l3lab/ntp-mathlib huggingface dataset:
protected theorem Prime.irreducible (hp : Prime p) : Irreducible p :=
  ⟨hp.not_unit, fun a b hab =>
    (show a * b ∣ a ∨ a * b ∣ b from hab ▸ hp.dvd_or_dvd (hab ▸ dvd_rfl)).elim
      (fun ⟨x, hx⟩ =>
        Or.inr
          (isUnit_iff_dvd_one.2
            ⟨x,
              mul_right_cancel₀ (show a ≠ 0 from fun h => by
                simp only [Prime, ne_eq, IsUnit.mul_iff] at *
                rw [h, zero_mul] at hab
                have := hp.left
                contradiction
                ) <| by
                conv =>
                    lhs
                    rw [hx]
                · simp [mul_comm, mul_assoc, mul_left_comm]
                ⟩))
      fun ⟨x, hx⟩ =>
      Or.inl
        (isUnit_iff_dvd_one.2
          ⟨x,
            mul_right_cancel₀ (show b ≠ 0 from fun h => by
            simp only [Prime, ne_eq, IsUnit.mul_iff] at *
            rw [h, mul_zero] at hab
            have := hp.left
            contradiction
            ) <| by
              

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
hab : p = a * b
x✝ : a * b ∣ b
x : α
hx : b = a * b * x
⊢ 1 * b = a * x * b

Next tactic:
conv =>
                  lhs
                  rw [hx]
protected theorem Prime.irreducible (hp : Prime p) : Irreducible p :=
  ⟨hp.not_unit, fun a b hab =>
    (show a * b ∣ a ∨ a * b ∣ b from hab ▸ hp.dvd_or_dvd (hab ▸ dvd_rfl)).elim
      (fun ⟨x, hx⟩ =>
        Or.inr
          (isUnit_iff_dvd_one.2
            ⟨x,
              mul_right_cancel₀ (show a ≠ 0 from fun h => by
                simp only [Prime, ne_eq, IsUnit.mul_iff] at *
                rw [h, zero_mul] at hab
                have := hp.left
                contradiction
                ) <| by
                conv =>
                    lhs
                    rw [hx]
                · simp [mul_comm, mul_assoc, mul_left_comm]
                ⟩))
      fun ⟨x, hx⟩ =>
      Or.inl
        (isUnit_iff_dvd_one.2
          ⟨x,
            mul_right_cancel₀ (show b ≠ 0 from fun h => by
            simp only [Prime, ne_eq, IsUnit.mul_iff] at *
            rw [h, mul_zero] at hab
            have := hp.left
            contradiction
            ) <| by
              conv =>
                  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
hab : p = a * b
x✝ : a * b ∣ b
x : α
hx : b = a * b * x
| 1 * b = a * x * b

Next tactic:
lhs
                  rw [hx]
protected theorem Prime.irreducible (hp : Prime p) : Irreducible p :=
  ⟨hp.not_unit, fun a b hab =>
    (show a * b ∣ a ∨ a * b ∣ b from hab ▸ hp.dvd_or_dvd (hab ▸ dvd_rfl)).elim
      (fun ⟨x, hx⟩ =>
        Or.inr
          (isUnit_iff_dvd_one.2
            ⟨x,
              mul_right_cancel₀ (show a ≠ 0 from fun h => by
                simp only [Prime, ne_eq, IsUnit.mul_iff] at *
                rw [h, zero_mul] at hab
                have := hp.left
                contradiction
                ) <| by
                conv =>
                    lhs
                    rw [hx]
                · simp [mul_comm, mul_assoc, mul_left_comm]
                ⟩))
      fun ⟨x, hx⟩ =>
      Or.inl
        (isUnit_iff_dvd_one.2
          ⟨x,
            mul_right_cancel₀ (show b ≠ 0 from fun h => by
            simp only [Prime, ne_eq, IsUnit.mul_iff] at *
            rw [h, mul_zero] at hab
            have := hp.left
            contradiction
            ) <| by
              conv =>
                  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
hab : p = a * b
x✝ : a * b ∣ b
x : α
hx : b = a * b * x
| 1 * b = a * x * b

