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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
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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.  *)
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Require Export Ensembles.
Require Export Relations_1.
Section Partial_orders.
Variable U : Type.
Definition Carrier := Ensemble U.
Definition Rel := Relation U.
Record PO : Type := Definition_of_PO
{ Carrier_of : Ensemble U;
Rel_of : Relation U;
PO_cond1 : Inhabited U Carrier_of;
PO_cond2 : Order U Rel_of }.
Variable p : PO.
Definition Strict_Rel_of : Rel := fun x y:U => Rel_of p x y /\ x <> y.
Inductive covers (y x:U) : Prop :=
Definition_of_covers :
Strict_Rel_of x y ->
~ (exists z : _, Strict_Rel_of x z /\ Strict_Rel_of z y) ->
covers y x.
End Partial_orders.
#[global]
Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets.
#[global]
Hint Resolve Definition_of_covers: sets.
Section Partial_order_facts.
Variable U : Type.
Variable D : PO U.

(* Specifications: *)
Lemma Strict_Rel_Transitive_with_Rel :
forall x y z:U,
Strict_Rel_of U D x y -> @Rel_of U D y z -> Strict_Rel_of U D x z.
Lemma Strict_Rel_Transitive_with_Rel_left :
forall x y z:U,
@Rel_of U D x y -> Strict_Rel_of U D y z -> Strict_Rel_of U D x z.
Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D).
red.