26 lines
1 KiB
Text
26 lines
1 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 classical hilbert bit decidable
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using bit decidable
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-- Excluded middle + Hilbert implies every proposition is decidable
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definition to_bit (a : Bool) : bit
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:= epsilon (λ b : bit, (b = '1) ↔ a)
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theorem to_bit_def (a : Bool) : (to_bit a) = '1 ↔ a
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:= or_elim (em a)
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(assume Hp, epsilon_ax (exists_intro '1 (iff_intro (assume H, Hp) (assume H, refl b1))))
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(assume Hn, epsilon_ax (exists_intro '0 (iff_intro (assume H, absurd_elim a H b0_ne_b1) (assume H, absurd_elim ('0 = '1) H Hn))))
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theorem bool_decidable [instance] (a : Bool) : decidable a
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:= bit_rec
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(assume H0 : to_bit a = '0,
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have e1 : ¬ to_bit a = '1, from subst (symm H0) b0_ne_b1,
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have Hna : ¬a, from iff_mp_left (iff_flip_sign (to_bit_def a)) e1,
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inr Hna)
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(assume H1 : to_bit a = '1,
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have Ha : a, from iff_mp_left (to_bit_def a) H1,
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inl Ha)
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(to_bit a)
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(refl (to_bit a))
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