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diff --git a/theories/Logic/ClassicalEpsilon.v b/theories/Logic/ClassicalEpsilon.v new file mode 100644 index 00000000..b7293bec --- /dev/null +++ b/theories/Logic/ClassicalEpsilon.v @@ -0,0 +1,90 @@ +(************************************************************************) +(* 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: ClassicalEpsilon.v 8933 2006-06-09 14:08:38Z herbelin $ 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), + (ex P) -> { 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 | ex P -> 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 => ex P -> 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) : + (ex P) -> P (epsilon i P) + := proj2_sig (classical_indefinite_description P i). + +Opaque epsilon. + +(** 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]). *) + +(** Remark: we use [ex P] rather than [exists x, P x] (which is [ex + (fun x => P x)] to ease unification *) + +(** *** 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 (H x))). +intro x. +apply (proj2_sig (constructive_indefinite_description (H x))). +Qed. + |