proposition: Lemma gpaco8_dist clo r rg (CMP: wcompatible8 gf clo) (DIST: forall r1 r2, clo (r1 \8/ r2) <8= (clo r1 \8/ clo r2)): gpaco8 gf clo r rg <8= (paco8 gf (rclo8 clo (rg \8/ r)) \8/ rclo8 clo r).
proof: Proof. intros. apply gpaco8_unfold in PR; [|apply gf_mon]. apply rclo8_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 x3 x4 x5 x6 x7 H. pcofix CIH; intros. apply rclo8_wcompat in H0; [|apply gf_mon|apply CMP]. pstep. eapply gf_mon. apply H0. clear x0 x1 x2 x3 x4 x5 x6 x7 H0. intros. apply gpaco8_unfold in PR; [|apply gf_mon]. apply rclo8_compose in PR. apply rclo8_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo8_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco8_gupaco. apply gf_mon. apply gpaco8_gen_rclo. apply gf_mon. eapply gupaco8_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo8 clo (rclo8 clo (gf (gupaco8 gf clo ((rg \8/ r) \8/ (rg \8/ r))) \8/ (rg \8/ r))) x0 x1 x2 x3 x4 x5 x6 x7). { eapply rclo8_mon. apply H. intros. apply gpaco8_unfold in PR. apply PR. apply gf_mon. } apply rclo8_rclo in REL. apply rclo8_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo8_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco8_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco8 pacotac)