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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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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Require Import Relation_Definitions.
Require Import Relation_Operators.

Section Wf_Transitive_Closure.
  Variable A : Type.
  Variable R : relation A.

  Notation trans_clos := (clos_trans A R).

  Lemma incl_clos_trans : inclusion A R trans_clos.
A:Type
R:relation A
inclusion A R trans_clos
    red; auto with sets.
  Qed.

  Lemma Acc_clos_trans : forall x:A, Acc R x -> Acc trans_clos x.
A:Type
R:relation A
forall x : A, Acc R x -> Acc trans_clos x
    induction 1 as [x0 _ H1].
A:Type
R:relation A
x0:A
H1:forall y : A, R y x0 -> Acc trans_clos y
Acc trans_clos x0
    apply Acc_intro.
A:Type
R:relation A
x0:A
H1:forall y : A, R y x0 -> Acc trans_clos y
forall y : A, trans_clos y x0 -> Acc trans_clos y
    intros y H2.
A:Type
R:relation A
x0:A
H1:forall y0 : A, R y0 x0 -> Acc trans_clos y0
y:A
H2:trans_clos y x0
Acc trans_clos y
    induction H2; auto with sets.
A:Type
R:relation A
z:A
H1:forall y0 : A, R y0 z -> Acc trans_clos y0
x, y:A
H2_:trans_clos x y
H2_0:trans_clos y z
IHclos_trans1:(forall y0 : A,
 R y0 y -> Acc trans_clos y0) ->
Acc trans_clos x
IHclos_trans2:(forall y0 : A,
 R y0 z -> Acc trans_clos y0) ->
Acc trans_clos y
Acc trans_clos x
    apply Acc_inv with y; auto with sets.
  Defined.

  Hint Resolve Acc_clos_trans : core.

  Lemma Acc_inv_trans : forall x y:A, trans_clos y x -> Acc R x -> Acc R y.
A:Type
R:relation A
forall x y : A, trans_clos y x -> Acc R x -> Acc R y
  Proof.
A:Type
R:relation A
forall x y : A, trans_clos y x -> Acc R x -> Acc R y
    induction 1 as [| x y]; auto with sets.
A:Type
R:relation A
x, y:A
H:R x y
Acc R y -> Acc R x
    intro; apply Acc_inv with y; assumption.
  Qed.

  Theorem wf_clos_trans : well_founded R -> well_founded trans_clos.
A:Type
R:relation A
well_founded R -> well_founded trans_clos
  Proof.
A:Type
R:relation A
well_founded R -> well_founded trans_clos
    unfold well_founded; auto with sets.
  Defined.

End Wf_Transitive_Closure.