l3lab/ntp-mathlib huggingface dataset:
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
  · rintro _ ⟨x, rfl⟩ _ a_dvd_bx
    apply Units.dvd_mul_right.mp a_dvd_bx
  · intro c p _ hp ih no_factors a_dvd_bpc
    apply ih fun {q} dvd_a dvd_c hq => no_factors dvd_a (dvd_c.mul_left _) hq
    

Proof state:
case refine'_3
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c✝ : R
ha : a ≠ 0
c p : R
a✝ : c ≠ 0
hp : Prime p
ih : (∀ {d : R}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b
no_factors : ∀ {d : R}, d ∣ a → d ∣ p * c → ¬Prime d
a_dvd_bpc : a ∣ b * (p * c)
⊢ a ∣ b * c

Next tactic:
rw [mul_left_comm] at a_dvd_bpc
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
  · rintro _ ⟨x, rfl⟩ _ a_dvd_bx
    apply Units.dvd_mul_right.mp a_dvd_bx
  · intro c p _ hp ih no_factors a_dvd_bpc
    apply ih fun {q} dvd_a dvd_c hq => no_factors dvd_a (dvd_c.mul_left _) hq
    rw [mul_left_comm] at a_dvd_bpc
    

Proof state:
case refine'_3
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c✝ : R
ha : a ≠ 0
c p : R
a✝ : c ≠ 0
hp : Prime p
ih : (∀ {d : R}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b
no_factors : ∀ {d : R}, d ∣ a → d ∣ p * c → ¬Prime d
a_dvd_bpc : a ∣ p * (b * c)
⊢ a ∣ b * c

Next tactic:
refine' Or.resolve_left (hp.left_dvd_or_dvd_right_of_dvd_mul a_dvd_bpc) fun h => _
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `c` have no common prime factors, `a ∣ b`.
Compare `IsCoprime.dvd_of_dvd_mul_left`. -/
theorem dvd_of_dvd_mul_left_of_no_prime_factors {a b c : R} (ha : a ≠ 0) :
    (∀ {d}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b := by
  refine' induction_on_prime c _ _ _
  · intro no_factors
    simp only [dvd_zero, mul_zero, forall_prop_of_true]
    haveI := Classical.propDecidable
    exact
      isUnit_iff_forall_dvd.mp
        (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _
  · rintro _ ⟨x, rfl⟩ _ a_dvd_bx
    apply Units.dvd_mul_right.mp a_dvd_bx
  · intro c p _ hp ih no_factors a_dvd_bpc
    apply ih fun {q} dvd_a dvd_c hq => no_factors dvd_a (dvd_c.mul_left _) hq
    rw [mul_left_comm] at a_dvd_bpc
    refine' Or.resolve_left (hp.left_dvd_or_dvd_right_of_dvd_mul a_dvd_bpc) fun h => _
    

Proof state:
case refine'_3
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c✝ : R
ha : a ≠ 0
c p : R
a✝ : c ≠ 0
hp : Prime p
ih : (∀ {d : R}, d ∣ a → d ∣ c → ¬Prime d) → a ∣ b * c → a ∣ b
no_factors : ∀ {d : R}, d ∣ a → d ∣ p * c → ¬Prime d
a_dvd_bpc : a ∣ p * (b * c)
h : p ∣ a
⊢ False

Next tactic:
exact no_factors h (dvd_mul_right p c) hp
/-- Euclid's lemma: if `a ∣ b * c` and `a` and `b` have no common prime factors, `a ∣ c`.
Compare `IsCoprime.dvd_of_dvd_mul_right`. -/
theorem dvd_of_dvd_mul_right_of_no_prime_factors {a b c : R} (ha : a ≠ 0)
    (no_factors : ∀ {d}, d ∣ a → d ∣ b → ¬Prime d) : a ∣ b * c → a ∣ c := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a b c : R
ha : a ≠ 0
no_factors : ∀ {d : R}, d ∣ a → d ∣ b → ¬Prime d
⊢ a ∣ b * c → a ∣ c

Next tactic:
simpa [mul_comm b c] using dvd_of_dvd_mul_left_of_no_prime_factors ha @no_factors
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
⊢ ∀ (a : R), a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b

