15 lines
464 B
Text
15 lines
464 B
Text
open nat prod
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open decidable
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constant modulo' (x : ℕ) (y : ℕ) : ℕ
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infixl `mod`:70 := modulo'
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constant gcd_aux : ℕ × ℕ → ℕ
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noncomputable definition gcd' (x y : ℕ) : ℕ := gcd_aux (x, y)
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theorem gcd_def (x y : ℕ) : gcd' x y = @ite (y = 0) (nat.decidable_eq (snd (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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eq.trans (gcd_def _ _) (if_neg (nat.succ_ne_zero _))
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