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Diffstat (limited to 'theories7/Sets/Constructive_sets.v')
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diff --git a/theories7/Sets/Constructive_sets.v b/theories7/Sets/Constructive_sets.v new file mode 100755 index 00000000..35c88e9d --- /dev/null +++ b/theories7/Sets/Constructive_sets.v @@ -0,0 +1,162 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Constructive_sets.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +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. +Red; NewDestruct 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. |