proposition: Lemma compatible'1_companion clo (RES: compatible'1 clo): clo <2= cpn1 gf.
proof: Proof. assert (MON: monotone1 gf'1). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <2= cpn1 gf'1). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn1_mon|]; intros. assert (PR1: cpn1 gf'1 (gf r) <1= cpn1 gf'1 (gf'1 (cpn1 gf r))). { intros. eapply cpn1_mon. apply PR1. intros. assert (TMP: gf (cpn1 gf r) <1= (cpn1 gf r /1\ gf (cpn1 gf r))). { split; [apply cpn1_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn1_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat1_compat with (gf:=gf'1) in PR0; [|apply cpn1_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn1_cpn; [apply MON|]. eapply cpn1_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat1_compatible', cpn1_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco1 pacotac)