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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(****************************************************************************)
(* *)
(* 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. *)
(****************************************************************************)
Section Relations_1.
Variable U : Type.
Definition Relation := U -> U -> Prop.
Variable R : Relation.
Definition Reflexive : Prop := forall x:U, R x x.
Definition Transitive : Prop := forall x y z:U, R x y -> R y z -> R x z.
Definition Symmetric : Prop := forall x y:U, R x y -> R y x.
Definition Antisymmetric : Prop := forall x y:U, R x y -> R y x -> x = y.
Definition contains (R R':Relation) : Prop :=
forall x y:U, R' x y -> R x y.
Definition same_relation (R R':Relation) : Prop :=
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.
Hint Unfold Reflexive Transitive Antisymmetric Symmetric contains
same_relation: sets.
Hint Resolve Definition_of_preorder Definition_of_order
Definition_of_equivalence Definition_of_PER: sets.
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