proposition: Lemma compatible'12_companion clo (RES: compatible'12 clo): clo <13= cpn12 gf.
proof: Proof. assert (MON: monotone12 gf'12). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <13= cpn12 gf'12). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn12_mon|]; intros. assert (PR1: cpn12 gf'12 (gf r) <12= cpn12 gf'12 (gf'12 (cpn12 gf r))). { intros. eapply cpn12_mon. apply PR1. intros. assert (TMP: gf (cpn12 gf r) <12= (cpn12 gf r /12\ gf (cpn12 gf r))). { split; [apply cpn12_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn12_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat12_compat with (gf:=gf'12) in PR0; [|apply cpn12_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn12_cpn; [apply MON|]. eapply cpn12_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat12_compatible', cpn12_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco12 pacotac)