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(************************************************************************)
(* 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: Relations_1.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*)
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.
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