proposition: Lemma compatible'11_companion clo (RES: compatible'11 clo): clo <12= cpn11 gf.
proof: Proof. assert (MON: monotone11 gf'11). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <12= cpn11 gf'11). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn11_mon|]; intros. assert (PR1: cpn11 gf'11 (gf r) <11= cpn11 gf'11 (gf'11 (cpn11 gf r))). { intros. eapply cpn11_mon. apply PR1. intros. assert (TMP: gf (cpn11 gf r) <11= (cpn11 gf r /11\ gf (cpn11 gf r))). { split; [apply cpn11_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn11_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat11_compat with (gf:=gf'11) in PR0; [|apply cpn11_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn11_cpn; [apply MON|]. eapply cpn11_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat11_compatible', cpn11_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco11 pacotac)