l3lab/ntp-mathlib huggingface dataset:
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
    rw [hb]
    

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 0) a

Next tactic:
exact gcd_zero_right' a
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
    rw [hb]
    exact gcd_zero_right' a
  

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
hb : ¬b = 0
⊢ ∃ d, Associated (d ^ k) a

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
    · 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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · 

Proof state:
case neg.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
hb : ¬b = 0
h : a * b = c ^ 0
⊢ ∃ d, Associated (d ^ 0) 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
    rw [one_pow]
    apply (associated_one_iff_isUnit.mpr hab).symm.trans
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · 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)
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
h : a * b = c ^ 0
⊢ Associated (1 ^ 0) a

Next tactic:
rw [pow_zero] at 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
      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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    

Proof state:
case 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
hb : ¬b = 0
h : a * b = 1
⊢ Associated 1 a

Next tactic:
use Units.mkOfMulEqOne _ _ 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
      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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    

Proof state:
case 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
hb : ¬b = 0
h : a * b = 1
⊢ 1 * ↑(Units.mkOfMulEqOne a b h) = a

Next tactic:
rw [Units.val_mkOfMulEqOne, one_mul]
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  

Proof state:
case neg.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
hb : ¬b = 0
hk : k > 0
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := 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
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
⊢ c ∣ a * b

Next tactic:
rw [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
      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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    

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
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
⊢ c ∣ c ^ k

Next tactic:
exact dvd_pow_self _ hk.ne'
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  

Proof state:
case neg.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
hb : ¬b = 0
hk : k > 0
hc : c ∣ a * b
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  

Proof state:
case neg.inr.intro.intro.intro.intro
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
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₁ * d₂
⊢ ∃ d, Associated (d ^ k) a

Next tactic:
use d₁
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  

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
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₁ * d₂
⊢ Associated (d₁ ^ k) a

Next tactic:
obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  

Proof state:
case h.intro.intro
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
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₁ * d₂
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
⊢ Associated (d₁ ^ k) a

Next tactic:
rw [mul_comm] at h hc
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  

Proof state:
case h.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : b * a = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
⊢ Associated (d₁ ^ k) a

Next tactic:
rw [(gcd_comm' a b).isUnit_iff] at hab
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  rw [(gcd_comm' a b).isUnit_iff] at hab
  

Proof state:
case h.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h : b * a = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
⊢ Associated (d₁ ^ k) a

Next tactic:
obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  rw [(gcd_comm' a b).isUnit_iff] at hab
  obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
  

Proof state:
case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h : b * a = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
hb' : b = d₂ ^ k * b'
⊢ Associated (d₁ ^ k) a

Next tactic:
rw [ha', hb', hc, mul_pow] at 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
      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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  rw [(gcd_comm' a b).isUnit_iff] at hab
  obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
  rw [ha', hb', hc, mul_pow] at h
  

Proof state:
case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
h : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k
hb' : b = d₂ ^ k * b'
⊢ Associated (d₁ ^ k) a

Next tactic:
have h' : a' * b' = 1 := by
    apply (mul_right_inj' h0₁).mp
    rw [mul_one]
    apply (mul_right_inj' h0₂).mp
    rw [← h]
    rw [mul_assoc, mul_comm a', ← mul_assoc _ b', ← mul_assoc b', mul_comm b']
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  rw [(gcd_comm' a b).isUnit_iff] at hab
  obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
  rw [ha', hb', hc, mul_pow] at h
  have h' : a' * b' = 1 := by
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
h : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k
hb' : b = d₂ ^ k * b'
⊢ a' * b' = 1

Next tactic:
apply (mul_right_inj' h0₁).mp
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
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  rw [(gcd_comm' a b).isUnit_iff] at hab
  obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
  rw [ha', hb', hc, mul_pow] at h
  have h' : a' * b' = 1 := by
    apply (mul_right_inj' h0₁).mp
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
h : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k
hb' : b = d₂ ^ k * b'
⊢ d₁ ^ k * (a' * b') = d₁ ^ k * 1

Next tactic:
rw [mul_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
    · rw [zero_pow hk]
  by_cases hb : b = 0
  · use 1
    rw [one_pow]
    apply (associated_one_iff_isUnit.mpr hab).symm.trans
    rw [hb]
    exact gcd_zero_right' a
  obtain rfl | hk := k.eq_zero_or_pos
  · use 1
    rw [pow_zero] at h ⊢
    use Units.mkOfMulEqOne _ _ h
    rw [Units.val_mkOfMulEqOne, one_mul]
  have hc : c ∣ a * b := by
    rw [h]
    exact dvd_pow_self _ hk.ne'
  obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc
  use d₁
  obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁
  rw [mul_comm] at h hc
  rw [(gcd_comm' a b).isUnit_iff] at hab
  obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂
  rw [ha', hb', hc, mul_pow] at h
  have h' : a' * b' = 1 := by
    apply (mul_right_inj' h0₁).mp
    rw [mul_one]
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
h : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k
hb' : b = d₂ ^ k * b'
⊢ d₁ ^ k * (a' * b') = d₁ ^ k

Next tactic:
apply (mul_right_inj' h0₂).mp