# Library Coq.Logic.ClassicalUniqueChoice

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This file provides classical logic and unique choice
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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 : ((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). destruct (f P).   left.   destruct HfP as [[Ha _]| [_ Hfalse]].     assumption.     discriminate.   right.   destruct HfP as [[_ Hfalse]| [Hna _]].     discriminate.     assumption. Qed. ```