proposition: Lemma compatible'13_companion clo (RES: compatible'13 clo): clo <14= cpn13 gf.
proof: Proof. assert (MON: monotone13 gf'13). { econstructor. apply LE, IN. eapply gf_mon, LE. apply IN. } assert (CPN: clo <14= cpn13 gf'13). { intros. econstructor. apply RES. apply PR. } intros. apply CPN in PR. econstructor; [|apply PR]. econstructor; [apply cpn13_mon|]; intros. assert (PR1: cpn13 gf'13 (gf r) <13= cpn13 gf'13 (gf'13 (cpn13 gf r))). { intros. eapply cpn13_mon. apply PR1. intros. assert (TMP: gf (cpn13 gf r) <13= (cpn13 gf r /13\ gf (cpn13 gf r))). { split; [apply cpn13_step; [apply gf_mon|]|]; assumption. } apply TMP. eapply gf_mon. apply PR2. intros. apply cpn13_base; assumption. } apply PR1 in PR0. clear PR1. eapply compat13_compat with (gf:=gf'13) in PR0; [|apply cpn13_compat, MON]. eapply gf_mon; [apply PR0|]. intros. eapply cpn13_cpn; [apply MON|]. eapply cpn13_mon; [apply PR1|]. intros. econstructor; [|apply PR2]. apply compat13_compatible', cpn13_compat, gf_mon. Qed.
imports: Program.Basics Paco(paco13 pacotac)