diff options
author | herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2006-06-04 18:04:53 +0000 |
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committer | herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2006-06-04 18:04:53 +0000 |
commit | 158ea581a82fa8fda6cc13c3653bddc1147f5c79 (patch) | |
tree | ba8637a27c790ce041b5519f3d4fa825ac5b160f | |
parent | 03c392f24a204be29093166b9c42fa5c485e627c (diff) |
Ajout exists! et restructuration/extension des fichiers sur la
description et le choix
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@8893 85f007b7-540e-0410-9357-904b9bb8a0f7
-rw-r--r-- | theories/Logic/ClassicalEpsilon.v | 84 | ||||
-rw-r--r-- | theories/Logic/ClassicalUniqueChoice.v | 79 |
2 files changed, 163 insertions, 0 deletions
diff --git a/theories/Logic/ClassicalEpsilon.v b/theories/Logic/ClassicalEpsilon.v new file mode 100644 index 000000000..b3efa5fad --- /dev/null +++ b/theories/Logic/ClassicalEpsilon.v @@ -0,0 +1,84 @@ +(************************************************************************) +(* 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 $Id:$ i*) + +(** This file provides classical logic and indefinite description + (Hilbert's epsilon operator) *) + +(** Classical epsilon's operator (i.e. indefinite description) implies + excluded-middle in [Set] and leads to a classical world populated + with non computable functions. It conflicts with the + impredicativity of [Set] *) + +Require Export Classical. +Require Import ChoiceFacts. + +Set Implicit Arguments. + +Notation Local "'inhabited' A" := A (at level 200, only parsing). + +Axiom constructive_indefinite_description : + forall (A : Type) (P : A->Prop), + (exists x : A, P x) -> { x : A | P x }. + +Lemma constructive_definite_description : + forall (A : Type) (P : A->Prop), + (exists! x : A, P x) -> { x : A | P x }. +Proof. +intros; apply constructive_indefinite_description; firstorder. +Qed. + +Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}. +Proof. +apply + (constructive_definite_descr_excluded_middle + constructive_definite_description classic). +Qed. + +Theorem classical_indefinite_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_indefinite_description with (P:= fun x => (exists x, P x) -> P x). + destruct Hex as (x,Hx). + exists x; intros _; exact Hx. + firstorder. +Qed. + +(** Hilbert's epsilon operator *) + +Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A + := proj1_sig (classical_indefinite_description P i). + +Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) : + (exists x:A, P x) -> P (epsilon i P) + := proj2_sig (classical_indefinite_description P i). + +(** Open question: is classical_indefinite_description constructively + provable from [relational_choice] and + [constructive_definite_description] (at least, using the fact that + [functional_choice] is provable from [relational_choice] and + [unique_choice], we know that the double negation of + [classical_indefinite_description] is provable (see + [relative_non_contradiction_of_indefinite_desc]). *) + +(** Weaker lemmas (compatibility lemmas) *) + +Theorem 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 R H. +exists (fun x => proj1_sig (constructive_indefinite_description (R x) (H x))). +intro x. +apply (proj2_sig (constructive_indefinite_description (R x) (H x))). +Qed. diff --git a/theories/Logic/ClassicalUniqueChoice.v b/theories/Logic/ClassicalUniqueChoice.v new file mode 100644 index 000000000..2be5a0eb6 --- /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 $Id$ 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. + |