l3lab/ntp-mathlib huggingface dataset:
theorem max_power_factor {a₀ : R} {x : R} (h : a₀ ≠ 0) (hx : Irreducible x) :
    ∃ n : ℕ, ∃ a : R, ¬x ∣ a ∧ a₀ = x ^ n * a := by
  classical
    let n := (normalizedFactors a₀).count (normalize x)
    

Proof state:
α : Type u_1
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : Nontrivial R
inst✝ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
a₀ x : R
h : a₀ ≠ 0
hx : Irreducible x
n : ℕ := count (normalize x) (normalizedFactors a₀)
⊢ ∃ n a, ¬x ∣ a ∧ a₀ = x ^ n * a

Next tactic:
obtain ⟨a, ha1, ha2⟩ := @exists_eq_pow_mul_and_not_dvd R _ _ x a₀
      (ne_top_iff_finite.mp (PartENat.ne_top_iff.mpr
        -- Porting note: this was a hole that was filled at the end of the proof with `use`:
        ⟨n, multiplicity_eq_count_normalizedFactors hx h⟩))
theorem max_power_factor {a₀ : R} {x : R} (h : a₀ ≠ 0) (hx : Irreducible x) :
    ∃ n : ℕ, ∃ a : R, ¬x ∣ a ∧ a₀ = x ^ n * a := by
  classical
    let n := (normalizedFactors a₀).count (normalize x)
    obtain ⟨a, ha1, ha2⟩ := @exists_eq_pow_mul_and_not_dvd R _ _ x a₀
      (ne_top_iff_finite.mp (PartENat.ne_top_iff.mpr
        -- Porting note: this was a hole that was filled at the end of the proof with `use`:
        ⟨n, multiplicity_eq_count_normalizedFactors hx h⟩))
    

Proof state:
case intro.intro
α : Type u_1
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : Nontrivial R
inst✝ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
a₀ x : R
h : a₀ ≠ 0
hx : Irreducible x
n : ℕ := count (normalize x) (normalizedFactors a₀)
a : R
ha1 : a₀ = x ^ Part.get (multiplicity x a₀) (_ : multiplicity.Finite x a₀) * a
ha2 : ¬x ∣ a
⊢ ∃ n a, ¬x ∣ a ∧ a₀ = x ^ n * a

Next tactic:
simp_rw [← (multiplicity_eq_count_normalizedFactors hx h).symm] at ha1
theorem max_power_factor {a₀ : R} {x : R} (h : a₀ ≠ 0) (hx : Irreducible x) :
    ∃ n : ℕ, ∃ a : R, ¬x ∣ a ∧ a₀ = x ^ n * a := by
  classical
    let n := (normalizedFactors a₀).count (normalize x)
    obtain ⟨a, ha1, ha2⟩ := @exists_eq_pow_mul_and_not_dvd R _ _ x a₀
      (ne_top_iff_finite.mp (PartENat.ne_top_iff.mpr
        -- Porting note: this was a hole that was filled at the end of the proof with `use`:
        ⟨n, multiplicity_eq_count_normalizedFactors hx h⟩))
    simp_rw [← (multiplicity_eq_count_normalizedFactors hx h).symm] at ha1
    

Proof state:
case intro.intro
α : Type u_1
R : Type u_2
inst✝³ : CancelCommMonoidWithZero R
inst✝² : UniqueFactorizationMonoid R
inst✝¹ : Nontrivial R
inst✝ : NormalizationMonoid R
dec_dvd : DecidableRel Dvd.dvd
a₀ x : R
h : a₀ ≠ 0
hx : Irreducible x
n : ℕ := count (normalize x) (normalizedFactors a₀)
a : R
ha2 : ¬x ∣ a
ha1 :
  a₀ =
    x ^
        Part.get ↑(count (normalize x) (normalizedFactors a₀))
          (_ : (↑(count (normalize x) (normalizedFactors a₀))).Dom) *
      a
⊢ ∃ n a, ¬x ∣ a ∧ a₀ = x ^ n * a

Next tactic:
use n, a, ha2, ha1
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
⊢ ∀ (q : α), q ∣ p ^ i p → q ∣ ∏ p' in s, p' ^ i p' → IsUnit q

Next tactic:
have hp := is_prime _ (Finset.mem_insert_self _ _)
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
⊢ ∀ (q : α), q ∣ p ^ i p → q ∣ ∏ p' in s, p' ^ i p' → IsUnit q

