proposition: Theorem step_ordered_pt_ext_mapped_simulation_1 : forall net net' tr, @step_ordered _ multi_fst net net' tr -> (exists tr', @step_ordered _ multi_snd (pt_ext_map_onet net) (pt_ext_map_onet net') tr') \/ pt_ext_map_onet net' = pt_ext_map_onet net.
proof: Proof. Proof using name_map_bijective multi_map_congr. move => net net' tr. case => {net net' tr}. - move => net net' tr m ms out d l from to H_eq H_hnd H_eq' H_eq_tr. destruct (pt_map_msg m) eqn:?. left. destruct (net_handlers (tot_map_name to) (tot_map_name from) m0 (pt_ext_map_data (onwState net to) to)) eqn:?. destruct p as [out' d']. exists (map2fst (tot_map_name to) (map inr out')). rewrite H_eq' /= /pt_ext_map_onet /=. apply (@StepOrdered_deliver _ _ _ _ _ m0 (filterMap pt_map_msg ms) out' (pt_ext_map_data d to) (filterMap pt_map_name_msg l) (tot_map_name from) (tot_map_name to)) => //=. * rewrite 2!tot_map_name_inv_inverse H_eq /=. case H_m1: pt_map_msg => [m1|]; last by rewrite Heqo in H_m1. rewrite H_m1 in Heqo. by inversion Heqo. * rewrite tot_map_name_inv_inverse. have H_q := @pt_ext_net_handlers_some _ _ _ _ _ _ _ multi_map_congr _ _ _ _ _ _ _ _ Heqo Heqp. rewrite /pt_ext_mapped_net_handlers /= in H_q. by repeat break_let; repeat tuple_inversion. * by rewrite /= pt_ext_map_update_eq collate_pt_map_update2_eq. right. have [H_eq_d H_ms] := pt_ext_net_handlers_none _ _ _ _ Heqo H_hnd. rewrite H_eq' /pt_ext_map_onet /=. rewrite pt_ext_map_update_eq /= H_eq_d. rewrite collate_pt_map_eq H_ms /=. set nwS1 := update _ _ _ _. set nwS2 := fun n => pt_ext_map_data _ _. set nwP1 := fun _ _ => _. set nwP2 := fun _ _ => _. have H_eq_s: nwS1 = nwS2. rewrite /nwS1 /nwS2 {nwS1 nwS2}. apply functional_extensionality => n. rewrite /update /=. case (name_eq_dec _ _) => H_dec //. by rewrite H_dec tot_map_name_inv_inverse. have H_eq_p: nwP1 = nwP2. rewrite /nwP1 /nwP2 /=. apply functional_extensionality => src. apply functional_extensionality => dst. rewrite /update2 /=. case (sumbool_and _ _ _ _) => H_dec //. move: H_dec => [H_eq_from H_eq_to]. rewrite -H_eq_from -H_eq_to H_eq /=. case H_m': (pt_map_msg _) => [m'|] //. by rewrite H_m' in Heqo. by rewrite H_eq_s H_eq_p. - move => h net net' tr out inp d l H_hnd H_eq H_eq_tr. destruct (pt_ext_map_input inp h (onwState net h)) eqn:?. left. destruct (input_handlers (tot_map_name h) i (pt_ext_map_data (onwState net h) h)) eqn:?. destruct p as [out' d']. exists ((tot_map_name h, inl i) :: map2fst (tot_map_name h) (map inr out')). apply (@StepOrdered_input _ _ (tot_map_name h) _ _ _ out' i (pt_ext_map_data d h) (filterMap pt_map_name_msg l)) => //=; last by rewrite H_eq /pt_ext_map_onet /= pt_ext_map_update_eq collate_pt_map_eq. have H_q := @pt_ext_input_handlers_some _ _ _ _ _ _ _ multi_map_congr h inp (onwState net h) _ _ _ _ Heqo Heqp. rewrite /pt_ext_mapped_input_handlers /= in H_q. rewrite tot_map_name_inv_inverse. by repeat break_let; repeat tuple_inversion. right. rewrite /=. have [H_d H_l] := pt_ext_input_handlers_none h inp (onwState net h) Heqo H_hnd. rewrite H_eq /= /pt_ext_map_onet /=. rewrite pt_ext_map_update_eq /= H_d. rewrite collate_pt_map_eq H_l /=. set nwS1 := update _ _ _ _. set nwS2 := fun n => pt_ext_map_data _ _. have H_eq_n: nwS1 = nwS2. rewrite /nwS1 /nwS2 /=. apply functional_extensionality => n. rewrite /update /=. case (name_eq_dec _ _) => H_dec //. by rewrite H_dec tot_map_name_inv_inverse. by rewrite H_eq_n. Qed.
imports: Verdi(Verdi DynamicNetLemmas) Verdi(TotalMapSimulations PartialMapSimulations) Coq(FunctionalExtensionality Sumbool) Coq(Sorting.Permutation) Verdi(Ssrexport)