proposition: Lemma compatible'4_companion clo (RES: compatible'4 clo): clo <5= cpn4 gf.
proof: Proof. assert (MON: monotone4 gf'4). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <5= cpn4 gf'4). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn4_mon|]; intros. assert (PR1: cpn4 gf'4 (gf r) <4= cpn4 gf'4 (gf'4 (cpn4 gf r))). { intros. eapply cpn4_mon. apply PR1. intros. assert (TMP: gf (cpn4 gf r) <4= (cpn4 gf r /4\ gf (cpn4 gf r))). { split; [apply cpn4_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn4_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat4_compat with (gf:=gf'4) in PR0; [|apply cpn4_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn4_cpn; [apply MON|]. eapply cpn4_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat4_compatible', cpn4_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco4 pacotac)