proposition: Lemma compatible'9_companion clo (RES: compatible'9 clo): clo <10= cpn9 gf.
proof: Proof. assert (MON: monotone9 gf'9). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <10= cpn9 gf'9). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn9_mon|]; intros. assert (PR1: cpn9 gf'9 (gf r) <9= cpn9 gf'9 (gf'9 (cpn9 gf r))). { intros. eapply cpn9_mon. apply PR1. intros. assert (TMP: gf (cpn9 gf r) <9= (cpn9 gf r /9\ gf (cpn9 gf r))). { split; [apply cpn9_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn9_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat9_compat with (gf:=gf'9) in PR0; [|apply cpn9_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn9_cpn; [apply MON|]. eapply cpn9_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat9_compatible', cpn9_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco9 pacotac)