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(************************************************************************)
(* 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 $Id$ 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.
Axiom constructive_indefinite_description :
forall (A : Type) (P : A->Prop),
(exists x, P x) -> { x : A | P x }.
Lemma constructive_definite_description :
forall (A : Type) (P : A->Prop),
(exists! x, 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 | (exists x, P x) -> 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 => (exists x, P x) -> P x).
destruct Hex as (x,Hx).
exists x; intros _; exact Hx.
assert {x : A | True} as (a,_).
apply constructive_indefinite_description with (P := fun _ : A => True).
destruct i as (a); firstorder.
firstorder.
Defined.
(** 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) :
(exists x, P x) -> P (epsilon i P)
:= proj2_sig (classical_indefinite_description P i).
(** 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]). *)
(** A proof that if [P] is inhabited, [epsilon a P] does not depend on
the actual proof that the domain of [P] is inhabited
(proof idea kindly provided by Pierre Castéran) *)
Lemma epsilon_inh_irrelevance :
forall (A:Type) (i j : inhabited A) (P:A->Prop),
(exists x, P x) -> epsilon i P = epsilon j P.
Proof.
intros.
unfold epsilon, classical_indefinite_description.
destruct (excluded_middle_informative (exists x : A, P x)) as [|[]]; trivial.
Qed.
Opaque epsilon.
(** *** 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.
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