58 lines
1.6 KiB
Text
58 lines
1.6 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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-/
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import logic.axioms.hilbert logic.eq
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open eq.ops
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/-
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Diaconescu’s theorem: excluded middle follows from Hilbert's choice operator, function extensionality,
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and Prop extensionality
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-/
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section
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parameter p : Prop
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private definition U (x : Prop) : Prop := x = true ∨ p
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private definition V (x : Prop) : Prop := x = false ∨ p
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private noncomputable definition u := epsilon U
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private noncomputable definition v := epsilon V
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private lemma u_def : U u :=
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epsilon_spec (exists.intro true (or.inl rfl))
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private lemma v_def : V v :=
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epsilon_spec (exists.intro false (or.inl rfl))
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private lemma not_uv_or_p : ¬(u = v) ∨ p :=
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or.elim u_def
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(assume Hut : u = true,
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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 : U = V, 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 (iff.intro Hl Hr)),
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have H' : epsilon U = epsilon V, from Hpred ▸ rfl,
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show u = v, from H'
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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 not_uv_or_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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