blob: 2a501eca0c3375d7fe742228ed41f07ce17fc96b (
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
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
|
(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
(* We show that the functional formulation of the axiom of Choice
(usual formulation in type theory) is equivalent to its relational
formulation (only formulation of set theory) + the axiom of
(parametric) finite description (aka as axiom of unique choice) *)
(* This shows that the axiom of choice can be assumed (under its
relational formulation) without known inconsistency with classical logic,
though finite description conflicts with classical logic *)
Definition RelationalChoice :=
(A:Type;B:Set;R: A->B->Prop)
((x:A)(EX y:B|(R x y)))
-> (EXT R':A->B->Prop |
((x:A)(EX y:B|(R x y)/\(R' x y)/\ ((y':B) (R' x y') -> y=y')))).
Definition FunctionalChoice :=
(A:Type;B:Set;R: A->B->Prop)
((x:A)(EX y:B|(R x y))) -> (EX f:A->B | (x:A)(R x (f x))).
Definition ParamFiniteDescription :=
(A:Type;B:Set;R: A->B->Prop)
((x:A)(EX y:B|(R x y)/\ ((y':B)(R x y') -> y=y')))
-> (EX f:A->B | (x:A)(R x (f x))).
Lemma lem1 : ParamFiniteDescription->RelationalChoice->FunctionalChoice.
Intros Descr RelCh.
Red; Intros A B R H.
NewDestruct (RelCh A B R H) as [R' H0].
NewDestruct (Descr A B R') as [f H1].
Intro x.
Elim (H0 x); Intros y [H2 [H3 H4]]; Exists y; Split; [Exact H3 | Exact H4].
Exists f; Intro x.
Elim (H0 x); Intros y [H2 [H3 H4]].
Rewrite <- (H4 (f x) (H1 x)).
Exact H2.
Qed.
Lemma lem2 : FunctionalChoice->RelationalChoice.
Intros FunCh.
Red; Intros A B R H.
NewDestruct (FunCh A B R H) as [f H0].
Exists [x,y]y=(f x).
Intro x; Exists (f x);
Split; [Apply H0| Split;[Reflexivity| Intros y H1; Symmetry; Exact H1]].
Qed.
Lemma lem3 : FunctionalChoice->ParamFiniteDescription.
Intros FunCh.
Red; Intros A B R H.
NewDestruct (FunCh A B R) as [f H0].
(* 1 *)
Intro x.
Elim (H x); Intros y [H0 H1].
Exists y; Exact H0.
(* 2 *)
Exists f; Exact H0.
Qed.
Theorem FunChoice_Equiv_RelChoice_and_ParamFinDescr :
FunctionalChoice <-> RelationalChoice /\ ParamFiniteDescription.
Split.
Intro H; Split; [Exact (lem2 H) | Exact (lem3 H)].
Intros [H H0]; Exact (lem1 H0 H).
Qed.
|