Next tactic:
lhs
protected theorem Prime.irreducible (hp : Prime p) : Irreducible p :=
  ⟨hp.not_unit, fun a b hab =>
    (show a * b ∣ a ∨ a * b ∣ b from hab ▸ hp.dvd_or_dvd (hab ▸ dvd_rfl)).elim
      (fun ⟨x, hx⟩ =>
        Or.inr
          (isUnit_iff_dvd_one.2
            ⟨x,
              mul_right_cancel₀ (show a ≠ 0 from fun h => by
                simp only [Prime, ne_eq, IsUnit.mul_iff] at *
                rw [h, zero_mul] at hab
                have := hp.left
                contradiction
                ) <| by
                conv =>
                    lhs
                    rw [hx]
                · simp [mul_comm, mul_assoc, mul_left_comm]
                ⟩))
      fun ⟨x, hx⟩ =>
      Or.inl
        (isUnit_iff_dvd_one.2
          ⟨x,
            mul_right_cancel₀ (show b ≠ 0 from fun h => by
            simp only [Prime, ne_eq, IsUnit.mul_iff] at *
            rw [h, mul_zero] at hab
            have := hp.left
            contradiction
            ) <| by
              conv =>
                  lhs
                  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
hab : p = a * b
x✝ : a * b ∣ b
x : α
hx : b = a * b * x
| 1 * b

Next tactic:
rw [hx]
protected theorem Prime.irreducible (hp : Prime p) : Irreducible p :=
  ⟨hp.not_unit, fun a b hab =>
    (show a * b ∣ a ∨ a * b ∣ b from hab ▸ hp.dvd_or_dvd (hab ▸ dvd_rfl)).elim
      (fun ⟨x, hx⟩ =>
        Or.inr
          (isUnit_iff_dvd_one.2
            ⟨x,
              mul_right_cancel₀ (show a ≠ 0 from fun h => by
                simp only [Prime, ne_eq, IsUnit.mul_iff] at *
                rw [h, zero_mul] at hab
                have := hp.left
                contradiction
                ) <| by
                conv =>
                    lhs
                    rw [hx]
                · simp [mul_comm, mul_assoc, mul_left_comm]
                ⟩))
      fun ⟨x, hx⟩ =>
      Or.inl
        (isUnit_iff_dvd_one.2
          ⟨x,
            mul_right_cancel₀ (show b ≠ 0 from fun h => by
            simp only [Prime, ne_eq, IsUnit.mul_iff] at *
            rw [h, mul_zero] at hab
            have := hp.left
            contradiction
            ) <| by
              conv =>
                  lhs
                  rw [hx]
              · 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
hab : p = a * b
x✝ : a * b ∣ b
x : α
hx : b = a * b * x
⊢ 1 * (a * b * x) = a * x * b

Next tactic:
simp [mul_comm, mul_assoc, mul_left_comm]
theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : Prime p) {a b : α} {k l : ℕ} :
    p ^ k ∣ a → p ^ l ∣ b → p ^ (k + l + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b :=
  fun ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩ =>
  have h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z) := by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
k l : ℕ
x✝² : p ^ k ∣ a
x✝¹ : p ^ l ∣ b
x✝ : p ^ (k + l + 1) ∣ a * b
x : α
hx : a = p ^ k * x
y : α
hy : b = p ^ l * y
z : α
hz : a * b = p ^ (k + l + 1) * z
⊢ p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z)

Next tactic:
simpa [mul_comm, pow_add, hx, hy, mul_assoc, mul_left_comm] using hz
theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : Prime p) {a b : α} {k l : ℕ} :
    p ^ k ∣ a → p ^ l ∣ b → p ^ (k + l + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b :=
  fun ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩ =>
  have h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z) := by
    simpa [mul_comm, pow_add, hx, hy, mul_assoc, mul_left_comm] using hz
  have hp0 : p ^ (k + l) ≠ 0 := pow_ne_zero _ hp.ne_zero
  have hpd : p ∣ x * y := ⟨z, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
k l : ℕ
x✝² : p ^ k ∣ a
x✝¹ : p ^ l ∣ b
x✝ : p ^ (k + l + 1) ∣ a * b
x : α
hx : a = p ^ k * x
y : α
hy : b = p ^ l * y
z : α
hz : a * b = p ^ (k + l + 1) * z
h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z)
hp0 : p ^ (k + l) ≠ 0
⊢ x * y = p * z

