diff options
Diffstat (limited to 'theories/Sets')
| -rw-r--r-- | theories/Sets/Powerset_facts.v | 91 |
1 files changed, 91 insertions, 0 deletions
diff --git a/theories/Sets/Powerset_facts.v b/theories/Sets/Powerset_facts.v index 2dd559a95..209c22f71 100644 --- a/theories/Sets/Powerset_facts.v +++ b/theories/Sets/Powerset_facts.v @@ -40,6 +40,11 @@ Section Sets_as_an_algebra. auto 6 with sets. Qed. + Theorem Empty_set_zero_right : forall X:Ensemble U, Union U X (Empty_set U) = X. + Proof. + auto 6 with sets. + Qed. + Theorem Empty_set_zero' : forall x:U, Add U (Empty_set U) x = Singleton U x. Proof. unfold Add at 1; auto using Empty_set_zero with sets. @@ -131,6 +136,17 @@ Section Sets_as_an_algebra. elim H'; intros x0 H'0; elim H'0; auto with sets. Qed. + 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). + Proof. + intros A B C. + rewrite Intersection_commutative. + rewrite Distributivity. + f_equal; apply Intersection_commutative. + Qed. + Theorem Distributivity' : forall A B C:Ensemble U, Union U A (Intersection U B C) = @@ -251,6 +267,81 @@ Section Sets_as_an_algebra. intros; apply Definition_of_covers; auto with sets. Qed. + Lemma Disjoint_Intersection: + forall A s1 s2, Disjoint A s1 s2 -> Intersection A s1 s2 = Empty_set A. + Proof. + intros. apply Extensionality_Ensembles. split. + * destruct H. + intros x H1. unfold In in *. exfalso. intuition. apply (H _ H1). + * intuition. + Qed. + + Lemma Intersection_Empty_set_l: + forall A s, Intersection A (Empty_set A) s = Empty_set A. + Proof. + intros. auto with sets. + Qed. + + Lemma Intersection_Empty_set_r: + forall A s, Intersection A s (Empty_set A) = Empty_set A. + Proof. + intros. auto with sets. + Qed. + + Lemma Seminus_Empty_set_l: + forall A s, Setminus A (Empty_set A) s = Empty_set A. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. destruct H1. unfold In in *. assumption. + * intuition. + Qed. + + Lemma Seminus_Empty_set_r: + forall A s, Setminus A s (Empty_set A) = s. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. destruct H1. unfold In in *. assumption. + * intuition. + Qed. + + 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). + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H. inversion H. inversion H0; intuition. + * intros x H. constructor; inversion H; inversion H0; intuition. + Qed. + + Lemma Setminus_Union_r: + forall A s1 s2 s3, + Setminus A s1 (Union A s2 s3) = Setminus A (Setminus A s1 s2) s3. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H. inversion H. constructor. intuition. contradict H1. intuition. + * intros x H. inversion H. inversion H0. constructor; intuition. inversion H4; intuition. + Qed. + + Lemma Setminus_Disjoint_noop: + forall A s1 s2, + Intersection A s1 s2 = Empty_set A -> Setminus A s1 s2 = s1. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. inversion_clear H1. intuition. + * intros x H1. constructor; intuition. contradict H. + apply Inhabited_not_empty. + exists x. intuition. + Qed. + + Lemma Setminus_Included_empty: + forall A s1 s2, + Included A s1 s2 -> Setminus A s1 s2 = Empty_set A. + Proof. + intros. apply Extensionality_Ensembles. split. + * intros x H1. inversion_clear H1. contradiction H2. intuition. + * intuition. + Qed. + End Sets_as_an_algebra. Hint Resolve Empty_set_zero Empty_set_zero' Union_associative Union_add |