372 lines
15 KiB
Text
372 lines
15 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jeremy Avigad, Leonardo de Moura
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Definitions and properties of gcd, lcm, and coprime.
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-/
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import .div
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open eq.ops well_founded decidable prod
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open algebra
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namespace nat
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/- gcd -/
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private definition pair_nat.lt : nat × nat → nat × nat → Prop := measure pr₂
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private definition pair_nat.lt.wf : well_founded pair_nat.lt :=
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intro_k (measure.wf pr₂) 20 -- we use intro_k to be able to execute gcd efficiently in the kernel
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local attribute pair_nat.lt.wf [instance] -- instance will not be saved in .olean
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local infixl ` ≺ `:50 := pair_nat.lt
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private definition gcd.lt.dec (x y₁ : nat) : (succ y₁, x % succ y₁) ≺ (x, succ y₁) :=
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!mod_lt (succ_pos y₁)
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definition gcd.F : Π (p₁ : nat × nat), (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat
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| (x, 0) f := x
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| (x, succ y) f := f (succ y, x % succ y) !gcd.lt.dec
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definition gcd (x y : nat) := fix gcd.F (x, y)
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theorem gcd_zero_right (x : nat) : gcd x 0 = x := rfl
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theorem gcd_succ (x y : nat) : gcd x (succ y) = gcd (succ y) (x % succ y) :=
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well_founded.fix_eq gcd.F (x, succ y)
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theorem gcd_one_right (n : ℕ) : gcd n 1 = 1 :=
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calc gcd n 1 = gcd 1 (n % 1) : gcd_succ
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... = gcd 1 0 : mod_one
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theorem gcd_def (x : ℕ) : Π (y : ℕ), gcd x y = if y = 0 then x else gcd y (x % y)
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| 0 := !gcd_zero_right
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| (succ y) := !gcd_succ ⬝ (if_neg !succ_ne_zero)⁻¹
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theorem gcd_self : Π (n : ℕ), gcd n n = n
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| 0 := rfl
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| (succ n₁) := calc
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gcd (succ n₁) (succ n₁) = gcd (succ n₁) (succ n₁ % succ n₁) : gcd_succ
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... = gcd (succ n₁) 0 : mod_self
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theorem gcd_zero_left : Π (n : ℕ), gcd 0 n = n
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| 0 := rfl
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| (succ n₁) := calc
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gcd 0 (succ n₁) = gcd (succ n₁) (0 % succ n₁) : gcd_succ
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... = gcd (succ n₁) 0 : zero_mod
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theorem gcd_of_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m % n) :=
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gcd_def m n ⬝ if_neg (ne_zero_of_pos H)
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theorem gcd_rec (m n : ℕ) : gcd m n = gcd n (m % n) :=
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by_cases_zero_pos n
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(calc
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m = gcd 0 m : gcd_zero_left
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... = gcd 0 (m % 0) : mod_zero)
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(take n, assume H : 0 < n, gcd_of_pos m H)
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theorem gcd.induction {P : ℕ → ℕ → Prop}
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(m n : ℕ)
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(H0 : ∀m, P m 0)
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(H1 : ∀m n, 0 < n → P n (m % n) → P m n) :
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P m n :=
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induction (m, n) (prod.rec (λm, nat.rec (λ IH, H0 m)
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(λ n₁ v (IH : ∀p₂, p₂ ≺ (m, succ n₁) → P (pr₁ p₂) (pr₂ p₂)),
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H1 m (succ n₁) !succ_pos (IH _ !gcd.lt.dec))))
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theorem gcd_dvd (m n : ℕ) : (gcd m n ∣ m) ∧ (gcd m n ∣ n) :=
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gcd.induction m n
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(take m, and.intro (!one_mul ▸ !dvd_mul_left) !dvd_zero)
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(take m n (npos : 0 < n), and.rec
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(assume (IH₁ : gcd n (m % n) ∣ n) (IH₂ : gcd n (m % n) ∣ (m % n)),
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have H : (gcd n (m % n) ∣ (m / n * n + m % n)), from
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dvd_add (dvd.trans IH₁ !dvd_mul_left) IH₂,
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have H1 : (gcd n (m % n) ∣ m), from !eq_div_mul_add_mod⁻¹ ▸ H,
