lean2/library/logic/axioms/examples/diaconescu.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import logic.axioms.hilbert logic.axioms.funext
open eq_ops nonempty inhabited
-- Diaconescus 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 (λx, x = true p)
definition v [private] := epsilon (λx, x = false p)
lemma u_def [private] : u = true p :=
epsilon_spec (exists_intro true (or_inl (refl true)))
lemma v_def [private] : v = false p :=
epsilon_spec (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 mt p_implies_uv,
or_elim uv_implies_p
(assume Hne : ¬(u = v), or_inr (H Hne))
(assume Hp : p, or_inl Hp)
end