blob: 59068623ae8156aa25e0806b3197fdf8fb971caa (
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
|
(************************************************************************)
(* * 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) *)
(************************************************************************)
(** Author: Bruno Barras *)
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.
red; 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.
Defined.
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; auto with sets.
Defined.
End Wf_Transitive_Closure.
|
