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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: Rstar.v,v 1.8.2.1 2004/07/16 19:31:16 herbelin Exp $ 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
fun (x:A) (P:A -> A -> Prop) (h1:forall u:A, P u u)
(h2:forall u v w:A, R u v -> P v w -> P u w) =>
h1 x.
Theorem Rstar_R : forall x y z:A, R x y -> Rstar y z -> Rstar x z.
Proof
fun (x y z:A) (t1:R x y) (t2:Rstar y z) (P:A -> A -> Prop)
(h1:forall u:A, P u u) (h2:forall u v w:A, R u v -> P v w -> P u w) =>
h2 x y z t1 (t2 P h1 h2).
(** We conclude with transitivity of [Rstar] : *)
Theorem Rstar_transitive :
forall x y z:A, Rstar x y -> Rstar y z -> Rstar x z.
Proof
fun (x y z:A) (h:Rstar x y) =>
h (fun u v:A => Rstar v z -> Rstar u z) (fun (u:A) (t:Rstar u z) => t)
(fun (u v w:A) (t1:R u v) (t2:Rstar w z -> Rstar v z)
(t3:Rstar w z) => Rstar_R u v z t1 (t2 t3)).
(** 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
fun (x:A) (P:A -> A -> Prop) (h:P x x)
(h':forall u:A, R x u -> Rstar u x -> P x x) => h.
Theorem Rstar'_R : forall x y z:A, R x z -> Rstar z y -> Rstar' x y.
Proof
fun (x y z:A) (t1:R x z) (t2:Rstar z y) (P:A -> A -> Prop)
(h1:P x x) (h2:forall u:A, R x u -> Rstar u y -> P x y) =>
h2 z t1 t2.
(** Equivalence of the two definitions: *)
Theorem Rstar'_Rstar : forall x y:A, Rstar' x y -> Rstar x y.
Proof
fun (x y:A) (h:Rstar' x y) =>
h Rstar (Rstar_reflexive x) (fun u:A => Rstar_R x u y).
Theorem Rstar_Rstar' : forall x y:A, Rstar x y -> Rstar' x y.
Proof
fun (x y:A) (h:Rstar x y) =>
h Rstar' (fun u:A => Rstar'_reflexive u)
(fun (u v w:A) (h1:R u v) (h2:Rstar' v w) =>
Rstar'_R u w v h1 (Rstar'_Rstar v w h2)).
(** Property of Commutativity of two relations *)
Definition commut (A:Set) (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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