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(* 02110-1301 USA                                                     *)
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(** * The intuitionistic Hilbert epsilon operator  *)
(***************************************************)
(** The following axioms mimick (a specialized version of) the intuitionistic Hilbert epsilon operator described in the author's LICS'03 submitted paper. *)
(** We first introduce a type of codes, that could be defined as a primitive type of natural numbers by following the paper.  But since the actual structure of its inhabitants will never be used in the forthcoming proofs, we prefer to keep it abstract. *)

(* Specifications: *)
Axiom Code : Typ1.  (* We could set Code := nat. *)
(** We now introduce a specialized version of the epsilon operator described in the paper.  This version only works for predicates over the (large) type Typ1, which is sufficent for what we want to prove (i.e. the collection scheme). *)
Axiom eps : (Typ1 -> Prop) -> Code -> Typ1.
Axiom
choice : forall P : Typ1 -> Prop, exT P -> ex Code (fun c => P (eps P c)).