summaryrefslogtreecommitdiff
path: root/theories/Wellfounded/Transitive_Closure.v
blob: 2e9d497b48a984e51c4053d27f55982a1e1a774b (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(*i $Id: Transitive_Closure.v,v 1.7.2.1 2004/07/16 19:31:19 herbelin Exp $ i*)

(** Author: Bruno Barras *)

Require Import Relation_Definitions.
Require Import Relation_Operators.

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

  Notation trans_clos := (clos_trans A R).
 
  Lemma incl_clos_trans : inclusion A R trans_clos.
    red in |- *; auto with sets.
  Qed.

  Lemma Acc_clos_trans : forall x:A, Acc R x -> Acc trans_clos x.
    induction 1 as [x0 _ H1].
    apply Acc_intro.
    intros y H2.
    induction H2; auto with sets.
    apply Acc_inv with y; auto with sets.
  Qed.

  Hint Resolve Acc_clos_trans.

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

  Theorem wf_clos_trans : well_founded R -> well_founded trans_clos.
  Proof.
    unfold well_founded in |- *; auto with sets.
  Qed.

End Wf_Transitive_Closure.