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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: ClassicalDescription.v 5920 2004-07-16 20:01:26Z herbelin $ i*)
(** This file provides classical logic and definite description *)
(** Classical logic and definite description, as shown in [1],
implies the double-negation of excluded-middle in Set, hence it
implies a strongly classical world. Especially it conflicts with
impredicativity of Set, knowing that true<>false in Set.
[1] 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_description :
forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop),
(forall x:A,
exists y : B x, R x y /\ (forall y':B x, R x y' -> y = y')) ->
exists f : forall x:A, B x, (forall x:A, R x (f x)).
(** Principle of definite descriptions (aka axiom of unique choice) *)
Theorem description :
forall (A B:Type) (R:A -> B -> Prop),
(forall x:A, exists y : B, R x y /\ (forall y':B, R x y' -> y = y')) ->
exists f : A -> B, (forall x:A, R x (f x)).
Proof.
intros A B.
apply (dependent_description A (fun _ => B)).
Qed.
(** The followig proof comes from [1] *)
Theorem classic_set : ((forall P:Prop, {P} + {~ P}) -> False) -> False.
Proof.
intro 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 description.
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.
|
