Library SCHROEDER.Sums
Require Import Ensembles.
Section Set_Sums.
Variable U : Type.
Inductive Set_Sum (D : Ensemble (Ensemble U)) : U -> Prop :=
Set_Sum_intro :
forall B : Ensemble U,
In (Ensemble U) D B -> forall x : U, In U B x -> Set_Sum D x.
Lemma Set_Sum_is_majoring :
forall (D : Ensemble (Ensemble U)) (A : Ensemble U),
(forall C : Ensemble U, D C -> Included U C A) -> Included U (Set_Sum D) A.
intros.
red in |- *; intros x x_in_Set_Sum_D.
red in x_in_Set_Sum_D.
elim x_in_Set_Sum_D.
assumption.
Qed.
Lemma Set_Sum_is_minoring :
forall (D : Ensemble (Ensemble U)) (A : Ensemble U),
In (Ensemble U) D A -> Included U A (Set_Sum D).
do 2 intro; red in |- *; intros A_in_D x x_in_A.
red in |- *; apply Set_Sum_intro with A; assumption.
Qed.
End Set_Sums.