l3lab/ntp-mathlib huggingface dataset:
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
h : ¬m = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
h : ¬m = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
rw [r] at h'
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ m
h : ¬m = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
have : m * d ≤ m * 1 := by simpa using h'
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ m
h : ¬m = 0
⊢ m * d ≤ m * 1

Next tactic:
simpa using h'
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ m
h : ¬m = 0
this : m * d ≤ m * 1
⊢ m = 1 ∨ m = m * d

Next tactic:
have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ m
h : ¬m = 0
this✝ : m * d ≤ m * 1
this : d ≤ 1
⊢ m = 1 ∨ m = m * d

Next tactic:
have : d = 1 := bot_unique this
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ m
h : ¬m = 0
this✝¹ : m * d ≤ m * 1
this✝ : d ≤ 1
this : d = 1
⊢ m = 1 ∨ m = m * d

Next tactic:
simp [this]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
⊢ m = 1 ∨ m = m * d

Next tactic:
by_cases h : d = 0
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        

Proof state:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : d = 0
⊢ m = 1 ∨ m = m * d
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : ¬d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
rw [r] at hp0
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
hp0 : m * d ≠ 0
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
have : m * d = 0 := by rw [h]; simp
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
hp0 : m * d ≠ 0
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : d = 0
⊢ m * d = 0

Next tactic:
rw [h]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
hp0 : m * d ≠ 0
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : d = 0
⊢ m * 0 = 0

Next tactic:
simp
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
hp0 : m * d ≠ 0
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : d = 0
this : m * d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
contradiction
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
        

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : ¬d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case neg =>
          rw [r] at h'
          have : d * m ≤ d * 1 := by simpa [mul_comm] using h'
          exact Or.inl <| bot_unique <| Associates.le_of_mul_le_mul_left d m 1 ‹d ≠ 0› this
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : ¬d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case neg =>
          rw [r] at h'
          have : d * m ≤ d * 1 := by simpa [mul_comm] using h'
          exact Or.inl <| bot_unique <| Associates.le_of_mul_le_mul_left d m 1 ‹d ≠ 0› this
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
        case neg =>
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ d
h : ¬d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
rw [r] at h'
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
        case neg =>
          rw [r] at h'
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ d
h : ¬d = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
have : d * m ≤ d * 1 := by simpa [mul_comm] using h'
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
        case neg =>
          rw [r] at h'
          have : d * m ≤ d * 1 := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ d
h : ¬d = 0
⊢ d * m ≤ d * 1

Next tactic:
simpa [mul_comm] using h'
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
        case neg =>
          rw [r] at h'
          have : d * m ≤ d * 1 := by simpa [mul_comm] using h'
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : m * d ∣ d
h : ¬d = 0
this : d * m ≤ d * 1
⊢ m = 1 ∨ m = m * d

Next tactic:
exact Or.inl <| bot_unique <| Associates.le_of_mul_le_mul_left d m 1 ‹d ≠ 0› this
theorem le_one_iff {p : Associates α} : p ≤ 1 ↔ p = 1 := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p : Associates α
⊢ p ≤ 1 ↔ p = 1

Next tactic:
rw [← Associates.bot_eq_one, le_bot_iff]
theorem DvdNotUnit.isUnit_of_irreducible_right [CommMonoidWithZero α] {p q : α}
    (h : DvdNotUnit p q) (hq : Irreducible q) : IsUnit p := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
p q : α
h : DvdNotUnit p q
hq : Irreducible q
⊢ IsUnit p

Next tactic:
obtain ⟨_, x, hx, hx'⟩ := h
theorem DvdNotUnit.isUnit_of_irreducible_right [CommMonoidWithZero α] {p q : α}
    (h : DvdNotUnit p q) (hq : Irreducible q) : IsUnit p := by
  obtain ⟨_, x, hx, hx'⟩ := h
  

Proof state:
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
p q : α
hq : Irreducible q
left✝ : p ≠ 0
x : α
hx : ¬IsUnit x
hx' : q = p * x
⊢ IsUnit p

