l3lab/ntp-mathlib huggingface dataset:
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      

Proof state:
case hu.a
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
⊢ IsUnit (gcd d₁ b)

Next tactic:
apply isUnit_of_dvd_one
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      

Proof state:
case hu.a.h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
⊢ gcd d₁ b ∣ 1

Next tactic:
apply dvd_trans _ hab.dvd
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
⊢ gcd d₁ b ∣ gcd a b

Next tactic:
apply gcd_dvd_gcd hd₁ (dvd_refl b)
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
⊢ d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a

Next tactic:
have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
⊢ d₁ ^ k ∣ a * b

Next tactic:
use d₂ ^ k
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
⊢ a * b = d₁ ^ k * d₂ ^ k

Next tactic:
rw [h, hc]
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
⊢ (d₁ * d₂) ^ k = d₁ ^ k * d₂ ^ k

Next tactic:
exact mul_pow d₁ d₂ k
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ a * b
⊢ d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a

Next tactic:
rw [mul_comm] at h2
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
⊢ d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a

Next tactic:
have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    rw [IsUnit.mul_left_dvd _ _ _ h1]
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
⊢ d₁ ^ k ∣ a

Next tactic:
apply (dvd_gcd_mul_of_dvd_mul h2).trans
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
⊢ gcd (d₁ ^ k) b * a ∣ a

Next tactic:
rw [IsUnit.mul_left_dvd _ _ _ h1]
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    rw [IsUnit.mul_left_dvd _ _ _ h1]
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
h3 : d₁ ^ k ∣ a
⊢ d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a

Next tactic:
have h4 : d₁ ^ k ≠ 0 := by
    intro hdk
    rw [hdk] at h3
    apply absurd (zero_dvd_iff.mp h3) ha
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    rw [IsUnit.mul_left_dvd _ _ _ h1]
  have h4 : d₁ ^ k ≠ 0 := by
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
h3 : d₁ ^ k ∣ a
⊢ d₁ ^ k ≠ 0

Next tactic:
intro hdk
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    rw [IsUnit.mul_left_dvd _ _ _ h1]
  have h4 : d₁ ^ k ≠ 0 := by
    intro hdk
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
h3 : d₁ ^ k ∣ a
hdk : d₁ ^ k = 0
⊢ False

Next tactic:
rw [hdk] at h3
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    rw [IsUnit.mul_left_dvd _ _ _ h1]
  have h4 : d₁ ^ k ≠ 0 := by
    intro hdk
    rw [hdk] at h3
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
h3 : 0 ∣ a
hdk : d₁ ^ k = 0
⊢ False

Next tactic:
apply absurd (zero_dvd_iff.mp h3) ha
theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a := by
  have h1 : IsUnit (gcd (d₁ ^ k) b) := by
    apply isUnit_of_dvd_one
    trans gcd d₁ b ^ k
    · exact gcd_pow_left_dvd_pow_gcd
    · apply IsUnit.dvd
      apply IsUnit.pow
      apply isUnit_of_dvd_one
      apply dvd_trans _ hab.dvd
      apply gcd_dvd_gcd hd₁ (dvd_refl b)
  have h2 : d₁ ^ k ∣ a * b := by
    use d₂ ^ k
    rw [h, hc]
    exact mul_pow d₁ d₂ k
  rw [mul_comm] at h2
  have h3 : d₁ ^ k ∣ a := by
    apply (dvd_gcd_mul_of_dvd_mul h2).trans
    rw [IsUnit.mul_left_dvd _ _ _ h1]
  have h4 : d₁ ^ k ≠ 0 := by
    intro hdk
    rw [hdk] at h3
    apply absurd (zero_dvd_iff.mp h3) ha
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
h3 : d₁ ^ k ∣ a
h4 : d₁ ^ k ≠ 0
⊢ d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a

Next tactic:
exact ⟨h4, h3⟩
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
cases subsingleton_or_nontrivial α
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · 

Proof state:
case inl
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Subsingleton α
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
use 0
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Subsingleton α
⊢ Associated (0 ^ k) a

Next tactic:
rw [Subsingleton.elim a (0 ^ k)]
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  

Proof state:
case inr
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
by_cases ha : a = 0
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · 

Proof state:
case pos
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : a = 0
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
use 0
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : a = 0
⊢ Associated (0 ^ k) a

Next tactic:
rw [ha]
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : a = 0
⊢ Associated (0 ^ k) 0

Next tactic:
obtain rfl | hk := k.eq_zero_or_pos
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · 

Proof state:
case h.inl
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
h✝ : Nontrivial α
ha : a = 0
h : a * b = c ^ 0
⊢ Associated (0 ^ 0) 0

Next tactic:
exfalso
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      

Proof state:
case h.inl.h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
h✝ : Nontrivial α
ha : a = 0
h : a * b = c ^ 0
⊢ False

Next tactic:
revert h
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      

Proof state:
case h.inl.h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
h✝ : Nontrivial α
ha : a = 0
⊢ a * b = c ^ 0 → False

Next tactic:
rw [ha, zero_mul, pow_zero]
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, zero_mul, pow_zero]
      

Proof state:
case h.inl.h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
h✝ : Nontrivial α
ha : a = 0
⊢ 0 = 1 → False

Next tactic:
apply zero_ne_one
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, zero_mul, pow_zero]
      apply zero_ne_one
    · 

Proof state:
case h.inr
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : a = 0
hk : k > 0
⊢ Associated (0 ^ k) 0

Next tactic:
rw [zero_pow hk]
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, zero_mul, pow_zero]
      apply zero_ne_one
    · rw [zero_pow hk]
  

Proof state:
case neg
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
by_cases hb : b = 0
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, zero_mul, pow_zero]
      apply zero_ne_one
    · rw [zero_pow hk]
  by_cases hb : b = 0
  · 

Proof state:
case pos
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : b = 0
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
use 1
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, zero_mul, pow_zero]
      apply zero_ne_one
    · rw [zero_pow hk]
  by_cases hb : b = 0
  · use 1
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : b = 0
⊢ Associated (1 ^ k) a

Next tactic:
rw [one_pow]
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, zero_mul, pow_zero]
      apply zero_ne_one
    · rw [zero_pow hk]
  by_cases hb : b = 0
  · use 1
    rw [one_pow]
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : b = 0
⊢ Associated 1 a

Next tactic:
apply (associated_one_iff_isUnit.mpr hab).symm.trans
theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a := by
  cases subsingleton_or_nontrivial α
  · use 0
    rw [Subsingleton.elim a (0 ^ k)]
  by_cases ha : a = 0
  · use 0
    rw [ha]
    obtain rfl | hk := k.eq_zero_or_pos
    · exfalso
      revert h
      rw [ha, zero_mul, pow_zero]
      apply zero_ne_one
    · rw [zero_pow hk]
  by_cases hb : b = 0
  · use 1
    rw [one_pow]
    apply (associated_one_iff_isUnit.mpr hab).symm.trans
    

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : b = 0
⊢ Associated (gcd a b) a

Next tactic:
rw [hb]