(**************************************************************************)
(*       ___                                                              *)
(*      ||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: Lexicographic_Product.v,v 1.12.2.1 2004/07/16 19:31:19 herbelin Exp $ i*)

(*#* Authors: Bruno Barras, Cristina Cornes *)

include "Logic/Eqdep.ma".

include "Relations/Relation_Operators.ma".

include "Wellfounded/Transitive_Closure.ma".

(*#*  From : Constructing Recursion Operators in Type Theory            
     L. Paulson  JSC (1986) 2, 325-355 *)

(* UNEXPORTED
Section WfLexicographic_Product
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/WfLexicographic_Product/A.var
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/WfLexicographic_Product/B.var
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/WfLexicographic_Product/leA.var
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/WfLexicographic_Product/leB.var
*)

(* NOTATION
Notation LexProd := (lexprod A B leA leB).
*)

(* UNEXPORTED
Hint Resolve t_step Acc_clos_trans wf_clos_trans.
*)

inline procedural "cic:/Coq/Wellfounded/Lexicographic_Product/acc_A_B_lexprod.con" as lemma.

inline procedural "cic:/Coq/Wellfounded/Lexicographic_Product/wf_lexprod.con" as theorem.

(* UNEXPORTED
End WfLexicographic_Product
*)

(* UNEXPORTED
Section Wf_Symmetric_Product
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/Wf_Symmetric_Product/A.var
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/Wf_Symmetric_Product/B.var
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/Wf_Symmetric_Product/leA.var
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/Wf_Symmetric_Product/leB.var
*)

(* NOTATION
Notation Symprod := (symprod A B leA leB).
*)

(*i
  Local sig_prod:=
         [x:A*B]<{_:A&B}>Case x of [a:A][b:B](existS A [_:A]B a b) end.

Lemma incl_sym_lexprod: (included (A*B) Symprod  
            (R_o_f (A*B) {_:A&B} sig_prod (lexprod A [_:A]B leA [_:A]leB))).
Proof.
 Red.
 Induction x.
 (Induction y1;Intros).
 Red.
 Unfold sig_prod .
 Inversion_clear H.
 (Apply left_lex;Auto with sets).

 (Apply right_lex;Auto with sets).
Qed.
i*)

inline procedural "cic:/Coq/Wellfounded/Lexicographic_Product/Acc_symprod.con" as lemma.

inline procedural "cic:/Coq/Wellfounded/Lexicographic_Product/wf_symprod.con" as lemma.

(* UNEXPORTED
End Wf_Symmetric_Product
*)

(* UNEXPORTED
Section Swap
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/Swap/A.var
*)

(* UNEXPORTED
cic:/Coq/Wellfounded/Lexicographic_Product/Swap/R.var
*)

(* NOTATION
Notation SwapProd := (swapprod A R).
*)

inline procedural "cic:/Coq/Wellfounded/Lexicographic_Product/swap_Acc.con" as lemma.

inline procedural "cic:/Coq/Wellfounded/Lexicographic_Product/Acc_swapprod.con" as lemma.

inline procedural "cic:/Coq/Wellfounded/Lexicographic_Product/wf_swapprod.con" as lemma.

(* UNEXPORTED
End Swap
*)