summaryrefslogtreecommitdiff
path: root/theories/Logic/Epsilon.v
blob: da3e5b086111eedcf771055966773ceafb0818f4 (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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(** This file provides indefinite description under the form of
    Hilbert's epsilon operator; it does not assume classical logic. *)

Require Import ChoiceFacts.

Set Implicit Arguments.

(** Hilbert's epsilon: operator and specification in one statement *)

Axiom epsilon_statement :
  forall (A : Type) (P : A->Prop), inhabited A ->
    { x : A | (exists x, P x) -> P x }.

Lemma constructive_indefinite_description :
  forall (A : Type) (P : A->Prop),
    (exists x, P x) -> { x : A | P x }.
Proof.
  apply epsilon_imp_constructive_indefinite_description.
  exact epsilon_statement.
Qed.

Lemma small_drinkers'_paradox :
  forall (A:Type) (P:A -> Prop), inhabited A ->
    exists x, (exists x, P x) -> P x.
Proof.
  apply epsilon_imp_small_drinker.
  exact epsilon_statement.
Qed.

Theorem iota_statement :
  forall (A : Type) (P : A->Prop), inhabited A ->
  { x : A | (exists! x : A, P x) -> P x }.
Proof.
  intros; destruct epsilon_statement with (P:=P); firstorder.
Qed.

Lemma constructive_definite_description :
  forall (A : Type) (P : A->Prop),
    (exists! x, P x) -> { x : A | P x }.
Proof.
  apply iota_imp_constructive_definite_description.
  exact iota_statement.
Qed.

(** Hilbert's epsilon operator and its specification *)

Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A
  := proj1_sig (epsilon_statement P i).

Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) :
  (exists x, P x) -> P (epsilon i P)
  := proj2_sig (epsilon_statement P i).

(** Church's iota operator and its specification *)

Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A
  := proj1_sig (iota_statement P i).

Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) :
  (exists! x:A, P x) -> P (iota i P)
  := proj2_sig (iota_statement P i).