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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Author: Bruno Barras *)
Require Import Relation_Operators.
Require Import Relation_Definitions.
Require Import Transitive_Closure.
Section WfUnion.
Variable A : Type.
Variables R1 R2 : relation A.
Notation Union := (union A R1 R2).
Remark strip_commut :
commut A R1 R2 ->
forall x y:A,
clos_trans A R1 y x ->
forall z:A, R2 z y -> exists2 y' : A, R2 y' x & clos_trans A R1 z y'.
Proof.
induction 2 as [x y| x y z H0 IH1 H1 IH2]; intros.
- elim H with y x z; auto with sets; intros x0 H2 H3.
exists x0; auto with sets.
- elim IH1 with z0; auto with sets; intros.
elim IH2 with x0; auto with sets; intros.
exists x1; auto with sets.
apply t_trans with x0; auto with sets.
Qed.
Lemma Acc_union :
commut A R1 R2 ->
(forall x:A, Acc R2 x -> Acc R1 x) -> forall a:A, Acc R2 a -> Acc Union a.
Proof.
induction 3 as [x H1 H2].
apply Acc_intro; intros.
elim H3; intros; auto with sets.
cut (clos_trans A R1 y x); auto with sets.
elimtype (Acc (clos_trans A R1) y); intros.
- apply Acc_intro; intros.
elim H8; intros.
+ apply H6; auto with sets.
apply t_trans with x0; auto with sets.
+ elim strip_commut with x x0 y0; auto with sets; intros.
apply Acc_inv_trans with x1; auto with sets.
unfold union.
elim H11; auto with sets; intros.
apply t_trans with y1; auto with sets.
- apply (Acc_clos_trans A).
apply Acc_inv with x; auto with sets.
apply H0.
apply Acc_intro; auto with sets.
Qed.
Theorem wf_union :
commut A R1 R2 -> well_founded R1 -> well_founded R2 -> well_founded Union.
Proof.
unfold well_founded.
intros.
apply Acc_union; auto with sets.
Qed.
End WfUnion.
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