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%%% %%% %%% For more details, please see the paper: %%% %%% "A lightweight formalization of the meta-theory of %%% bisimulation-up-to" by K. Chaudhuri, M. Cimini, and %%% D. Miller, 2014 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Processes Kind name,proc type. %% action prefixes Type taup proc -> proc. Type inp name -> (name -> proc) -> proc. Type oup name -> name -> proc -> proc. %% name restriction Type nu (name -> proc) -> proc. %% replication Type repl proc -> proc. %% core process algebra Type plus, par proc -> proc -> proc. Type null proc. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Labels of the LTS Kind label type. Type tau label. Type up, dn name -> name -> label. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% The steps of the LTS % The steps are defined by a mutually inductive definition of a relation % one for closed steps and oneb for bound steps, i.e., steps that contain % a name that is restricted (nu) or bound (inp) in an outer scope. Define one : proc -> label -> proc -> prop, oneb : proc -> (name -> label) -> (name -> proc) -> prop by /* internal steps */ one (taup P) tau P /* closed output */ ; one (oup X Y P) (up X Y) P /* core process algebra */ ; one (plus P Q) L R := one P L R ; one (plus P Q) L R := one Q L R ; one (par P Q) L (par R Q) := one P L R ; one (par P Q) L (par P R) := one Q L R ; one (repl P) L (par (repl P) R) := one P L R ; one (nu P) L (nu Q) := nabla x, one (P x) L (Q x) /* closed communication */ ; one (par P Q) tau (par PP QQ) := /* case of P getting input */ (exists X Y R, oneb P (dn X) R /\ one Q (up X Y) QQ /\ PP = R Y) /* case of Q getting input */ \/ (exists X Y R, one P (up X Y) PP /\ oneb Q (dn X) R /\ QQ = R Y) /* internal closed communication in a replicated process */ ; one (repl P) tau (par (repl P) (par PP QQ)) := exists X Y R, one P (up X Y) PP /\ oneb P (dn X) R /\ QQ = R Y /* bound communication */ ; one (par P Q) tau (nu y\ par (PP y) (QQ y)) := /* case of P getting input */ (exists X, oneb P (dn X) PP /\ oneb Q (up X) QQ) /* case of Q getting input */ \/ (exists X, oneb P (up X) PP /\ oneb Q (dn X) QQ) /* internal bound communication in a replicated process */ ; one (repl P) tau (par (repl P) (nu y\ par (PP y) (QQ y))) := exists X, oneb P (up X) PP /\ oneb P (dn X) QQ /* bound output */ ; oneb (nu P) (up X) R := nabla y, one (P y) (up X y) (R y) /* bound input */ ; oneb (inp X P) (dn X) P /* core process algebra for bound labels */ ; oneb (plus P Q) L R := oneb P L R ; oneb (plus P Q) L R := oneb Q L R ; oneb (par P Q) L (x\ par (R x) Q) := oneb P L R ; oneb (par P Q) L (x\ par P (R x)) := oneb Q L R ; oneb (repl P) L (x\ par (repl P) (R x)) := oneb P L R ; oneb (nu P) L (y\ nu x\ R x y) := nabla x, oneb (P x) L (R x). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Bisimilarity-up-to CoDefine bisim_up_to : (proc -> proc -> proc -> proc -> prop) -> proc -> proc -> prop by bisim_up_to Tech P Q := (forall L P1, one P L P1 -> exists Q1, one Q L Q1 /\ exists P2 Q2, Tech P1 P2 Q1 Q2 /\ bisim_up_to Tech P2 Q2) /\ (forall X P1, oneb P (dn X) P1 -> exists Q1, oneb Q (dn X) Q1 /\ exists P2 Q2, forall N, Tech (P1 N) (P2 N) (Q1 N) (Q2 N) /\ bisim_up_to Tech (P2 N) (Q2 N)) /\ (forall X P1, oneb P (up X) P1 -> exists Q1, oneb Q (up X) Q1 /\ exists P2 Q2, nabla x, Tech (P1 x) (P2 x) (Q1 x) (Q2 x) /\ bisim_up_to Tech (P2 x) (Q2 x)) /\ (forall L Q1, one Q L Q1 -> exists P1, one P L P1 /\ exists P2 Q2, Tech P1 P2 Q1 Q2 /\ bisim_up_to Tech P2 Q2) /\ (forall X Q1, oneb Q (dn X) Q1 -> exists P1, oneb P (dn X) P1 /\ exists P2 Q2, forall N, Tech (P1 N) (P2 N) (Q1 N) (Q2 N) /\ bisim_up_to Tech (P2 N) (Q2 N)) /\ (forall X Q1, oneb Q (up X) Q1 -> exists P1, oneb P (up X) P1 /\ exists P2 Q2, nabla x, Tech (P1 x) (P2 x) (Q1 x) (Q2 x) /\ bisim_up_to Tech (P2 x) (Q2 x)). Define