l3lab/ntp-mathlib huggingface dataset:
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    

Proof state:
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ ¬p ^ (k + 1) ∣ b
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ p ^ (k + 1) ∣ a

Next tactic:
intro h_dvd
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · 

Proof state:
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ False

Next tactic:
apply multiplicity.is_greatest _ h_dvd
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · apply multiplicity.is_greatest _ h_dvd
      

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ multiplicity p b < ↑(k + 1)

Next tactic:
rw [hk, ← Nat.succ_eq_add_one]
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · apply multiplicity.is_greatest _ h_dvd
      rw [hk, ← Nat.succ_eq_add_one]
      

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ ↑k < ↑(succ k)

Next tactic:
norm_cast
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · apply multiplicity.is_greatest _ h_dvd
      rw [hk, ← Nat.succ_eq_add_one]
      norm_cast
      

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
h_dvd : p ^ (k + 1) ∣ b
⊢ k < succ k

Next tactic:
apply Nat.lt_succ_self k
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · apply multiplicity.is_greatest _ h_dvd
      rw [hk, ← Nat.succ_eq_add_one]
      norm_cast
      apply Nat.lt_succ_self k
    · 

Proof state:
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ p ^ (k + 1) ∣ a

Next tactic:
rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · apply multiplicity.is_greatest _ h_dvd
      rw [hk, ← Nat.succ_eq_add_one]
      norm_cast
      apply Nat.lt_succ_self k
    · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
      

Proof state:
case a.h.intro
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
k : ℕ
hk : multiplicity p b = ↑k
⊢ multiplicity p b + 1 ≤ multiplicity p a

Next tactic:
exact PartENat.add_one_le_of_lt h
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · apply multiplicity.is_greatest _ h_dvd
      rw [hk, ← Nat.succ_eq_add_one]
      norm_cast
      apply Nat.lt_succ_self k
    · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
      exact PartENat.add_one_le_of_lt h
  · 

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p b ≤ multiplicity p (a + b)

Next tactic:
have := @min_le_multiplicity_add α _ _ p a b
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a + b) = multiplicity p b := by
  apply le_antisymm
  · apply PartENat.le_of_lt_add_one
    cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
    rw [hk]
    rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
    intro h_dvd
    · apply multiplicity.is_greatest _ h_dvd
      rw [hk, ← Nat.succ_eq_add_one]
      norm_cast
      apply Nat.lt_succ_self k
    · rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
      exact PartENat.add_one_le_of_lt h
  · have := @min_le_multiplicity_add α _ _ p a b
    

Proof state:
case a
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
this : min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)
⊢ multiplicity p b ≤ multiplicity p (a + b)

Next tactic:
rwa [← min_eq_right (le_of_lt h)]
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a - b) = multiplicity p b := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (a - b) = multiplicity p b

Next tactic:
rw [sub_eq_add_neg, multiplicity_add_of_gt]
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a - b) = multiplicity p b := by
  rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (-b) = multiplicity p b

Next tactic:
rw [multiplicity.neg]
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a - b) = multiplicity p b := by
  rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p (-b) < multiplicity p a

Next tactic:
rw [multiplicity.neg]
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
    multiplicity p (a - b) = multiplicity p b := by
  rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p b < multiplicity p a
⊢ multiplicity p b < multiplicity p a

Next tactic:
assumption
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
    multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)

Next tactic:
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
    multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
  rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
  · 

Proof state:
case inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p a < multiplicity p b
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)

Next tactic:
rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
    multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
  rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
  · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
    

Proof state:
case inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p a < multiplicity p b
⊢ multiplicity p a ≤ multiplicity p b

Next tactic:
exact le_of_lt hab
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
    multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
  rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
  · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
    exact le_of_lt hab
  · 

Proof state:
case inr.inl
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p a = multiplicity p b
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)

Next tactic:
contradiction
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
    multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
  rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
  · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
    exact le_of_lt hab
  · contradiction
  · 

Proof state:
case inr.inr
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p b < multiplicity p a
⊢ multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b)

Next tactic:
rw [multiplicity_add_of_gt hab, min_eq_right]
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
    multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
  rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
  · rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
    exact le_of_lt hab
  · contradiction
  · rw [multiplicity_add_of_gt hab, min_eq_right]
    

Proof state:
case inr.inr
α : Type u_1
β : Type u_2
inst✝¹ : Ring α
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
p a b : α
h : multiplicity p a ≠ multiplicity p b
hab : multiplicity p b < multiplicity p a
⊢ multiplicity p b ≤ multiplicity p a

Next tactic:
exact le_of_lt hab
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 

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✝ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
⊢ a * b = p * (p ^ (n + m) * s)

Next tactic:
simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]
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 

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 ∣ a
x : α
hx : a = p * x
hn0 : n = 0
⊢ False

Next tactic:
simp [hx, hn0] at ha
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
                  

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✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
x✝ : p ^ (n - 1 + 1) ∣ x
y : α
hy : x = p ^ (n - 1 + 1) * y
⊢ p * x * p = p ^ (n + 1) * y * p

Next tactic:
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy
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;
                    

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✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
x✝ : p ^ (n - 1 + 1) ∣ x
y : α
hy : x = p ^ n * y
⊢ p * x * p = p ^ (n + 1) * y * p

Next tactic:
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]
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
                

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 ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
hpx : ¬p ^ (n - 1 + 1) ∣ x
this : 1 ≤ n + m
⊢ x * b * p = p ^ (n - 1 + m + 1) * s * p

Next tactic:
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
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]
                

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 ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
hpx : ¬p ^ (n - 1 + 1) ∣ x
this : 1 ≤ n + m
⊢ x * b * p = p ^ (n + m) * s * p

Next tactic:
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add]
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 

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 : m = 0
⊢ False

Next tactic:
simp [hx, hm0] at hb
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
                  

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✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
x✝ : p ^ (m - 1 + 1) ∣ x
y : α
hy : x = p ^ (m - 1 + 1) * y
⊢ p * x * p = p ^ (m + 1) * y * p

Next tactic:
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy
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;
                    

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✝² : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this : p ∣ a * b
x✝¹ : p ∣ b
x : α
hx : b = p * x
hm0 : 0 < m
x✝ : p ^ (m - 1 + 1) ∣ x
y : α
hy : x = p ^ m * y
⊢ p * x * p = p ^ (m + 1) * y * p

Next tactic:
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]