proposition: Lemma gpaco7_dist clo r rg (CMP: wcompatible7 gf clo) (DIST: forall r1 r2, clo (r1 \7/ r2) <7= (clo r1 \7/ clo r2)): gpaco7 gf clo r rg <7= (paco7 gf (rclo7 clo (rg \7/ r)) \7/ rclo7 clo r).
proof: Proof. intros. apply gpaco7_unfold in PR; [|apply gf_mon]. apply rclo7_dist in PR; [|apply CMP|apply DIST]. destruct PR; [|right; apply H]. left. revert x0 x1 x2 x3 x4 x5 x6 H. pcofix CIH; intros. apply rclo7_wcompat in H0; [|apply gf_mon|apply CMP]. pstep. eapply gf_mon. apply H0. clear x0 x1 x2 x3 x4 x5 x6 H0. intros. apply gpaco7_unfold in PR; [|apply gf_mon]. apply rclo7_compose in PR. apply rclo7_dist in PR; [|apply CMP|apply DIST]. destruct PR. - right. apply CIH. eapply rclo7_mon. apply H. intros. eapply gf_mon. apply PR. intros. apply gpaco7_gupaco. apply gf_mon. apply gpaco7_gen_rclo. apply gf_mon. eapply gupaco7_mon. apply PR0. intros. destruct PR1; apply H0. - assert (REL: @rclo7 clo (rclo7 clo (gf (gupaco7 gf clo ((rg \7/ r) \7/ (rg \7/ r))) \7/ (rg \7/ r))) x0 x1 x2 x3 x4 x5 x6). { eapply rclo7_mon. apply H. intros. apply gpaco7_unfold in PR. apply PR. apply gf_mon. } apply rclo7_rclo in REL. apply rclo7_dist in REL; [|apply CMP|apply DIST]. right. destruct REL; cycle 1. + apply CIH0, H0. + apply CIH. eapply rclo7_mon. apply H0. intros. eapply gf_mon. apply PR. intros. eapply gupaco7_mon. apply PR0. intros. destruct PR1; apply H1. Qed.
imports: Program.Basics Paco(paco7 pacotac)