23 lines
1.1 KiB
Text
23 lines
1.1 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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-- Author: Leonardo de Moura
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import logic
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variable epsilon {A : Type} {H : inhabited A} (P : A → Bool) : A
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axiom epsilon_ax {A : Type} {P : A → Bool} (Hex : ∃ a, P a) : P (@epsilon A (inhabited_exists Hex) P)
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theorem epsilon_singleton {A : Type} (a : A) : @epsilon A (inhabited_intro a) (λ x, x = a) = a
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:= epsilon_ax (exists_intro a (refl a))
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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)
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:= let f [inline] := λ x, @epsilon _ (inhabited_exists (H x)) (λ y, R x y),
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H [inline] := take x, epsilon_ax (H x)
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in exists_intro f H
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theorem skolem {A : Type} {B : A → Type} {P : Π x, B x → Bool} : (∀ 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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