theories/Sets

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+(****************************************************************************)
+(*                                                                          *)
+(*                         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.  *)
+(****************************************************************************)
+
+(* $Id$ *)
+
+Section Relations_1.
+   Variable U: Type.
+   
+   Definition Relation :=  U -> U -> Prop.
+   Variable R: Relation.
+   
+   Definition Reflexive : Prop :=  (x: U) (R x x).
+   
+   Definition Transitive : Prop :=  (x,y,z: U) (R x y) -> (R y z) -> (R x z).
+   
+   Definition Symmetric : Prop :=  (x,y: U) (R x y) -> (R y x).
+   
+   Definition Antisymmetric : Prop :=
+      (x: U) (y: U) (R x y) -> (R y x) -> x == y.
+   
+   Definition contains : Relation -> Relation -> Prop :=
+      [R,R': Relation] (x: U) (y: U) (R' x y) -> (R x y).
+   
+   Definition same_relation : Relation -> Relation -> Prop :=
+      [R,R': Relation] (contains R R') /\ (contains R' R).
+   
+   Inductive Preorder : Prop :=
+         Definition_of_preorder: Reflexive -> Transitive -> Preorder.
+   
+   Inductive Order : Prop :=
+         Definition_of_order: Reflexive -> Transitive -> Antisymmetric -> Order.
+   
+   Inductive Equivalence : Prop :=
+         Definition_of_equivalence:
+           Reflexive -> Transitive -> Symmetric -> Equivalence.
+   
+   Inductive PER : Prop :=
+         Definition_of_PER: Symmetric -> Transitive -> PER.
+   
+End Relations_1.
+Hints Unfold Reflexive Transitive Antisymmetric Symmetric contains 
+	same_relation : sets v62.
+Hints Resolve Definition_of_preorder Definition_of_order 
+	Definition_of_equivalence Definition_of_PER : sets v62.