l3lab/ntp-mathlib huggingface dataset:
/-- An arbitrary choice of factors of `x : M` is exactly the (unique) normalized set of factors,
if `M` has a trivial group of units. -/
@[simp]
theorem factors_eq_normalizedFactors {M : Type*} [CancelCommMonoidWithZero M] [DecidableEq M]
    [UniqueFactorizationMonoid M] [Unique Mˣ] (x : M) : factors x = normalizedFactors x := by
  unfold normalizedFactors
  convert (Multiset.map_id (factors x)).symm
  

Proof state:
case h.e'_3.a.h.e
α : Type u_1
inst✝⁷ : CancelCommMonoidWithZero α
inst✝⁶ : DecidableEq α
inst✝⁵ : NormalizationMonoid α
inst✝⁴ : UniqueFactorizationMonoid α
M : Type u_2
inst✝³ : CancelCommMonoidWithZero M
inst✝² : DecidableEq M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Unique Mˣ
x x✝ : M
a✝ : x✝ ∈ factors x
⊢ ⇑normalize = id

Next tactic:
ext p
/-- An arbitrary choice of factors of `x : M` is exactly the (unique) normalized set of factors,
if `M` has a trivial group of units. -/
@[simp]
theorem factors_eq_normalizedFactors {M : Type*} [CancelCommMonoidWithZero M] [DecidableEq M]
    [UniqueFactorizationMonoid M] [Unique Mˣ] (x : M) : factors x = normalizedFactors x := by
  unfold normalizedFactors
  convert (Multiset.map_id (factors x)).symm
  ext p
  

Proof state:
case h.e'_3.a.h.e.h
α : Type u_1
inst✝⁷ : CancelCommMonoidWithZero α
inst✝⁶ : DecidableEq α
inst✝⁵ : NormalizationMonoid α
inst✝⁴ : UniqueFactorizationMonoid α
M : Type u_2
inst✝³ : CancelCommMonoidWithZero M
inst✝² : DecidableEq M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Unique Mˣ
x x✝ : M
a✝ : x✝ ∈ factors x
p : M
⊢ normalize p = id p

Next tactic:
exact normalize_eq p
theorem normalizedFactors_prod {a : α} (ane0 : a ≠ 0) :
    Associated (normalizedFactors a).prod a := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Multiset.prod (normalizedFactors a) ~ᵤ a

Next tactic:
rw [normalizedFactors, factors, dif_neg ane0]
theorem normalizedFactors_prod {a : α} (ane0 : a ≠ 0) :
    Associated (normalizedFactors a).prod a := by
  rw [normalizedFactors, factors, dif_neg ane0]
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Multiset.prod (Multiset.map (⇑normalize) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) ~ᵤ a

Next tactic:
refine' Associated.trans _ (Classical.choose_spec (exists_prime_factors a ane0)).2
theorem normalizedFactors_prod {a : α} (ane0 : a ≠ 0) :
    Associated (normalizedFactors a).prod a := by
  rw [normalizedFactors, factors, dif_neg ane0]
  refine' Associated.trans _ (Classical.choose_spec (exists_prime_factors a ane0)).2
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Multiset.prod (Multiset.map (⇑normalize) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) ~ᵤ
    Multiset.prod (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))

Next tactic:
rw [← Associates.mk_eq_mk_iff_associated, ← Associates.prod_mk, ← Associates.prod_mk,
    Multiset.map_map]
theorem normalizedFactors_prod {a : α} (ane0 : a ≠ 0) :
    Associated (normalizedFactors a).prod a := by
  rw [normalizedFactors, factors, dif_neg ane0]
  refine' Associated.trans _ (Classical.choose_spec (exists_prime_factors a ane0)).2
  rw [← Associates.mk_eq_mk_iff_associated, ← Associates.prod_mk, ← Associates.prod_mk,
    Multiset.map_map]
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Multiset.prod
      (Multiset.map (Associates.mk ∘ ⇑normalize)
        (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) =
    Multiset.prod (Multiset.map Associates.mk (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)))

