Library Coq.Sets.Constructive_sets


Require Export Ensembles.




Section Ensembles_facts.

  Variable U : Type.




  Lemma Extension : forall B C:Ensemble U, B = C -> Same_set U B C.

  Proof.

    intros B C H'; rewrite H'; auto with sets.

  Qed.




  Lemma Noone_in_empty : forall x:U, ~ In U (Empty_set U) x.

  Proof.

    red in |- *; destruct 1.

  Qed.




  Lemma Included_Empty : forall A:Ensemble U, Included U (Empty_set U) A.

  Proof.

    intro; red in |- *.

    intros x H; elim (Noone_in_empty x); auto with sets.

  Qed.




  Lemma Add_intro1 :

    forall (A:Ensemble U) (x y:U), In U A y -> In U (Add U A x) y.

  Proof.

    unfold Add at 1 in |- *; auto with sets.

  Qed.




  Lemma Add_intro2 : forall (A:Ensemble U) (x:U), In U (Add U A x) x.

  Proof.

    unfold Add at 1 in |- *; auto with sets.

  Qed.




  Lemma Inhabited_add : forall (A:Ensemble U) (x:U), Inhabited U (Add U A x).

  Proof.

    intros A x.

    apply Inhabited_intro with (x := x); auto using Add_intro2 with sets.

  Qed.




  Lemma Inhabited_not_empty :

    forall X:Ensemble U, Inhabited U X -> X <> Empty_set U.

  Proof.

    intros X H'; elim H'.

    intros x H'0; red in |- *; intro H'1.

    absurd (In U X x); auto with sets.

    rewrite H'1; auto using Noone_in_empty with sets.

  Qed.




  Lemma Add_not_Empty : forall (A:Ensemble U) (x:U), Add U A x <> Empty_set U.

  Proof.

    intros A x; apply Inhabited_not_empty; apply Inhabited_add.

  Qed.




  Lemma not_Empty_Add : forall (A:Ensemble U) (x:U), Empty_set U <> Add U A x.

  Proof.

    intros; red in |- *; intro H; generalize (Add_not_Empty A x); auto with sets.

  Qed.




  Lemma Singleton_inv : forall x y:U, In U (Singleton U x) y -> x = y.

  Proof.

    intros x y H'; elim H'; trivial with sets.

  Qed.




  Lemma Singleton_intro : forall x y:U, x = y -> In U (Singleton U x) y.

  Proof.

    intros x y H'; rewrite H'; trivial with sets.

  Qed.




  Lemma Union_inv :

    forall (B C:Ensemble U) (x:U), In U (Union U B C) x -> In U B x \/ In U C x.

  Proof.

    intros B C x H'; elim H'; auto with sets.

  Qed.




  Lemma Add_inv :

    forall (A:Ensemble U) (x y:U), In U (Add U A x) y -> In U A y \/ x = y.

  Proof.

    intros A x y H'; induction H'.

      left; assumption.

      right; apply Singleton_inv; assumption.

  Qed.




  Lemma Intersection_inv :

    forall (B C:Ensemble U) (x:U),

      In U (Intersection U B C) x -> In U B x /\ In U C x.

  Proof.

    intros B C x H'; elim H'; auto with sets.

  Qed.




  Lemma Couple_inv : forall x y z:U, In U (Couple U x y) z -> z = x \/ z = y.

  Proof.

    intros x y z H'; elim H'; auto with sets.

  Qed.




  Lemma Setminus_intro :

    forall (A B:Ensemble U) (x:U),

      In U A x -> ~ In U B x -> In U (Setminus U A B) x.

  Proof.

    unfold Setminus at 1 in |- *; red in |- *; auto with sets.

  Qed.




  Lemma Strict_Included_intro :

    forall X Y:Ensemble U, Included U X Y /\ X <> Y -> Strict_Included U X Y.

  Proof.

    auto with sets.

  Qed.




  Lemma Strict_Included_strict : forall X:Ensemble U, ~ Strict_Included U X X.

  Proof.

    intro X; red in |- *; intro H'; elim H'.

    intros H'0 H'1; elim H'1; auto with sets.

  Qed.




End Ensembles_facts.




Hint Resolve Singleton_inv Singleton_intro Add_intro1 Add_intro2

  Intersection_inv Couple_inv Setminus_intro Strict_Included_intro

  Strict_Included_strict Noone_in_empty Inhabited_not_empty Add_not_Empty

  not_Empty_Add Inhabited_add Included_Empty: sets v62.