multi_params (pSrc p) (pDst p) body)
end.
Definition deserialize_net (net: @network _ serialized_multi_params) :  (@network _ orig_multi_params) :=
@mkNetwork _ orig_multi_params (filterMap deserialize_packet (nwPackets net)) net.(nwState).
Definition deserialize_onet (net: @ordered_network _ serialized_multi_params) :  (@ordered_network _ orig_multi_params) :=
@mkONetwork _ orig_multi_params (fun src dst => filterMap (fun m => match deserialize_top deserialize m with Some data => Some data | None => None end) (net.(onwPackets) src dst)) net.(onwState).
Definition deserialize_odnet (net: @ordered_dynamic_network _ serialized_multi_params) :  (@ordered_dynamic_network _ orig_multi_params) :=
@mkODNetwork _ orig_multi_params net.(odnwNodes) (fun src dst => filterMap (fun m => match deserialize_top deserialize m with Some data => Some data | None => None end) (net.(odnwPackets) src dst)) net.(odnwState).
Proof using.
case => ps nwS.
rewrite /deserialize_net /serialize_net /=.
set ps' := filterMap _ _.
have H_p: ps' = ps.
rewrite /ps'.
clear.
rewrite /deserialize_packet.
elim: ps => [|p ps IH] //=.
rewrite serialize_deserialize_top_id IH.
by case: p.
by rewrite H_p.
Qed.
Proof using.
case => ps s.
rewrite /deserialize_onet /serialize_onet /=.
set ps' := fun _ _ => filterMap _ _.
have H_p: ps' = ps.
rewrite /ps'.
clear.
apply functional_extensionality => src.
apply functional_extensionality => dst.
elim: (ps src dst) => [|m ms IH] //=.
by rewrite serialize_deserialize_top_id IH.
by rewrite H_p.
Qed.
Proof using.
case => ns ps s.
rewrite /deserialize_odnet /serialize_odnet /=.
set ps' := fun _ _ => filterMap _ _.
have H_p: ps' = ps.
rewrite /ps'.
clear.
apply functional_extensionality => src.
apply functional_extensionality => dst.
elim: (ps src dst) => [|m ms IH] //=.
by rewrite serialize_deserialize_top_id IH.
by rewrite H_p.
Qed.
Instance multi_params_orig_name_tot_map :
MultiParamsNameTotalMap serialized_multi_params orig_multi_params :=
{
tot_map_name := id ;
tot_map_name_inv := id
}.
Instance multi_params_orig_name_tot_map_bijective :
MultiParamsNameTotalMapBijective multi_params_orig_name_tot_map :=
{
tot_map_name_inv_inverse := fun _ => eq_refl _;
tot_map_name_inverse_inv := fun _ => eq_refl _
}.
Instance multi_orig_params_pt_msg_map : MultiParamsMsgPartialMap serialized_multi_params orig_multi_params :=
{
pt_map_msg := fun m =>
match deserialize_top deserialize m with
| Some data => Some data
| None => None
end
}.
Instance multi_orig_base_params_pt_map : BaseParamsPartialMap serialized_base_params orig_base_params :=
{
pt_map_data := id;
pt_map_input := fun i => Some i;
pt_map_output := fun o => Some o
}.
Instance multi_orig_failure_params_pt_map_congruency : FailureParamsPartialMapCongruency serialized_failure_params orig_failure_params multi_orig_base_params_pt_map :=
{
pt_reboot_eq := fun _ => eq_refl _
}.
Instance multi_orig_name_overlay_params_tot_map_congruency : NameOverlayParamsTotalMapCongruency serialized_name_overlay_params orig_name_overlay_params multi_params_orig_name_tot_map :=
{
tot_adjacent_to_fst_snd := fun _ _ => conj (fun H => H) (fun H => H)
}.
Program Instance multi_orig_fail_msg_params_pt_map_congruency : FailMsgParamsPartialMapCongruency serialized_fail_msg_params orig_fail_msg_params multi_orig_params_pt_msg_map :=
{
pt_fail_msg_fst_snd := _
}.
Next Obligation.
by rewrite serialize_deserialize_top_id.
Qed.
