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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(** Authors: Bruno Barras, Cristina Cornes *)
(**  From : Constructing Recursion Operators in Type Theory L. Paulson  JSC (1986) 2, 325-355 *)
Section WfLexicographic_Product.
Variable A : Type.
Variable B : A -> Type.
Variable leA : A -> A -> Prop.
Variable leB : forall x : A, B x -> B x -> Prop.
Notation LexProd := (lexprod A B leA leB).

(* Specifications: *)
Lemma acc_A_B_lexprod :
forall x : A,
Acc leA x ->
(forall x0 : A, clos_trans A leA x0 x -> well_founded (leB x0)) ->
forall y : B x, Acc (leB x) y -> Acc LexProd (x; y).