Extended GCD and divisibility over ℤ #

Main definitions #

• Given x y : ℕ, xgcd x y computes the pair of integers (a, b) such that gcd x y = x * a + y * b. gcdA x y and gcdB x y are defined to be a and b, respectively.

Main statements #

• gcd_eq_gcd_ab: Bézout's lemma, given x y : ℕ, gcd x y = x * gcdA x y + y * gcdB x y.

Tags #

Bézout's lemma, Bezout's lemma

Extended Euclidean algorithm #

def Nat.xgcdAux :
×

Helper function for the extended GCD algorithm (Nat.xgcd).

Equations
• Nat.xgcdAux 0 x✝³ x✝² x✝¹ x✝ x = (x✝¹, x✝, x)
• Nat.xgcdAux () x✝³ x✝² x✝¹ x✝ x = let_fun this := ; let q := x✝¹ / ; Nat.xgcdAux (x✝¹ % ) (x✝ - q * x✝³) (x - q * x✝²) () x✝³ x✝²
Instances For
theorem Nat.xgcdAux_zero {s : } {t : } {r' : } {s' : } {t' : } :
Nat.xgcdAux 0 s t r' s' t' = (r', s', t')
theorem Nat.xgcdAux_succ {k : } {s : } {t : } {r' : } {s' : } {t' : } :
Nat.xgcdAux () s t r' s' t' = Nat.xgcdAux (r' % ) (s' - r' / () * s) (t' - r' / () * t) () s t
@[simp]
theorem Nat.xgcd_zero_left {s : } {t : } {r' : } {s' : } {t' : } :
Nat.xgcdAux 0 s t r' s' t' = (r', s', t')
theorem Nat.xgcdAux_rec {r : } {s : } {t : } {r' : } {s' : } {t' : } (h : 0 < r) :
Nat.xgcdAux r s t r' s' t' = Nat.xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t
def Nat.xgcd (x : ) (y : ) :

Use the extended GCD algorithm to generate the a and b values satisfying gcd x y = x * a + y * b.

Equations
Instances For
def Nat.gcdA (x : ) (y : ) :

The extended GCD a value in the equation gcd x y = x * a + y * b.

Equations
Instances For
def Nat.gcdB (x : ) (y : ) :

The extended GCD b value in the equation gcd x y = x * a + y * b.

Equations
Instances For
@[simp]
theorem Nat.gcdA_zero_left {s : } :
Nat.gcdA 0 s = 0
@[simp]
theorem Nat.gcdB_zero_left {s : } :
Nat.gcdB 0 s = 1
@[simp]
theorem Nat.gcdA_zero_right {s : } (h : s 0) :
Nat.gcdA s 0 = 1
@[simp]
theorem Nat.gcdB_zero_right {s : } (h : s 0) :
Nat.gcdB s 0 = 0
@[simp]
theorem Nat.xgcdAux_fst (x : ) (y : ) (s : ) (t : ) (s' : ) (t' : ) :
(Nat.xgcdAux x s t y s' t').1 = Nat.gcd x y
theorem Nat.xgcdAux_val (x : ) (y : ) :
Nat.xgcdAux x 1 0 y 0 1 = (Nat.gcd x y, Nat.xgcd x y)
theorem Nat.xgcd_val (x : ) (y : ) :
Nat.xgcd x y = (Nat.gcdA x y, Nat.gcdB x y)
theorem Nat.xgcdAux_P (x : ) (y : ) {r : } {r' : } {s : } {t : } {s' : } {t' : } :
Nat.P x y (r, s, t)Nat.P x y (r', s', t')Nat.P x y (Nat.xgcdAux r s t r' s' t')
theorem Nat.gcd_eq_gcd_ab (x : ) (y : ) :
(Nat.gcd x y) = x * Nat.gcdA x y + y * Nat.gcdB x y

Bézout's lemma: given x y : ℕ, gcd x y = x * a + y * b, where a = gcd_a x y and b = gcd_b x y are computed by the extended Euclidean algorithm.

theorem Nat.exists_mul_emod_eq_gcd {k : } {n : } (hk : Nat.gcd n k < k) :
∃ (m : ), n * m % k = Nat.gcd n k
theorem Nat.exists_mul_emod_eq_one_of_coprime {k : } {n : } (hkn : ) (hk : 1 < k) :
∃ (m : ), n * m % k = 1

Divisibility over ℤ #

theorem Int.gcd_def (i : ) (j : ) :
Int.gcd i j = Nat.gcd () ()
theorem Int.coe_nat_gcd (m : ) (n : ) :
Int.gcd m n = Nat.gcd m n
def Int.gcdA :

The extended GCD a value in the equation gcd x y = x * a + y * b.

