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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: Disjoint_Union.v 10681 2008-03-16 13:40:45Z msozeau $ i*)
(** Author: Cristina Cornes
From : Constructing Recursion Operators in Type Theory
L. Paulson JSC (1986) 2, 325-355 *)
Require Import Relation_Operators.
Section Wf_Disjoint_Union.
Variables A B : Type.
Variable leA : A -> A -> Prop.
Variable leB : B -> B -> Prop.
Notation Le_AsB := (le_AsB A B leA leB).
Lemma acc_A_sum : forall x:A, Acc leA x -> Acc Le_AsB (inl B x).
Proof.
induction 1.
apply Acc_intro; intros y H2.
inversion_clear H2.
auto with sets.
Qed.
Lemma acc_B_sum :
well_founded leA -> forall x:B, Acc leB x -> Acc Le_AsB (inr A x).
Proof.
induction 2.
apply Acc_intro; intros y H3.
inversion_clear H3; auto with sets.
apply acc_A_sum; auto with sets.
Qed.
Lemma wf_disjoint_sum :
well_founded leA -> well_founded leB -> well_founded Le_AsB.
Proof.
intros.
unfold well_founded in |- *.
destruct a as [a| b].
apply (acc_A_sum a).
apply (H a).
apply (acc_B_sum H b).
apply (H0 b).
Qed.
End Wf_Disjoint_Union.
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