l3lab/ntp-mathlib huggingface dataset:
theorem totient_gcd_mul_totient_mul (a b : ℕ) : φ (a.gcd b) * φ (a * b) = φ a * φ b * a.gcd b := by
  have shuffle :
    ∀ a1 a2 b1 b2 c1 c2 : ℕ,
      b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2) := by
    intro a1 a2 b1 b2 c1 c2 h1 h2
    calc
      a1 / b1 * c1 * (a2 / b2 * c2) = a1 / b1 * (a2 / b2) * (c1 * c2) := by apply mul_mul_mul_comm
      _ = a1 * a2 / (b1 * b2) * (c1 * c2) := by
        congr 1
        exact div_mul_div_comm h1 h2
  simp only [totient_eq_div_primeFactors_mul]
  rw [shuffle, shuffle]
  rotate_left
  repeat' 

Proof state:
a b : ℕ
shuffle : ∀ (a1 a2 b1 b2 c1 c2 : ℕ), b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2)
⊢ gcd a b * (a * b) / ((∏ p in (gcd a b).primeFactors, p) * ∏ p in (a * b).primeFactors, p) *
      ((∏ p in (gcd a b).primeFactors, (p - 1)) * ∏ p in (a * b).primeFactors, (p - 1)) =
    a * b / ((∏ p in a.primeFactors, p) * ∏ p in b.primeFactors, p) *
        ((∏ p in a.primeFactors, (p - 1)) * ∏ p in b.primeFactors, (p - 1)) *
      gcd a b

Next tactic:
apply prod_primeFactors_dvd
theorem totient_gcd_mul_totient_mul (a b : ℕ) : φ (a.gcd b) * φ (a * b) = φ a * φ b * a.gcd b := by
  have shuffle :
    ∀ a1 a2 b1 b2 c1 c2 : ℕ,
      b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2) := by
    intro a1 a2 b1 b2 c1 c2 h1 h2
    calc
      a1 / b1 * c1 * (a2 / b2 * c2) = a1 / b1 * (a2 / b2) * (c1 * c2) := by apply mul_mul_mul_comm
      _ = a1 * a2 / (b1 * b2) * (c1 * c2) := by
        congr 1
        exact div_mul_div_comm h1 h2
  simp only [totient_eq_div_primeFactors_mul]
  rw [shuffle, shuffle]
  rotate_left
  repeat' apply prod_primeFactors_dvd
  · 

Proof state:
a b : ℕ
shuffle : ∀ (a1 a2 b1 b2 c1 c2 : ℕ), b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2)
⊢ gcd a b * (a * b) / ((∏ p in (gcd a b).primeFactors, p) * ∏ p in (a * b).primeFactors, p) *
      ((∏ p in (gcd a b).primeFactors, (p - 1)) * ∏ p in (a * b).primeFactors, (p - 1)) =
    a * b / ((∏ p in a.primeFactors, p) * ∏ p in b.primeFactors, p) *
        ((∏ p in a.primeFactors, (p - 1)) * ∏ p in b.primeFactors, (p - 1)) *
      gcd a b

Next tactic:
simp only [prod_primeFactors_gcd_mul_prod_primeFactors_mul]
theorem totient_gcd_mul_totient_mul (a b : ℕ) : φ (a.gcd b) * φ (a * b) = φ a * φ b * a.gcd b := by
  have shuffle :
    ∀ a1 a2 b1 b2 c1 c2 : ℕ,
      b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2) := by
    intro a1 a2 b1 b2 c1 c2 h1 h2
    calc
      a1 / b1 * c1 * (a2 / b2 * c2) = a1 / b1 * (a2 / b2) * (c1 * c2) := by apply mul_mul_mul_comm
      _ = a1 * a2 / (b1 * b2) * (c1 * c2) := by
        congr 1
        exact div_mul_div_comm h1 h2
  simp only [totient_eq_div_primeFactors_mul]
  rw [shuffle, shuffle]
  rotate_left
  repeat' apply prod_primeFactors_dvd
  · simp only [prod_primeFactors_gcd_mul_prod_primeFactors_mul]
    