Next tactic:
haveI := Classical.propDecidable
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
⊢ ∀ (a : R), a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b

Next tactic:
intro a
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a : R
⊢ a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b

Next tactic:
refine' induction_on_prime a _ _ _
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · 

Proof state:
case refine'_1
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a : R
⊢ 0 ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = 0 ∧ c' * b' = b

Next tactic:
intros
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    

Proof state:
case refine'_1
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a : R
a✝ : 0 ≠ 0
b✝ : R
⊢ ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = 0 ∧ c' * b' = b✝

Next tactic:
contradiction
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · 

Proof state:
case refine'_2
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a : R
⊢ ∀ (x : R),
    IsUnit x → x ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = x ∧ c' * b' = b

Next tactic:
intro a a_unit _ b
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    

Proof state:
case refine'_2
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝¹ a : R
a_unit : IsUnit a
a✝ : a ≠ 0
b : R
⊢ ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b

Next tactic:
use a, b, 1
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    

Proof state:
case h
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝¹ a : R
a_unit : IsUnit a
a✝ : a ≠ 0
b : R
⊢ (∀ {d : R}, d ∣ a → d ∣ b → IsUnit d) ∧ 1 * a = a ∧ 1 * b = b

Next tactic:
constructor
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · 

Proof state:
case h.left
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝¹ a : R
a_unit : IsUnit a
a✝ : a ≠ 0
b : R
⊢ ∀ {d : R}, d ∣ a → d ∣ b → IsUnit d

Next tactic:
intro p p_dvd_a _
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      

Proof state:
case h.left
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝² a : R
a_unit : IsUnit a
a✝¹ : a ≠ 0
b p : R
p_dvd_a : p ∣ a
a✝ : p ∣ b
⊢ IsUnit p

Next tactic:
exact isUnit_of_dvd_unit p_dvd_a a_unit
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · 

Proof state:
case h.right
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝¹ a : R
a_unit : IsUnit a
a✝ : a ≠ 0
b : R
⊢ 1 * a = a ∧ 1 * b = b

Next tactic:
simp
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · 

Proof state:
case refine'_3
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a : R
⊢ ∀ (a p : R),
    a ≠ 0 →
      Prime p →
        (a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b) →
          p * a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = p * a ∧ c' * b' = b

Next tactic:
intro a p a_ne_zero p_prime ih_a pa_ne_zero b
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    

Proof state:
case refine'_3
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝ a p : R
a_ne_zero : a ≠ 0
p_prime : Prime p
ih_a : a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b
pa_ne_zero : p * a ≠ 0
b : R
⊢ ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = p * a ∧ c' * b' = b

Next tactic:
by_cases h : p ∣ b
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    by_cases h : p ∣ b
    · 

Proof state:
case pos
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝ a p : R
a_ne_zero : a ≠ 0
p_prime : Prime p
ih_a : a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b
pa_ne_zero : p * a ≠ 0
b : R
h : p ∣ b
⊢ ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = p * a ∧ c' * b' = b

Next tactic:
rcases h with ⟨b, rfl⟩
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    by_cases h : p ∣ b
    · rcases h with ⟨b, rfl⟩
      

Proof state:
case pos.intro
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝ a p : R
a_ne_zero : a ≠ 0
p_prime : Prime p
ih_a : a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b
pa_ne_zero : p * a ≠ 0
b : R
⊢ ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = p * a ∧ c' * b' = p * b

Next tactic:
obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    by_cases h : p ∣ b
    · rcases h with ⟨b, rfl⟩
      obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b
      

Proof state:
case pos.intro.intro.intro.intro.intro.intro
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝ a p : R
a_ne_zero : a ≠ 0
p_prime : Prime p
ih_a : a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b
pa_ne_zero : p * a ≠ 0
b a' b' c' : R
no_factor : ∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d
ha' : c' * a' = a
hb' : c' * b' = b
⊢ ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = p * a ∧ c' * b' = p * b

Next tactic:
refine' ⟨a', b', p * c', @no_factor, _, _⟩
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    by_cases h : p ∣ b
    · rcases h with ⟨b, rfl⟩
      obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b
      refine' ⟨a', b', p * c', @no_factor, _, _⟩
      · 