Program Instance multi_orig_new_msg_params_pt_map_congruency : NewMsgParamsPartialMapCongruency serialized_new_msg_params orig_new_msg_params multi_orig_params_pt_msg_map :=
{
pt_new_msg_fst_snd := _
}.
Next Obligation.
by rewrite serialize_deserialize_top_id.
Qed.
Program Instance multi_orig_params_pt_map_congruency : MultiParamsPartialMapCongruency multi_orig_base_params_pt_map multi_params_orig_name_tot_map multi_orig_params_pt_msg_map :=
{
pt_init_handlers_eq := fun _ => eq_refl ;
pt_net_handlers_some := _ ;
pt_net_handlers_none := _ ;
pt_input_handlers_some := _ ;
pt_input_handlers_none := _;
}.
Next Obligation.
rewrite /pt_mapped_net_handlers.
repeat break_let.
rewrite /tot_map_name /= /id.
rewrite /pt_map_msg /= in H.
rewrite /net_handlers /= /serialized_net_handlers in Heqp.
move: H Heqp.
case H_m: (deserialize_top deserialize m) => [m0|] => H_eq //.
find_injection.
rewrite /serialize_handler_result in Heqp.
repeat break_let.
find_injection.
set sl2 := filterMap (fun _ => _) _.
set sl1 := filterMap _ _.
have H_eq: sl2 = l0.
rewrite /sl2.
clear.
elim: l0 => //=.
move => o l IH.
by rewrite IH.
have H_eq': sl1 = l1.
rewrite /sl1 /pt_map_name_msg /tot_map_name /id /= /id /serialize_name_msg_tuple.
clear.
elim: l1 => //=.
case => [n m] => /=.
move => l IH.
by rewrite serialize_deserialize_top_id IH.
by rewrite H_eq H_eq'.
Qed.
Next Obligation.
rewrite /net_handlers /= /serialized_net_handlers /= in H0.
rewrite /pt_map_msg /= in H.
move: H H0.
case H_eq_m: (deserialize_top deserialize m) => [m'|] H_eq H_eq'; first by repeat break_let.
by find_injection.
Qed.
Next Obligation.
rewrite /pt_mapped_input_handlers.
repeat break_let.
destruct p.
rewrite /tot_map_name /= /id.
rewrite /input_handlers /= /serialized_input_handlers in Heqp.
find_injection.
rewrite /serialize_handler_result in Heqp.
repeat break_let.
find_injection.
set sl2 := filterMap (fun _ => _) _.
set sl1 := filterMap _ _.
have H_eq: sl2 = l0.
rewrite /sl2.
clear.
elim: l0 => //=.
move => o l IH.
by rewrite IH.
have H_eq': sl1 = l1.
rewrite /sl1 /pt_map_name_msg /tot_map_name /id /= /id /serialize_name_msg_tuple.
clear.
elim: l1 => //=.
case => [n m] => /=.
move => l IH.
by rewrite serialize_deserialize_top_id IH.
by rewrite H_eq H_eq'.
Qed.
Proof using.
move => net.
rewrite /deserialize_net.
rewrite /pt_map_net /pt_map_data /= /id /= /pt_map_packet.
set fm := filterMap _ _.
set fm' := filterMap _ _.
suff H_eq: fm = fm' by rewrite H_eq.
rewrite /fm /fm'.
clear.
rewrite /pt_map_packet /tot_map_name /= /deserialize_packet.
elim (nwPackets net) => //=.
case => [src dst body] /= l IH.
rewrite IH.
by case (deserialize_top deserialize _).
Qed.
Proof using. by []. Qed.
Proof using.
move => net.
rewrite /deserialize_odnet.
rewrite /pt_map_odnet /pt_map_data /= /id /= /pt_map_msg.
set fm := fun _ _ => filterMap _ _.
rewrite map_id.
set s := fun _ => match _ with _ => _ end.
have H_eq: odnwState net = s.
rewrite /s.
apply functional_extensionality => n.
by case: odnwState.
by rewrite H_eq.
Qed.

Lemma step_async_deserialized_simulation :
forall net net' tr,
@step_async _ serialized_multi_params net net' tr ->
@step_async _ orig_multi_params (deserialize_net net) (deserialize_net net') tr \/
(deserialize_net net' = deserialize_net net).