Section Inverse_Image.
Variables (A B : Type) (R : B -> B -> Prop) (f : A -> B).
Definition Rof (x y : A) : Prop := R (f x) (f y).
Remark ACC_lemma :
forall y : B, ACC B R y -> forall x : A, y = f x -> ACC A Rof x.
simple induction 1; intros.
constructor; intros.
apply (H1 (f y0)); trivial.
elim H2 using eq_ind_r; trivial.
Qed.

Lemma ACC_inverse_image : forall x : A, ACC B R (f x) -> ACC A Rof x.