proposition: Lemma gpaco9_dist clo r rg (CMP: wcompatible9 gf clo) (DIST: forall r1 r2, clo (r1 \9/ r2) <9= (clo r1 \9/ clo r2)): gpaco9 gf clo r rg <9= (paco9 gf (rclo9 clo (rg \9/ r)) \9/ rclo9 clo r).
proof: Proof. intros. apply gpaco9_unfold in PR; [|apply gf_mon]. apply rclo9_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 x3 x4 x5 x6 x7 x8 H. pcofix CIH; intros. apply rclo9_wcompat in H0; [|apply gf_mon|apply CMP]. pstep. eapply gf_mon. apply H0. clear x0 x1 x2 x3 x4 x5 x6 x7 x8 H0. intros. apply gpaco9_unfold in PR; [|apply gf_mon]. apply rclo9_compose in PR. apply rclo9_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo9_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco9_gupaco. apply gf_mon. apply gpaco9_gen_rclo. apply gf_mon. eapply gupaco9_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo9 clo (rclo9 clo (gf (gupaco9 gf clo ((rg \9/ r) \9/ (rg \9/ r))) \9/ (rg \9/ r))) x0 x1 x2 x3 x4 x5 x6 x7 x8). { eapply rclo9_mon. apply H. intros. apply gpaco9_unfold in PR. apply PR. apply gf_mon. } apply rclo9_rclo in REL. apply rclo9_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo9_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco9_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco9 pacotac)