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show (gcd m n ∣ m) ∧ (gcd m n ∣ n), from !gcd_rec⁻¹ ▸ (and.intro H1 IH₁)))
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theorem gcd_dvd_left (m n : ℕ) : gcd m n ∣ m := and.left !gcd_dvd
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theorem gcd_dvd_right (m n : ℕ) : gcd m n ∣ n := and.right !gcd_dvd
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theorem dvd_gcd {m n k : ℕ} : k ∣ m → k ∣ n → k ∣ gcd m n :=
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gcd.induction m n (take m, imp.intro)
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(take m n (npos : n > 0)
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(IH : k ∣ n → k ∣ m % n → k ∣ gcd n (m % n))
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(H1 : k ∣ m) (H2 : k ∣ n),
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have H3 : k ∣ m / n * n + m % n, from !eq_div_mul_add_mod ▸ H1,
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have H4 : k ∣ m % n, from nat.dvd_of_dvd_add_left H3 (dvd.trans H2 !dvd_mul_left),
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!gcd_rec⁻¹ ▸ IH H2 H4)
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theorem gcd.comm (m n : ℕ) : gcd m n = gcd n m :=
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dvd.antisymm
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(dvd_gcd !gcd_dvd_right !gcd_dvd_left)
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(dvd_gcd !gcd_dvd_right !gcd_dvd_left)
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theorem gcd.assoc (m n k : ℕ) : gcd (gcd m n) k = gcd m (gcd n k) :=
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dvd.antisymm
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(dvd_gcd
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(dvd.trans !gcd_dvd_left !gcd_dvd_left)
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(dvd_gcd (dvd.trans !gcd_dvd_left !gcd_dvd_right) !gcd_dvd_right))
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(dvd_gcd
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(dvd_gcd !gcd_dvd_left (dvd.trans !gcd_dvd_right !gcd_dvd_left))
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(dvd.trans !gcd_dvd_right !gcd_dvd_right))
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theorem gcd_one_left (m : ℕ) : gcd 1 m = 1 :=
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!gcd.comm ⬝ !gcd_one_right
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theorem gcd_mul_left (m n k : ℕ) : gcd (m * n) (m * k) = m * gcd n k :=
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gcd.induction n k
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(take n, calc gcd (m * n) (m * 0) = gcd (m * n) 0 : mul_zero)
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(take n k,
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assume H : 0 < k,
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assume IH : gcd (m * k) (m * (n % k)) = m * gcd k (n % k),
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calc
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gcd (m * n) (m * k) = gcd (m * k) (m * n % (m * k)) : !gcd_rec
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... = gcd (m * k) (m * (n % k)) : mul_mod_mul_left
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... = m * gcd k (n % k) : IH
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... = m * gcd n k : !gcd_rec)
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theorem gcd_mul_right (m n k : ℕ) : gcd (m * n) (k * n) = gcd m k * n :=
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calc
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gcd (m * n) (k * n) = gcd (n * m) (k * n) : mul.comm
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... = gcd (n * m) (n * k) : mul.comm
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... = n * gcd m k : gcd_mul_left
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... = gcd m k * n : mul.comm
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theorem gcd_pos_of_pos_left {m : ℕ} (n : ℕ) (mpos : m > 0) : gcd m n > 0 :=
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pos_of_dvd_of_pos !gcd_dvd_left mpos
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theorem gcd_pos_of_pos_right (m : ℕ) {n : ℕ} (npos : n > 0) : gcd m n > 0 :=
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pos_of_dvd_of_pos !gcd_dvd_right npos
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theorem eq_zero_of_gcd_eq_zero_left {m n : ℕ} (H : gcd m n = 0) : m = 0 :=
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or.elim (eq_zero_or_pos m)
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(assume H1, H1)
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(assume H1 : m > 0, absurd H⁻¹ (ne_of_lt (!gcd_pos_of_pos_left H1)))
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theorem eq_zero_of_gcd_eq_zero_right {m n : ℕ} (H : gcd m n = 0) : n = 0 :=
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eq_zero_of_gcd_eq_zero_left (!gcd.comm ▸ H)
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theorem gcd_div {m n k : ℕ} (H1 : k ∣ m) (H2 : k ∣ n) :
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gcd (m / k) (n / k) = gcd m n / k :=
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or.elim (eq_zero_or_pos k)
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(assume H3 : k = 0, by subst k; rewrite *nat.div_zero)
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(assume H3 : k > 0, (nat.div_eq_of_eq_mul_left H3 (calc
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gcd m n = gcd m (n / k * k) : nat.div_mul_cancel H2
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... = gcd (m / k * k) (n / k * k) : nat.div_mul_cancel H1
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... = gcd (m / k) (n / k) * k : gcd_mul_right))⁻¹)
