Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Logic.IndefiniteDescription

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 }.
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)).