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
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
|
(************************************************************************)
(* 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: ClassicalEpsilon.v 8933 2006-06-09 14:08:38Z herbelin $ i*)
(** *** This file provides classical logic and indefinite description
(Hilbert's epsilon operator) *)
(** Classical epsilon's operator (i.e. indefinite description) implies
excluded-middle in [Set] and leads to a classical world populated
with non computable functions. It conflicts with the
impredicativity of [Set] *)
Require Export Classical.
Require Import ChoiceFacts.
Set Implicit Arguments.
Notation Local "'inhabited' A" := A (at level 200, only parsing).
Axiom constructive_indefinite_description :
forall (A : Type) (P : A->Prop),
(ex P) -> { x : A | P x }.
Lemma constructive_definite_description :
forall (A : Type) (P : A->Prop),
(exists! x : A, P x) -> { x : A | P x }.
Proof.
intros; apply constructive_indefinite_description; firstorder.
Qed.
Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}.
Proof.
apply
(constructive_definite_descr_excluded_middle
constructive_definite_description classic).
Qed.
Theorem classical_indefinite_description :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | ex P -> P x }.
Proof.
intros A P i.
destruct (excluded_middle_informative (exists x, P x)) as [Hex|HnonP].
apply constructive_indefinite_description with (P:= fun x => ex P -> P x).
destruct Hex as (x,Hx).
exists x; intros _; exact Hx.
firstorder.
Qed.
(** Hilbert's epsilon operator *)
Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A
:= proj1_sig (classical_indefinite_description P i).
Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) :
(ex P) -> P (epsilon i P)
:= proj2_sig (classical_indefinite_description P i).
Opaque epsilon.
(** Open question: is classical_indefinite_description constructively
provable from [relational_choice] and
[constructive_definite_description] (at least, using the fact that
[functional_choice] is provable from [relational_choice] and
[unique_choice], we know that the double negation of
[classical_indefinite_description] is provable (see
[relative_non_contradiction_of_indefinite_desc]). *)
(** Remark: we use [ex P] rather than [exists x, P x] (which is [ex
(fun x => P x)] to ease unification *)
(** *** Weaker lemmas (compatibility lemmas) *)
Theorem choice :
forall (A B : Type) (R : A->B->Prop),
(forall x : A, exists y : B, R x y) ->
(exists f : A->B, forall x : A, R x (f x)).
Proof.
intros A B R H.
exists (fun x => proj1_sig (constructive_indefinite_description (H x))).
intro x.
apply (proj2_sig (constructive_indefinite_description (H x))).
Qed.
|
