# Library Coq.Sets.Powerset

Require Export Ensembles.
Require Export Relations_1.
Require Export Relations_1_facts.
Require Export Partial_Order.
Require Export Cpo.
Section The_power_set_partial_order.
Variable U : Type.
Inductive Power_set (A:Ensemble U) : Ensemble (Ensemble U) :=
Definition_of_Power_set :
forall X:Ensemble U, Included U X A -> In (Ensemble U) (Power_set A) X.
Hint Resolve Definition_of_Power_set.
Theorem Empty_set_minimal : forall X:Ensemble U, Included U (Empty_set U) X.
Hint Resolve Empty_set_minimal.
Theorem Power_set_Inhabited :
forall X:Ensemble U, Inhabited (Ensemble U) (Power_set X).
Hint Resolve Power_set_Inhabited.
Theorem Inclusion_is_an_order : Order (Ensemble U) (Included U).
Hint Resolve Inclusion_is_an_order.
Theorem Inclusion_is_transitive : Transitive (Ensemble U) (Included U).
Hint Resolve Inclusion_is_transitive.
Definition Power_set_PO : Ensemble U -> PO (Ensemble U).
Defined.
Hint Unfold Power_set_PO.
Theorem Strict_Rel_is_Strict_Included :
same_relation (Ensemble U) (Strict_Included U)
(Strict_Rel_of (Ensemble U) (Power_set_PO (Full_set U))).
Hint Resolve Strict_Rel_Transitive Strict_Rel_is_Strict_Included.
Lemma Strict_inclusion_is_transitive_with_inclusion :
forall x y z:Ensemble U,
Strict_Included U x y -> Included U y z -> Strict_Included U x z.
Lemma Strict_inclusion_is_transitive_with_inclusion_left :
forall x y z:Ensemble U,
Included U x y -> Strict_Included U y z -> Strict_Included U x z.
Lemma Strict_inclusion_is_transitive :
Transitive (Ensemble U) (Strict_Included U).
Theorem Empty_set_is_Bottom :
forall A:Ensemble U, Bottom (Ensemble U) (Power_set_PO A) (Empty_set U).
Hint Resolve Empty_set_is_Bottom.
Theorem Union_minimal :
forall a b X:Ensemble U,
Included U a X -> Included U b X -> Included U (Union U a b) X.
Hint Resolve Union_minimal.
Theorem Intersection_maximal :
forall a b X:Ensemble U,
Included U X a -> Included U X b -> Included U X (Intersection U a b).
Theorem Union_increases_l : forall a b:Ensemble U, Included U a (Union U a b).
Theorem Union_increases_r : forall a b:Ensemble U, Included U b (Union U a b).
Theorem Intersection_decreases_l :
forall a b:Ensemble U, Included U (Intersection U a b) a.
Theorem Intersection_decreases_r :
forall a b:Ensemble U, Included U (Intersection U a b) b.
Hint Resolve Union_increases_l Union_increases_r Intersection_decreases_l
Intersection_decreases_r.
Theorem Union_is_Lub :
forall A a b:Ensemble U,
Included U a A ->
Included U b A ->
Lub (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Union U a b).
Theorem Intersection_is_Glb :
forall A a b:Ensemble U,
Included U a A ->
Included U b A ->
Glb (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b)
(Intersection U a b).
End The_power_set_partial_order.
Hint Resolve Empty_set_minimal: sets v62.
Hint Resolve Power_set_Inhabited: sets v62.
Hint Resolve Inclusion_is_an_order: sets v62.
Hint Resolve Inclusion_is_transitive: sets v62.
Hint Resolve Union_minimal: sets v62.
Hint Resolve Union_increases_l: sets v62.
Hint Resolve Union_increases_r: sets v62.
Hint Resolve Intersection_decreases_l: sets v62.
Hint Resolve Intersection_decreases_r: sets v62.
Hint Resolve Empty_set_is_Bottom: sets v62.
Hint Resolve Strict_inclusion_is_transitive: sets v62.