Section completeness.
Variables (X : Type) (M' : fo_model Σ' X).
Let M : fo_model Σ X.
Let convert_complete A φ : fol_sem M φ A <-> fol_sem M' φ (convert A).
Hypothesis (Xfin : finite_t X)
(M'dec : fo_model_dec M')
(phi : nat -> X)
(A : fol_form Σ)
(HA : fol_sem M' phi (convert A)).
Local Fact convert_completeness : fo_form_fin_dec_SAT A.
End completeness.

Theorem Σn1_Σ_correct A :
fo_form_fin_dec_SAT A
<-> fo_form_fin_dec_SAT (Σn1_Σ A).