Next tactic:
rwa [mul_right_inj' hp0] at h
theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : Prime p) {a b : α} {k l : ℕ} :
    p ^ k ∣ a → p ^ l ∣ b → p ^ (k + l + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b :=
  fun ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩ =>
  have h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z) := by
    simpa [mul_comm, pow_add, hx, hy, mul_assoc, mul_left_comm] using hz
  have hp0 : p ^ (k + l) ≠ 0 := pow_ne_zero _ hp.ne_zero
  have hpd : p ∣ x * y := ⟨z, by rwa [mul_right_inj' hp0] at h⟩
  (hp.dvd_or_dvd hpd).elim
    (fun ⟨d, hd⟩ => Or.inl ⟨d, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
k l : ℕ
x✝³ : p ^ k ∣ a
x✝² : p ^ l ∣ b
x✝¹ : p ^ (k + l + 1) ∣ a * b
x : α
hx : a = p ^ k * x
y : α
hy : b = p ^ l * y
z : α
hz : a * b = p ^ (k + l + 1) * z
h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z)
hp0 : p ^ (k + l) ≠ 0
hpd : p ∣ x * y
x✝ : p ∣ x
d : α
hd : x = p * d
⊢ a = p ^ (k + 1) * d

Next tactic:
simp [*, pow_succ, mul_comm, mul_left_comm, mul_assoc]
theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : Prime p) {a b : α} {k l : ℕ} :
    p ^ k ∣ a → p ^ l ∣ b → p ^ (k + l + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b :=
  fun ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩ =>
  have h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z) := by
    simpa [mul_comm, pow_add, hx, hy, mul_assoc, mul_left_comm] using hz
  have hp0 : p ^ (k + l) ≠ 0 := pow_ne_zero _ hp.ne_zero
  have hpd : p ∣ x * y := ⟨z, by rwa [mul_right_inj' hp0] at h⟩
  (hp.dvd_or_dvd hpd).elim
    (fun ⟨d, hd⟩ => Or.inl ⟨d, by simp [*, pow_succ, mul_comm, mul_left_comm, mul_assoc]⟩)
    fun ⟨d, hd⟩ => Or.inr ⟨d, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a✝ p : α
hp : Prime p
a b : α
k l : ℕ
x✝³ : p ^ k ∣ a
x✝² : p ^ l ∣ b
x✝¹ : p ^ (k + l + 1) ∣ a * b
x : α
hx : a = p ^ k * x
y : α
hy : b = p ^ l * y
z : α
hz : a * b = p ^ (k + l + 1) * z
h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z)
hp0 : p ^ (k + l) ≠ 0
hpd : p ∣ x * y
x✝ : p ∣ y
d : α
hd : y = p * d
⊢ b = p ^ (l + 1) * d

Next tactic:
simp [*, pow_succ, mul_comm, mul_left_comm, mul_assoc]
@[refl]
protected theorem refl [Monoid α] (x : α) : x ~ᵤ x :=
  ⟨1, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
x : α
⊢ x * ↑1 = x

Next tactic:
simp
@[symm]
protected theorem symm [Monoid α] : ∀ {x y : α}, x ~ᵤ y → y ~ᵤ x
  | x, _, ⟨u, rfl⟩ => ⟨u⁻¹, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
x : α
u : αˣ
⊢ x * ↑u * ↑u⁻¹ = x

Next tactic:
rw [mul_assoc, Units.mul_inv, mul_one]
@[trans]
protected theorem trans [Monoid α] : ∀ {x y z : α}, x ~ᵤ y → y ~ᵤ z → x ~ᵤ z
  | x, _, _, ⟨u, rfl⟩, ⟨v, rfl⟩ => ⟨u * v, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
x : α
u v : αˣ
⊢ x * ↑(u * v) = x * ↑u * ↑v

Next tactic:
rw [Units.val_mul, mul_assoc]
theorem associated_one_iff_isUnit [Monoid α] {a : α} : (a : α) ~ᵤ 1 ↔ IsUnit a :=
  Iff.intro
    (fun h =>
      let ⟨c, h⟩ := h.symm
      h ▸ ⟨c, (one_mul _).symm⟩)
    fun ⟨c, h⟩ => Associated.symm ⟨c, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
a : α
x✝ : IsUnit a
c : αˣ
h : ↑c = a
⊢ 1 * ↑c = a

Next tactic:
simp [h]
theorem associated_zero_iff_eq_zero [MonoidWithZero α] (a : α) : a ~ᵤ 0 ↔ a = 0 :=
  Iff.intro
    (fun h => by
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MonoidWithZero α
a : α
h : a ~ᵤ 0
⊢ a = 0

