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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** Authors: Bruno Barras, Cristina Cornes *)

Require Import Eqdep.
Require Import Relation_Operators.
Require Import Transitive_Closure.
Require Import Inclusion.
Require Import Inverse_Image.

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

Section WfLexicographic_Product.

  Import SigTNotations.

  Variable A : Type.
  Variable B : A -> Type.
  Variable leA : A -> A -> Prop.
  Variable leB : forall x : A, B x -> B x -> Prop.

  Notation LexProd := (lexprod A B leA leB).

  Lemma acc_A_B_lexprod :
    forall x : A,
      Acc leA x ->
      (forall x0 : A, clos_trans A leA x0 x -> well_founded (leB x0)) ->
      forall y : B x, Acc (leB x) y -> Acc LexProd (x; y).
  Proof.
    induction 1 as [x _ IHAcc]; intros H2 y.
    induction 1 as [x0 H IHAcc0]; intros.
    apply Acc_intro.
    destruct y as [x2 y1]; intro H6.
    simple inversion H6; intro.
    - cut (leA x2 x); intros.
      + apply IHAcc; auto with sets.
        * intros.
          apply H2.
          apply t_trans with x2; auto with sets.

        * red in H2.
          apply H2.
          auto with sets.

      + injection H1 as [= <- _].
        injection H3 as [= <- _]; auto with sets.

    - rewrite <- H1.
      injection H3 as [= -> H3].
      apply IHAcc0.
      elim inj_pair2 with A B x y' x0; assumption.
  Defined.

  Theorem wf_lexprod :
    well_founded leA ->
    (forall x : A, well_founded (leB x)) -> well_founded LexProd.
  Proof.
    intros wfA wfB; unfold well_founded.
    destruct a.
    apply acc_A_B_lexprod; auto with sets; intros.
    red in wfB.
    auto with sets.
  Defined.


End WfLexicographic_Product.

Section WfSimple_Lexicographic_Product.

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

  Notation LexProd := (slexprod A B leA leB).

  Theorem wf_slexprod :
    well_founded leA -> well_founded leB -> well_founded LexProd.
  Proof.
    intros; eapply wf_incl.
    - intros x y; apply slexprod_lexprod.
    - now apply wf_inverse_image, wf_lexprod.
  Qed.

End WfSimple_Lexicographic_Product.

Section Wf_Symmetric_Product.
  Variable A : Type.
  Variable B : Type.
  Variable leA : A -> A -> Prop.
  Variable leB : B -> B -> Prop.

  Notation Symprod := (symprod A B leA leB).

  Lemma Acc_symprod :
    forall x:A, Acc leA x -> forall y:B, Acc leB y -> Acc Symprod (x, y).
  Proof.
    induction 1 as [x _ IHAcc]; intros y H2.
    induction H2 as [x1 H3 IHAcc1].
    apply Acc_intro; intros y H5.
    inversion_clear H5; auto with sets.
    apply IHAcc; auto.
    apply Acc_intro; trivial.
  Defined.


  Lemma wf_symprod :
    well_founded leA -> well_founded leB -> well_founded Symprod.
  Proof.
    red.
    destruct a.
    apply Acc_symprod; auto with sets.
  Defined.

End Wf_Symmetric_Product.


Section Swap.

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

  Notation SwapProd := (swapprod A R).


  Lemma swap_Acc : forall x y:A, Acc SwapProd (x, y) -> Acc SwapProd (y, x).
  Proof.
    intros.
    inversion_clear H.
    apply Acc_intro.
    destruct y0; intros.
    inversion_clear H; inversion_clear H1; apply H0.
    - apply sp_swap.
      apply right_sym; auto with sets.

    - apply sp_swap.
      apply left_sym; auto with sets.

    - apply sp_noswap.
      apply right_sym; auto with sets.

    - apply sp_noswap.
      apply left_sym; auto with sets.
  Defined.


  Lemma Acc_swapprod :
    forall x y:A, Acc R x -> Acc R y -> Acc SwapProd (x, y).
  Proof.
    induction 1 as [x0 _ IHAcc0]; intros H2.
    cut (forall y0:A, R y0 x0 -> Acc SwapProd (y0, y)).
    - clear IHAcc0.
      induction H2 as [x1 _ IHAcc1]; intros H4.
      cut (forall y:A, R y x1 -> Acc SwapProd (x0, y)).
      + clear IHAcc1.
        intro.
        apply Acc_intro.
        destruct y; intro H5.
        inversion_clear H5.
        * inversion_clear H0; auto with sets.

        * apply swap_Acc.
          inversion_clear H0; auto with sets.

      + intros.
        apply IHAcc1; auto with sets; intros.
        apply Acc_inv with (y0, x1); auto with sets.
        apply sp_noswap.
        apply right_sym; auto with sets.

    - auto with sets.
  Defined.


  Lemma wf_swapprod : well_founded R -> well_founded SwapProd.
  Proof.
    red.
    destruct a; intros.
    apply Acc_swapprod; auto with sets.
  Defined.

End Swap.
