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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 *)
(************************************************************************)
(*i $Id$ i*)
(** Properties of a binary relation [R] on type [A] *)
Section Rstar.
Variable A : Type.
Variable R : A -> A -> Prop.
(** Definition of the reflexive-transitive closure [R*] of [R] *)
(** Smallest reflexive [P] containing [R o P] *)
Definition Rstar (x y:A) :=
forall P:A -> A -> Prop,
(forall u:A, P u u) -> (forall u v w:A, R u v -> P v w -> P u w) -> P x y.
Theorem Rstar_reflexive : forall x:A, Rstar x x.
Proof.
unfold Rstar. intros x P P_refl RoP. apply P_refl.
Qed.
Theorem Rstar_R : forall x y z:A, R x y -> Rstar y z -> Rstar x z.
Proof.
intros x y z R_xy Rstar_yz.
unfold Rstar.
intros P P_refl RoP. apply RoP with (v:=y).
assumption.
apply Rstar_yz; assumption.
Qed.
(** We conclude with transitivity of [Rstar] : *)
Theorem Rstar_transitive :
forall x y z:A, Rstar x y -> Rstar y z -> Rstar x z.
Proof.
intros x y z Rstar_xy; unfold Rstar in Rstar_xy.
apply Rstar_xy; trivial.
intros u v w R_uv fz Rstar_wz.
apply Rstar_R with (y:=v); auto.
Qed.
(** Another characterization of [R*] *)
(** Smallest reflexive [P] containing [R o R*] *)
Definition Rstar' (x y:A) :=
forall P:A -> A -> Prop,
P x x -> (forall u:A, R x u -> Rstar u y -> P x y) -> P x y.
Theorem Rstar'_reflexive : forall x:A, Rstar' x x.
Proof.
unfold Rstar'; intros; assumption.
Qed.
Theorem Rstar'_R : forall x y z:A, R x z -> Rstar z y -> Rstar' x y.
Proof.
unfold Rstar'. intros x y z Rxz Rstar_zy P Pxx RoP.
apply RoP with (u:=z); trivial.
Qed.
(** Equivalence of the two definitions: *)
Theorem Rstar'_Rstar : forall x y:A, Rstar' x y -> Rstar x y.
Proof.
intros x z Rstar'_xz; unfold Rstar' in Rstar'_xz.
apply Rstar'_xz.
exact (Rstar_reflexive x).
intro y; generalize x y z; exact Rstar_R.
Qed.
Theorem Rstar_Rstar' : forall x y:A, Rstar x y -> Rstar' x y.
Proof.
intros.
apply H.
exact Rstar'_reflexive.
intros u v w R_uv Rs'_vw. apply Rstar'_R with (z:=v).
assumption.
apply Rstar'_Rstar; assumption.
Qed.
(** Property of Commutativity of two relations *)
Definition commut (A:Type) (R1 R2:A -> A -> Prop) :=
forall x y:A,
R1 y x -> forall z:A, R2 z y -> exists2 y' : A, R2 y' x & R1 z y'.
End Rstar.
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