52 lines
1.7 KiB
Text
52 lines
1.7 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||
-- Authors: Leonardo de Moura, Jeremy Avigad
|
||
import logic hilbert funext
|
||
using eq_proofs
|
||
|
||
-- Diaconescu’s theorem
|
||
-- Show that Excluded middle follows from
|
||
-- Hilbert's choice operator, function extensionality and Prop extensionality
|
||
section
|
||
hypothesis propext {a b : Prop} : (a → b) → (b → a) → a = b
|
||
|
||
parameter p : Prop
|
||
|
||
definition u [private] := epsilon (λ x, x = true ∨ p)
|
||
|
||
definition v [private] := epsilon (λ x, x = false ∨ p)
|
||
|
||
lemma u_def [private] : u = true ∨ p
|
||
:= epsilon_ax (exists_intro true (or_inl (refl true)))
|
||
|
||
lemma v_def [private] : v = false ∨ p
|
||
:= epsilon_ax (exists_intro false (or_inl (refl false)))
|
||
|
||
lemma uv_implies_p [private] : ¬(u = v) ∨ p
|
||
:= or_elim u_def
|
||
(assume Hut : u = true, or_elim v_def
|
||
(assume Hvf : v = false,
|
||
have Hne : ¬(u = v), from Hvf⁻¹ ▸ Hut⁻¹ ▸ true_ne_false,
|
||
or_inl Hne)
|
||
(assume Hp : p, or_inr Hp))
|
||
(assume Hp : p, or_inr Hp)
|
||
|
||
lemma p_implies_uv [private] : p → u = v
|
||
:= assume Hp : p,
|
||
have Hpred : (λ x, x = true ∨ p) = (λ x, x = false ∨ p), from
|
||
funext (take x : Prop,
|
||
have Hl : (x = true ∨ p) → (x = false ∨ p), from
|
||
assume A, or_inr Hp,
|
||
have Hr : (x = false ∨ p) → (x = true ∨ p), from
|
||
assume A, or_inr Hp,
|
||
show (x = true ∨ p) = (x = false ∨ p), from
|
||
propext Hl Hr),
|
||
show u = v, from
|
||
Hpred ▸ (refl (epsilon (λ x, x = true ∨ p)))
|
||
|
||
theorem em : p ∨ ¬p
|
||
:= have H : ¬(u = v) → ¬p, from contrapos p_implies_uv,
|
||
or_elim uv_implies_p
|
||
(assume Hne : ¬(u = v), or_inr (H Hne))
|
||
(assume Hp : p, or_inl Hp)
|
||
end
|