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(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: ClassicalUniqueChoice.v 13323 2010-07-24 15:57:30Z herbelin $ i*)
(** This file provides classical logic and unique choice; this is
weaker than providing iota operator and classical logic as the
definite descriptions provided by the axiom of unique choice can
be used only in a propositional context (especially, they cannot
be used to build functions outside the scope of a theorem proof) *)
(** Classical logic and unique choice, as shown in
[[ChicliPottierSimpson02]], implies the double-negation of
excluded-middle in [Set], hence it implies a strongly classical
world. Especially it conflicts with the impredicativity of [Set].
[[ChicliPottierSimpson02]] Laurent Chicli, Loïc Pottier, Carlos
Simpson, Mathematical Quotients and Quotient Types in Coq,
Proceedings of TYPES 2002, Lecture Notes in Computer Science 2646,
Springer Verlag. *)
Require Export Classical.
Axiom
dependent_unique_choice :
forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop),
(forall x : A, exists! y : B x, R x y) ->
(exists f : (forall x:A, B x), forall x:A, R x (f x)).
(** Unique choice reifies functional relations into functions *)
Theorem unique_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.
apply (dependent_unique_choice A (fun _ => B)).
Qed.
(** The following proof comes from [[ChicliPottierSimpson02]] *)
Require Import Setoid.
Theorem classic_set_in_prop_context :
forall C:Prop, ((forall P:Prop, {P} + {~ P}) -> C) -> C.
Proof.
intros C HnotEM.
set (R := fun A b => A /\ true = b \/ ~ A /\ false = b).
assert (H : exists f : Prop -> bool, (forall A:Prop, R A (f A))).
apply unique_choice.
intro A.
destruct (classic A) as [Ha| Hnota].
exists true; split.
left; split; [ assumption | reflexivity ].
intros y [[_ Hy]| [Hna _]].
assumption.
contradiction.
exists false; split.
right; split; [ assumption | reflexivity ].
intros y [[Ha _]| [_ Hy]].
contradiction.
assumption.
destruct H as [f Hf].
apply HnotEM.
intro P.
assert (HfP := Hf P).
(* Elimination from Hf to Set is not allowed but from f to Set yes ! *)
destruct (f P).
left.
destruct HfP as [[Ha _]| [_ Hfalse]].
assumption.
discriminate.
right.
destruct HfP as [[_ Hfalse]| [Hna _]].
discriminate.
assumption.
Qed.
Corollary not_not_classic_set :
((forall P:Prop, {P} + {~ P}) -> False) -> False.
Proof.
apply classic_set_in_prop_context.
Qed.
(* Compatibility *)
Notation classic_set := not_not_classic_set (only parsing).
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