# Library Coq.Wellfounded.Inverse_Image

Author: Bruno Barras
Section Inverse_Image.
Variables A B : Type.
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.
Lemma Acc_inverse_image : forall x:A, Acc R (f x) -> Acc Rof x.
Theorem wf_inverse_image : well_founded R -> well_founded Rof.
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.
Theorem wf_inverse_rel : well_founded R -> well_founded RoF.
End Inverse_Image.