Section PartialMapExecutionSimulations.
Context {base_fst : BaseParams}.
Context {base_snd : BaseParams}.
Context {labeled_multi_fst : LabeledMultiParams base_fst}.
Context {labeled_multi_snd : LabeledMultiParams base_snd}.
Context {base_map : BaseParamsPartialMap base_fst base_snd}.
Context {name_map : MultiParamsNameTotalMap (@unlabeled_multi_params _ labeled_multi_fst) (@unlabeled_multi_params _ labeled_multi_snd)}.
Context {msg_map : MultiParamsMsgPartialMap (@unlabeled_multi_params _ labeled_multi_fst) (@unlabeled_multi_params _ labeled_multi_snd)}.
Context {label_map : LabeledMultiParamsLabelTotalMap labeled_multi_fst labeled_multi_snd}.
Context {name_map_bijective : MultiParamsNameTotalMapBijective name_map}.
Context {multi_map_congr : MultiParamsPartialMapCongruency base_map name_map msg_map}.
Context {multi_map_lb_congr : LabeledMultiParamsPartialMapCongruency base_map name_map msg_map label_map}.
Hypothesis label_eq_dec : forall x y : label, {x = y} + {x <> y}.
Hypothesis tot_map_label_injective :
forall l l', tot_map_label l = tot_map_label l' -> l = l'.
Hypothesis label_tot_mapped :
forall l, exists l', l = tot_map_label l'.
(* lb_step_failure *)
Proof using name_map_bijective multi_map_lb_congr multi_map_congr.
move => net net' failed failed' lb tr H_neq H_step.
have H_neq': lb <> label_silent.
rewrite -pt_lb_label_silent_fst_snd in H_neq.
move => H_eq.
by rewrite H_eq in H_neq.
invcs H_step => //=.
- destruct (pt_map_packet p) eqn:?; last first.
destruct p.
simpl in *.
break_match => //.
have H_q := @pt_lb_net_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr pDst pSrc _ (nwState net pDst) Heqo0.
rewrite /tot_mapped_lb_net_handlers_label in H_q.
repeat break_let.
by tuple_inversion.
have H_eq_n: tot_map_name (pDst p) = pDst p0.
destruct p.
simpl in *.
break_match => //.
by find_injection.
rewrite H_eq_n.
apply (@LabeledStepFailure_deliver _ _ _ _ _ _ (filterMap pt_map_packet xs) (filterMap pt_map_packet ys) (filterMap pt_map_output out) (pt_map_data d) (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l)).
* rewrite /pt_map_net /=.
find_rewrite.
by rewrite filterMap_app /= Heqo.
* rewrite -H_eq_n.
exact: not_in_failed_not_in.
* rewrite /pt_map_net /= -{2}H_eq_n tot_map_name_inv_inverse.
destruct p, p0.
simpl in *.
break_match => //.
find_injection.
clean.
have H_q := @pt_net_handlers_some _ _ _ _ _ _ _ multi_map_congr pDst pSrc pBody (nwState net pDst) _ Heqo0.
rewrite /pt_mapped_net_handlers /net_handlers /= /unlabeled_net_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_net_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ _ Heqo0 Heqp1.
rewrite /tot_mapped_lb_net_handlers_label in H_q'.
repeat break_let.
break_and.
by repeat tuple_inversion.
* rewrite /pt_map_net /= 2!filterMap_app.
by rewrite (filterMap_pt_map_packet_map_eq_some _ _ Heqo) (pt_map_update_eq_some _ _ _ Heqo).
- case H_i: pt_map_input => [inp'|]; last first.
have H_q := @pt_lb_input_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr h _ (nwState net h) H_i.
rewrite /tot_mapped_lb_input_handlers_label /= in H_q.
repeat break_let.
by tuple_inversion.
apply (@LabeledStepFailure_input _ _ _ _ _ _ _ _ (pt_map_data d) (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l)).
* exact: not_in_failed_not_in.
* have H_q := @pt_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h _ (nwState net h) _ H_i.
rewrite /pt_mapped_input_handlers /input_handlers /= /unlabeled_input_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_input_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ H_i Heqp1.
break_and.
unfold tot_mapped_lb_input_handlers_label in *.
repeat break_let.
repeat tuple_inversion.
by rewrite /pt_map_net /= tot_map_name_inv_inverse.