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theorem gcd_dvd_gcd_mul_left (m n k : ℕ) : gcd m n ∣ gcd (k * m) n :=
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dvd_gcd (dvd.trans !gcd_dvd_left !dvd_mul_left) !gcd_dvd_right
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theorem gcd_dvd_gcd_mul_right (m n k : ℕ) : gcd m n ∣ gcd (m * k) n :=
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!mul.comm ▸ !gcd_dvd_gcd_mul_left
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theorem gcd_dvd_gcd_mul_left_right (m n k : ℕ) : gcd m n ∣ gcd m (k * n) :=
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dvd_gcd !gcd_dvd_left (dvd.trans !gcd_dvd_right !dvd_mul_left)
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theorem gcd_dvd_gcd_mul_right_right (m n k : ℕ) : gcd m n ∣ gcd m (n * k) :=
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!mul.comm ▸ !gcd_dvd_gcd_mul_left_right
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/- lcm -/
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definition lcm (m n : ℕ) : ℕ := m * n / (gcd m n)
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theorem lcm.comm (m n : ℕ) : lcm m n = lcm n m :=
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calc
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lcm m n = m * n / gcd m n : rfl
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... = n * m / gcd m n : mul.comm
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... = n * m / gcd n m : gcd.comm
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... = lcm n m : rfl
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theorem lcm_zero_left (m : ℕ) : lcm 0 m = 0 :=
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calc
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lcm 0 m = 0 * m / gcd 0 m : rfl
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... = 0 / gcd 0 m : zero_mul
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... = 0 : nat.zero_div
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theorem lcm_zero_right (m : ℕ) : lcm m 0 = 0 := !lcm.comm ▸ !lcm_zero_left
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theorem lcm_one_left (m : ℕ) : lcm 1 m = m :=
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calc
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lcm 1 m = 1 * m / gcd 1 m : rfl
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... = m / gcd 1 m : one_mul
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... = m / 1 : gcd_one_left
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... = m : nat.div_one
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theorem lcm_one_right (m : ℕ) : lcm m 1 = m := !lcm.comm ▸ !lcm_one_left
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theorem lcm_self (m : ℕ) : lcm m m = m :=
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have H : m * m / m = m, from
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by_cases_zero_pos m !nat.div_zero (take m, assume H1 : m > 0, !nat.mul_div_cancel H1),
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calc
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lcm m m = m * m / gcd m m : rfl
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... = m * m / m : gcd_self
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... = m : H
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theorem dvd_lcm_left (m n : ℕ) : m ∣ lcm m n :=
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have H : lcm m n = m * (n / gcd m n), from nat.mul_div_assoc _ !gcd_dvd_right,
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dvd.intro H⁻¹
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theorem dvd_lcm_right (m n : ℕ) : n ∣ lcm m n :=
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!lcm.comm ▸ !dvd_lcm_left
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theorem gcd_mul_lcm (m n : ℕ) : gcd m n * lcm m n = m * n :=
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eq.symm (nat.eq_mul_of_div_eq_right (dvd.trans !gcd_dvd_left !dvd_mul_right) rfl)
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theorem lcm_dvd {m n k : ℕ} (H1 : m ∣ k) (H2 : n ∣ k) : lcm m n ∣ k :=
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or.elim (eq_zero_or_pos k)
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(assume kzero : k = 0, !kzero⁻¹ ▸ !dvd_zero)
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(assume kpos : k > 0,
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have mpos : m > 0, from pos_of_dvd_of_pos H1 kpos,
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have npos : n > 0, from pos_of_dvd_of_pos H2 kpos,
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have gcd_pos : gcd m n > 0, from !gcd_pos_of_pos_left mpos,
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obtain p (km : k = m * p), from exists_eq_mul_right_of_dvd H1,
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obtain q (kn : k = n * q), from exists_eq_mul_right_of_dvd H2,
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have ppos : p > 0, from pos_of_mul_pos_left (km ▸ kpos),
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have qpos : q > 0, from pos_of_mul_pos_left (kn ▸ kpos),
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have H3 : p * q * (m * n * gcd p q) = p * q * (gcd m n * k), from
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calc
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p * q * (m * n * gcd p q)
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= m * p * (n * q * gcd p q) : by rewrite [*mul.assoc, *mul.left_comm q,
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mul.left_comm p]
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... = k * (k * gcd p q) : by rewrite [-kn, -km]
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... = k * gcd (k * p) (k * q) : by rewrite gcd_mul_left
