53 lines
1.7 KiB
Text
53 lines
1.7 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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-- Author: Leonardo de Moura
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import logic.axioms.hilbert logic.axioms.funext
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open eq.ops nonempty inhabited
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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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private definition u := epsilon (λx, x = true ∨ p)
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private definition v := epsilon (λx, x = false ∨ p)
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private lemma u_def : u = true ∨ p :=
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epsilon_spec (exists_intro true (or.inl rfl))
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private lemma v_def : v = false ∨ p :=
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epsilon_spec (exists_intro false (or.inl rfl))
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private lemma uv_implies_p : ¬(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.inl Hne)
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(assume Hp : p, or.inr Hp))
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(assume Hp : p, or.inr Hp)
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private lemma p_implies_uv : 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.inr Hp,
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have Hr : (x = false ∨ p) → (x = true ∨ p), from
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assume A, or.inr 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 ▸ (eq.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 mt p_implies_uv,
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or.elim uv_implies_p
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(assume Hne : ¬(u = v), or.inr (H Hne))
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(assume Hp : p, or.inl Hp)
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end
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