2014-08-22 00:56:18 +00:00
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import logic data.nat data.prod
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2014-09-03 23:00:38 +00:00
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open nat prod
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open decidable
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2014-08-22 00:56:18 +00:00
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variable modulo (x : ℕ) (y : ℕ) : ℕ
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infixl `mod`:70 := modulo
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variable gcd_aux : ℕ × ℕ → ℕ
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definition gcd (x y : ℕ) : ℕ := gcd_aux (pair x y)
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theorem gcd_def (x y : ℕ) : gcd x y = @ite (y = 0) (decidable_eq (pr2 (pair x y)) 0) nat x (gcd y (x mod y)) :=
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sorry
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theorem gcd_succ (m n : ℕ) : gcd m (succ n) = gcd (succ n) (m mod succ n) :=
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2014-09-05 01:41:06 +00:00
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eq.trans (gcd_def _ _) (if_neg succ_ne_zero)
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