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... = k * gcd (n * q * p) (m * p * q) : by rewrite [-kn, -km]
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... = k * (gcd n m * (p * q)) : by rewrite [*mul.assoc, mul.comm q, gcd_mul_right]
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... = p * q * (gcd m n * k) : by rewrite [mul.comm, mul.comm (gcd n m), gcd.comm,
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*mul.assoc],
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have H4 : m * n * gcd p q = gcd m n * k,
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from !eq_of_mul_eq_mul_left (mul_pos ppos qpos) H3,
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have H5 : gcd m n * (lcm m n * gcd p q) = gcd m n * k,
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from !mul.assoc ▸ !gcd_mul_lcm⁻¹ ▸ H4,
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have H6 : lcm m n * gcd p q = k,
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from !eq_of_mul_eq_mul_left gcd_pos H5,
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dvd.intro H6)
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theorem lcm.assoc (m n k : ℕ) : lcm (lcm m n) k = lcm m (lcm n k) :=
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dvd.antisymm
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(lcm_dvd
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(lcm_dvd !dvd_lcm_left (dvd.trans !dvd_lcm_left !dvd_lcm_right))
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(dvd.trans !dvd_lcm_right !dvd_lcm_right))
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(lcm_dvd
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(dvd.trans !dvd_lcm_left !dvd_lcm_left)
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(lcm_dvd (dvd.trans !dvd_lcm_right !dvd_lcm_left) !dvd_lcm_right))
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/- coprime -/
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definition coprime [reducible] (m n : ℕ) : Prop := gcd m n = 1
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lemma gcd_eq_one_of_coprime {m n : ℕ} : coprime m n → gcd m n = 1 :=
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λ h, h
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theorem coprime_swap {m n : ℕ} (H : coprime n m) : coprime m n :=
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!gcd.comm ▸ H
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theorem dvd_of_coprime_of_dvd_mul_right {m n k : ℕ} (H1 : coprime k n) (H2 : k ∣ m * n) : k ∣ m :=
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have H3 : gcd (m * k) (m * n) = m, from
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calc
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gcd (m * k) (m * n) = m * gcd k n : gcd_mul_left
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... = m * 1 : H1
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... = m : mul_one,
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have H4 : (k ∣ gcd (m * k) (m * n)), from dvd_gcd !dvd_mul_left H2,
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H3 ▸ H4
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theorem dvd_of_coprime_of_dvd_mul_left {m n k : ℕ} (H1 : coprime k m) (H2 : k ∣ m * n) : k ∣ n :=
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dvd_of_coprime_of_dvd_mul_right H1 (!mul.comm ▸ H2)
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theorem gcd_mul_left_cancel_of_coprime {k : ℕ} (m : ℕ) {n : ℕ} (H : coprime k n) :
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gcd (k * m) n = gcd m n :=
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have H1 : coprime (gcd (k * m) n) k, from
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calc
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gcd (gcd (k * m) n) k
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= gcd (k * gcd 1 m) n : by rewrite [-gcd_mul_left, mul_one, gcd.comm, gcd.assoc]
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... = 1 : by rewrite [gcd_one_left, mul_one, ↑coprime at H, H],
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dvd.antisymm
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(dvd_gcd (dvd_of_coprime_of_dvd_mul_left H1 !gcd_dvd_left) !gcd_dvd_right)
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(dvd_gcd (dvd.trans !gcd_dvd_left !dvd_mul_left) !gcd_dvd_right)
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theorem gcd_mul_right_cancel_of_coprime (m : ℕ) {k n : ℕ} (H : coprime k n) :
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gcd (m * k) n = gcd m n :=
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!mul.comm ▸ !gcd_mul_left_cancel_of_coprime H
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theorem gcd_mul_left_cancel_of_coprime_right {k m : ℕ} (n : ℕ) (H : coprime k m) :
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gcd m (k * n) = gcd m n :=
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!gcd.comm ▸ !gcd.comm ▸ !gcd_mul_left_cancel_of_coprime H
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theorem gcd_mul_right_cancel_of_coprime_right {k m : ℕ} (n : ℕ) (H : coprime k m) :
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gcd m (n * k) = gcd m n :=
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!gcd.comm ▸ !gcd.comm ▸ !gcd_mul_right_cancel_of_coprime H
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theorem coprime_div_gcd_div_gcd {m n : ℕ} (H : gcd m n > 0) :
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coprime (m / gcd m n) (n / gcd m n) :=
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calc
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gcd (m / gcd m n) (n / gcd m n) = gcd m n / gcd m n : gcd_div !gcd_dvd_left !gcd_dvd_right
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... = 1 : nat.div_self H
