Section completeness.
Variable (M' : fo_model Σ' X)
(φ : nat -> X).
Let M : fo_model Σ X.
Local Fact Σrem_constants_complete n A ψ :
ψ n = φ 0
-> fol_sem M ψ A <-> fol_sem M' ψ (Σrem_constants n A).
Hypothesis (Xfin : finite_t X)
(M'dec : fo_model_dec M')
(A : fol_form Σ)
(HA : fol_sem M' φ (Σrem_constants 0 A)).
Local Lemma Σrem_constants_completeness : fo_form_fin_dec_SAT_in A X.
End completeness.

Theorem Σrem_constants_correct A :
fo_form_fin_dec_SAT_in A X
<-> fo_form_fin_dec_SAT_in (Σrem_constants 0 A) X.