Require
Export
Ensembles.
Section
Ensembles_facts.
Variable
U: Type.
Lemma
Extension: (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: (x: U) ~ (In U (Empty_set U) x).
Proof
.
Intro x; Red; Induction 1.
Qed
.
Hints
Resolve Noone_in_empty.
Lemma
Included_Empty: (A: (Ensemble U))(Included U (Empty_set U) A).
Proof
.
Intro; Red.
Intros x H; Elim (Noone_in_empty x); Auto with sets.
Qed
.
Hints
Resolve Included_Empty.
Lemma
Add_intro1:
(A: (Ensemble U)) (x, y: U) (In U A y) -> (In U (Add
U A x) y).
Proof
.
Unfold 1 Add
; Auto with sets.
Qed
.
Hints
Resolve Add_intro1.
Lemma
Add_intro2: (A: (Ensemble U)) (x: U) (In U (Add
U A x) x).
Proof
.
Unfold 1 Add
; Auto with sets.
Qed
.
Hints
Resolve Add_intro2.
Lemma
Inhabited_add: (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
.
Hints
Resolve Inhabited_add.
Lemma
Inhabited_not_empty:
(X: (Ensemble U)) (Inhabited U X) -> ~ X == (Empty_set U).
Proof
.
Intros X H'; Elim H'.
Intros x H'0; Red; Intro H'1.
Absurd (In U X x); Auto with sets.
Rewrite H'1; Auto with sets.
Qed
.
Hints
Resolve Inhabited_not_empty.
Lemma
Add_not_Empty :
(A: (Ensemble U)) (x: U) ~ (Add
U A x) == (Empty_set U).
Proof
.
Auto with sets.
Qed
.
Hints
Resolve Add_not_Empty.
Lemma
not_Empty_Add :
(A: (Ensemble U)) (x: U) ~ (Empty_set U) == (Add
U A x).
Proof
.
Intros; Red; Intro H; Generalize (Add_not_Empty A x); Auto with sets.
Qed
.
Hints
Resolve not_Empty_Add.
Lemma
Singleton_inv: (x, y: U) (In U (Singleton U x) y) -> x == y.
Proof
.
Intros x y H'; Elim H'; Trivial with sets.
Qed
.
Hints
Resolve Singleton_inv.
Lemma
Singleton_intro: (x, y: U) x == y -> (In U (Singleton U x) y).
Proof
.
Intros x y H'; Rewrite H'; Trivial with sets.
Qed
.
Hints
Resolve Singleton_intro.
Lemma
Union_inv: (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:
(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:
(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
.
Hints
Resolve Intersection_inv.
Lemma
Couple_inv: (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
.
Hints
Resolve Couple_inv.
Lemma
Setminus_intro:
(A, B: (Ensemble U)) (x: U) (In U A x) -> ~ (In U B x) ->
(In U (Setminus U A B) x).
Proof
.
Unfold 1 Setminus; Red; Auto with sets.
Qed
.
Hints
Resolve Setminus_intro.
Lemma
Strict_Included_intro:
(X, Y: (Ensemble U)) (Included U X Y) /\ ~ X == Y ->
(Strict_Included U X Y).
Proof
.
Auto with sets.
Qed
.
Hints
Resolve Strict_Included_intro.
Lemma
Strict_Included_strict: (X: (Ensemble U)) ~ (Strict_Included U X X).
Proof
.
Intro X; Red; Intro H'; Elim H'.
Intros H'0 H'1; Elim H'1; Auto with sets.
Qed
.
Hints
Resolve Strict_Included_strict.
End
Ensembles_facts.
Hints
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.