78 lines
2.3 KiB
Text
78 lines
2.3 KiB
Text
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import data.fintype data.list data.sum data.nat
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open option list nat
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structure countable [class] (A : Type) :=
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(pickle : A → nat) (unpickle : nat → option A) (picklek : ∀ a, unpickle (pickle a) = some a)
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open countable
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definition countable_fintype [instance] {A : Type} [h₁ : fintype A] [h₂ : decidable_eq A] : countable A :=
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countable.mk
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(λ a, find a (elements_of A))
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(λ n, nth (elements_of A) n)
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(λ a, find_nth (fintype.complete a))
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definition countable_nat [instance] : countable nat :=
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countable.mk (λ a, a) (λ n, some n) (λ a, rfl)
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definition countable_option [instance] {A : Type} [h : countable A] : countable (option A) :=
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countable.mk
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(λ o, match o with
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| some a := succ (pickle a)
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| none := 0
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end)
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(λ n, if n = 0 then some none else some (unpickle A (pred n)))
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(λ o,
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begin
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cases o with [a],
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begin esimp end,
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begin esimp, rewrite [if_neg !succ_ne_zero, pred_succ, countable.picklek] end
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end)
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section sum
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variables {A B : Type}
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variables [h₁ : countable A] [h₂ : countable B]
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include h₁ h₂
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definition pickle_sum : sum A B → nat
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| (sum.inl a) := 2 * pickle a
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| (sum.inr b) := 2 * pickle b + 1
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definition unpickle_sum (n : nat) : option (sum A B) :=
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if n mod 2 = 0 then
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match unpickle A (n div 2) with
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| some a := some (sum.inl a)
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| none := none
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end
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else
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match unpickle B ((n - 1) div 2) with
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| some b := some (sum.inr b)
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| none := none
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end
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open decidable
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theorem unpickle_pickle_sum : ∀ s : sum A B, unpickle_sum (pickle_sum s) = some s
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| (sum.inl a) :=
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assert aux : 2 > 0, from dec_trivial,
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begin
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esimp [pickle_sum, unpickle_sum],
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rewrite [mul_mod_right, if_pos (eq.refl 0), mul_div_cancel_left _ aux, countable.picklek]
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end
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| (sum.inr b) :=
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assert aux₁ : 2 > 0, from dec_trivial,
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assert aux₂ : 1 mod 2 = 1, by rewrite [modulo_def],
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assert aux₃ : 1 ≠ 0, from dec_trivial,
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begin
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esimp [pickle_sum, unpickle_sum],
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rewrite [add.comm, add_mul_mod_self_left aux₁, aux₂, if_neg aux₃, add_sub_cancel_left,
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mul_div_cancel_left _ aux₁, countable.picklek]
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end
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definition countable_sum [instance] {A B : Type} [h₁ : countable A] [h₂ : countable B] : countable (sum A B) :=
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countable.mk
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(λ s, pickle_sum s)
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(λ n, unpickle_sum n)
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(λ s, unpickle_pickle_sum s)
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end sum
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