* rewrite /pt_map_net /=.
rewrite filterMap_app filterMap_pt_map_packet_map_eq.
by rewrite -(@pt_map_update_eq  _ _ _ _ _ _ name_map_bijective).
Qed.
Proof using multi_map_lb_congr multi_map_congr.
move => net net' failed failed' lb tr H_eq H_step.
invcs H_step => //=.
- destruct (pt_map_packet p) eqn:?.
destruct p, p0.
simpl in *.
break_match_hyp => //.
find_injection.
have H_q := @pt_net_handlers_some _ _ _ _ _ _ _ multi_map_congr pDst pSrc pBody (nwState net pDst) _ Heqo0.
rewrite /pt_mapped_net_handlers /net_handlers /= /unlabeled_net_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_net_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ _ Heqo0 Heqp1.
break_and.
unfold tot_mapped_lb_net_handlers_label in *.
repeat break_let.
by repeat tuple_inversion.
destruct p.
simpl in *.
break_match_hyp => //.
have H_q := @pt_net_handlers_none _ _ _ _ _ _ _ multi_map_congr pDst pSrc pBody (nwState net pDst) out d l Heqo0.
rewrite /net_handlers /= /unlabeled_net_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
concludes.
break_and.
have H_q' := @pt_lb_net_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr pDst pSrc _ (nwState net pDst) Heqo0.
rewrite /tot_mapped_lb_net_handlers_label in H_q'.
repeat break_let.
repeat tuple_inversion.
rewrite /pt_map_net /=.
rewrite filterMap_app.
rewrite filterMap_pt_map_name_msg_empty_eq //=.
rewrite H3.
rewrite filterMap_app /=.
repeat break_match => //.
rewrite -filterMap_app.
set s1 := fun _ => _.
set s2 := fun _ => _.
have H_eq_s: s1 = s2.
rewrite /s1 /s2.
apply functional_extensionality => n.
rewrite /update.
by break_if; first by rewrite H e.
rewrite -H_eq_s /s1 {s1 s2 H_eq_s}.
split => //.
exact: LabeledStepFailure_stutter.
- case H_i: (pt_map_input inp) => [inp'|].
have H_q := @pt_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h _ (nwState net h) _ H_i.
rewrite /pt_mapped_input_handlers /input_handlers /= /unlabeled_input_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_input_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ H_i Heqp1.
break_and.
unfold tot_mapped_lb_input_handlers_label in *.
repeat break_let.
by tuple_inversion.
have H_q := @pt_input_handlers_none _ _ _ _ _ _ _ multi_map_congr h _ (nwState net h) out d l H_i.
rewrite /input_handlers /= /unlabeled_input_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
concludes.
break_and.
have H_q' := @pt_lb_input_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr h _ (nwState net h) H_i.
rewrite /tot_mapped_lb_input_handlers_label in H_q'.
repeat break_let.
repeat tuple_inversion.
rewrite /pt_map_net /=.
rewrite filterMap_app.
rewrite filterMap_pt_map_name_msg_empty_eq //=.
set s1 := fun _ => _.
set s2 := fun _ => _.
have H_eq_s: s1 = s2.
rewrite /s1 /s2.
apply functional_extensionality => n.
rewrite /update.
by break_if; first by rewrite H e.
rewrite -H_eq_s /s1 {s1 s2 H_eq_s}.
split; first exact: LabeledStepFailure_stutter.
by repeat find_rewrite.
- split => //; exact: LabeledStepFailure_stutter.
Qed.
(* lb_step_ordered_failure *)
Proof using name_map_bijective multi_map_lb_congr multi_map_congr.
move => net net' failed failed' lb tr H_neq H_step.
have H_neq': lb <> label_silent.
rewrite -pt_lb_label_silent_fst_snd in H_neq.
move => H_eq.
by rewrite H_eq in H_neq.
invcs H_step => //=.
