48 lines
1.4 KiB
Text
48 lines
1.4 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.rat.basic data.countable data.encodable data.int.countable
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open encodable nat int option quot
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namespace rat
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definition encode_prerat : prerat → nat
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| (prerat.mk n d h) := mkpair (encode n) (encode d)
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definition decode_prerat (a : nat) : option prerat :=
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match unpair a with
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| (cn, cd) :=
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match decode int cn with
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| some n :=
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match decode int cd with
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| some d := if h : d > 0 then some (prerat.mk n d h) else none
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| none := none
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end
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| none := none
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end
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end
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lemma decode_encode_prerat (p : prerat) : decode_prerat (encode_prerat p) = some p :=
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begin
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cases p with n d h, unfold [encode_prerat, decode_prerat],
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rewrite unpair_mkpair, esimp, rewrite [*encodek, dif_pos h]
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end
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definition encodable_prerat [instance] : encodable prerat :=
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encodable.mk
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encode_prerat
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decode_prerat
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decode_encode_prerat
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definition decidable_equiv [instance] : ∀ a b : prerat, decidable (prerat.equiv a b)
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| (prerat.mk n₁ d₁ h₁) (prerat.mk n₂ d₂ h₂) := begin unfold prerat.equiv, exact _ end
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definition encodable_rat [instance] : encodable rat :=
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@encodable_quot _ _ decidable_equiv encodable_prerat
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lemma countable_rat : countable rat :=
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countable_of_encodable encodable_rat
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end rat
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