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(** Author: Cristina Cornes.
    From: Constructing Recursion Operators in Type Theory
    L. Paulson  JSC (1986) 2, 325-355 *)

Require Import Eqdep.

#[universes(template)]
Inductive WO (A : Type) (B : A -> Type) : Type :=
  sup : forall (a:A) (f:B a -> WO A B), WO A B.

Section WellOrdering.
  Variable A : Type.
  Variable B : A -> Type.

  Notation WO := (WO A B).

  Inductive le_WO : WO -> WO -> Prop :=
    le_sup : forall (a:A) (f:B a -> WO) (v:B a), le_WO (f v) (sup _ _ a f).

  Theorem wf_WO : well_founded le_WO.
  Proof.
    unfold well_founded; intro.
    apply Acc_intro.
    elim a.
    intros.
    inversion H0.
    apply Acc_intro.
    generalize H4; generalize H1; generalize f0; generalize v.
    rewrite H3.
    intros.
    apply (H v0 y0).
    cut (f = f1).
    - intros E; rewrite E; auto.
    - symmetry .
      apply (inj_pair2 A (fun a0:A => B a0 -> WO) a0 f1 f H5).
  Qed.

End WellOrdering.


Section Characterisation_wf_relations.

  (** Wellfounded relations are the inverse image of wellordering types *)
  (*  in course of development                                          *)


  Variable A : Type.
  Variable leA : A -> A -> Prop.

  Definition B (a:A) := {x : A | leA x a}.

  Definition wof : well_founded leA -> A -> WO A B.
  Proof.
    intros.
    apply (well_founded_induction_type H (fun a:A => WO A B)); auto.
    intros x H1.
    apply (sup A B x).
    unfold B at 1.
    destruct 1 as [x0].
    apply (H1 x0); auto.
Qed.

End Characterisation_wf_relations.