Equations
Instances For
def Int.gcdB :

The extended GCD b value in the equation gcd x y = x * a + y * b.

Equations
Instances For
theorem Int.gcd_eq_gcd_ab (x : ) (y : ) :
(Int.gcd x y) = x * Int.gcdA x y + y * Int.gcdB x y

Bézout's lemma

theorem Int.natAbs_ediv (a : ) (b : ) (H : b a) :
Int.natAbs (a / b) =
theorem Int.dvd_of_mul_dvd_mul_left {i : } {j : } {k : } (k_non_zero : k 0) (H : k * i k * j) :
i j

special case of mul_dvd_mul_iff_right for ℤ. Duplicated here to keep simple imports for this file.

theorem Int.dvd_of_mul_dvd_mul_right {i : } {j : } {k : } (k_non_zero : k 0) (H : i * k j * k) :
i j

special case of mul_dvd_mul_iff_right for ℤ. Duplicated here to keep simple imports for this file.

theorem Int.lcm_def (i : ) (j : ) :
Int.lcm i j = Nat.lcm () ()
theorem Int.coe_nat_lcm (m : ) (n : ) :
Int.lcm m n = Nat.lcm m n
theorem Int.dvd_gcd {i : } {j : } {k : } (h1 : k i) (h2 : k j) :
k (Int.gcd i j)
theorem Int.gcd_mul_lcm (i : ) (j : ) :
Int.gcd i j * Int.lcm i j = Int.natAbs (i * j)
theorem Int.gcd_comm (i : ) (j : ) :
Int.gcd i j = Int.gcd j i
theorem Int.gcd_assoc (i : ) (j : ) (k : ) :
Int.gcd ((Int.gcd i j)) k = Int.gcd i (Int.gcd j k)
@[simp]
theorem Int.gcd_self (i : ) :
Int.gcd i i =
@[simp]
theorem Int.gcd_zero_left (i : ) :
Int.gcd 0 i =
@[simp]
theorem Int.gcd_zero_right (i : ) :
Int.gcd i 0 =
theorem Int.gcd_mul_left (i : ) (j : ) (k : ) :
Int.gcd (i * j) (i * k) = * Int.gcd j k
theorem Int.gcd_mul_right (i : ) (j : ) (k : ) :
Int.gcd (i * j) (k * j) = Int.gcd i k *
theorem Int.gcd_pos_of_ne_zero_left {i : } (j : ) (hi : i 0) :
0 < Int.gcd i j
theorem Int.gcd_pos_of_ne_zero_right (i : ) {j : } (hj : j 0) :
0 < Int.gcd i j
theorem Int.gcd_eq_zero_iff {i : } {j : } :
Int.gcd i j = 0 i = 0 j = 0
theorem Int.gcd_pos_iff {i : } {j : } :
0 < Int.gcd i j i 0 j 0
theorem Int.gcd_div {i : } {j : } {k : } (H1 : k i) (H2 : k j) :
Int.gcd (i / k) (j / k) = Int.gcd i j /
theorem Int.gcd_div_gcd_div_gcd {i : } {j : } (H : 0 < Int.gcd i j) :
Int.gcd (i / (Int.gcd i j)) (j / (Int.gcd i j)) = 1
theorem Int.gcd_dvd_gcd_of_dvd_left {i : } {k : } (j : ) (H : i k) :
theorem Int.gcd_dvd_gcd_of_dvd_right {i : } {k : } (j : ) (H : i k) :
theorem Int.gcd_dvd_gcd_mul_left (i : ) (j : ) (k : ) :
Int.gcd i j Int.gcd (k * i) j
theorem Int.gcd_dvd_gcd_mul_right (i : ) (j : ) (k : ) :
Int.gcd i j Int.gcd (i * k) j
theorem Int.gcd_dvd_gcd_mul_left_right (i : ) (j : ) (k : ) :
Int.gcd i j Int.gcd i (k * j)
theorem Int.gcd_dvd_gcd_mul_right_right (i : ) (j : ) (k : ) :
Int.gcd i j Int.gcd i (j * k)
theorem Int.gcd_eq_left {i : } {j : } (H : i j) :
Int.gcd i j =
theorem Int.gcd_eq_right {i : } {j : } (H : j i) :
Int.gcd i j =
theorem Int.ne_zero_of_gcd {x : } {y : } (hc : Int.gcd x y 0) :
x 0 y 0
theorem Int.exists_gcd_one {m : } {n : } (H : 0 < Int.gcd m n) :
∃ (m' : ) (n' : ), Int.gcd m' n' = 1 m = m' * (Int.gcd m n) n = n' * (Int.gcd m n)
theorem Int.exists_gcd_one' {m : } {n : } (H : 0 < Int.gcd m n) :
∃ (g : ) (m' : ) (n' : ), 0 < g Int.gcd m' n' = 1 m = m' * g n = n' * g
theorem Int.pow_dvd_pow_iff {m : } {n : } {k : } (k0 : k 0) :
m ^ k n ^ k m n
theorem Int.gcd_dvd_iff {a : } {b : } {n : } :
Int.gcd a b n ∃ (x : ) (y : ), n = a * x + b * y
theorem Int.gcd_greatest {a : } {b : } {d : } (hd_pos : 0 d) (hda : d a) (hdb : d b) (hd : ∀ (e : ), e ae be d) :
d = (Int.gcd a b)
theorem Int.dvd_of_dvd_mul_left_of_gcd_one {a : } {b : } {c : } (habc : a b * c) (hab : Int.gcd a c = 1) :
a b

