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
.
Hint
Resolve Noone_in_empty.
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
.
Hint
Resolve Included_Empty.
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
.
Hint
Resolve Add_intro1.
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
.
Hint
Resolve Add_intro2.
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 with sets.
Qed
.
Hint
Resolve Inhabited_add.
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 with sets.
Qed
.
Hint
Resolve Inhabited_not_empty.
Lemma
Add_not_Empty : forall (A:Ensemble U) (x:U), Add
U A x <> Empty_set U.
Proof
.
auto with sets.
Qed
.
Hint
Resolve Add_not_Empty.
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
.
Hint
Resolve not_Empty_Add.
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
.
Hint
Resolve Singleton_inv.
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
.
Hint
Resolve Singleton_intro.
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'; elim H'; auto with sets.
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
.
Hint
Resolve Intersection_inv.
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
.
Hint
Resolve Couple_inv.
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
.
Hint
Resolve Setminus_intro.
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
.
Hint
Resolve Strict_Included_intro.
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
.
Hint
Resolve Strict_Included_strict.
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.
Index
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