Section CONSTRUCTIVE_FACTS.
Variant Th (X : ensemble formula) (b : formula) : Prop :=
| In_Th
(INFERS : X ⊢ b)
: b \in Th X.
#[local] Hint Constructors Th : core.
#[global]
Add Parametric Morphism :
Th with signature (eqProp ==> eqProp)
as Th_lifts_eqProp.
Proof with reflexivity || eauto.
eapply isFilter_intro.
- exists trueB. econstructor.
eapply ImplicationI, ByAssumption; done!.
- intros x1 x2 [INFERS1] [INFERS2].
econstructor. eapply ConjunctionI; done!.
- intros x [INFERS] x' x_le_x'. apply leB_iff in x_le_x'.
econstructor. eapply Cut_property. exact INFERS.
eapply extend_infers with (Gamma := E.singleton x); done!.
Qed.
Proof with eauto.
intros b [xs ?]; des. apply andsB_le_iff in andsB_LE.
destruct andsB_LE as [X' [xs_repr_X' INFERS]].
econstructor. eapply extend_infers...
intros z z_in. eapply FINITE_SUBSET, xs_repr_X'...
Qed.

Theorem inference_is_finite (X : ensemble formula) (b : formula)
(INFERS : X ⊢ b)
: exists xs, exists X', L.is_finsubset_of xs X /\ L.is_listrep_of xs X' /\ X' ⊢ b.