Next tactic:
refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ : α
⊢ ∀ {d : α}, d ∣ p ^ i p → d ∣ ∏ p' in s, p' ^ i p' → ¬Prime d

Next tactic:
intro d hdp hdprod hd
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdp : d ∣ p ^ i p
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
⊢ False

Next tactic:
apply hps
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  apply hps
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdp : d ∣ p ^ i p
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
⊢ p ∈ s

Next tactic:
replace hdp := hd.dvd_of_dvd_pow hdp
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  apply hps
  replace hdp := hd.dvd_of_dvd_pow hdp
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
⊢ p ∈ s

Next tactic:
obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  apply hps
  replace hdp := hd.dvd_of_dvd_pow hdp
  obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod
  

Proof state:
case intro.intro
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem' : q ∈ Multiset.map (fun p' => p' ^ i p') s.val
hdq : d ∣ q
⊢ p ∈ s

Next tactic:
obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem'
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  apply hps
  replace hdp := hd.dvd_of_dvd_pow hdp
  obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod
  obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem'
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : q ^ i q ∈ Multiset.map (fun p' => p' ^ i p') s.val
hdq : d ∣ q ^ i q
⊢ p ∈ s

Next tactic:
replace hdq := hd.dvd_of_dvd_pow hdq
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  apply hps
  replace hdp := hd.dvd_of_dvd_pow hdp
  obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod
  obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem'
  replace hdq := hd.dvd_of_dvd_pow hdq
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : q ^ i q ∈ Multiset.map (fun p' => p' ^ i p') s.val
hdq : d ∣ q
⊢ p ∈ s

Next tactic:
have : p ∣ q := dvd_trans (hd.irreducible.dvd_symm hp.irreducible hdp) hdq
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  apply hps
  replace hdp := hd.dvd_of_dvd_pow hdp
  obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod
  obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem'
  replace hdq := hd.dvd_of_dvd_pow hdq
  have : p ∣ q := dvd_trans (hd.irreducible.dvd_symm hp.irreducible hdp) hdq
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : q ^ i q ∈ Multiset.map (fun p' => p' ^ i p') s.val
hdq : d ∣ q
this : p ∣ q
⊢ p ∈ s

Next tactic:
convert q_mem
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    ∀ q : α, q ∣ p ^ i p → (q ∣ ∏ p' in s, p' ^ i p') → IsUnit q := by
  have hp := is_prime _ (Finset.mem_insert_self _ _)
  refine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _
  intro d hdp hdprod hd
  apply hps
  replace hdp := hd.dvd_of_dvd_pow hdp
  obtain ⟨q, q_mem', hdq⟩ := hd.exists_mem_multiset_dvd hdprod
  obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem'
  replace hdq := hd.dvd_of_dvd_pow hdq
  have : p ∣ q := dvd_trans (hd.irreducible.dvd_symm hp.irreducible hdp) hdq
  convert q_mem
  

Proof state:
case a
α : Type u_1
R : Type u_2
inst✝⁵ : CancelCommMonoidWithZero R
inst✝⁴ : UniqueFactorizationMonoid R
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero β
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
x✝ d : α
hdprod : d ∣ ∏ p' in s, p' ^ i p'
hd : Prime d
hdp : d ∣ p
q : α
q_mem : q ∈ s.val
q_mem' : q ^ i q ∈ Multiset.map (fun p' => p' ^ i p') s.val
hdq : d ∣ q
this : p ∣ q
⊢ p ∈ s ↔ q ∈ s.val

Next tactic:
rw [Finset.mem_val,
    is_coprime _ (Finset.mem_insert_self p s) _ (Finset.mem_insert_of_mem q_mem) this]
/-- If `P` holds for units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on a product of powers of distinct primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem induction_on_prime_power {P : α → Prop} (s : Finset α) (i : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) :
    P (∏ p in s, p ^ i p) := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
s : Finset α
i : α → ℕ
is_prime : ∀ p ∈ s, Prime p
is_coprime : ∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
⊢ P (∏ p in s, p ^ i p)

Next tactic:
letI := Classical.decEq α
/-- If `P` holds for units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on a product of powers of distinct primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem induction_on_prime_power {P : α → Prop} (s : Finset α) (i : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) :
    P (∏ p in s, p ^ i p) := by
  letI := Classical.decEq α
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
s : Finset α
i : α → ℕ
is_prime : ∀ p ∈ s, Prime p
is_coprime : ∀ p ∈ s, ∀ q ∈ s, p ∣ q → p = q
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
⊢ P (∏ p in s, p ^ i p)

