proposition: Lemma gpaco5_dist clo r rg (CMP: wcompatible5 gf clo) (DIST: forall r1 r2, clo (r1 \5/ r2) <5= (clo r1 \5/ clo r2)): gpaco5 gf clo r rg <5= (paco5 gf (rclo5 clo (rg \5/ r)) \5/ rclo5 clo r).
proof: Proof. intros. apply gpaco5_unfold in PR; [|apply gf_mon]. apply rclo5_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 x3 x4 H. pcofix CIH; intros. apply rclo5_wcompat in H0; [|apply gf_mon|apply CMP]. pstep. eapply gf_mon. apply H0. clear x0 x1 x2 x3 x4 H0. intros. apply gpaco5_unfold in PR; [|apply gf_mon]. apply rclo5_compose in PR. apply rclo5_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo5_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco5_gupaco. apply gf_mon. apply gpaco5_gen_rclo. apply gf_mon. eapply gupaco5_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo5 clo (rclo5 clo (gf (gupaco5 gf clo ((rg \5/ r) \5/ (rg \5/ r))) \5/ (rg \5/ r))) x0 x1 x2 x3 x4). { eapply rclo5_mon. apply H. intros. apply gpaco5_unfold in PR. apply PR. apply gf_mon. } apply rclo5_rclo in REL. apply rclo5_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo5_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco5_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco5 pacotac)