proposition: Lemma gpaco10_dist clo r rg (CMP: wcompatible10 gf clo) (DIST: forall r1 r2, clo (r1 \10/ r2) <10= (clo r1 \10/ clo r2)): gpaco10 gf clo r rg <10= (paco10 gf (rclo10 clo (rg \10/ r)) \10/ rclo10 clo r).
proof: Proof. intros. apply gpaco10_unfold in PR; [|apply gf_mon]. apply rclo10_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 H. pcofix CIH; intros. apply rclo10_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 H0. intros. apply gpaco10_unfold in PR; [|apply gf_mon]. apply rclo10_compose in PR. apply rclo10_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo10_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco10_gupaco. apply gf_mon. apply gpaco10_gen_rclo. apply gf_mon. eapply gupaco10_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo10 clo (rclo10 clo (gf (gupaco10 gf clo ((rg \10/ r) \10/ (rg \10/ r))) \10/ (rg \10/ r))) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9). { eapply rclo10_mon. apply H. intros. apply gpaco10_unfold in PR. apply PR. apply gf_mon. } apply rclo10_rclo in REL. apply rclo10_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo10_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco10_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco10 pacotac)