Section completeness.
Hypothesis (M' : fo_model Σ' X) (phi : nat -> X).
Let M : fo_model Σ X.
Hypothesis (Xf : finite_t X)
(M'd : fo_model_dec M')
(H : fol_sem M' phi (Σ_no_sym A HA)).
Local Fact Σ_no_sym_completeness : fo_form_fin_dec_SAT_in A X.
End completeness.
End semantics.

Theorem Σ_no_sym_correct : { B : fol_form Σ' | fo_form_fin_dec_SAT A <-> fo_form_fin_dec_SAT B }.