32 lines
638 B
Text
32 lines
638 B
Text
import data.nat
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open nat
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inductive Parity : nat → Type :=
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| even : ∀ n : nat, Parity (2 * n)
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| odd : ∀ n : nat, Parity (2 * n + 1)
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open Parity
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definition parity : Π (n : nat), Parity n
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| parity 0 := even 0
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| parity (n+1) :=
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begin
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have aux : Parity n, from parity n,
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cases aux with [k, k],
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begin
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apply (odd k)
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end,
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begin
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change (Parity (2*k + 2*1)),
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rewrite -mul.left_distrib,
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apply (even (k+1))
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end
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end
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print definition parity
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definition half (n : nat) : nat :=
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match ⟨n, parity n⟩ with
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| ⟨⌞2 * k⌟, even k⟩ := k
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| ⟨⌞2 * k + 1⌟, odd k⟩ := k
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end
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