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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: Newman.v 9245 2006-10-17 12:53:34Z notin $ i*)
-
-Require Import Rstar.
-
-Section Newman.
-
-Variable A : Type. 
-Variable R : A -> A -> Prop.
-
-Let Rstar := Rstar A R. 
-Let Rstar_reflexive := Rstar_reflexive A R.
-Let Rstar_transitive := Rstar_transitive A R.
-Let Rstar_Rstar' := Rstar_Rstar' A R.
-
-Definition coherence (x y:A) := ex2 (Rstar x) (Rstar y).
-
-Theorem coherence_intro :
-  forall x y z:A, Rstar x z -> Rstar y z -> coherence x y.
-Proof fun (x y z:A) (h1:Rstar x z) (h2:Rstar y z) =>
-  ex_intro2 (Rstar x) (Rstar y) z h1 h2.
-  
-(** A very simple case of coherence : *)
-
-Lemma Rstar_coherence : forall x y:A, Rstar x y -> coherence x y.
-Proof
-  fun (x y:A) (h:Rstar x y) => coherence_intro x y y h (Rstar_reflexive y).  
-  
-(** coherence is symmetric *)
-Lemma coherence_sym : forall x y:A, coherence x y -> coherence y x.
-Proof
-  fun (x y:A) (h:coherence x y) =>
-    ex2_ind
-      (fun (w:A) (h1:Rstar x w) (h2:Rstar y w) =>
-        coherence_intro y x w h2 h1) h.
-
-Definition confluence (x:A) :=
-  forall y z:A, Rstar x y -> Rstar x z -> coherence y z.  
-  
-Definition local_confluence (x:A) :=
-  forall y z:A, R x y -> R x z -> coherence y z.  
-  
-Definition noetherian :=
-  forall (x:A) (P:A -> Prop),
-    (forall y:A, (forall z:A, R y z -> P z) -> P y) -> P x.  
-  
-Section Newman_section.
-
-  (** The general hypotheses of the theorem *)
-
-  Hypothesis Hyp1 : noetherian.
-  Hypothesis Hyp2 : forall x:A, local_confluence x.  
-  
-  (** The induction hypothesis *)
-
-  Section Induct.
-    Variable x : A.
-    Hypothesis hyp_ind : forall u:A, R x u -> confluence u.  
-  
-    (** Confluence in [x] *)
-
-    Variables y z : A.  
-    Hypothesis h1 : Rstar x y.  
-    Hypothesis h2 : Rstar x z.  
-  
-    (** particular case [x->u] and [u->*y]   *)
-    Section Newman_.
-      Variable u : A.  
-      Hypothesis t1 : R x u.  
-      Hypothesis t2 : Rstar u y.  
-      
-      (** In the usual diagram, we assume also [x->v] and [v->*z] *)
-      
-      Theorem Diagram : forall (v:A) (u1:R x v) (u2:Rstar v z), coherence y z.
-      Proof
-       (* We draw the diagram ! *)
-      fun (v:A) (u1:R x v) (u2:Rstar v z) =>
-	ex2_ind
-	 (* local confluence in x for u,v *)
-	 (* gives w, u->*w and v->*w *)
-	(fun (w:A) (s1:Rstar u w) (s2:Rstar v w) =>
-          ex2_ind
-              (* confluence in u => coherence(y,w) *)
-              (* gives a, y->*a and z->*a *)
-          (fun (a:A) (v1:Rstar y a) (v2:Rstar w a) =>
-            ex2_ind
-              (* confluence in v => coherence(a,z) *)
-              (* gives b, a->*b and z->*b *)
-            (fun (b:A) (w1:Rstar a b) (w2:Rstar z b) =>
-              coherence_intro y z b (Rstar_transitive y a b v1 w1) w2)
-            (hyp_ind v u1 a z (Rstar_transitive v w a s2 v2) u2))
-          (hyp_ind u t1 y w t2 s1)) (Hyp2 x u v t1 u1).
-  
-      Theorem caseRxy : coherence y z.
-      Proof
-        Rstar_Rstar' x z h2 (fun v w:A => coherence y w)
-	(coherence_sym x y (Rstar_coherence x y h1)) (*i case x=z i*)
-	Diagram.                                     (*i case x->v->*z i*)
-    End Newman_.
-
-    Theorem Ind_proof : coherence y z.
-    Proof
-      Rstar_Rstar' x y h1 (fun u v:A => coherence v z)
-      (Rstar_coherence x z h2) (*i case x=y i*)
-      caseRxy.                                   (*i case x->u->*z i*)
-  End Induct.
-
-  Theorem Newman : forall x:A, confluence x.
-  Proof fun x:A => Hyp1 x confluence Ind_proof.  
-
-End Newman_section.
-
-
-End Newman.