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
Hilbert's epsilon operator
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)
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)).