aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Wellfounded/Disjoint_Union.v
blob: 76c9ad443a363f184c1390a6c77fac13f76ec0e1 (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
48
49
50
51
52
53
54
55
(************************************************************************)
(*  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 $Id$ i*)

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

Require Import Relation_Operators.

Section Wf_Disjoint_Union.
Variables A B : Set.
Variable leA : A -> A -> Prop.
Variable leB : B -> B -> Prop.

Notation Le_AsB := (le_AsB A B leA leB).

Lemma acc_A_sum : forall x:A, Acc leA x -> Acc Le_AsB (inl B x).
Proof.
 induction 1.
 apply Acc_intro; intros y H2.
 inversion_clear H2.
 auto with sets.
Qed.

Lemma acc_B_sum :
 well_founded leA -> forall x:B, Acc leB x -> Acc Le_AsB (inr A x).
Proof.
 induction 2.
 apply Acc_intro; intros y H3.
 inversion_clear H3; auto with sets.
 apply acc_A_sum; auto with sets.
Qed.


Lemma wf_disjoint_sum :
 well_founded leA -> well_founded leB -> well_founded Le_AsB.
Proof.
 intros.
 unfold well_founded in |- *.
 destruct a as [a| b].
 apply (acc_A_sum a).
 apply (H a).

 apply (acc_B_sum H b).
 apply (H0 b).
Qed.

End Wf_Disjoint_Union.