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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. *)
(****************************************************************************)
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.
Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets.
Hint Resolve Definition_of_covers: sets.
Section Partial_order_facts.
Variable U : Type.
Variable D : PO U.
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.
Proof.
unfold Strict_Rel_of at 1.
red.
elim D; simpl.
intros C R H' H'0; elim H'0.
intros H'1 H'2 H'3 x y z H'4 H'5; split.
- apply H'2 with (y := y); tauto.
- red; intro H'6.
elim H'4; intros H'7 H'8; apply H'8; clear H'4.
apply H'3; auto.
rewrite H'6; tauto.
Qed.
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.
Proof.
unfold Strict_Rel_of at 1.
red.
elim D; simpl.
intros C R H' H'0; elim H'0.
intros H'1 H'2 H'3 x y z H'4 H'5; split.
- apply H'2 with (y := y); tauto.
- red; intro H'6.
elim H'5; intros H'7 H'8; apply H'8; clear H'5.
apply H'3; auto.
rewrite <- H'6; auto.
Qed.
Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D).
red.
intros x y z H' H'0.
apply Strict_Rel_Transitive_with_Rel with (y := y);
[ intuition | unfold Strict_Rel_of in H', H'0; intuition ].
Qed.
End Partial_order_facts.
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