refl_t : proc -> proc -> proc -> proc -> prop by refl_t P P Q Q. %%% The ordinary bisimilarity CoDefine bisim : proc -> proc -> prop by bisim P Q := bisim_up_to refl_t P Q. Define is_sound : (proc -> proc -> proc -> proc -> prop) -> prop by is_sound Tech := forall P Q, bisim_up_to Tech P Q -> bisim_up_to refl_t P Q. Theorem bisim_reflexive : forall P, bisim_up_to refl_t P P.coinduction. intros. unfold. intros. witness P1. split. search. witness P1. witness P1. split. search. backchain CH. intros. witness P1. split. search. witness P1. witness P1. intros. split. search. backchain CH. intros. witness P1. split. search. witness P1. witness P1. intros. split. search. backchain CH. intros. witness Q1. split. search. witness Q1. witness Q1. intros. split. search. backchain CH. intros. witness Q1. split. search. witness Q1. witness Q1. intros. split. search. backchain CH. intros. witness Q1. split. search. witness Q1. witness Q1. intros. split. search. backchain CH.Theorem bisim_eq_1L : forall P1 P2 Q, bisim_up_to refl_t P1 Q -> P1 = P2 -> bisim_up_to refl_t P2 Q. Theorem bisim_eq_1R : forall P1 P2 Q, bisim_up_to refl_t P2 Q -> P1 = P2 -> bisim_up_to refl_t P1 Q. Theorem bisim_eq_2L : forall P Q1 Q2, bisim_up_to refl_t P Q1 -> Q1 = Q2 -> bisim_up_to refl_t P Q2. Theorem bisim_eq_2R : forall P Q1 Q2, bisim_up_to refl_t P Q2 -> Q1 = Q2 -> bisim_up_to refl_t P Q1. Theorem bisim_symmetric : forall P Q, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q P.coinduction. intros. Bis1 : case H1. unfold. intros. apply Bis4 to H2. witness P2. split. search. witness P1. witness P2. case H4. split. search. backchain CH. intros. apply Bis5 to H2. witness P2. split. search. witness P1. witness P2. intros. apply H4 with N = N. case H5. split. search. backchain CH. apply bisim_eq_1R to H6 H8. apply bisim_eq_2R to H9 H7. search. intros. apply Bis6 to H2. witness P2. split. search. witness P1. witness P2. intros. case H4. split. search. backchain CH. intros. apply Bis1 to H2. witness Q2. split. search. witness Q2. witness Q1. case H4. split. search. backchain CH. intros. apply Bis2 to H2. witness Q2. split. search. witness Q2. witness Q1. intros. apply H4 with N = N. case H5. split. search. backchain CH. apply bisim_eq_1R to H6 H8. apply bisim_eq_2R to H9 H7. search. intros. apply Bis3 to H2. witness Q2. split. search. witness Q2. witness Q1. intros. case H4. split. search. backchain CH.Theorem bisim_transitive : forall P Q R, bisim_up_to refl_t P Q -> bisim_up_to refl_t Q R -> bisim_up_to refl_t P R.coinduction. intros. Left1 : case H1. Right1 : case H2. unfold. intros. apply Left1 to H3. apply Right1 to H4. witness Q1. split. search. witness P1. witness Q1. split. search. case H5. case H8. backchain CH. intros. apply Left2 to H3. apply Right2 to H4. witness Q1. split. search. witness P1. witness Q1. intros. split. search. apply H5 with N = N. apply H7 with N = N. case H8. case H10. apply bisim_eq_1R to H9 H13. apply bisim_eq_2R to H16 H12. apply bisim_eq_2L to H17 H15. apply bisim_eq_2R to H11 H14. clear H9 H11 H16 H17. backchain CH. intros. apply Left3 to H3. apply Right3 to H4. witness Q1. split. search. witness P1. witness Q1. intros. split. search. case H5. case H8. backchain CH. intros. apply Right4 to H3. apply Left4 to H4. witness P1. split. search. witness P1. witness Q1. split. search. case H5. case H8. backchain CH. intros. apply Right5 to H3. apply Left5 to H4. witness P1. split. search. witness P1. witness Q1. intros. split. search. apply H5 with N = N. apply H7 with N = N. case H8. case H10. apply bisim_eq_1R to H9 H13. apply bisim_eq_2R to H16 H12. apply bisim_eq_1R to H11 H15. apply bisim_eq_2R to H18 H14. clear H9 H11 H16 H18. backchain CH. intros. apply Right6 to H3. apply Left6 to H4. witness P1. split. search. witness P1. witness Q1. intros. split. search. case H5. case H8. backchain CH.