Section filter_from_basis.
Variable X:Type.
Variable B:Family X.
Hypothesis B_nonempty: Inhabited B.
Hypothesis B_empty: ~ In B Empty_set.
Hypothesis B_basis_cond: forall S1 S2:Ensemble X,
In B S1 -> In B S2 -> exists T:Ensemble X,
In B T /\ Included T (Intersection S1 S2).
Definition Build_Filter_from_basis : Filter X.
refine (Build_Filter X [ S:Ensemble X | exists T:Ensemble X,
In B T /\ Included T S ] _ _ _ _).
- intros.
destruct H.
destruct H0.
destruct H as [T1 []].
destruct H0 as [T2 []].
destruct (B_basis_cond T1 T2 H H0) as [T' []].
constructor.
exists T'; split; trivial.
red; intros.
apply H4 in H5.
destruct H5.
constructor; auto.
- intros.
destruct H.
destruct H as [T []].
constructor.
exists T; split; auto with sets.
- constructor.
destruct B_nonempty as [T].
exists T; split; trivial.
red; intros; constructor.
- red; intro.
destruct H.
destruct H as [T []].
assert (T = Empty_set).
{ apply Extensionality_Ensembles; split; auto with sets. }
destruct H1.
contradiction B_empty.
Defined.

Lemma filter_from_basis_basis: filter_basis Build_Filter_from_basis B.