l3lab/ntp-mathlib huggingface dataset:
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
    ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
  | n, m => fun ha hb ⟨s, hs⟩ =>
    have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
    (hp.2.2 a b this).elim
      (fun ⟨x, hx⟩ =>
        have hn0 : 0 < n :=
          Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
        have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
          ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
                  rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
                    simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
        have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
        finite_mul_aux hp hpx hb
          ⟨s, mul_right_cancel₀ hp.1 (by
                rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
                simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
      fun ⟨x, hx⟩ =>
        have hm0 : 0 < m :=
          Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
        have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
          hb
            (hx.symm ▸
              ⟨y,
                mul_right_cancel₀ hp.1 <| by
                  rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
                    simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
        finite_mul_aux hp ha hpx
        ⟨s, mul_right_cancel₀ hp.1 (by
              

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
hpx : ¬p ^ (m - 1 + 1) ∣ x
⊢ a * x * p = p ^ (n + (m - 1) + 1) * s * p

Next tactic:
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
    ∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
  | n, m => fun ha hb ⟨s, hs⟩ =>
    have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
    (hp.2.2 a b this).elim
      (fun ⟨x, hx⟩ =>
        have hn0 : 0 < n :=
          Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
        have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
          ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
                  rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
                    simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
        have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
        finite_mul_aux hp hpx hb
          ⟨s, mul_right_cancel₀ hp.1 (by
                rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
                simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
      fun ⟨x, hx⟩ =>
        have hm0 : 0 < m :=
          Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
        have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
          hb
            (hx.symm ▸
              ⟨y,
                mul_right_cancel₀ hp.1 <| by
                  rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
                    simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
        finite_mul_aux hp ha hpx
        ⟨s, mul_right_cancel₀ hp.1 (by
              rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
              

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
hpx : ¬p ^ (m - 1 + 1) ∣ x
⊢ a * x * p = p ^ (n + m) * s * p

Next tactic:
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add]
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
  | 0, _ => ⟨0, by 

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p a : α
hp : Prime p
x✝ : Finite p a
⊢ ¬p ^ (0 + 1) ∣ a ^ 0

Next tactic:
simp [mt isUnit_iff_dvd_one.2 hp.2.1]
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
  | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
  | k + 1, ha => by 

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p a : α
hp : Prime p
k : ℕ
ha : Finite p a
⊢ Finite p (a ^ (k + 1))

Next tactic:
rw [_root_.pow_succ]
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
  | 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
  | k + 1, ha => by rw [_root_.pow_succ]; 

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CancelCommMonoidWithZero α
p a : α
hp : Prime p
k : ℕ
ha : Finite p a
⊢ Finite p (a * a ^ k)

Next tactic:
exact finite_mul hp ha (finite_pow hp ha)
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
⊢ multiplicity a a = 1

Next tactic:
rw [← Nat.cast_one]
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
  rw [← Nat.cast_one]
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
⊢ multiplicity a a = ↑1

Next tactic:
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
            ⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
  rw [← Nat.cast_one]
  exact eq_coe_iff.2 ⟨by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
⊢ a ^ 1 ∣ a

Next tactic:
simp
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
  rw [← Nat.cast_one]
  exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
            ⟨b, mul_left_cancel₀ ha0 <| by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : ¬IsUnit a
ha0 : a ≠ 0
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : a = a ^ (1 + 1) * b
⊢ a * 1 = a * (a * b)

Next tactic:
simpa [_root_.pow_succ, mul_assoc] using hb
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
  PartENat.get_eq_iff_eq_coe.2
    (eq_coe_iff.2
      ⟨by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
⊢ a ^ 1 ∣ a

Next tactic:
simp
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
  PartENat.get_eq_iff_eq_coe.2
    (eq_coe_iff.2
      ⟨by simp, fun ⟨b, hb⟩ => by
        

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : a = a ^ (1 + 1) * b
⊢ False

Next tactic:
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
            mul_right_inj' (ne_zero_of_finite ha)] at hb
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
  PartENat.get_eq_iff_eq_coe.2
    (eq_coe_iff.2
      ⟨by simp, fun ⟨b, hb⟩ => by
        rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
            mul_right_inj' (ne_zero_of_finite ha)] at hb;
          

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : 1 = 1 * (a * 1 * b)
⊢ False

Next tactic:
exact
            mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
  PartENat.get_eq_iff_eq_coe.2
    (eq_coe_iff.2
      ⟨by simp, fun ⟨b, hb⟩ => by
        rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
            mul_right_inj' (ne_zero_of_finite ha)] at hb;
          exact
            mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a : α
ha : Finite a a
x✝ : a ^ (1 + 1) ∣ a
b : α
hb : 1 = 1 * (a * 1 * b)
⊢ 1 = a * b

Next tactic:
simp_all
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
⊢ Part.get (multiplicity p (a * b)) h =
    Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)

Next tactic:
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
⊢ Part.get (multiplicity p (a * b)) h =
    Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)

Next tactic:
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
⊢ Part.get (multiplicity p (a * b)) h =
    Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)

Next tactic:
have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by simp [pow_add]
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
⊢ p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) =
    p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)

Next tactic:
simp [pow_add]
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by simp [pow_add]
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
  p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) =
    p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)
⊢ Part.get (multiplicity p (a * b)) h =
    Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)

Next tactic:
have hdiv :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
      a * b :=
    by rw [hpoweq]; apply mul_dvd_mul <;> assumption
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by simp [pow_add]
  have hdiv :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
      a * b :=
    by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
  p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) =
    p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)
⊢ p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) ∣ a * b

Next tactic:
rw [hpoweq]
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by simp [pow_add]
  have hdiv :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
      a * b :=
    by rw [hpoweq]; 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
  p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) =
    p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)
⊢ p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ a * b

Next tactic:
apply mul_dvd_mul
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by simp [pow_add]
  have hdiv :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
      a * b :=
    by rw [hpoweq]; apply mul_dvd_mul <;> 

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
  p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) =
    p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)
⊢ p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a

Next tactic:
assumption
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by simp [pow_add]
  have hdiv :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
      a * b :=
    by rw [hpoweq]; apply mul_dvd_mul <;> 

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
  p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) =
    p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)
⊢ p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b

Next tactic:
assumption
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
    get (multiplicity p (a * b)) h =
      get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
        get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
  have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
  have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
  have hpoweq :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
      p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
        p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
    by simp [pow_add]
  have hdiv :
    p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
          get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
      a * b :=
    by rw [hpoweq]; apply mul_dvd_mul <;> assumption
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
hp : Prime p
h : (multiplicity p (a * b)).Dom
hdiva : p ^ Part.get (multiplicity p a) (_ : Finite p a) ∣ a
hdivb : p ^ Part.get (multiplicity p b) (_ : Finite p b) ∣ b
hpoweq :
  p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) =
    p ^ Part.get (multiplicity p a) (_ : Finite p a) * p ^ Part.get (multiplicity p b) (_ : Finite p b)
hdiv : p ^ (Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)) ∣ a * b
⊢ Part.get (multiplicity p (a * b)) h =
    Part.get (multiplicity p a) (_ : Finite p a) + Part.get (multiplicity p b) (_ : Finite p b)

Next tactic:
have hsucc :
    ¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
              get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
            1) ∣
        a * b :=
    fun h =>
    not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
      (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)