proposition: Lemma gpaco4_dist clo r rg (CMP: wcompatible4 gf clo) (DIST: forall r1 r2, clo (r1 \4/ r2) <4= (clo r1 \4/ clo r2)): gpaco4 gf clo r rg <4= (paco4 gf (rclo4 clo (rg \4/ r)) \4/ rclo4 clo r).
proof: Proof. intros. apply gpaco4_unfold in PR; [|apply gf_mon]. apply rclo4_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 x3 H. pcofix CIH; intros. apply rclo4_wcompat in H0; [|apply gf_mon|apply CMP]. pstep. eapply gf_mon. apply H0. clear x0 x1 x2 x3 H0. intros. apply gpaco4_unfold in PR; [|apply gf_mon]. apply rclo4_compose in PR. apply rclo4_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo4_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco4_gupaco. apply gf_mon. apply gpaco4_gen_rclo. apply gf_mon. eapply gupaco4_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo4 clo (rclo4 clo (gf (gupaco4 gf clo ((rg \4/ r) \4/ (rg \4/ r))) \4/ (rg \4/ r))) x0 x1 x2 x3). { eapply rclo4_mon. apply H. intros. apply gpaco4_unfold in PR. apply PR. apply gf_mon. } apply rclo4_rclo in REL. apply rclo4_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo4_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco4_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco4 pacotac)