Epsilon.v

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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 }.forall (A : Type) (P : A -> Prop),
(exists x : A, P x) -> {x : A | P x}
Proof.forall (A : Type) (P : A -> Prop),
(exists x : A, P x) -> {x : A | P x}
  apply epsilon_imp_constructive_indefinite_description.EpsilonStatement
  exact epsilon_statement.
Qed.

Lemma small_drinkers'_paradox :
  forall (A:Type) (P:A -> Prop), inhabited A ->
    exists x, (exists x, P x) -> P x.forall (A : Type) (P : A -> Prop),
inhabited A ->
exists x : A, (exists x0 : A, P x0) -> P x
Proof.forall (A : Type) (P : A -> Prop),
inhabited A ->
exists x : A, (exists x0 : A, P x0) -> P x
  apply epsilon_imp_small_drinker.EpsilonStatement
  exact epsilon_statement.
Qed.

Theorem iota_statement :
  forall (A : Type) (P : A->Prop), inhabited A ->
  { x : A | (exists! x : A, P x) -> P x }.forall (A : Type) (P : A -> Prop),
inhabited A ->
{x : A | (exists ! x0 : A, P x0) -> P x}
Proof.forall (A : Type) (P : A -> Prop),
inhabited A ->
{x : A | (exists ! x0 : A, P x0) -> P x}
  intros; destruct epsilon_statement with (P:=P); firstorder.
Qed.

Lemma constructive_definite_description :
  forall (A : Type) (P : A->Prop),
    (exists! x, P x) -> { x : A | P x }.forall (A : Type) (P : A -> Prop),
(exists ! x : A, P x) -> {x : A | P x}
Proof.forall (A : Type) (P : A -> Prop),
(exists ! x : A, P x) -> {x : A | P x}
  apply iota_imp_constructive_definite_description.IotaStatement
  exact iota_statement.
Qed.

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).