aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Sets
diff options
context:
space:
mode:
Diffstat (limited to 'theories/Sets')
-rw-r--r--theories/Sets/Powerset_facts.v91
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