(* -*- coding: utf-8 -*- *) (************************************************************************) (* * 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 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).