Proof state:
a b : ℕ
shuffle : ∀ (a1 a2 b1 b2 c1 c2 : ℕ), b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2)
⊢ gcd a b * (a * b) / ((∏ p in a.primeFactors, p) * ∏ p in b.primeFactors, p) *
      ((∏ p in a.primeFactors, (p - 1)) * ∏ p in b.primeFactors, (p - 1)) =
    a * b / ((∏ p in a.primeFactors, p) * ∏ p in b.primeFactors, p) *
        ((∏ p in a.primeFactors, (p - 1)) * ∏ p in b.primeFactors, (p - 1)) *
      gcd a b

Next tactic:
rw [eq_comm, mul_comm, ← mul_assoc, ← Nat.mul_div_assoc]
theorem totient_gcd_mul_totient_mul (a b : ℕ) : φ (a.gcd b) * φ (a * b) = φ a * φ b * a.gcd b := by
  have shuffle :
    ∀ a1 a2 b1 b2 c1 c2 : ℕ,
      b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2) := by
    intro a1 a2 b1 b2 c1 c2 h1 h2
    calc
      a1 / b1 * c1 * (a2 / b2 * c2) = a1 / b1 * (a2 / b2) * (c1 * c2) := by apply mul_mul_mul_comm
      _ = a1 * a2 / (b1 * b2) * (c1 * c2) := by
        congr 1
        exact div_mul_div_comm h1 h2
  simp only [totient_eq_div_primeFactors_mul]
  rw [shuffle, shuffle]
  rotate_left
  repeat' apply prod_primeFactors_dvd
  · simp only [prod_primeFactors_gcd_mul_prod_primeFactors_mul]
    rw [eq_comm, mul_comm, ← mul_assoc, ← Nat.mul_div_assoc]
    

Proof state:
case H
a b : ℕ
shuffle : ∀ (a1 a2 b1 b2 c1 c2 : ℕ), b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2)
⊢ (∏ p in a.primeFactors, p) * ∏ p in b.primeFactors, p ∣ a * b

Next tactic:
exact mul_dvd_mul (prod_primeFactors_dvd a) (prod_primeFactors_dvd b)
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
  

Proof state:
a b : ℕ
⊢ φ a * φ b ≤ φ (a * b)

Next tactic:
let d := a.gcd b
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
  let d := a.gcd b
  

Proof state:
a b : ℕ
d : ℕ := gcd a b
⊢ φ a * φ b ≤ φ (a * b)

Next tactic:
rcases (zero_le a).eq_or_lt with (rfl | ha0)
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
  let d := a.gcd b
  rcases (zero_le a).eq_or_lt with (rfl | ha0)
  · 

Proof state:
case inl
b : ℕ
d : ℕ := gcd 0 b
⊢ φ 0 * φ b ≤ φ (0 * b)

Next tactic:
simp
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
  let d := a.gcd b
  rcases (zero_le a).eq_or_lt with (rfl | ha0)
  · simp
  

Proof state:
case inr
a b : ℕ
d : ℕ := gcd a b
ha0 : 0 < a
⊢ φ a * φ b ≤ φ (a * b)

Next tactic:
have hd0 : 0 < d := Nat.gcd_pos_of_pos_left _ ha0
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
  let d := a.gcd b
  rcases (zero_le a).eq_or_lt with (rfl | ha0)
  · simp
  have hd0 : 0 < d := Nat.gcd_pos_of_pos_left _ ha0
  

Proof state:
case inr
a b : ℕ
d : ℕ := gcd a b
ha0 : 0 < a
hd0 : 0 < d
⊢ φ a * φ b ≤ φ (a * b)

Next tactic:
apply le_of_mul_le_mul_right _ hd0
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
  let d := a.gcd b
  rcases (zero_le a).eq_or_lt with (rfl | ha0)
  · simp
  have hd0 : 0 < d := Nat.gcd_pos_of_pos_left _ ha0
  apply le_of_mul_le_mul_right _ hd0
  

Proof state:
a b : ℕ
d : ℕ := gcd a b
ha0 : 0 < a
hd0 : 0 < d
⊢ φ a * φ b * d ≤ φ (a * b) * d

