proposition: Lemma gpaco13_dist clo r rg (CMP: wcompatible13 gf clo) (DIST: forall r1 r2, clo (r1 \13/ r2) <13= (clo r1 \13/ clo r2)): gpaco13 gf clo r rg <13= (paco13 gf (rclo13 clo (rg \13/ r)) \13/ rclo13 clo r).
proof: Proof. intros. apply gpaco13_unfold in PR; [|apply gf_mon]. apply rclo13_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 H. pcofix CIH; intros. apply rclo13_wcompat in H0; [|apply gf_mon|apply CMP]. pstep. eapply gf_mon. apply H0. clear x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 H0. intros. apply gpaco13_unfold in PR; [|apply gf_mon]. apply rclo13_compose in PR. apply rclo13_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo13_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco13_gupaco. apply gf_mon. apply gpaco13_gen_rclo. apply gf_mon. eapply gupaco13_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo13 clo (rclo13 clo (gf (gupaco13 gf clo ((rg \13/ r) \13/ (rg \13/ r))) \13/ (rg \13/ r))) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12). { eapply rclo13_mon. apply H. intros. apply gpaco13_unfold in PR. apply PR. apply gf_mon. } apply rclo13_rclo in REL. apply rclo13_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo13_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco13_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco13 pacotac)