# Library Coq.Sets.Powerset_facts

Require Export Ensembles.
Require Export Constructive_sets.
Require Export Relations_1.
Require Export Relations_1_facts.
Require Export Partial_Order.
Require Export Cpo.
Require Export Powerset.

Local Ltac Tauto.intuition_solver ::= auto with sets.

Section Sets_as_an_algebra.
Variable U : Type.

Theorem Empty_set_zero : forall X:Ensemble U, Union U (Empty_set U) X = X.

Theorem Empty_set_zero_right : forall X:Ensemble U, Union U X (Empty_set U) = X.

Theorem Empty_set_zero' : forall x:U, Add U (Empty_set U) x = Singleton U x.

Lemma less_than_empty :
forall X:Ensemble U, Included U X (Empty_set U) -> X = Empty_set U.

Theorem Union_commutative : forall A B:Ensemble U, Union U A B = Union U B A.

Theorem Union_associative :
forall A B C:Ensemble U, Union U (Union U A B) C = Union U A (Union U B C).

Theorem Union_idempotent : forall A:Ensemble U, Union U A A = A.

Lemma Union_absorbs :
forall A B:Ensemble U, Included U B A -> Union U A B = A.

Theorem Couple_as_union :
forall x y:U, Union U (Singleton U x) (Singleton U y) = Couple U x y.

Theorem Triple_as_union :
forall x y z:U,
Union U (Union U (Singleton U x) (Singleton U y)) (Singleton U z) =
Triple U x y z.

Theorem Triple_as_Couple : forall x y:U, Couple U x y = Triple U x x y.

Theorem Triple_as_Couple_Singleton :
forall x y z:U, Triple U x y z = Union U (Couple U x y) (Singleton U z).

Theorem Intersection_commutative :
forall A B:Ensemble U, Intersection U A B = Intersection U B A.

Theorem Distributivity :
forall A B C:Ensemble U,
Intersection U A (Union U B C) =
Union U (Intersection U A B) (Intersection U A C).

Lemma Distributivity_l
: forall (A B C : Ensemble U),
Intersection U (Union U A B) C =
Union U (Intersection U A C) (Intersection U B C).

Theorem Distributivity' :
forall A B C:Ensemble U,
Union U A (Intersection U B C) =
Intersection U (Union U A B) (Union U A C).

forall (A B:Ensemble U) (x:U), Add U (Union U A B) x = Union U A (Add U B x).

Theorem Non_disjoint_union :
forall (X:Ensemble U) (x:U), In U X x -> Add U X x = X.

Theorem Non_disjoint_union' :
forall (X:Ensemble U) (x:U), ~ In U X x -> Subtract U X x = X.

Lemma singlx : forall x y:U, In U (Add U (Empty_set U) x) y -> x = y.

forall (A B:Ensemble U) (x:U),
Included U A B -> Included U (Add U A x) (Add U B x).

forall (A B:Ensemble U) (x:U),
~ In U A x -> Included U (Add U A x) (Add U B x) -> Included U A B.

forall (A:Ensemble U) (x y z:U),

forall (A B:Ensemble U) (x y:U),
Included U B A -> Add U (Add U A x) y = Union U (Add U A x) (Add U B y).

Lemma setcover_intro :
forall (U:Type) (A x y:Ensemble U),
Strict_Included U x y ->
~ (exists z : _, Strict_Included U x z /\ Strict_Included U z y) ->
covers (Ensemble U) (Power_set_PO U A) y x.

Lemma Disjoint_Intersection:
forall A s1 s2, Disjoint A s1 s2 -> Intersection A s1 s2 = Empty_set A.

Lemma Intersection_Empty_set_l:
forall A s, Intersection A (Empty_set A) s = Empty_set A.

Lemma Intersection_Empty_set_r:
forall A s, Intersection A s (Empty_set A) = Empty_set A.

Lemma Seminus_Empty_set_l:
forall A s, Setminus A (Empty_set A) s = Empty_set A.

Lemma Seminus_Empty_set_r:
forall A s, Setminus A s (Empty_set A) = s.

Lemma Setminus_Union_l:
forall A s1 s2 s3,
Setminus A (Union A s1 s2) s3 = Union A (Setminus A s1 s3) (Setminus A s2 s3).

Lemma Setminus_Union_r:
forall A s1 s2 s3,
Setminus A s1 (Union A s2 s3) = Setminus A (Setminus A s1 s2) s3.

Lemma Setminus_Disjoint_noop:
forall A s1 s2,
Intersection A s1 s2 = Empty_set A -> Setminus A s1 s2 = s1.

Lemma Setminus_Included_empty:
forall A s1 s2,
Included A s1 s2 -> Setminus A s1 s2 = Empty_set A.

End Sets_as_an_algebra.

#[global]
Hint Resolve Empty_set_zero Empty_set_zero' Union_associative Union_add