Next tactic:
rw [← totient_gcd_mul_totient_mul a b, mul_comm]
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
  let d := a.gcd b
  rcases (zero_le a).eq_or_lt with (rfl | ha0)
  · simp
  have hd0 : 0 < d := Nat.gcd_pos_of_pos_left _ ha0
  apply le_of_mul_le_mul_right _ hd0
  rw [← totient_gcd_mul_totient_mul a b, mul_comm]
  

Proof state:
a b : ℕ
d : ℕ := gcd a b
ha0 : 0 < a
hd0 : 0 < d
⊢ φ (a * b) * φ (gcd a b) ≤ φ (a * b) * d

Next tactic:
apply mul_le_mul_left' (Nat.totient_le d)
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  

Proof state:
a b : ℕ
h : a ∣ b
⊢ φ a ∣ φ b

Next tactic:
rcases eq_or_ne a 0 with (rfl | ha0)
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  rcases eq_or_ne a 0 with (rfl | ha0)
  · 

Proof state:
case inl
b : ℕ
h : 0 ∣ b
⊢ φ 0 ∣ φ b

Next tactic:
simp [zero_dvd_iff.1 h]
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  rcases eq_or_ne a 0 with (rfl | ha0)
  · simp [zero_dvd_iff.1 h]
  

Proof state:
case inr
a b : ℕ
h : a ∣ b
ha0 : a ≠ 0
⊢ φ a ∣ φ b

Next tactic:
rcases eq_or_ne b 0 with (rfl | hb0)
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  rcases eq_or_ne a 0 with (rfl | ha0)
  · simp [zero_dvd_iff.1 h]
  rcases eq_or_ne b 0 with (rfl | hb0)
  · 

Proof state:
case inr.inl
a : ℕ
ha0 : a ≠ 0
h : a ∣ 0
⊢ φ a ∣ φ 0

Next tactic:
simp
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  rcases eq_or_ne a 0 with (rfl | ha0)
  · simp [zero_dvd_iff.1 h]
  rcases eq_or_ne b 0 with (rfl | hb0)
  · simp
  

Proof state:
case inr.inr
a b : ℕ
h : a ∣ b
ha0 : a ≠ 0
hb0 : b ≠ 0
⊢ φ a ∣ φ b

Next tactic:
have hab' := primeFactors_mono h hb0
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  rcases eq_or_ne a 0 with (rfl | ha0)
  · simp [zero_dvd_iff.1 h]
  rcases eq_or_ne b 0 with (rfl | hb0)
  · simp
  have hab' := primeFactors_mono h hb0
  

Proof state:
case inr.inr
a b : ℕ
h : a ∣ b
ha0 : a ≠ 0
hb0 : b ≠ 0
hab' : a.primeFactors ⊆ b.primeFactors
⊢ φ a ∣ φ b

Next tactic:
rw [totient_eq_prod_factorization ha0, totient_eq_prod_factorization hb0]
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  rcases eq_or_ne a 0 with (rfl | ha0)
  · simp [zero_dvd_iff.1 h]
  rcases eq_or_ne b 0 with (rfl | hb0)
  · simp
  have hab' := primeFactors_mono h hb0
  rw [totient_eq_prod_factorization ha0, totient_eq_prod_factorization hb0]
  

Proof state:
case inr.inr
a b : ℕ
h : a ∣ b
ha0 : a ≠ 0
hb0 : b ≠ 0
hab' : a.primeFactors ⊆ b.primeFactors
⊢ (Finsupp.prod (factorization a) fun p k => p ^ (k - 1) * (p - 1)) ∣
    Finsupp.prod (factorization b) fun p k => p ^ (k - 1) * (p - 1)

Next tactic:
refine' Finsupp.prod_dvd_prod_of_subset_of_dvd hab' fun p _ => mul_dvd_mul _ dvd_rfl
theorem totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := by
  rcases eq_or_ne a 0 with (rfl | ha0)
  · simp [zero_dvd_iff.1 h]
  rcases eq_or_ne b 0 with (rfl | hb0)
  · simp
  have hab' := primeFactors_mono h hb0
  rw [totient_eq_prod_factorization ha0, totient_eq_prod_factorization hb0]
  refine' Finsupp.prod_dvd_prod_of_subset_of_dvd hab' fun p _ => mul_dvd_mul _ dvd_rfl
  

