(** 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) :
  (exists x, P x) -> P (epsilon i P)
  := proj2_sig (classical_indefinite_description P i).

Lemma epsilon_inh_irrelevance : forall (A:Type) (i j : inhabited A) (P:A->Prop), (exists x, P x) -> epsilon i P = epsilon j P.