l3lab/ntp-mathlib huggingface dataset:
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
⊢ a = p ^ count p (factors a)

Next tactic:
have ha := ne_zero_of_dvd_ne_zero hph h
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
⊢ a = p ^ count p (factors a)

Next tactic:
apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
⊢ ∀ (p_1 : Associates α), Irreducible p_1 → count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))

Next tactic:
have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      symm
      rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
⊢ ?m.1363735 q ≤ 0

Next tactic:
convert count_le_count_of_le hph hq h
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      

Proof state:
case h.e'_4
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
⊢ 0 = count q (factors (p ^ n))

Next tactic:
symm
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      symm
      

Proof state:
case h.e'_4
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
⊢ count q (factors (p ^ n)) = 0

Next tactic:
rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      symm
      rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0
⊢ ∀ (p_1 : Associates α), Irreducible p_1 → count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))

Next tactic:
intro q hq
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      symm
      rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
  intro q hq
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0
q : Associates α
hq : Irreducible q
⊢ count q (factors a) = count q (factors (p ^ count p (factors a)))

Next tactic:
rw [count_pow hp.ne_zero hq]
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      symm
      rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
  intro q hq
  rw [count_pow hp.ne_zero hq]
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0
q : Associates α
hq : Irreducible q
⊢ count q (factors a) = count p (factors a) * count q (factors p)

Next tactic:
by_cases h : q = p
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      symm
      rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
  intro q hq
  rw [count_pow hp.ne_zero hq]
  by_cases h : q = p
  · 

Proof state:
case pos
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h✝ : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0
q : Associates α
hq : Irreducible q
h : q = p
⊢ count q (factors a) = count p (factors a) * count q (factors p)

Next tactic:
rw [h, count_self hp, mul_one]
/-- The only divisors of prime powers are prime powers. See `eq_pow_find_of_dvd_irreducible_pow`
for an explicit expression as a p-power (without using `count`). -/
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α} (hp : Irreducible p) {n : ℕ}
    (h : a ∣ p ^ n) : a = p ^ p.count a.factors := by
  nontriviality α
  have hph := pow_ne_zero n hp.ne_zero
  have ha := ne_zero_of_dvd_ne_zero hph h
  apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
  have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert count_le_count_of_le hph hq h
      symm
      rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
  intro q hq
  rw [count_pow hp.ne_zero hq]
  by_cases h : q = p
  · rw [h, count_self hp, mul_one]
  · 

Proof state:
case neg
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
p a : Associates α
hp : Irreducible p
n : ℕ
h✝ : a ∣ p ^ n
✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → count q (factors a) = 0
q : Associates α
hq : Irreducible q
h : ¬q = p
⊢ count q (factors a) = count p (factors a) * count q (factors p)

Next tactic:
rw [count_eq_zero_of_ne hq hp h, mul_zero, eq_zero_of_ne q hq h]
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) = count p (factors a)

Next tactic:
apply le_antisymm
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · 

Proof state:
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≤ count p (factors a)

Next tactic:
refine' Nat.find_le ⟨1, _⟩
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    

Proof state:
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ p ^ count p (factors a) = a * 1

Next tactic:
rw [mul_one]
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    rw [mul_one]
    

Proof state:
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ p ^ count p (factors a) = a

Next tactic:
symm
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    rw [mul_one]
    symm
    

Proof state:
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ a = p ^ count p (factors a)

Next tactic:
exact eq_pow_count_factors_of_dvd_pow hp h
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    rw [mul_one]
    symm
    exact eq_pow_count_factors_of_dvd_pow hp h
  · 

Proof state:
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)

Next tactic:
have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    rw [mul_one]
    symm
    exact eq_pow_count_factors_of_dvd_pow hp h
  · have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
    

Proof state:
case a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)

Next tactic:
cases' subsingleton_or_nontrivial α with hα hα
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    rw [mul_one]
    symm
    exact eq_pow_count_factors_of_dvd_pow hp h
  · have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
    cases' subsingleton_or_nontrivial α with hα hα
    · 

Proof state:
case a.inl
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Subsingleton α
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)

Next tactic:
simp [eq_iff_true_of_subsingleton] at hph
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    rw [mul_one]
    symm
    exact eq_pow_count_factors_of_dvd_pow hp h
  · have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
    cases' subsingleton_or_nontrivial α with hα hα
    · simp [eq_iff_true_of_subsingleton] at hph
    

Proof state:
case a.inr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Nontrivial α
⊢ count p (factors a) ≤ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)

Next tactic:
convert count_le_count_of_le hph hp (@Nat.find_spec (fun n => a ∣ p ^ n) _ ⟨n, h⟩)
theorem count_factors_eq_find_of_dvd_pow {a p : Associates α} (hp : Irreducible p)
    [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
    @Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by
  apply le_antisymm
  · refine' Nat.find_le ⟨1, _⟩
    rw [mul_one]
    symm
    exact eq_pow_count_factors_of_dvd_pow hp h
  · have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
    cases' subsingleton_or_nontrivial α with hα hα
    · simp [eq_iff_true_of_subsingleton] at hph
    convert count_le_count_of_le hph hp (@Nat.find_spec (fun n => a ∣ p ^ n) _ ⟨n, h⟩)
    

Proof state:
case h.e'_4
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : Associates α
hp : Irreducible p
inst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) ≠ 0
hα : Nontrivial α
⊢ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n) = count p (factors (p ^ Nat.find (_ : ∃ n, (fun n => a ∣ p ^ n) n)))

Next tactic:
rw [count_pow hp.ne_zero hp, count_self hp, mul_one]
theorem eq_pow_of_mul_eq_pow [Nontrivial α] {a b c : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ} (h : a * b = c ^ k) :
    ∃ d : Associates α, a = d ^ k := by
  

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
⊢ ∃ d, a = d ^ k

Next tactic:
classical
    by_cases hk0 : k = 0
    · use 1
      rw [hk0, pow_zero] at h ⊢
      apply (mul_eq_one_iff.1 h).1
    · refine' is_pow_of_dvd_count ha _
      intro p hp
      apply dvd_count_of_dvd_count_mul hb hp hab
      rw [h]
      apply dvd_count_pow _ hp
      rintro rfl
      rw [zero_pow' _ hk0] at h
      cases mul_eq_zero.mp h <;> contradiction
theorem eq_pow_of_mul_eq_pow [Nontrivial α] {a b c : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ} (h : a * b = c ^ k) :
    ∃ d : Associates α, a = d ^ k := by
  classical
    

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
⊢ ∃ d, a = d ^ k

Next tactic:
by_cases hk0 : k = 0
theorem eq_pow_of_mul_eq_pow [Nontrivial α] {a b c : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ} (h : a * b = c ^ k) :
    ∃ d : Associates α, a = d ^ k := by
  classical
    by_cases hk0 : k = 0
    · 

Proof state:
case pos
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
hk0 : k = 0
⊢ ∃ d, a = d ^ k

Next tactic:
use 1
theorem eq_pow_of_mul_eq_pow [Nontrivial α] {a b c : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ} (h : a * b = c ^ k) :
    ∃ d : Associates α, a = d ^ k := by
  classical
    by_cases hk0 : k = 0
    · use 1
      

Proof state:
case h
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
hk0 : k = 0
⊢ a = 1 ^ k

Next tactic:
rw [hk0, pow_zero] at h ⊢