# Library Coq.Wellfounded.Inverse_Image

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Author: Bruno Barras
``` Section Inverse_Image.   Variables A B : Set.   Variable R : B -> B -> Prop.   Variable f : A -> B.   Let Rof (x y:A) : Prop := R (f x) (f y).   Remark Acc_lemma : forall y:B, Acc R y -> forall x:A, y = f x -> Acc Rof x.   Proof.     induction 1 as [y _ IHAcc]; intros x H.     apply Acc_intro; intros y0 H1.     apply (IHAcc (f y0)); try trivial.     rewrite H; trivial.   Qed.   Lemma Acc_inverse_image : forall x:A, Acc R (f x) -> Acc Rof x.   Proof.     intros; apply (Acc_lemma (f x)); trivial.   Qed.   Theorem wf_inverse_image : well_founded R -> well_founded Rof.   Proof.     red in |- *; intros; apply Acc_inverse_image; auto.   Qed.   Variable F : A -> B -> Prop.   Let RoF (x y:A) : Prop :=     exists2 b : B, F x b & (forall c:B, F y c -> R b c).   Lemma Acc_inverse_rel : forall b:B, Acc R b -> forall x:A, F x b -> Acc RoF x.   Proof.     induction 1 as [x _ IHAcc]; intros x0 H2.     constructor; intros y H3.     destruct H3.     apply (IHAcc x1); auto.   Qed.   Theorem wf_inverse_rel : well_founded R -> well_founded RoF.   Proof.     red in |- *; constructor; intros.     case H0; intros.     apply (Acc_inverse_rel x); auto.   Qed. End Inverse_Image. ```