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theorem not_coprime_of_dvd_of_dvd {m n d : ℕ} (dgt1 : d > 1) (Hm : d ∣ m) (Hn : d ∣ n) :
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¬ coprime m n :=
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assume co : coprime m n,
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assert d ∣ gcd m n, from dvd_gcd Hm Hn,
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have d ∣ 1, by rewrite [↑coprime at co, co at this]; apply this,
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have d ≤ 1, from le_of_dvd dec_trivial this,
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show false, from not_lt_of_ge `d ≤ 1` `d > 1`
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theorem exists_coprime {m n : ℕ} (H : gcd m n > 0) :
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exists m' n', coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n :=
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have H1 : m = (m / gcd m n) * gcd m n, from (nat.div_mul_cancel !gcd_dvd_left)⁻¹,
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have H2 : n = (n / gcd m n) * gcd m n, from (nat.div_mul_cancel !gcd_dvd_right)⁻¹,
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exists.intro _ (exists.intro _ (and.intro (coprime_div_gcd_div_gcd H) (and.intro H1 H2)))
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theorem coprime_mul {m n k : ℕ} (H1 : coprime m k) (H2 : coprime n k) : coprime (m * n) k :=
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calc
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gcd (m * n) k = gcd n k : !gcd_mul_left_cancel_of_coprime H1
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... = 1 : H2
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theorem coprime_mul_right {k m n : ℕ} (H1 : coprime k m) (H2 : coprime k n) : coprime k (m * n) :=
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coprime_swap (coprime_mul (coprime_swap H1) (coprime_swap H2))
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theorem coprime_of_coprime_mul_left {k m n : ℕ} (H : coprime (k * m) n) : coprime m n :=
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have H1 : (gcd m n ∣ gcd (k * m) n), from !gcd_dvd_gcd_mul_left,
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eq_one_of_dvd_one (H ▸ H1)
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theorem coprime_of_coprime_mul_right {k m n : ℕ} (H : coprime (m * k) n) : coprime m n :=
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coprime_of_coprime_mul_left (!mul.comm ▸ H)
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theorem coprime_of_coprime_mul_left_right {k m n : ℕ} (H : coprime m (k * n)) : coprime m n :=
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coprime_swap (coprime_of_coprime_mul_left (coprime_swap H))
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theorem coprime_of_coprime_mul_right_right {k m n : ℕ} (H : coprime m (n * k)) : coprime m n :=
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coprime_of_coprime_mul_left_right (!mul.comm ▸ H)
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theorem comprime_one_left : ∀ n, coprime 1 n :=
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λ n, !gcd_one_left
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theorem comprime_one_right : ∀ n, coprime n 1 :=
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λ n, !gcd_one_right
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theorem exists_eq_prod_and_dvd_and_dvd {m n k : nat} (H : k ∣ m * n) :
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∃ m' n', k = m' * n' ∧ m' ∣ m ∧ n' ∣ n :=
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or.elim (eq_zero_or_pos (gcd k m))
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(assume H1 : gcd k m = 0,
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have H2 : k = 0, from eq_zero_of_gcd_eq_zero_left H1,
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have H3 : m = 0, from eq_zero_of_gcd_eq_zero_right H1,
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have H4 : k = 0 * n, from H2 ⬝ !zero_mul⁻¹,
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have H5 : 0 ∣ m, from H3⁻¹ ▸ !dvd.refl,
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have H6 : n ∣ n, from !dvd.refl,
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exists.intro _ (exists.intro _ (and.intro H4 (and.intro H5 H6))))
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(assume H1 : gcd k m > 0,
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have H2 : gcd k m ∣ k, from !gcd_dvd_left,
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have H3 : k / gcd k m ∣ (m * n) / gcd k m, from nat.div_dvd_div H2 H,
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have H4 : (m * n) / gcd k m = (m / gcd k m) * n, from
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calc
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m * n / gcd k m = n * m / gcd k m : mul.comm
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... = n * (m / gcd k m) : !nat.mul_div_assoc !gcd_dvd_right
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... = m / gcd k m * n : mul.comm,
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have H5 : k / gcd k m ∣ (m / gcd k m) * n, from H4 ▸ H3,
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have H6 : coprime (k / gcd k m) (m / gcd k m), from coprime_div_gcd_div_gcd H1,
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have H7 : k / gcd k m ∣ n, from dvd_of_coprime_of_dvd_mul_left H6 H5,
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have H8 : k = gcd k m * (k / gcd k m), from (nat.mul_div_cancel' H2)⁻¹,
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exists.intro _ (exists.intro _ (and.intro H8 (and.intro !gcd_dvd_right H7))))
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end nat
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