Library hilbert.ClassicalEpsilonModified

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

slight modification : some propositions "exists x, P x" are changed into "ex P"

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).

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 (H x))).
intro x.
apply (proj2_sig (constructive_indefinite_description (H x))).
Qed.

Opaque epsilon.