Library Coq.Logic.ClassicalEpsilon
This file provides classical logic and indefinite description under
the form of Hilbert's epsilon operator
Hilbert's epsilon operator and classical logic 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.
Axiom constructive_indefinite_description :
forall (A : Type) (P : A->Prop),
(exists x, P x) -> { x : A | P x }.
Lemma constructive_definite_description :
forall (A : Type) (P : A->Prop),
(exists! x, P x) -> { x : A | P x }.
Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}.
Theorem classical_indefinite_description :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists x, P x) -> P x }.
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, 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).
A proof that if P is inhabited, epsilon a P does not depend on
the actual proof that the domain of P is inhabited
(proof idea kindly provided by Pierre Castéran)