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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
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(*i $Id: Rstar.v 8642 2006-03-17 10:09:02Z notin $ 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.