Next tactic:
congr 2
theorem normalizedFactors_prod {a : α} (ane0 : a ≠ 0) :
    Associated (normalizedFactors a).prod a := by
  rw [normalizedFactors, factors, dif_neg ane0]
  refine' Associated.trans _ (Classical.choose_spec (exists_prime_factors a ane0)).2
  rw [← Associates.mk_eq_mk_iff_associated, ← Associates.prod_mk, ← Associates.prod_mk,
    Multiset.map_map]
  congr 2
  

Proof state:
case e_a.e_f
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Associates.mk ∘ ⇑normalize = Associates.mk

Next tactic:
ext
theorem normalizedFactors_prod {a : α} (ane0 : a ≠ 0) :
    Associated (normalizedFactors a).prod a := by
  rw [normalizedFactors, factors, dif_neg ane0]
  refine' Associated.trans _ (Classical.choose_spec (exists_prime_factors a ane0)).2
  rw [← Associates.mk_eq_mk_iff_associated, ← Associates.prod_mk, ← Associates.prod_mk,
    Multiset.map_map]
  congr 2
  ext
  

Proof state:
case e_a.e_f.h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
x✝ : α
⊢ (Associates.mk ∘ ⇑normalize) x✝ = Associates.mk x✝

Next tactic:
rw [Function.comp_apply, Associates.mk_normalize]
theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ x ∈ normalizedFactors a, Prime x

Next tactic:
rw [normalizedFactors, factors]
theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by
  rw [normalizedFactors, factors]
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀
    x ∈
      Multiset.map (⇑normalize)
        (if h : a = 0 then 0 else Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)),
    Prime x

Next tactic:
split_ifs with ane0
theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by
  rw [normalizedFactors, factors]
  split_ifs with ane0; · 

Proof state:
case pos
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a = 0
⊢ ∀ x ∈ Multiset.map (fun x => normalize x) 0, Prime x

Next tactic:
simp
theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by
  rw [normalizedFactors, factors]
  split_ifs with ane0; · simp
  

Proof state:
case neg
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
⊢ ∀ x ∈ Multiset.map (fun x => normalize x) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)),
    Prime x

Next tactic:
intro x hx
theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by
  rw [normalizedFactors, factors]
  split_ifs with ane0; · simp
  intro x hx; 

Proof state:
case neg
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
x : α
hx : x ∈ Multiset.map (fun x => normalize x) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))
⊢ Prime x

Next tactic:
rcases Multiset.mem_map.1 hx with ⟨y, ⟨hy, rfl⟩⟩
theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by
  rw [normalizedFactors, factors]
  split_ifs with ane0; · simp
  intro x hx; rcases Multiset.mem_map.1 hx with ⟨y, ⟨hy, rfl⟩⟩
  

Proof state:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
y : α
hy : y ∈ Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)
hx :
  normalize y ∈
    Multiset.map (fun x => normalize x) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))
⊢ Prime (normalize y)

Next tactic:
rw [(normalize_associated _).prime_iff]
theorem prime_of_normalized_factor {a : α} : ∀ x : α, x ∈ normalizedFactors a → Prime x := by
  rw [normalizedFactors, factors]
  split_ifs with ane0; · simp
  intro x hx; rcases Multiset.mem_map.1 hx with ⟨y, ⟨hy, rfl⟩⟩
  rw [(normalize_associated _).prime_iff]
  

Proof state:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
y : α
hy : y ∈ Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)
hx :
  normalize y ∈
    Multiset.map (fun x => normalize x) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))
⊢ Prime y

Next tactic:
exact (Classical.choose_spec (UniqueFactorizationMonoid.exists_prime_factors a ane0)).1 y hy
theorem normalize_normalized_factor {a : α} :
    ∀ x : α, x ∈ normalizedFactors a → normalize x = x := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ x ∈ normalizedFactors a, normalize x = x

Next tactic:
rw [normalizedFactors, factors]
theorem normalize_normalized_factor {a : α} :
    ∀ x : α, x ∈ normalizedFactors a → normalize x = x := by
  rw [normalizedFactors, factors]
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀
    x ∈
      Multiset.map (⇑normalize)
        (if h : a = 0 then 0 else Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)),
    normalize x = x

