proposition: Lemma FST_numeral_trans T n x y : FSTeq <<= T -> T ⊢ x ∈ tnumeral n → y ∈ x → y ∈ tnumeral n.
proof: Proof. intros HT. induction n; cbn. - apply II, Exp. eapply IE. apply FST_eset'. all: auto. - repeat apply II. apply FST_adj; auto. assert2 H. apply FST_adj in H; auto. apply (DE H). + apply DI2. eapply FST_eq_elem; auto. apply FST_refl'; auto. + apply DI2. eapply IE. eapply IE. apply (Weak IHn); auto. auto. auto. Qed.
imports: Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Deduction.FullNDFacts Undecidability.FOL.Sets.FST Undecidability.FOL.FST Undecidability.FOL.Reductions.PCPb_to_FST Undecidability.PCP(PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb) Undecidability(Shared.ListAutomation)