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.