proposition: Lemma HF_numeral T n : HFeq <<= T -> T ⊢ htransitive (tnumeral n).
proof: Proof. intros HT. induction n; apply CI; prv_all x; apply II. - apply Exp. apply imps. apply HF_eset'. auto. - apply Exp. apply imps. apply HF_eset'. auto. - prv_all y. apply -> imps. try apply (@HF_numeral_trans T (S n)); auto. - eapply DE; try apply HF_sig_iff; auto. + apply CE2 in IHn. apply (AllE x) in IHn. apply imps. cbn in IHn. subsimpl_in IHn. eapply Weak; eauto. + prv_all y. apply II. prv_all z. apply II. eapply HF_eq_elem. * auto. * apply HF_refl'. auto. * apply HF_sym'; auto. * apply imps. eapply IE; try apply HF_numeral_trans; auto. eapply HF_eq_elem; auto. apply HF_refl'. auto. Qed.
imports: Undecidability.FOL.ZF Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Sets.ZF Undecidability.FOL.Reductions.PCPb_to_HF Undecidability.PCP(PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb) Undecidability(Shared.ListAutomation)