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fix(library/data/int/gcd): remove sorry

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Leonardo de Moura 2015-10-11 12:07:30 -07:00
parent a9f5735bb5
commit 8657ccfc04
1 changed files with 9 additions and 10 deletions
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19
library/data/int/gcd.lean
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@ -42,19 +42,18 @@ by rewrite [↑gcd, *nat_abs_abs]
theorem gcd_abs_abs (a b : ℤ) : gcd (abs a) (abs b) = gcd a b :=
by rewrite [↑gcd, *nat_abs_abs]
section
open nat
theorem gcd_of_ne_zero (a : ℤ) {b : ℤ} (H : b ≠ 0) : gcd a b = gcd b (abs a mod abs b) :=
sorry
/-
have nat_abs b ≠ nat.zero, from assume H', H (eq_zero_of_nat_abs_eq_zero H'),
have (#nat nat_abs b > nat.zero), from nat.pos_of_ne_zero this,
assert nat.gcd (nat_abs a) (nat_abs b) = (#nat nat.gcd (nat_abs b) (nat_abs a mod nat_abs b)),
have nat_abs b ≠ 0, from assume H', H (eq_zero_of_nat_abs_eq_zero H'),
have nat_abs b > 0, from pos_of_ne_zero this,
assert nat.gcd (nat_abs a) (nat_abs b) = (nat.gcd (nat_abs b) (nat_abs a mod nat_abs b)),
from @nat.gcd_of_pos (nat_abs a) (nat_abs b) this,
calc
gcd a b = nat.gcd (nat_abs b) (#nat nat_abs a mod nat_abs b) : by rewrite [↑gcd, this]
... = gcd (abs b) (abs a mod abs b) :
by rewrite [↑gcd, -*of_nat_nat_abs, of_nat_mod]
... = gcd b (abs a mod abs b) : by rewrite [↑gcd, *nat_abs_abs]
-/
gcd a b = nat.gcd (nat_abs b) (nat_abs a mod nat_abs b) : by rewrite [↑gcd, this]
... = gcd (abs b) (abs a mod abs b) : by rewrite [↑gcd, -*of_nat_nat_abs, of_nat_mod]
... = gcd b (abs a mod abs b) : by rewrite [↑gcd, *nat_abs_abs]
end
theorem gcd_of_pos (a : ℤ) {b : ℤ} (H : b > 0) : gcd a b = gcd b (abs a mod b) :=
by rewrite [!gcd_of_ne_zero (ne_of_gt H), abs_of_pos H]