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Diffstat (limited to 'theories/Logic/ClassicalUniqueChoice.v')
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diff --git a/theories/Logic/ClassicalUniqueChoice.v b/theories/Logic/ClassicalUniqueChoice.v new file mode 100644 index 00000000..79bef2af --- /dev/null +++ b/theories/Logic/ClassicalUniqueChoice.v @@ -0,0 +1,79 @@ +(************************************************************************) +(* 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: ClassicalUniqueChoice.v 8893 2006-06-04 18:04:53Z herbelin $ i*) + +(** This file provides classical logic and unique choice *) + +(** 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 followig proof comes from [ChicliPottierSimpson02] *) + +Require Import Setoid. + +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 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. + |