feat(library/standard): add hilbert's choice operator

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

library/standard/hilbert.lean

@@ -0,0 +1,22 @@
+-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Leonardo de Moura
+import logic
+
+variable epsilon {A : Type} {H : inhabited A} (P : A → Bool) : A
+axiom epsilon_ax {A : Type} {P : A → Bool} (Hex : ∃ a, P a) : P (@epsilon A (inhabited_exists Hex) P)
+
+theorem epsilon_singleton {A : Type} (a : A) : @epsilon A (inhabited_intro a) (λ x, x = a) = a
+:= epsilon_ax (exists_intro a (refl a))
+
+theorem axiom_of_choice {A : Type} {B : A → Type} {R : Π x, B x → Bool} (H : ∀ x, ∃ y, R x y) : ∃ f, ∀ x, R x (f x)
+:= let f [inline] := λ x, @epsilon _ (inhabited_exists (H x)) (λ y, R x y),
+       H [inline] := take x, epsilon_ax (H x)
+   in exists_intro f H
+
+theorem skolem {A : Type} {B : A → Type} {P : Π x, B x → Bool} : (∀ x, ∃ y, P x y) ↔ ∃ f, (∀ x, P x (f x))
+:= iff_intro
+      (assume H : (∀ x, ∃ y, P x y), axiom_of_choice H)
+      (assume H : (∃ f, (∀ x, P x (f x))),
+             take x, obtain (fw : ∀ x, B x) (Hw : ∀ x, P x (fw x)), from H,
+                  exists_intro (fw x) (Hw x))