proposition: Theorem Σuniformize_complete : fo_form_fin_dec_SAT_in Σuniformize X -> fo_form_fin_dec_SAT_in A X.
proof: Proof. Proof using e HA. intros (M' & H1 & H2 & phi & H3 & H4). rewrite fol_all_uniform_after_spec in H3. exists (fom_specialize M'), H1. exists. { intros ? ?; apply H2. } exists phi. revert H4. apply fol_uniformize_complete; auto. Qed.
imports: List Arith Bool Lia Eqdep_dec Undecidability.Shared.Libs.DLW.Utils(utils_tac utils_list utils_nat finite) Undecidability.Shared.Libs.DLW.Vec(pos vec) Undecidability.FOL.TRAKHTENBROT(notations utils fol_ops fo_sig fo_terms fo_logic fo_sat)