Library Coq.Wellfounded.Inverse_Image

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.