Next tactic:
let ⟨u, h⟩ := h.symm
theorem associated_zero_iff_eq_zero [MonoidWithZero α] (a : α) : a ~ᵤ 0 ↔ a = 0 :=
  Iff.intro
    (fun h => by
      let ⟨u, h⟩ := h.symm
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MonoidWithZero α
a : α
h✝ : a ~ᵤ 0
u : αˣ
h : 0 * ↑u = a
⊢ a = 0

Next tactic:
simpa using h.symm
theorem associated_one_of_associated_mul_one [CommMonoid α] {a b : α} : a * b ~ᵤ 1 → a ~ᵤ 1
  | ⟨u, h⟩ => associated_one_of_mul_eq_one (b * u) <| by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a b : α
u : αˣ
h : a * b * ↑u = 1
⊢ a * (b * ↑u) = 1

Next tactic:
simpa [mul_assoc] using h
theorem associated_unit_mul_left {β : Type*} [CommMonoid β] (a u : β) (hu : IsUnit u) :
    Associated (u * a) a := by
  

Proof state:
α : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
β : Type u_5
inst✝ : CommMonoid β
a u : β
hu : IsUnit u
⊢ u * a ~ᵤ a

Next tactic:
rw [mul_comm]
theorem associated_unit_mul_left {β : Type*} [CommMonoid β] (a u : β) (hu : IsUnit u) :
    Associated (u * a) a := by
  rw [mul_comm]
  

Proof state:
α : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
β : Type u_5
inst✝ : CommMonoid β
a u : β
hu : IsUnit u
⊢ a * u ~ᵤ a

Next tactic:
exact associated_mul_unit_left _ _ hu
theorem associated_isUnit_mul_left_iff {β : Type*} [CommMonoid β] {u a b : β} (hu : IsUnit u) :
    Associated (u * a) b ↔ Associated a b := by
  

Proof state:
α : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
β : Type u_5
inst✝ : CommMonoid β
u a b : β
hu : IsUnit u
⊢ u * a ~ᵤ b ↔ a ~ᵤ b

Next tactic:
rw [mul_comm]
theorem associated_isUnit_mul_left_iff {β : Type*} [CommMonoid β] {u a b : β} (hu : IsUnit u) :
    Associated (u * a) b ↔ Associated a b := by
  rw [mul_comm]
  

Proof state:
α : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
β : Type u_5
inst✝ : CommMonoid β
u a b : β
hu : IsUnit u
⊢ a * u ~ᵤ b ↔ a ~ᵤ b

Next tactic:
exact associated_mul_isUnit_left_iff hu
theorem Associated.mul_mul [CommMonoid α] {a₁ a₂ b₁ b₂ : α} :
    a₁ ~ᵤ b₁ → a₂ ~ᵤ b₂ → a₁ * a₂ ~ᵤ b₁ * b₂
  | ⟨c₁, h₁⟩, ⟨c₂, h₂⟩ => ⟨c₁ * c₂, by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a₁ a₂ b₁ b₂ : α
c₁ : αˣ
h₁ : a₁ * ↑c₁ = b₁
c₂ : αˣ
h₂ : a₂ * ↑c₂ = b₂
⊢ a₁ * a₂ * ↑(c₁ * c₂) = b₁ * b₂

Next tactic:
simp [h₁.symm, h₂.symm, mul_assoc, mul_comm, mul_left_comm]
theorem Associated.pow_pow [CommMonoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a b : α
n : ℕ
h : a ~ᵤ b
⊢ a ^ n ~ᵤ b ^ n

Next tactic:
induction' n with n ih
theorem Associated.pow_pow [CommMonoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
  induction' n with n ih;
  · 

Proof state:
case zero
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a b : α
h : a ~ᵤ b
⊢ a ^ Nat.zero ~ᵤ b ^ Nat.zero

Next tactic:
simp [h]
theorem Associated.pow_pow [CommMonoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
  induction' n with n ih;
  · simp [h]; 

Proof state:
case zero
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a b : α
h : a ~ᵤ b
⊢ 1 ~ᵤ 1

Next tactic:
rfl
theorem Associated.pow_pow [CommMonoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
  induction' n with n ih;
  · simp [h]; rfl
  

