import logic data.nat data.prod open nat prod open decidable constant modulo (x : ℕ) (y : ℕ) : ℕ infixl `mod`:70 := modulo constant gcd_aux : ℕ × ℕ → ℕ definition gcd (x y : ℕ) : ℕ := gcd_aux (pair x y) theorem gcd_def (x y : ℕ) : gcd x y = @ite (y = 0) (nat.has_decidable_eq (pr2 (pair x y)) 0) nat x (gcd y (x mod y)) := sorry theorem gcd_succ (m n : ℕ) : gcd m (succ n) = gcd (succ n) (m mod succ n) := eq.trans (gcd_def _ _) (if_neg !succ_ne_zero)