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