blob: 04930170dbc45259f587c5f56b181bf2ed6f5c65 (
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
56
|
(************************************************************************)
(* 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: Disjoint_Union.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*)
(** Author: Cristina Cornes
From : Constructing Recursion Operators in Type Theory
L. Paulson JSC (1986) 2, 325-355 *)
Require Relation_Operators.
Section Wf_Disjoint_Union.
Variable 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: (x:A)(Acc A leA x)->(Acc A+B Le_AsB (inl A B x)).
Proof.
NewInduction 1.
Apply Acc_intro;Intros y H2.
Inversion_clear H2.
Auto with sets.
Qed.
Lemma acc_B_sum: (well_founded A leA) ->(x:B)(Acc B leB x)
->(Acc A+B Le_AsB (inr A B x)).
Proof.
NewInduction 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 A leA)
-> (well_founded B leB) -> (well_founded A+B Le_AsB).
Proof.
Intros.
Unfold well_founded .
NewDestruct 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.
|
