lean2/library/logic/axioms/hilbert.lean
2015-05-23 20:52:23 +10:00

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/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
Follows Coq.Logic.ClassicalEpsilon (but our definition of "inhabited" is the constructive one).
-/
import logic.quantifiers
import data.subtype data.sum
open subtype inhabited nonempty
/- the axiom -/
-- In the presence of classical logic, we could prove this from a weaker statement:
-- axiom indefinite_description {A : Type} {P : A->Prop} (H : ∃x, P x) : {x : A, P x}
axiom strong_indefinite_description {A : Type} (P : A → Prop) (H : nonempty A) :
{ x | (∃y : A, P y) → P x}
theorem exists_true_of_nonempty {A : Type} (H : nonempty A) : ∃x : A, true :=
nonempty.elim H (take x, exists.intro x trivial)
theorem inhabited_of_nonempty {A : Type} (H : nonempty A) : inhabited A :=
let u : {x | (∃y : A, true) → true} := strong_indefinite_description (λa, true) H in
inhabited.mk (elt_of u)
theorem inhabited_of_exists {A : Type} {P : A → Prop} (H : ∃x, P x) : inhabited A :=
inhabited_of_nonempty (obtain w Hw, from H, nonempty.intro w)
/- the Hilbert epsilon function -/
definition epsilon {A : Type} [H : nonempty A] (P : A → Prop) : A :=
let u : {x | (∃y, P y) → P x} :=
strong_indefinite_description P H in
elt_of u
theorem epsilon_spec_aux {A : Type} (H : nonempty A) (P : A → Prop) (Hex : ∃y, P y) :
P (@epsilon A H P) :=
let u : {x | (∃y, P y) → P x} :=
strong_indefinite_description P H in
have aux : (∃y, P y) → P (elt_of (strong_indefinite_description P H)), from has_property u,
aux Hex
theorem epsilon_spec {A : Type} {P : A → Prop} (Hex : ∃y, P y) :
P (@epsilon A (nonempty_of_exists Hex) P) :=
epsilon_spec_aux (nonempty_of_exists Hex) P Hex
theorem epsilon_singleton {A : Type} (a : A) : @epsilon A (nonempty.intro a) (λx, x = a) = a :=
epsilon_spec (exists.intro a (eq.refl a))
definition some {A : Type} {P : A → Prop} (H : ∃x, P x) : A :=
@epsilon A (nonempty_of_exists H) P
theorem some_spec {A : Type} {P : A → Prop} (H : ∃x, P x) : P (some H) :=
epsilon_spec H
/- the axiom of choice -/
theorem axiom_of_choice {A : Type} {B : A → Type} {R : Πx, B x → Prop} (H : ∀x, ∃y, R x y) :
∃f, ∀x, R x (f x) :=
have H : ∀x, R x (some (H x)), from take x, some_spec (H x),
exists.intro _ H
theorem skolem {A : Type} {B : A → Type} {P : Πx, B x → Prop} :
(∀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))