proposition: Lemma compatible'7_companion clo (RES: compatible'7 clo): clo <8= cpn7 gf.
proof: Proof. assert (MON: monotone7 gf'7). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <8= cpn7 gf'7). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn7_mon|]; intros. assert (PR1: cpn7 gf'7 (gf r) <7= cpn7 gf'7 (gf'7 (cpn7 gf r))). { intros. eapply cpn7_mon. apply PR1. intros. assert (TMP: gf (cpn7 gf r) <7= (cpn7 gf r /7\ gf (cpn7 gf r))). { split; [apply cpn7_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn7_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat7_compat with (gf:=gf'7) in PR0; [|apply cpn7_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn7_cpn; [apply MON|]. eapply cpn7_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat7_compatible', cpn7_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco7 pacotac)