# Library Coq.Wellfounded.Union

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Author: Bruno Barras
``` Require Import Relation_Operators. Require Import Relation_Definitions. Require Import Transitive_Closure. Section WfUnion.   Variable A : Set.   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 in |- *.     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 in |- *.     intros.     apply Acc_union; auto with sets.   Qed. End WfUnion. ```