Proof state:
case pos.intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝ a p : R
a_ne_zero : a ≠ 0
p_prime : Prime p
ih_a : a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b
pa_ne_zero : p * a ≠ 0
b a' b' c' : R
no_factor : ∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d
ha' : c' * a' = a
hb' : c' * b' = b
⊢ p * c' * a' = p * a

Next tactic:
rw [mul_assoc, ha']
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    by_cases h : p ∣ b
    · rcases h with ⟨b, rfl⟩
      obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b
      refine' ⟨a', b', p * c', @no_factor, _, _⟩
      · rw [mul_assoc, ha']
      · 

Proof state:
case pos.intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝ a p : R
a_ne_zero : a ≠ 0
p_prime : Prime p
ih_a : a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b
pa_ne_zero : p * a ≠ 0
b a' b' c' : R
no_factor : ∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d
ha' : c' * a' = a
hb' : c' * b' = b
⊢ p * c' * b' = p * b

Next tactic:
rw [mul_assoc, hb']
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    by_cases h : p ∣ b
    · rcases h with ⟨b, rfl⟩
      obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b
      refine' ⟨a', b', p * c', @no_factor, _, _⟩
      · rw [mul_assoc, ha']
      · rw [mul_assoc, hb']
    · 

Proof state:
case neg
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a✝ a p : R
a_ne_zero : a ≠ 0
p_prime : Prime p
ih_a : a ≠ 0 → ∀ (b : R), ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b
pa_ne_zero : p * a ≠ 0
b : R
h : ¬p ∣ b
⊢ ∃ a' b' c', (∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = p * a ∧ c' * b' = b

Next tactic:
obtain ⟨a', b', c', coprime, rfl, rfl⟩ := ih_a a_ne_zero b
/-- If `a ≠ 0, b` are elements of a unique factorization domain, then dividing
out their common factor `c'` gives `a'` and `b'` with no factors in common. -/
theorem exists_reduced_factors :
    ∀ a ≠ (0 : R), ∀ b,
      ∃ a' b' c', (∀ {d}, d ∣ a' → d ∣ b' → IsUnit d) ∧ c' * a' = a ∧ c' * b' = b := by
  haveI := Classical.propDecidable
  intro a
  refine' induction_on_prime a _ _ _
  · intros
    contradiction
  · intro a a_unit _ b
    use a, b, 1
    constructor
    · intro p p_dvd_a _
      exact isUnit_of_dvd_unit p_dvd_a a_unit
    · simp
  · intro a p a_ne_zero p_prime ih_a pa_ne_zero b
    by_cases h : p ∣ b
    · rcases h with ⟨b, rfl⟩
      obtain ⟨a', b', c', no_factor, ha', hb'⟩ := ih_a a_ne_zero b
      refine' ⟨a', b', p * c', @no_factor, _, _⟩
      · rw [mul_assoc, ha']
      · rw [mul_assoc, hb']
    · obtain ⟨a', b', c', coprime, rfl, rfl⟩ := ih_a a_ne_zero b
      

Proof state:
case neg.intro.intro.intro.intro.intro
α : Type u_1
R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
this : (a : Prop) → Decidable a
a p : R
p_prime : Prime p
a' b' c' : R
coprime : ∀ {d : R}, d ∣ a' → d ∣ b' → IsUnit d
a_ne_zero : c' * a' ≠ 0
ih_a :
  c' * a' ≠ 0 →
    ∀ (b : R), ∃ a'_1 b' c'_1, (∀ {d : R}, d ∣ a'_1 → d ∣ b' → IsUnit d) ∧ c'_1 * a'_1 = c' * a' ∧ c'_1 * b' = b
pa_ne_zero : p * (c' * a') ≠ 0
h : ¬p ∣ c' * b'
⊢ ∃ a'_1 b'_1 c'_1, (∀ {d : R}, d ∣ a'_1 → d ∣ b'_1 → IsUnit d) ∧ c'_1 * a'_1 = p * (c' * a') ∧ c'_1 * b'_1 = c' * b'

Next tactic:
refine' ⟨p * a', b', c', _, mul_left_comm _ _ _, rfl⟩