Proof state:
case inr.inr
a b : ℕ
h : a ∣ b
ha0 : a ≠ 0
hb0 : b ≠ 0
hab' : a.primeFactors ⊆ b.primeFactors
p : ℕ
x✝ : p ∈ (factorization a).support
⊢ p ^ ((factorization a) p - 1) ∣ p ^ ((factorization b) p - 1)

Next tactic:
exact pow_dvd_pow p (tsub_le_tsub_right ((factorization_le_iff_dvd ha0 hb0).2 h p) 1)
theorem totient_mul_of_prime_of_dvd {p n : ℕ} (hp : p.Prime) (h : p ∣ n) :
    (p * n).totient = p * n.totient := by
  

Proof state:
p n : ℕ
hp : Prime p
h : p ∣ n
⊢ φ (p * n) = p * φ n

Next tactic:
have h1 := totient_gcd_mul_totient_mul p n
theorem totient_mul_of_prime_of_dvd {p n : ℕ} (hp : p.Prime) (h : p ∣ n) :
    (p * n).totient = p * n.totient := by
  have h1 := totient_gcd_mul_totient_mul p n
  

Proof state:
p n : ℕ
hp : Prime p
h : p ∣ n
h1 : φ (gcd p n) * φ (p * n) = φ p * φ n * gcd p n
⊢ φ (p * n) = p * φ n

Next tactic:
rw [gcd_eq_left h, mul_assoc] at h1
theorem totient_mul_of_prime_of_dvd {p n : ℕ} (hp : p.Prime) (h : p ∣ n) :
    (p * n).totient = p * n.totient := by
  have h1 := totient_gcd_mul_totient_mul p n
  rw [gcd_eq_left h, mul_assoc] at h1
  

Proof state:
p n : ℕ
hp : Prime p
h : p ∣ n
h1 : φ p * φ (p * n) = φ p * (φ n * p)
⊢ φ (p * n) = p * φ n

Next tactic:
simpa [(totient_pos hp.pos).ne', mul_comm] using h1
theorem totient_mul_of_prime_of_not_dvd {p n : ℕ} (hp : p.Prime) (h : ¬p ∣ n) :
    (p * n).totient = (p - 1) * n.totient := by
  

Proof state:
p n : ℕ
hp : Prime p
h : ¬p ∣ n
⊢ φ (p * n) = (p - 1) * φ n

Next tactic:
rw [totient_mul _, totient_prime hp]
theorem totient_mul_of_prime_of_not_dvd {p n : ℕ} (hp : p.Prime) (h : ¬p ∣ n) :
    (p * n).totient = (p - 1) * n.totient := by
  rw [totient_mul _, totient_prime hp]
  

Proof state:
p n : ℕ
hp : Prime p
h : ¬p ∣ n
⊢ Coprime p n

Next tactic:
simpa [h] using coprime_or_dvd_of_prime hp n
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  

Proof state:
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
⊢ FrobeniusNumber (m * n - m - n) {m, n}

Next tactic:
simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  

Proof state:
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
⊢ IsGreatest {k | ¬∃ m_1 n_1, m_1 • m + n_1 • n = k} (m * n - m - n)

Next tactic:
have hmn : m + n ≤ m * n := add_le_mul hm hn
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  

Proof state:
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
⊢ IsGreatest {k | ¬∃ m_1 n_1, m_1 • m + n_1 • n = k} (m * n - m - n)

Next tactic:
constructor
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · 

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
⊢ m * n - m - n ∈ {k | ¬∃ m_1 n_1, m_1 • m + n_1 • n = k}

Next tactic:
push_neg
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
⊢ m * n - m - n ∈ {k | ∀ (m_1 n_1 : ℕ), m_1 • m + n_1 • n ≠ k}

Next tactic:
intro a b h
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
a b : ℕ
h : a • m + b • n = m * n - m - n
⊢ False

Next tactic:
apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
a b : ℕ
h : a • m + b • n = m * n - m - n
⊢ (a + 1) * m + (b + 1) * n = m * n

Next tactic:
simp only [Nat.sub_sub, smul_eq_mul] at h
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
a b : ℕ
h : a * m + b * n = m * n - (m + n)
⊢ (a + 1) * m + (b + 1) * n = m * n

