Section MsgGhostVars.
Context {base_params : BaseParams}.
Context {multi_params : MultiParams base_params}.
Context {failure_params : FailureParams multi_params}.
Context {msg_ghost_params : MsgGhostMultiParams multi_params}.
Definition add_ghost_msg (me : name) (st : data) (ps : list (name * msg)) :
list (name * (ghost_msg * msg)) :=
map (fun m => (fst m, (write_ghost_msg me st, snd m))) ps.
Definition mgv_refined_net_handlers me src (m : ghost_msg * msg) st :=
let '(out, st', ps) := net_handlers me src (snd m) st in
(out, st', add_ghost_msg me st' ps).
Definition mgv_refined_input_handlers me inp st :=
let '(out, st', ps) := input_handlers me inp st in
(out, st', add_ghost_msg me st' ps).
Definition mgv_msg_eq_dec :
forall x y : ghost_msg * msg, {x = y} + {x <> y}.
Proof using.
intros.
decide equality; auto using msg_eq_dec, ghost_msg_eq_dec.
Qed.
Instance mgv_refined_base_params : BaseParams :=
{
data := data ;
input := input ;
output := output
}.
Instance mgv_refined_multi_params : MultiParams _ :=
{
name := name ;
msg := (ghost_msg * msg) ;
msg_eq_dec := mgv_msg_eq_dec ;
name_eq_dec := name_eq_dec ;
nodes := nodes ;
all_names_nodes := all_names_nodes ;
no_dup_nodes := no_dup_nodes ;
init_handlers := init_handlers;
net_handlers := mgv_refined_net_handlers ;
input_handlers := mgv_refined_input_handlers
}.
Instance mgv_refined_failure_params : FailureParams _ :=
{
reboot := (@reboot base_params multi_params failure_params)
}.
Definition mgv_deghost_packet p :=
@mkPacket _ multi_params
(@pSrc _ mgv_refined_multi_params p)
(pDst p)
(snd (pBody p)).
Definition mgv_deghost (net : @network _ mgv_refined_multi_params) : (@network _ multi_params).
refine (@mkNetwork _ multi_params
(map mgv_deghost_packet
(nwPackets net))
_
).
intros.
destruct net.
concludes.
auto.
Defined.
Arguments mgv_deghost_packet /_.
Instance mgv_refined_base_params_tot_map :
BaseParamsTotalMap mgv_refined_base_params base_params :=
{
tot_map_data := id ;
tot_map_input := id ;
tot_map_output := id
}.
Instance mgv_refined_multi_params_name_tot_map :
MultiParamsNameTotalMap mgv_refined_multi_params multi_params :=
{
tot_map_name := id ;
tot_map_name_inv := id ;
}.
Instance mgv_refined_multi_params_name_tot_map_bijective :
MultiParamsNameTotalMapBijective mgv_refined_multi_params_name_tot_map :=
{
tot_map_name_inv_inverse := fun _ => eq_refl ;
tot_map_name_inverse_inv := fun _ => eq_refl
}.
Instance mgv_refined_multi_params_tot_map :
MultiParamsMsgTotalMap mgv_refined_multi_params multi_params :=
{
tot_map_msg := snd ;
}.
Program Instance mgv_refined_multi_params_map_congruency :
MultiParamsTotalMapCongruency mgv_refined_base_params_tot_map
mgv_refined_multi_params_name_tot_map mgv_refined_multi_params_tot_map :=
{
tot_init_handlers_eq := fun _ => eq_refl ;
tot_net_handlers_eq := _ ;
tot_input_handlers_eq := _
}.
Next Obligation.
rewrite /tot_mapped_net_handlers /= /mgv_refined_net_handlers /= /tot_map_name_msgs /= /id /=.
repeat break_let.
find_inversion.
rewrite -/id map_id /= /add_ghost_msg /=.
elim l0 => //=.
case => n m' l IH.
find_inversion.
by find_rewrite; find_rewrite.
Qed.
Next Obligation.
rewrite /tot_mapped_input_handlers /=.
repeat break_let.
rewrite map_id /id /=.
unfold mgv_refined_input_handlers in *.
repeat break_let.
find_inversion.
elim l1 => //=.
case => n m l.
move => IH.
find_inversion.
by find_rewrite; find_rewrite.
Qed.
