Library Stdlib.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_simulation (Q : A -> A -> Prop) :
    forall b, Acc R b ->
    (forall a1 a2 b1, Q a2 a1 -> F a1 b1 -> exists b2, F a2 b2 /\ R b2 b1) ->
    forall a, F a b -> Acc Q a.

  Lemma wf_simulation (Q : A -> A -> Prop) :
    well_founded R ->
    (forall a1 a2, Q a2 a1 -> exists b2, F a2 b2) ->
    (forall a1 a2 b1, Q a2 a1 -> F a1 b1 -> exists b2, F a2 b2 /\ R b2 b1) ->
    well_founded Q.

  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.