proposition: Lemma gpaco3_dist clo r rg (CMP: wcompatible3 gf clo) (DIST: forall r1 r2, clo (r1 \3/ r2) <3= (clo r1 \3/ clo r2)): gpaco3 gf clo r rg <3= (paco3 gf (rclo3 clo (rg \3/ r)) \3/ rclo3 clo r).
proof: Proof. intros. apply gpaco3_unfold in PR; [|apply gf_mon]. apply rclo3_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 H. pcofix CIH; intros. apply rclo3_wcompat in H0; [|apply gf_mon|apply CMP]. pstep. eapply gf_mon. apply H0. clear x0 x1 x2 H0. intros. apply gpaco3_unfold in PR; [|apply gf_mon]. apply rclo3_compose in PR. apply rclo3_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo3_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco3_gupaco. apply gf_mon. apply gpaco3_gen_rclo. apply gf_mon. eapply gupaco3_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo3 clo (rclo3 clo (gf (gupaco3 gf clo ((rg \3/ r) \3/ (rg \3/ r))) \3/ (rg \3/ r))) x0 x1 x2). { eapply rclo3_mon. apply H. intros. apply gpaco3_unfold in PR. apply PR. apply gf_mon. } apply rclo3_rclo in REL. apply rclo3_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo3_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco3_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco3 pacotac)