Instance mgv_refined_failure_params_map_congruency :
FailureParamsTotalMapCongruency mgv_refined_failure_params
failure_params mgv_refined_base_params_tot_map :=
{
tot_reboot_eq := fun _ => eq_refl
}.
Proof using.
elim => //.
move => tr l IH.
rewrite /= IH.
break_let.
break_match => //.
by rewrite map_id.
Qed.
Proof using.
move => net net' failed failed' out H_step.
apply step_failure_tot_mapped_simulation_1 in H_step.
rewrite /tot_map_name /tot_map_net /= 2!map_id /id /= in H_step.
rewrite /tot_map_trace_occ /= /id /= in H_step.
rewrite /tot_map_packet /= /id /= in H_step.
rewrite /mgv_deghost /=.
rewrite -/id mgv_map_id_tr in H_step.
move: H_step.
set fp := fun p : packet => _.
set fp' := fun p : packet => _.
have H_eq: fp = fp' by rewrite /fp /fp'; apply functional_extensionality; case => /= src dst m.
rewrite H_eq {H_eq fp}.
set fs1 := fun n => _.
set fs2 := fun n => _.
set fs1' := fun n => _.
set fs2' := fun n => _.
have H_eq: fs1 = fs1' by rewrite /fs1 /fs1' {fs1 fs1'}; apply functional_extensionality => n; case: net.
rewrite H_eq {H_eq fs1}.
have H_eq: fs2 = fs2' by rewrite /fs2 /fs2' {fs2 fs2'}; apply functional_extensionality => n; case: net'.
by rewrite H_eq.
Qed.
Definition mgv_ghost_packet p :=
@mkPacket _ mgv_refined_multi_params
(@pSrc _ multi_params p)
(pDst p)
(ghost_msg_default, pBody p).
Definition mgv_reghost (net : @network _ multi_params) : @network _ mgv_refined_multi_params.
refine (@mkNetwork _ mgv_refined_multi_params
(map mgv_ghost_packet
(nwPackets net))
_
).
intros.
destruct net.
concludes.
auto.
Defined.
Arguments mgv_ghost_packet /_.
Proof using.
destruct net. unfold mgv_deghost, mgv_reghost. simpl in *. f_equal.
rewrite map_map. map_id.
Qed.
Proof using.
move => net net' failed failed' out gnet H_step H_eq.
eapply step_failure_tot_mapped_simulation_2 in H_step => //.
- move: H_step => [gnet' [H_step H_eq_net]].
exists gnet'.
split; eauto.
rewrite -H_eq_net {H_step H_eq_net}.
rewrite /mgv_deghost /tot_map_net /= /id /= /tot_map_packet /= /id /=.
set nwPf1 := fun p : packet => _.
set nwPf2 := fun p : packet => _.
have H_eq_p: nwPf1 = nwPf2 by rewrite /nwPf1 /nwPf2 {nwPf1 nwPf2}; apply functional_extensionality; case.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2 by rewrite /nwS1 /nwS2 {nwS1 nwS2}; apply functional_extensionality => n; case: gnet'.
by rewrite H_eq_p H_eq_s.
- rewrite -H_eq {H_step H_eq}.
rewrite /mgv_deghost /tot_map_net /= /id /= /tot_map_packet /= /id /=.
set nwPf1 := fun p : packet => _.
set nwPf2 := fun p : packet => _.
have H_eq_p: nwPf1 = nwPf2 by rewrite /nwPf1 /nwPf2 {nwPf1 nwPf2}; apply functional_extensionality; case.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2 by rewrite /nwS1 /nwS2 {nwS1 nwS2}; apply functional_extensionality => n; case: gnet.
by rewrite H_eq_p H_eq_s.
- by rewrite /tot_map_name /= map_id.
- by rewrite /tot_map_name /= map_id.
- move {H_step}.
elim: out => //.
case => n t out IH.
case: t => /=; first by move => inp; rewrite /id /= IH.
move => out'.
by rewrite {1}/id map_id /= IH.
Qed.
Proof using.
intros. eauto using mgv_ghost_simulation_1.
Qed.

Theorem mgv_ghost_invariant_lower :
forall P : _ -> Prop,
(forall net net' failed failed' out,
step_failure (failed, net) (failed', net') out ->
P (mgv_deghost net) ->
P (mgv_deghost net')) ->
(forall net net' failed failed' out,
@step_failure _ _ failure_params (failed, net) (failed', net') out ->
P net ->
P net').