proposition: Lemma compatible'2_companion clo (RES: compatible'2 clo): clo <3= cpn2 gf.
proof: Proof. assert (MON: monotone2 gf'2). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <3= cpn2 gf'2). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn2_mon|]; intros. assert (PR1: cpn2 gf'2 (gf r) <2= cpn2 gf'2 (gf'2 (cpn2 gf r))). { intros. eapply cpn2_mon. apply PR1. intros. assert (TMP: gf (cpn2 gf r) <2= (cpn2 gf r /2\ gf (cpn2 gf r))). { split; [apply cpn2_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn2_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat2_compat with (gf:=gf'2) in PR0; [|apply cpn2_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn2_cpn; [apply MON|]. eapply cpn2_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat2_compatible', cpn2_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco2 pacotac)