proposition: Lemma gpaco11_dist clo r rg (CMP: wcompatible11 gf clo) (DIST: forall r1 r2, clo (r1 \11/ r2) <11= (clo r1 \11/ clo r2)): gpaco11 gf clo r rg <11= (paco11 gf (rclo11 clo (rg \11/ r)) \11/ rclo11 clo r).
proof: Proof. intros. apply gpaco11_unfold in PR; [|apply gf_mon]. apply rclo11_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 H. pcofix CIH; intros. apply rclo11_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 H0. intros. apply gpaco11_unfold in PR; [|apply gf_mon]. apply rclo11_compose in PR. apply rclo11_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo11_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco11_gupaco. apply gf_mon. apply gpaco11_gen_rclo. apply gf_mon. eapply gupaco11_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo11 clo (rclo11 clo (gf (gupaco11 gf clo ((rg \11/ r) \11/ (rg \11/ r))) \11/ (rg \11/ r))) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10). { eapply rclo11_mon. apply H. intros. apply gpaco11_unfold in PR. apply PR. apply gf_mon. } apply rclo11_rclo in REL. apply rclo11_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo11_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco11_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco11 pacotac)