17 lines
472 B
Text
17 lines
472 B
Text
import logic data.nat data.prod
|
||
|
||
open nat prod
|
||
open decidable
|
||
|
||
variable modulo (x : ℕ) (y : ℕ) : ℕ
|
||
infixl `mod`:70 := modulo
|
||
|
||
variable 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) (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)
|