IndefiniteDescription.v

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This file provides a constructive form of indefinite description that allows building choice functions; this is weaker than Hilbert's epsilon operator (which implies weakly classical properties) but stronger than the axiom of choice (which cannot be used outside the context of a theorem proof).

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 }.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}
  intros; apply constructive_indefinite_description; firstorder.
Qed.

Lemma functional_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)).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.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)
  apply constructive_indefinite_descr_fun_choice.ConstructiveIndefiniteDescription
  exact constructive_indefinite_description.
Qed.