52 lines
1.9 KiB
Text
52 lines
1.9 KiB
Text
-- 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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-- Authors: Leonardo de Moura, Jeremy Avigad
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import logic hilbert funext
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using eq_proofs
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-- Diaconescu’s theorem
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-- Show that Excluded middle follows from
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-- Hilbert's choice operator, function extensionality and Prop extensionality
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section
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hypothesis propext ⦃a b : Prop⦄ : (a → b) → (b → a) → a = b
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parameter p : Prop
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definition u [private] := epsilon (λ x, x = true ∨ p)
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definition v [private] := epsilon (λ x, x = false ∨ p)
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lemma u_def [private] : u = true ∨ p
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:= epsilon_ax (exists_intro true (or_intro_left p (refl true)))
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lemma v_def [private] : v = false ∨ p
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:= epsilon_ax (exists_intro false (or_intro_left p (refl false)))
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lemma uv_implies_p [private] : ¬(u = v) ∨ p
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:= or_elim u_def
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(assume Hut : u = true, or_elim v_def
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(assume Hvf : v = false,
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have Hne : ¬(u = v), from Hvf⁻¹ ▸ Hut⁻¹ ▸ true_ne_false,
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or_intro_left p Hne)
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(assume Hp : p, or_intro_right (¬u = v) Hp))
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(assume Hp : p, or_intro_right (¬u = v) Hp)
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lemma p_implies_uv [private] : p → u = v
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:= assume Hp : p,
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have Hpred : (λ x, x = true ∨ p) = (λ x, x = false ∨ p), from
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funext (take x : Prop,
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have Hl : (x = true ∨ p) → (x = false ∨ p), from
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assume A, or_intro_right (x = false) Hp,
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have Hr : (x = false ∨ p) → (x = true ∨ p), from
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assume A, or_intro_right (x = true) Hp,
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show (x = true ∨ p) = (x = false ∨ p), from
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propext Hl Hr),
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show u = v, from
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Hpred ▸ (refl (epsilon (λ x, x = true ∨ p)))
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theorem em : p ∨ ¬ p
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:= have H : ¬(u = v) → ¬ p, from contrapos p_implies_uv,
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or_elim uv_implies_p
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(assume Hne : ¬(u = v), or_intro_right p (H Hne))
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(assume Hp : p, or_intro_left (¬p) Hp)
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end
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