proposition: Lemma FST_eq_sig T x y : FSTeq <<= T -> T ⊢ x ≡ y -> T ⊢ σ x ≡ σ y.
proof: Proof. intros HT Hxy. apply FST_ext'; auto. - prv_all z. apply II. apply FST_adj; auto. assert1 H. apply FST_adj in H; auto. apply (DE H). + apply DI1. apply FST_trans' with x; auto. apply (Weak Hxy); auto. + apply DI2. eapply FST_eq_elem; auto; try apply FST_refl'; auto. apply (Weak Hxy); auto. - prv_all z. apply II. apply FST_adj; auto. assert1 H. apply FST_adj in H; auto. apply (DE H). + apply DI1. apply FST_trans' with y; auto. apply FST_sym'; auto. apply (Weak Hxy); auto. + apply DI2. eapply FST_eq_elem; auto; try apply FST_refl'; auto. apply FST_sym'; auto. apply (Weak Hxy); 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)