(**************************************************************************)
(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
(**************************************************************************)

(* This file was automatically generated: do not edit *********************)

include "Coq.ma".

(*#***********************************************************************)

(*  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: Wf_Z.v,v 1.20.2.1 2004/07/16 19:31:20 herbelin Exp $ i*)

include "ZArith/BinInt.ma".

include "ZArith/Zcompare.ma".

include "ZArith/Zorder.ma".

include "ZArith/Znat.ma".

include "ZArith/Zmisc.ma".

include "Arith/Wf_nat.ma".

(* UNEXPORTED
Open Local Scope Z_scope.
*)

(*#* Our purpose is to write an induction shema for {0,1,2,...}
  similar to the [nat] schema (Theorem [Natlike_rec]). For that the
  following implications will be used :
<<
 (n:nat)(Q n)==(n:nat)(P (inject_nat n)) ===> (x:Z)`x > 0) -> (P x)

       	     /\
             ||
             ||

  (Q O) (n:nat)(Q n)->(Q (S n)) <=== (P 0) (x:Z) (P x) -> (P (Zs x))

      	       	       	       	<=== (inject_nat (S n))=(Zs (inject_nat n))

      	       	       	       	<=== inject_nat_complete
>>
  Then the  diagram will be closed and the theorem proved. *)

inline procedural "cic:/Coq/ZArith/Wf_Z/Z_of_nat_complete.con" as lemma.

inline procedural "cic:/Coq/ZArith/Wf_Z/ZL4_inf.con" as lemma.

inline procedural "cic:/Coq/ZArith/Wf_Z/Z_of_nat_complete_inf.con" as lemma.

inline procedural "cic:/Coq/ZArith/Wf_Z/Z_of_nat_prop.con" as lemma.

inline procedural "cic:/Coq/ZArith/Wf_Z/Z_of_nat_set.con" as lemma.

inline procedural "cic:/Coq/ZArith/Wf_Z/natlike_ind.con" as lemma.

inline procedural "cic:/Coq/ZArith/Wf_Z/natlike_rec.con" as lemma.

(* UNEXPORTED
Section Efficient_Rec
*)

(*#* [natlike_rec2] is the same as [natlike_rec], but with a different proof, designed 
     to give a better extracted term. *)

(* UNAVAILABLE OBJECT: cic:/Coq/ZArith/Wf_Z/Efficient_Rec/R.con ***********)

inline procedural "cic:/Coq/ZArith/Wf_Z/Efficient_Rec/R.con" "Efficient_Rec__" as definition.

(* UNAVAILABLE OBJECT: cic:/Coq/ZArith/Wf_Z/Efficient_Rec/R_wf.con ********)

inline procedural "cic:/Coq/ZArith/Wf_Z/Efficient_Rec/R_wf.con" "Efficient_Rec__" as definition.

inline procedural "cic:/Coq/ZArith/Wf_Z/natlike_rec2.con" as lemma.

(*#* A variant of the previous using [Zpred] instead of [Zs]. *)

inline procedural "cic:/Coq/ZArith/Wf_Z/natlike_rec3.con" as lemma.

(*#* A more general induction principal using [Zlt]. *)

inline procedural "cic:/Coq/ZArith/Wf_Z/Z_lt_rec.con" as lemma.

inline procedural "cic:/Coq/ZArith/Wf_Z/Z_lt_induction.con" as lemma.

(* UNEXPORTED
End Efficient_Rec
*)