Next tactic:
split_ifs with h
theorem normalize_normalized_factor {a : α} :
    ∀ x : α, x ∈ normalizedFactors a → normalize x = x := by
  rw [normalizedFactors, factors]
  split_ifs with h; · 

Proof state:
case pos
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
h : a = 0
⊢ ∀ x ∈ Multiset.map (fun x => normalize x) 0, normalize x = x

Next tactic:
simp
theorem normalize_normalized_factor {a : α} :
    ∀ x : α, x ∈ normalizedFactors a → normalize x = x := by
  rw [normalizedFactors, factors]
  split_ifs with h; · simp
  

Proof state:
case neg
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
h : ¬a = 0
⊢ ∀ x ∈ Multiset.map (fun x => normalize x) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)),
    normalize x = x

Next tactic:
intro x hx
theorem normalize_normalized_factor {a : α} :
    ∀ x : α, x ∈ normalizedFactors a → normalize x = x := by
  rw [normalizedFactors, factors]
  split_ifs with h; · simp
  intro x hx
  

Proof state:
case neg
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
h : ¬a = 0
x : α
hx : x ∈ Multiset.map (fun x => normalize x) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))
⊢ normalize x = x

Next tactic:
obtain ⟨y, _, rfl⟩ := Multiset.mem_map.1 hx
theorem normalize_normalized_factor {a : α} :
    ∀ x : α, x ∈ normalizedFactors a → normalize x = x := by
  rw [normalizedFactors, factors]
  split_ifs with h; · simp
  intro x hx
  obtain ⟨y, _, rfl⟩ := Multiset.mem_map.1 hx
  

Proof state:
case neg.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
h : ¬a = 0
y : α
left✝ : y ∈ Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)
hx :
  normalize y ∈
    Multiset.map (fun x => normalize x) (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))
⊢ normalize (normalize y) = normalize y

Next tactic:
apply normalize_idem
theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) :
    normalizedFactors a = {normalize a} := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
⊢ normalizedFactors a = {normalize a}

Next tactic:
obtain ⟨p, a_assoc, hp⟩ :=
    prime_factors_irreducible ha ⟨prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero⟩
theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) :
    normalizedFactors a = {normalize a} := by
  obtain ⟨p, a_assoc, hp⟩ :=
    prime_factors_irreducible ha ⟨prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero⟩
  

Proof state:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
⊢ normalizedFactors a = {normalize a}

Next tactic:
have p_mem : p ∈ normalizedFactors a := by
    rw [hp]
    exact Multiset.mem_singleton_self _
theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) :
    normalizedFactors a = {normalize a} := by
  obtain ⟨p, a_assoc, hp⟩ :=
    prime_factors_irreducible ha ⟨prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero⟩
  have p_mem : p ∈ normalizedFactors a := by
    

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
⊢ p ∈ normalizedFactors a

Next tactic:
rw [hp]
theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) :
    normalizedFactors a = {normalize a} := by
  obtain ⟨p, a_assoc, hp⟩ :=
    prime_factors_irreducible ha ⟨prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero⟩
  have p_mem : p ∈ normalizedFactors a := by
    rw [hp]
    

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
⊢ p ∈ {p}

Next tactic:
exact Multiset.mem_singleton_self _
theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) :
    normalizedFactors a = {normalize a} := by
  obtain ⟨p, a_assoc, hp⟩ :=
    prime_factors_irreducible ha ⟨prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero⟩
  have p_mem : p ∈ normalizedFactors a := by
    rw [hp]
    exact Multiset.mem_singleton_self _
  

Proof state:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
p_mem : p ∈ normalizedFactors a
⊢ normalizedFactors a = {normalize a}

Next tactic:
convert hp
theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) :
    normalizedFactors a = {normalize a} := by
  obtain ⟨p, a_assoc, hp⟩ :=
    prime_factors_irreducible ha ⟨prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero⟩
  have p_mem : p ∈ normalizedFactors a := by
    rw [hp]
    exact Multiset.mem_singleton_self _
  convert hp
  

Proof state:
case h.e'_3.h.e'_4
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
p_mem : p ∈ normalizedFactors a
⊢ normalize a = p