Proof state:
case succ
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a b : α
h : a ~ᵤ b
n : ℕ
ih : a ^ n ~ᵤ b ^ n
⊢ a ^ Nat.succ n ~ᵤ b ^ Nat.succ n

Next tactic:
convert h.mul_mul ih
theorem Associated.pow_pow [CommMonoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
  induction' n with n ih;
  · simp [h]; rfl
  convert h.mul_mul ih <;> 

Proof state:
case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a b : α
h : a ~ᵤ b
n : ℕ
ih : a ^ n ~ᵤ b ^ n
⊢ a ^ Nat.succ n = a * a ^ n

Next tactic:
rw [pow_succ]
theorem Associated.pow_pow [CommMonoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
  induction' n with n ih;
  · simp [h]; rfl
  convert h.mul_mul ih <;> 

Proof state:
case h.e'_4
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoid α
a b : α
h : a ~ᵤ b
n : ℕ
ih : a ^ n ~ᵤ b ^ n
⊢ b ^ Nat.succ n = b * b ^ n

Next tactic:
rw [pow_succ]
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a b : α
hab : a ∣ b
hba : b ∣ a
⊢ a ~ᵤ b

Next tactic:
rcases hab with ⟨c, rfl⟩
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  

Proof state:
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c : α
hba : a * c ∣ a
⊢ a ~ᵤ a * c

Next tactic:
rcases hba with ⟨d, a_eq⟩
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  

Proof state:
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
⊢ a ~ᵤ a * c

Next tactic:
by_cases ha0 : a = 0
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : a = 0
⊢ a ~ᵤ a * c

Next tactic:
simp_all
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; 

Proof state:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
ha0 : a = 0
⊢ 0 ~ᵤ 0

Next tactic:
rfl
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
⊢ a ~ᵤ a * c

Next tactic:
have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
⊢ a * c ≠ 0

Next tactic:
intro con
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
con : a * c = 0
⊢ False

Next tactic:
rw [con, zero_mul] at a_eq
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = 0
ha0 : ¬a = 0
con : a * c = 0
⊢ False

Next tactic:
apply ha0 a_eq
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
  

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
⊢ a ~ᵤ a * c

Next tactic:
have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
  have : a * (c * d) = a * 1 := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
⊢ a * (c * d) = a * 1

Next tactic:
rw [← mul_assoc, ← a_eq, mul_one]
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
  have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
  

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
⊢ a ~ᵤ a * c

Next tactic:
have hcd : c * d = 1 := mul_left_cancel₀ ha0 this
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
  have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
  have hcd : c * d = 1 := mul_left_cancel₀ ha0 this
  

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
hcd : c * d = 1
⊢ a ~ᵤ a * c

Next tactic:
have : a * c * (d * c) = a * c * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
  have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
  have hcd : c * d = 1 := mul_left_cancel₀ ha0 this
  have : a * c * (d * c) = a * c * 1 := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
hcd : c * d = 1
⊢ a * c * (d * c) = a * c * 1

Next tactic:
rw [← mul_assoc, ← a_eq, mul_one]
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
  have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
  have hcd : c * d = 1 := mul_left_cancel₀ ha0 this
  have : a * c * (d * c) = a * c * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
  

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this✝ : a * (c * d) = a * 1
hcd : c * d = 1
this : a * c * (d * c) = a * c * 1
⊢ a ~ᵤ a * c

Next tactic:
have hdc : d * c = 1 := mul_left_cancel₀ hac0 this
theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b := by
  rcases hab with ⟨c, rfl⟩
  rcases hba with ⟨d, a_eq⟩
  by_cases ha0 : a = 0
  · simp_all; rfl
  have hac0 : a * c ≠ 0 := by
    intro con
    rw [con, zero_mul] at a_eq
    apply ha0 a_eq
  have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
  have hcd : c * d = 1 := mul_left_cancel₀ ha0 this
  have : a * c * (d * c) = a * c * 1 := by rw [← mul_assoc, ← a_eq, mul_one]
  have hdc : d * c = 1 := mul_left_cancel₀ hac0 this
  

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelMonoidWithZero α
a c d : α
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this✝ : a * (c * d) = a * 1
hcd : c * d = 1
this : a * c * (d * c) = a * c * 1
hdc : d * c = 1
⊢ a ~ᵤ a * c

Next tactic:
exact ⟨⟨c, d, hcd, hdc⟩, rfl⟩