- rewrite {2}/pt_map_onet /=.
case H_m: (@pt_map_msg _ _ _ _ msg_map m) => [m'|]; last first.
have H_q := @pt_lb_net_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr to from _ (onwState net to) H_m.
rewrite /tot_mapped_lb_net_handlers_label in H_q.
repeat break_let.
by tuple_inversion.
apply (@LabeledStepOrderedFailure_deliver _ _ _ _ _ _ m' (filterMap (@pt_map_msg _ _ _ _ msg_map) ms) (filterMap pt_map_output out) (pt_map_data d) (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l) (@tot_map_name _ _ _ _ name_map from) (@tot_map_name _ _ _ _ name_map to)).
* by rewrite /= 2!tot_map_name_inv_inverse /= H3 /= H_m.
* exact: not_in_failed_not_in.
* rewrite /pt_map_onet /= tot_map_name_inv_inverse.
have H_q := @pt_net_handlers_some _ _ _ _ _ _ _ multi_map_congr to from m (onwState net to) _ H_m.
rewrite /pt_mapped_net_handlers /net_handlers /= /unlabeled_net_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_net_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ _ H_m Heqp1.
rewrite /tot_mapped_lb_net_handlers_label in H_q'.
repeat break_let.
break_and.
by repeat tuple_inversion.
* rewrite (@collate_pt_map_update2_eq _ _ _ _ name_map).
set f1 := fun _ => pt_map_data _.
set f2 := update _ _ _ _.
have H_eq_f: f1 = f2.
rewrite /f1 /f2.
apply functional_extensionality => n.
rewrite /update.
break_if; break_if => //=; first by rewrite -e tot_map_name_inverse_inv in n0.
by rewrite e tot_map_name_inv_inverse in n0.
by rewrite H_eq_f.
* by rewrite -filterMap_pt_map_trace_ev_outputs_eq.
- rewrite {2}/pt_map_onet /=.
case H_i: pt_map_input => [inp'|]; last first.
have H_q := @pt_lb_input_handlers_none _ _ _ _ _ _ _ _ multi_map_lb_congr h inp (onwState net h) H_i.
rewrite /tot_mapped_lb_input_handlers_label in H_q.
repeat break_let.
by tuple_inversion.
apply (@LabeledStepOrderedFailure_input _ _ (@tot_map_name _ _ _ _ name_map h) _ _ _ _ (filterMap pt_map_output out) inp' (pt_map_data d) (filterMap (@pt_map_name_msg _ _ _ _ _ msg_map) l)).
* exact: not_in_failed_not_in.
* rewrite /pt_map_onet /= tot_map_name_inv_inverse.
have H_q := @pt_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h inp (onwState net h) _ H_i.
rewrite /pt_mapped_input_handlers /input_handlers /= /unlabeled_input_handlers in H_q.
repeat break_let.
repeat tuple_inversion.
have H_q' := @pt_lb_input_handlers_some _ _ _ _ _ _ _ _ multi_map_lb_congr _ _ _ _ _ _ _ _ H_i Heqp1.
rewrite /tot_mapped_lb_input_handlers_label in H_q'.
repeat break_let.
break_and.
by repeat tuple_inversion.
* rewrite {2}/pt_map_onet /=.
rewrite (@collate_pt_map_eq _ _ _ _ name_map).
set f1 := fun _ => pt_map_data _.
set f2 := update _ _ _ _.
have H_eq_f: f1 = f2.
rewrite /f1 /f2.
apply functional_extensionality => n.
rewrite /update.
break_if; break_if => //=; first by rewrite -e tot_map_name_inverse_inv in n0.
by rewrite e tot_map_name_inv_inverse in n0.
by rewrite H_eq_f.
* by rewrite -(@filterMap_pt_map_trace_ev_outputs_eq _ _ _ _ _ name_map out h).
Qed.

Theorem lb_step_ordered_failure_pt_mapped_simulation_1_silent :
forall net net' failed failed' lb tr,
tot_map_label lb = label_silent ->
@lb_step_ordered_failure _ labeled_multi_fst (failed, net) lb (failed', net') tr ->
@lb_step_ordered_failure _ labeled_multi_snd (List.map tot_map_name failed, pt_map_onet net) label_silent (List.map tot_map_name failed', pt_map_onet net') [] /\ filterMap pt_map_trace_ev tr = [].