Next tactic:
rwa [← normalize_normalized_factor p p_mem, normalize_eq_normalize_iff, dvd_dvd_iff_associated]
theorem normalizedFactors_eq_of_dvd (a : α) :
    ∀ᵉ (p ∈ normalizedFactors a) (q ∈ normalizedFactors a), p ∣ q → p = q := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ p ∈ normalizedFactors a, ∀ q ∈ normalizedFactors a, p ∣ q → p = q

Next tactic:
intro p hp q hq hdvd
theorem normalizedFactors_eq_of_dvd (a : α) :
    ∀ᵉ (p ∈ normalizedFactors a) (q ∈ normalizedFactors a), p ∣ q → p = q := by
  intro p hp q hq hdvd
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
hp : p ∈ normalizedFactors a
q : α
hq : q ∈ normalizedFactors a
hdvd : p ∣ q
⊢ p = q

Next tactic:
convert normalize_eq_normalize hdvd
          ((prime_of_normalized_factor _ hp).irreducible.dvd_symm
            (prime_of_normalized_factor _ hq).irreducible hdvd)
theorem normalizedFactors_eq_of_dvd (a : α) :
    ∀ᵉ (p ∈ normalizedFactors a) (q ∈ normalizedFactors a), p ∣ q → p = q := by
  intro p hp q hq hdvd
  convert normalize_eq_normalize hdvd
          ((prime_of_normalized_factor _ hp).irreducible.dvd_symm
            (prime_of_normalized_factor _ hq).irreducible hdvd) <;>
    

Proof state:
case h.e'_2
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
hp : p ∈ normalizedFactors a
q : α
hq : q ∈ normalizedFactors a
hdvd : p ∣ q
⊢ p = normalize p

Next tactic:
apply (normalize_normalized_factor _ ‹_›).symm
theorem normalizedFactors_eq_of_dvd (a : α) :
    ∀ᵉ (p ∈ normalizedFactors a) (q ∈ normalizedFactors a), p ∣ q → p = q := by
  intro p hp q hq hdvd
  convert normalize_eq_normalize hdvd
          ((prime_of_normalized_factor _ hp).irreducible.dvd_symm
            (prime_of_normalized_factor _ hq).irreducible hdvd) <;>
    

Proof state:
case h.e'_3
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
hp : p ∈ normalizedFactors a
q : α
hq : q ∈ normalizedFactors a
hdvd : p ∣ q
⊢ q = normalize q

Next tactic:
apply (normalize_normalized_factor _ ‹_›).symm
theorem exists_mem_normalizedFactors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    p ∣ a → ∃ q ∈ normalizedFactors a, p ~ᵤ q := fun ⟨b, hb⟩ =>
  have hb0 : b ≠ 0 := fun hb0 => by 

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
x✝ : p ∣ a
b : α
hb : a = p * b
hb0 : b = 0
⊢ False

Next tactic:
simp_all
theorem exists_mem_normalizedFactors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    p ∣ a → ∃ q ∈ normalizedFactors a, p ~ᵤ q := fun ⟨b, hb⟩ =>
  have hb0 : b ≠ 0 := fun hb0 => by simp_all
  have : Multiset.Rel Associated (p ::ₘ normalizedFactors b) (normalizedFactors a) :=
    factors_unique
      (fun x hx =>
        (Multiset.mem_cons.1 hx).elim (fun h => h.symm ▸ hp) (irreducible_of_normalized_factor _))
      irreducible_of_normalized_factor
      (Associated.symm <|
        calc
          Multiset.prod (normalizedFactors a) ~ᵤ a := normalizedFactors_prod ha0
          _ = p * b := hb
          _ ~ᵤ Multiset.prod (p ::ₘ normalizedFactors b) := by
            

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
x✝ : p ∣ a
b : α
hb : a = p * b
hb0 : b ≠ 0
⊢ p * b ~ᵤ Multiset.prod (p ::ₘ normalizedFactors b)

