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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** 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).
|