Next tactic:
zify [hmn] at h ⊢
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
a b : ℕ
h : ↑a * ↑m + ↑b * ↑n = ↑m * ↑n - (↑m + ↑n)
⊢ (↑a + 1) * ↑m + (↑b + 1) * ↑n = ↑m * ↑n

Next tactic:
rw [← sub_eq_zero] at h ⊢
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
a b : ℕ
h : ↑a * ↑m + ↑b * ↑n - (↑m * ↑n - (↑m + ↑n)) = 0
⊢ (↑a + 1) * ↑m + (↑b + 1) * ↑n - ↑m * ↑n = 0

Next tactic:
rw [← h]
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    

Proof state:
case left
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
a b : ℕ
h : ↑a * ↑m + ↑b * ↑n - (↑m * ↑n - (↑m + ↑n)) = 0
⊢ (↑a + 1) * ↑m + (↑b + 1) * ↑n - ↑m * ↑n = ↑a * ↑m + ↑b * ↑n - (↑m * ↑n - (↑m + ↑n))

Next tactic:
ring
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    ring
  · 

Proof state:
case right
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
⊢ m * n - m - n ∈ upperBounds {k | ¬∃ m_1 n_1, m_1 • m + n_1 • n = k}

Next tactic:
intro k hk
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    ring
  · intro k hk
    

Proof state:
case right
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
k : ℕ
hk : k ∈ {k | ¬∃ m_1 n_1, m_1 • m + n_1 • n = k}
⊢ k ≤ m * n - m - n

Next tactic:
dsimp at hk
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    ring
  · intro k hk
    dsimp at hk
    

Proof state:
case right
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
k : ℕ
hk : ¬∃ m_1 n_1, m_1 * m + n_1 * n = k
⊢ k ≤ m * n - m - n

Next tactic:
contrapose! hk
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    ring
  · intro k hk
    dsimp at hk
    contrapose! hk
    

Proof state:
case right
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
k : ℕ
hk : m * n - m - n < k
⊢ ∃ m_1 n_1, m_1 * m + n_1 * n = k

Next tactic:
let x := chineseRemainder cop 0 k
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    ring
  · intro k hk
    dsimp at hk
    contrapose! hk
    let x := chineseRemainder cop 0 k
    

Proof state:
case right
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
k : ℕ
hk : m * n - m - n < k
x : { k_1 // k_1 ≡ 0 [MOD m] ∧ k_1 ≡ k [MOD n] } := chineseRemainder cop 0 k
⊢ ∃ m_1 n_1, m_1 * m + n_1 * n = k

Next tactic:
have hx : x.val < m * n := chineseRemainder_lt_mul cop 0 k (ne_bot_of_gt hm) (ne_bot_of_gt hn)
/-- The **Chicken McNugget theorem** stating that the Frobenius number
  of positive numbers `m` and `n` is `m * n - m - n`. -/
theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
    FrobeniusNumber (m * n - m - n) {m, n} := by
  simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
  have hmn : m + n ≤ m * n := add_le_mul hm hn
  constructor
  · push_neg
    intro a b h
    apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
    simp only [Nat.sub_sub, smul_eq_mul] at h
    zify [hmn] at h ⊢
    rw [← sub_eq_zero] at h ⊢
    rw [← h]
    ring
  · intro k hk
    dsimp at hk
    contrapose! hk
    let x := chineseRemainder cop 0 k
    have hx : x.val < m * n := chineseRemainder_lt_mul cop 0 k (ne_bot_of_gt hm) (ne_bot_of_gt hn)
    

Proof state:
case right
m n : ℕ
cop : Coprime m n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
k : ℕ
hk : m * n - m - n < k
x : { k_1 // k_1 ≡ 0 [MOD m] ∧ k_1 ≡ k [MOD n] } := chineseRemainder cop 0 k
hx : ↑x < m * n
⊢ ∃ m_1 n_1, m_1 * m + n_1 * n = k

Next tactic:
suffices key : x.1 ≤ k by
      obtain ⟨a, ha⟩ := modEq_zero_iff_dvd.mp x.2.1
      obtain ⟨b, hb⟩ := (modEq_iff_dvd' key).mp x.2.2
      exact ⟨a, b, by rw [mul_comm, ← ha, mul_comm, ← hb, Nat.add_sub_of_le key]⟩