proposition: Lemma gpaco12_dist clo r rg (CMP: wcompatible12 gf clo) (DIST: forall r1 r2, clo (r1 \12/ r2) <12= (clo r1 \12/ clo r2)): gpaco12 gf clo r rg <12= (paco12 gf (rclo12 clo (rg \12/ r)) \12/ rclo12 clo r).
proof: Proof. intros. apply gpaco12_unfold in PR; [|apply gf_mon]. apply rclo12_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 H. pcofix CIH; intros. apply rclo12_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 H0. intros. apply gpaco12_unfold in PR; [|apply gf_mon]. apply rclo12_compose in PR. apply rclo12_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo12_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco12_gupaco. apply gf_mon. apply gpaco12_gen_rclo. apply gf_mon. eapply gupaco12_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo12 clo (rclo12 clo (gf (gupaco12 gf clo ((rg \12/ r) \12/ (rg \12/ r))) \12/ (rg \12/ r))) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11). { eapply rclo12_mon. apply H. intros. apply gpaco12_unfold in PR. apply PR. apply gf_mon. } apply rclo12_rclo in REL. apply rclo12_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo12_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco12_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco12 pacotac)