Section completeness.
Variable (M' : fo_model Σ' X).
Local Definition fom_specialize : fo_model Σ X.
End uniform_after.
Local Definition fol_all_uniform_after : fol_form Σ' :=
fol_lconj (map fol_uniform_after lr).
Let uniform := forall r, In r lr -> forall (v : vec _ (ar_rels _ r)) e1 e2,
fom_rels M' r (vec_fill_tail n v e1)
<-> fom_rels M' r (vec_fill_tail n v e2).
Local Fact fol_all_uniform_after_spec φ :
fol_sem M' φ fol_all_uniform_after <-> uniform.
Hypothesis Hlr : uniform.
End completeness.
Variable (A : fol_form Σ) (HA : incl (fol_rels A) lr).
Definition Σuniformize := fol_all_uniform_after ⟑ fol_uniformize A.

Theorem Σuniformize_sound : fo_form_fin_dec_SAT_in A X
-> fo_form_fin_dec_SAT_in Σuniformize X.