Next tactic:
exact Or.resolve_right ((irreducible_iff.1 hq).right p x hx') hx
theorem DvdNotUnit.not_unit [CommMonoidWithZero α] {p q : α} (hp : DvdNotUnit p q) : ¬IsUnit q := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
p q : α
hp : DvdNotUnit p q
⊢ ¬IsUnit q

Next tactic:
obtain ⟨-, x, hx, rfl⟩ := hp
theorem DvdNotUnit.not_unit [CommMonoidWithZero α] {p q : α} (hp : DvdNotUnit p q) : ¬IsUnit q := by
  obtain ⟨-, x, hx, rfl⟩ := hp
  

Proof state:
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
p x : α
hx : ¬IsUnit x
⊢ ¬IsUnit (p * x)

Next tactic:
exact fun hc => hx (isUnit_iff_dvd_one.mpr (dvd_of_mul_left_dvd (isUnit_iff_dvd_one.mp hc)))
theorem dvdNotUnit_of_dvdNotUnit_associated [CommMonoidWithZero α] [Nontrivial α] {p q r : α}
    (h : DvdNotUnit p q) (h' : Associated q r) : DvdNotUnit p r := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : CommMonoidWithZero α
inst✝ : Nontrivial α
p q r : α
h : DvdNotUnit p q
h' : q ~ᵤ r
⊢ DvdNotUnit p r

Next tactic:
obtain ⟨u, rfl⟩ := Associated.symm h'
theorem dvdNotUnit_of_dvdNotUnit_associated [CommMonoidWithZero α] [Nontrivial α] {p q r : α}
    (h : DvdNotUnit p q) (h' : Associated q r) : DvdNotUnit p r := by
  obtain ⟨u, rfl⟩ := Associated.symm h'
  

Proof state:
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : CommMonoidWithZero α
inst✝ : Nontrivial α
p r : α
u : αˣ
h : DvdNotUnit p (r * ↑u)
h' : r * ↑u ~ᵤ r
⊢ DvdNotUnit p r

Next tactic:
obtain ⟨hp, x, hx⟩ := h
theorem dvdNotUnit_of_dvdNotUnit_associated [CommMonoidWithZero α] [Nontrivial α] {p q r : α}
    (h : DvdNotUnit p q) (h' : Associated q r) : DvdNotUnit p r := by
  obtain ⟨u, rfl⟩ := Associated.symm h'
  obtain ⟨hp, x, hx⟩ := h
  

Proof state:
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : CommMonoidWithZero α
inst✝ : Nontrivial α
p r : α
u : αˣ
h' : r * ↑u ~ᵤ r
hp : p ≠ 0
x : α
hx : ¬IsUnit x ∧ r * ↑u = p * x
⊢ DvdNotUnit p r

Next tactic:
refine' ⟨hp, x * ↑u⁻¹, DvdNotUnit.not_unit ⟨u⁻¹.ne_zero, x, hx.left, mul_comm _ _⟩, _⟩
theorem dvdNotUnit_of_dvdNotUnit_associated [CommMonoidWithZero α] [Nontrivial α] {p q r : α}
    (h : DvdNotUnit p q) (h' : Associated q r) : DvdNotUnit p r := by
  obtain ⟨u, rfl⟩ := Associated.symm h'
  obtain ⟨hp, x, hx⟩ := h
  refine' ⟨hp, x * ↑u⁻¹, DvdNotUnit.not_unit ⟨u⁻¹.ne_zero, x, hx.left, mul_comm _ _⟩, _⟩
  

Proof state:
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : CommMonoidWithZero α
inst✝ : Nontrivial α
p r : α
u : αˣ
h' : r * ↑u ~ᵤ r
hp : p ≠ 0
x : α
hx : ¬IsUnit x ∧ r * ↑u = p * x
⊢ r = p * (x * ↑u⁻¹)

Next tactic:
rw [← mul_assoc, ← hx.right, mul_assoc, Units.mul_inv, mul_one]
theorem isUnit_of_associated_mul [CancelCommMonoidWithZero α] {p b : α} (h : Associated (p * b) p)
    (hp : p ≠ 0) : IsUnit b := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p b : α
h : p * b ~ᵤ p
hp : p ≠ 0
⊢ IsUnit b

Next tactic:
cases' h with a ha