Section Σfull_mon_red'_complete.
Variable (M : fo_model Σ' K)
(Kfin : finite_t K)
(Mdec : fo_model_dec M)
(φ : nat -> K)
(HA : fol_sem M φ Σfull_mon_red').
Let R x (v : vec _ n) := fol_sem M (env_vlift x·φ v) (fol_lconj (map Eq' lwY)).
Let Rreif : { f : K -> vec K n | forall x, R x (f x) }.
Let g := proj1_sig Rreif.
Let Hg x : fol_sem M (env_vlift x·φ (g x)) (fol_lconj (map Eq' lwY)).
End Σfull_mon_red'_complete.
(* 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.