From e978da8c41d8a3c19a29036d9c569fbe2a4616b0 Mon Sep 17 00:00:00 2001 From: Samuel Mimram Date: Fri, 16 Jun 2006 14:41:51 +0000 Subject: Imported Upstream version 8.0pl3+8.1beta --- theories/Logic/ClassicalDescription.v | 124 ++++++++++++++++++++-------------- 1 file changed, 73 insertions(+), 51 deletions(-) (limited to 'theories/Logic/ClassicalDescription.v') diff --git a/theories/Logic/ClassicalDescription.v b/theories/Logic/ClassicalDescription.v index ce3e279c..7053266a 100644 --- a/theories/Logic/ClassicalDescription.v +++ b/theories/Logic/ClassicalDescription.v @@ -6,73 +6,95 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: ClassicalDescription.v 5920 2004-07-16 20:01:26Z herbelin $ i*) +(*i $Id: ClassicalDescription.v 8892 2006-06-04 17:59:53Z 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. +(** Classical definite description operator (i.e. iota) implies + excluded-middle in [Set] and leads to a classical world populated + with non computable functions. It conflicts with the + impredicativity of [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. -*) +Set Implicit Arguments. Require Export Classical. +Require Import ChoiceFacts. -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)). +Notation Local "'inhabited' A" := A (at level 200, only parsing). + +Axiom constructive_definite_description : + forall (A : Type) (P : A->Prop), (exists! x : A, P x) -> { x : A | P x }. + +(** The idea for the following proof comes from [ChicliPottierSimpson02] *) + +Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}. +Proof. +apply + (constructive_definite_descr_excluded_middle + constructive_definite_description classic). +Qed. + +Theorem classical_definite_description : + forall (A : Type) (P : A->Prop), inhabited A -> + { x : A | (exists! x : A, P x) -> P x }. +Proof. +intros A P i. +destruct (excluded_middle_informative (exists! x, P x)) as [Hex|HnonP]. + apply constructive_definite_description with (P:= fun x => (exists! x : A, P x) -> P x). + destruct Hex as (x,(Hx,Huni)). + exists x; split. + intros _; exact Hx. + firstorder. +exists i; tauto. +Qed. + +(** Church's iota operator *) -(** Principle of definite descriptions (aka axiom of unique choice) *) +Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A + := proj1_sig (classical_definite_description P i). + +Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) : + (exists! x:A, P x) -> P (iota i P) + := proj2_sig (classical_definite_description P i). + +(** Weaker lemmas (compatibility lemmas) *) + +Unset Implicit Arguments. + +Lemma 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) -> + (exists f : (forall x:A, B x), forall x:A, R x (f x)). +Proof. +intros A B R H. +assert (Hexuni:forall x, exists! y, R x y). + intro x. apply H. +exists (fun x => proj1_sig (constructive_definite_description (R x) (Hexuni x))). +intro x. +apply (proj2_sig (constructive_definite_description (R x) (Hexuni x))). +Qed. 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)). + (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_description A (fun _ => B)). Qed. -(** The followig proof comes from [1] *) +(** Axiom of unique "choice" (functional reification of functional relations) *) + +Set Implicit Arguments. -Theorem classic_set : ((forall P:Prop, {P} + {~ P}) -> False) -> False. +Require Import Setoid. + +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. -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. +intros A B R H. +apply (description A B). +intro x. apply H. Qed. - -- cgit v1.2.3