Next tactic:
induction' s using Finset.induction_on with p f' hpf' ih
/-- If `P` holds for units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on a product of powers of distinct primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem induction_on_prime_power {P : α → Prop} (s : Finset α) (i : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) :
    P (∏ p in s, p ^ i p) := by
  letI := Classical.decEq α
  induction' s using Finset.induction_on with p f' hpf' ih
  · 

Proof state:
case empty
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
is_prime : ∀ p ∈ ∅, Prime p
is_coprime : ∀ p ∈ ∅, ∀ q ∈ ∅, p ∣ q → p = q
⊢ P (∏ p in ∅, p ^ i p)

Next tactic:
simpa using h1 isUnit_one
/-- If `P` holds for units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on a product of powers of distinct primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem induction_on_prime_power {P : α → Prop} (s : Finset α) (i : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) :
    P (∏ p in s, p ^ i p) := by
  letI := Classical.decEq α
  induction' s using Finset.induction_on with p f' hpf' ih
  · simpa using h1 isUnit_one
  

Proof state:
case insert
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
p : α
f' : Finset α
hpf' : p ∉ f'
ih : (∀ p ∈ f', Prime p) → (∀ p ∈ f', ∀ q ∈ f', p ∣ q → p = q) → P (∏ p in f', p ^ i p)
is_prime : ∀ p_1 ∈ insert p f', Prime p_1
is_coprime : ∀ p_1 ∈ insert p f', ∀ q ∈ insert p f', p_1 ∣ q → p_1 = q
⊢ P (∏ p in insert p f', p ^ i p)

Next tactic:
rw [Finset.prod_insert hpf']
/-- If `P` holds for units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on a product of powers of distinct primes. -/
-- @[elab_as_elim] Porting note: commented out
theorem induction_on_prime_power {P : α → Prop} (s : Finset α) (i : α → ℕ)
    (is_prime : ∀ p ∈ s, Prime p) (is_coprime : ∀ᵉ (p ∈ s) (q ∈ s), p ∣ q → p = q)
    (h1 : ∀ {x}, IsUnit x → P x) (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) :
    P (∏ p in s, p ^ i p) := by
  letI := Classical.decEq α
  induction' s using Finset.induction_on with p f' hpf' ih
  · simpa using h1 isUnit_one
  rw [Finset.prod_insert hpf']
  

Proof state:
case insert
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
i : α → ℕ
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
p : α
f' : Finset α
hpf' : p ∉ f'
ih : (∀ p ∈ f', Prime p) → (∀ p ∈ f', ∀ q ∈ f', p ∣ q → p = q) → P (∏ p in f', p ^ i p)
is_prime : ∀ p_1 ∈ insert p f', Prime p_1
is_coprime : ∀ p_1 ∈ insert p f', ∀ q ∈ insert p f', p_1 ∣ q → p_1 = q
⊢ P (p ^ i p * ∏ x in f', x ^ i x)

Next tactic:
exact
    hcp (prime_pow_coprime_prod_of_coprime_insert i p hpf' is_prime is_coprime)
      (hpr (i p) (is_prime _ (Finset.mem_insert_self _ _)))
      (ih (fun q hq => is_prime _ (Finset.mem_insert_of_mem hq)) fun q hq q' hq' =>
        is_coprime _ (Finset.mem_insert_of_mem hq) _ (Finset.mem_insert_of_mem hq'))
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
⊢ P a

Next tactic:
letI := Classical.decEq α
/-- If `P` holds for `0`, units and powers of primes,
and `P x ∧ P y` for coprime `x, y` implies `P (x * y)`,
then `P` holds on all `a : α`. -/
@[elab_as_elim]
theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
    (hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
    (hcp : ∀ {x y}, (∀ p, p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)) : P a := by
  letI := Classical.decEq α
  

Proof state:
α : Type u_1
R : Type u_2
inst✝⁴ : CancelCommMonoidWithZero R
inst✝³ : UniqueFactorizationMonoid R
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CancelCommMonoidWithZero β
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, (∀ (p : α), p ∣ x → p ∣ y → IsUnit p) → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
⊢ P a

Next tactic:
have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
    rintro x y ⟨u, rfl⟩ hx
    exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)