# Library Coq.Logic.Epsilon

This file provides indefinite description under the form of Hilbert's epsilon operator; it does not assume classical logic.

Require Import ChoiceFacts.

Set Implicit Arguments.

Hilbert's epsilon: operator and specification in one statement

Axiom epsilon_statement :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists x, P x) -> P x }.

Lemma constructive_indefinite_description :
forall (A : Type) (P : A->Prop),
(exists x, P x) -> { x : A | P x }.

Lemma small_drinkers'_paradox :
forall (A:Type) (P:A -> Prop), inhabited A ->
exists x, (exists x, P x) -> P x.

Theorem iota_statement :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists! x : A, P x) -> P x }.

Lemma constructive_definite_description :
forall (A : Type) (P : A->Prop),
(exists! x, P x) -> { x : A | P x }.

Hilbert's epsilon operator and its specification

Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A
:= proj1_sig (epsilon_statement P i).

Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) :
(exists x, P x) -> P (epsilon i P)
:= proj2_sig (epsilon_statement P i).

Church's iota operator and its specification

Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A
:= proj1_sig (iota_statement P i).

Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) :
(exists! x:A, P x) -> P (iota i P)
:= proj2_sig (iota_statement P i).