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.

Theorem fol_uniformize_complete A φ :
incl (fol_rels A) lr
-> fol_sem M φ A <-> fol_sem M' φ (fol_uniformize A).