proposition: Theorem step_ordered_dynamic_failure_tot_mapped_simulation_1 : forall net net' failed failed' tr, NoDup (odnwNodes net) -> @step_ordered_dynamic_failure _ _ overlay_fst new_msg_fst fail_msg_fst (failed, net) (failed', net') tr -> @step_ordered_dynamic_failure _ _ overlay_snd new_msg_snd fail_msg_snd (map tot_map_name failed, tot_map_odnet net) (map tot_map_name failed', tot_map_odnet net') (map tot_map_trace tr).
proof: Proof. Proof using overlay_map_congr new_msg_map_congr name_map_bijective multi_map_congr fail_msg_map_congr. move => net net' failed failed' tr H_nd H_step. invcs H_step. - rewrite /tot_map_odnet /=. apply (@StepOrderedDynamicFailure_start _ _ _ _ _ _ _ _ (tot_map_name h)) => /=; first exact: not_in_failed_not_in. set p1 := fun _ _ => _. set p2 := collate_ls _ _ _ _ _. set s1 := fun _ => _. set s2 := update _ _ _ _. have H_eq_s: s1 = s2. rewrite /s1 /s2 /update {s1 s2}. apply functional_extensionality => n. rewrite -tot_init_handlers_eq. break_match_goal. break_if; break_if; try by congruence. - by repeat find_rewrite; repeat find_rewrite_lem tot_map_name_inv_inverse. - by find_reverse_rewrite; find_rewrite_lem tot_map_name_inverse_inv. - by find_rewrite. break_if; break_if; (try by congruence); last by find_rewrite. by repeat find_rewrite; repeat find_rewrite_lem tot_map_name_inv_inverse. rewrite H_eq_s /s2 {s1 s2 H_eq_s}. have H_eq_p: p1 = p2. rewrite /p1 /p2 {p1 p2}. rewrite collate_ls_tot_map_eq /=. rewrite collate_tot_map_eq. rewrite tot_new_msg_fst_snd. set f1 := fun _ _ => _. set c1 := collate _ _ _ _. set c2 := collate _ _ _ _. set f'1 := map tot_map_name _. set f'2 := filter_rel _ (tot_map_name h) _. have H_c: c1 = c2. rewrite /c1 /c2 {c1 c2}. apply: NoDup_Permutation_collate_eq; last first. rewrite /tot_map_name_msgs. by apply: nodup_perm_map_map_pair_perm; last exact: Permutation_refl. rewrite /tot_map_name_msgs /=. apply: NoDup_map_snd_fst => //. apply (@nodup_tot_map msg_new); first exact: in_map2snd_snd. apply: NoDup_map2snd. exact: NoDup_remove_all. move => nm nm' H_in H_in'. apply tot_map_in_snd in H_in. apply tot_map_in_snd in H_in'. by rewrite H_in H_in'. rewrite H_c {H_c}. suff H_suff: f'1 = f'2 by rewrite H_suff. rewrite /f'1 /f'2. elim (odnwNodes net) => /=; first by rewrite 2!remove_all_nil. move => n ns. set mn := tot_map_name n. set mns := map _ ns. set mfailed' := map _ failed'. move => IH. have H_cn := remove_all_cons name_eq_dec failed' n ns. have H_cn' := remove_all_cons name_eq_dec mfailed' mn mns. unfold mn, mns, mfailed' in *. repeat break_or_hyp; repeat break_and; repeat find_rewrite => //=. * by find_apply_lem_hyp not_in_failed_not_in. * by find_apply_lem_hyp in_failed_in. * case: adjacent_to_dec => H_dec; case adjacent_to_dec => H_dec' => //=. + by rewrite IH. + by find_apply_lem_hyp tot_adjacent_to_fst_snd. + by find_apply_lem_hyp tot_adjacent_to_fst_snd. by rewrite H_eq_p. - rewrite /tot_map_odnet /=. apply (@StepOrderedDynamicFailure_deliver _ _ _ _ _ _ _ _ _ (tot_map_msg m) (map tot_map_msg ms) (map tot_map_output out) (tot_map_data d) (tot_map_data d') (tot_map_name_msgs l) (tot_map_name from) (tot_map_name to)) => //=. * exact: not_in_failed_not_in. * exact: in_failed_in. * rewrite tot_map_name_inv_inverse. by break_match; congruence. * rewrite 2!tot_map_name_inv_inverse. by find_rewrite. * rewrite -tot_net_handlers_eq /tot_mapped_net_handlers. repeat break_let. by repeat tuple_inversion. * set u1 := fun _ => match _ with | _ => _ end. set u2 := update _ _ _ _. rewrite collate_tot_map_update2_eq. suff H_suff: u1 = u2 by rewrite H_suff. rewrite /u1 /u2 /update /=. apply functional_extensionality => n. repeat break_if; try by congruence. rewrite -(tot_map_name_inverse_inv n) in n0. by rewrite e in n0. find_rewrite. by find_rewrite_lem tot_map_name_inv_inverse. * by rewrite map_tot_map_trace_eq. - rewrite /tot_map_odnet /=. apply (@StepOrderedDynamicFailure_input _ _ _ _ _ (tot_map_name h) _ _ _ _ (map tot_map_output out) (tot_map_input inp) (tot_map_data d) (tot_map_data d') (tot_map_name_msgs l)) => //=. * exact: not_in_failed_not_in. * exact: in_failed_in. * rewrite tot_map_name_inv_inverse. by break_match; congruence. * rewrite /= -tot_input_handlers_eq /tot_mapped_input_handlers. repeat break_let. by repeat tuple_inversion. * rewrite collate_tot_map_eq. set u1 := fun _ => match _ with | _ => _ end. set u2 := update _ _ _ _. suff H_suff: u1 = u2 by rewrite H_suff. rewrite /u1 /u2 /update /=. apply functional_extensionality => n. repeat break_if; try by congruence. rewrite -(tot_map_name_inverse_inv n) in n0. by rewrite e in n0. find_rewrite. by find_rewrite_lem tot_map_name_inv_inverse. * by rewrite map_tot_map_trace_eq. - rewrite /tot_map_odnet /=. set l := map2snd _ _. have H_nd': NoDup (map (fun nm => fst nm) (tot_map_name_msgs l)). rewrite /tot_map_name_msgs /=. rewrite /l {l}. apply NoDup_map_snd_fst. apply (@nodup_tot_map msg_fail); first exact: in_map2snd_snd. apply NoDup_map2snd. exact: NoDup_remove_all. move => nm nm' H_in H_in'. have H_fail := tot_map_in_snd _ _ _ _ H_in. have H_fail' := tot_map_in_snd _ _ _ _ H_in'. by rewrite H_fail H_fail'. have H_pm := map_msg_fail_eq_nodup h failed H_nd (Permutation_refl (map tot_map_name (odnwNodes net))). have H_eq := NoDup_Permutation_collate_eq _ _ name_eq_dec _ _ _ _ H_nd' H_pm. rewrite /l /tot_map_name_msgs in H_eq. apply: StepOrderedDynamicFailure_fail. * exact: not_in_failed_not_in. * exact: in_failed_in. * rewrite /=. rewrite /l collate_tot_map_eq /tot_map_name_msgs. by rewrite H_eq. Qed.
imports: Verdi(Verdi DynamicNetLemmas) Coq(FunctionalExtensionality) Coq(Sorting.Permutation Sumbool) Verdi(Ssrexport)