Euclid's lemma: if a ∣ b * c and gcd a c = 1 then a ∣ b. Compare with IsCoprime.dvd_of_dvd_mul_left and UniqueFactorizationMonoid.dvd_of_dvd_mul_left_of_no_prime_factors

theorem Int.dvd_of_dvd_mul_right_of_gcd_one {a : } {b : } {c : } (habc : a b * c) (hab : Int.gcd a b = 1) :
a c

Euclid's lemma: if a ∣ b * c and gcd a b = 1 then a ∣ c. Compare with IsCoprime.dvd_of_dvd_mul_right and UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors

theorem Int.gcd_least_linear {a : } {b : } (ha : a 0) :
IsLeast {n : | 0 < n ∃ (x : ) (y : ), n = a * x + b * y} (Int.gcd a b)

For nonzero integers a and b, gcd a b is the smallest positive natural number that can be written in the form a * x + b * y for some pair of integers x and y

lcm #

theorem Int.lcm_comm (i : ) (j : ) :
Int.lcm i j = Int.lcm j i
theorem Int.lcm_assoc (i : ) (j : ) (k : ) :
Int.lcm ((Int.lcm i j)) k = Int.lcm i (Int.lcm j k)
@[simp]
theorem Int.lcm_zero_left (i : ) :
Int.lcm 0 i = 0
@[simp]
theorem Int.lcm_zero_right (i : ) :
Int.lcm i 0 = 0
@[simp]
theorem Int.lcm_one_left (i : ) :
Int.lcm 1 i =
@[simp]
theorem Int.lcm_one_right (i : ) :
Int.lcm i 1 =
theorem Int.lcm_dvd {i : } {j : } {k : } :
i kj k(Int.lcm i j) k
theorem Int.lcm_mul_left {m : } {n : } {k : } :
Int.lcm (m * n) (m * k) = * Int.lcm n k
theorem Int.lcm_mul_right {m : } {n : } {k : } :
Int.lcm (m * n) (k * n) = Int.lcm m k *
theorem gcd_nsmul_eq_zero {M : Type u_1} [] (x : M) {m : } {n : } (hm : m x = 0) (hn : n x = 0) :
Nat.gcd m n x = 0
theorem pow_gcd_eq_one {M : Type u_1} [] (x : M) {m : } {n : } (hm : x ^ m = 1) (hn : x ^ n = 1) :
x ^ Nat.gcd m n = 1
theorem Commute.pow_eq_pow_iff_of_coprime {α : Type u_1} [] {a : α} {b : α} {m : } {n : } (hab : Commute a b) (hmn : ) :
a ^ m = b ^ n ∃ (c : α), a = c ^ n b = c ^ m
theorem pow_eq_pow_iff_of_coprime {α : Type u_1} {a : α} {b : α} {m : } {n : } (hmn : ) :
a ^ m = b ^ n ∃ (c : α), a = c ^ n b = c ^ m
theorem pow_mem_range_pow_of_coprime {α : Type u_1} {m : } {n : } (hmn : ) (a : α) :
(a ^ m Set.range fun (x : α) => x ^ n) a Set.range fun (x : α) => x ^ n