Section completeness.
Variable (M' : fo_model Σ0 unit).
Let M : fo_model Σ unit.
Hypothesis (M'dec : fo_model_dec M')
(psi : nat -> unit)
(A : fol_form Σ)
(HA : fol_sem M' psi (Σ_Σ0 A)).
Local Lemma Σ_Σ0_completeness : fo_form_fin_dec_SAT_in A unit.
End completeness.

Theorem Σ_Σ0_correct A : fo_form_fin_dec_SAT A <-> fo_form_fin_dec_SAT_in (Σ_Σ0 A) unit.