lean2/library/data/nat/gcd.lean

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