Section completeness.
Variable (M' : fo_model Σ' K).
Let M : fo_model Σ K.
Fact Σfull_mon_rec_complete φ :
fol_sem M' φ (Σfull_mon_rec A) <-> fol_sem M φ A.
End Σfull_mon_rec_complete.
Variable (Kfin : finite_t K)
(M'dec : fo_model_dec M')
(φ : nat -> K)
(HA : fol_sem M' φ Σfull_mon_red).
End completeness.
(* The non-skolemized reduction is correct *)

Theorem Σfull_mon_red_correct : fo_form_fin_dec_SAT_in A K
<-> fo_form_fin_dec_SAT_in Σfull_mon_red K.