31 lines
1.2 KiB
Text
31 lines
1.2 KiB
Text
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/-
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Copyright (c) 2015 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 data.equiv data.int.basic data.encodable data.countable
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open equiv bool sum
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namespace int
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definition int_equiv_bool_nat : int ≃ (bool × nat) :=
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equiv.mk
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(λ i, match i with of_nat a := (tt, a) | neg_succ_of_nat a := (ff, a) end)
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(λ p, match p with (tt, a) := of_nat a | (ff, a) := neg_succ_of_nat a end)
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(λ i, begin cases i, repeat reflexivity end)
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(λ p, begin cases p with b a, cases b, repeat reflexivity end)
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definition int_equiv_nat : int ≃ nat :=
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calc int ≃ (bool × nat) : int_equiv_bool_nat
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... ≃ ((unit + unit) × nat) : prod_congr bool_equiv_unit_sum_unit !_root_.equiv.refl
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... ≃ (unit × nat) + (unit × nat) : sum_prod_distrib
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... ≃ nat + nat : sum_congr !prod_unit_left !prod_unit_left
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... ≃ nat : nat_sum_nat_equiv_nat
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definition encodable_int [instance] : encodable int :=
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encodable_of_equiv (_root_.equiv.symm int_equiv_nat)
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lemma countable_int : countable int :=
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countable_of_encodable encodable_int
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end int
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