proposition: Lemma compatible'0_companion clo (RES: compatible'0 clo): clo <1= cpn0 gf.
proof: Proof. assert (MON: monotone0 gf'0). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <1= cpn0 gf'0). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn0_mon|]; intros. assert (PR1: cpn0 gf'0 (gf r) <0= cpn0 gf'0 (gf'0 (cpn0 gf r))). { intros. eapply cpn0_mon. apply PR1. intros. assert (TMP: gf (cpn0 gf r) <0= (cpn0 gf r /0\ gf (cpn0 gf r))). { split; [apply cpn0_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn0_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat0_compat with (gf:=gf'0) in PR0; [|apply cpn0_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn0_cpn; [apply MON|]. eapply cpn0_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat0_compatible', cpn0_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco0 pacotac)