Next tactic:
rw [Multiset.prod_cons]
theorem exists_mem_normalizedFactors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    p ∣ a → ∃ q ∈ normalizedFactors a, p ~ᵤ q := fun ⟨b, hb⟩ =>
  have hb0 : b ≠ 0 := fun hb0 => by simp_all
  have : Multiset.Rel Associated (p ::ₘ normalizedFactors b) (normalizedFactors a) :=
    factors_unique
      (fun x hx =>
        (Multiset.mem_cons.1 hx).elim (fun h => h.symm ▸ hp) (irreducible_of_normalized_factor _))
      irreducible_of_normalized_factor
      (Associated.symm <|
        calc
          Multiset.prod (normalizedFactors a) ~ᵤ a := normalizedFactors_prod ha0
          _ = p * b := hb
          _ ~ᵤ Multiset.prod (p ::ₘ normalizedFactors b) := by
            rw [Multiset.prod_cons]
            

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
x✝ : p ∣ a
b : α
hb : a = p * b
hb0 : b ≠ 0
⊢ p * b ~ᵤ p * Multiset.prod (normalizedFactors b)

Next tactic:
exact (normalizedFactors_prod hb0).symm.mul_left _
theorem exists_mem_normalizedFactors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    p ∣ a → ∃ q ∈ normalizedFactors a, p ~ᵤ q := fun ⟨b, hb⟩ =>
  have hb0 : b ≠ 0 := fun hb0 => by simp_all
  have : Multiset.Rel Associated (p ::ₘ normalizedFactors b) (normalizedFactors a) :=
    factors_unique
      (fun x hx =>
        (Multiset.mem_cons.1 hx).elim (fun h => h.symm ▸ hp) (irreducible_of_normalized_factor _))
      irreducible_of_normalized_factor
      (Associated.symm <|
        calc
          Multiset.prod (normalizedFactors a) ~ᵤ a := normalizedFactors_prod ha0
          _ = p * b := hb
          _ ~ᵤ Multiset.prod (p ::ₘ normalizedFactors b) := by
            rw [Multiset.prod_cons]
            exact (normalizedFactors_prod hb0).symm.mul_left _
          )
  Multiset.exists_mem_of_rel_of_mem this (by 

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
x✝ : p ∣ a
b : α
hb : a = p * b
hb0 : b ≠ 0
this : Multiset.Rel Associated (p ::ₘ normalizedFactors b) (normalizedFactors a)
⊢ p ∈ p ::ₘ normalizedFactors b

Next tactic:
simp
theorem exists_mem_normalizedFactors {x : α} (hx : x ≠ 0) (h : ¬IsUnit x) :
    ∃ p, p ∈ normalizedFactors x := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
h : ¬IsUnit x
⊢ ∃ p, p ∈ normalizedFactors x

Next tactic:
obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx
theorem exists_mem_normalizedFactors {x : α} (hx : x ≠ 0) (h : ¬IsUnit x) :
    ∃ p, p ∈ normalizedFactors x := by
  obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx
  

Proof state:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
h : ¬IsUnit x
p' : α
hp' : Irreducible p'
hp'x : p' ∣ x
⊢ ∃ p, p ∈ normalizedFactors x

Next tactic:
obtain ⟨p, hp, _⟩ := exists_mem_normalizedFactors_of_dvd hx hp' hp'x
theorem exists_mem_normalizedFactors {x : α} (hx : x ≠ 0) (h : ¬IsUnit x) :
    ∃ p, p ∈ normalizedFactors x := by
  obtain ⟨p', hp', hp'x⟩ := WfDvdMonoid.exists_irreducible_factor h hx
  obtain ⟨p, hp, _⟩ := exists_mem_normalizedFactors_of_dvd hx hp' hp'x
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x : α
hx : x ≠ 0
h : ¬IsUnit x
p' : α
hp' : Irreducible p'
hp'x : p' ∣ x
p : α
hp : p ∈ normalizedFactors x
right✝ : p' ~ᵤ p
⊢ ∃ p, p ∈ normalizedFactors x

Next tactic:
exact ⟨p, hp⟩
@[simp]
theorem normalizedFactors_zero : normalizedFactors (0 : α) = 0 := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
⊢ normalizedFactors 0 = 0

Next tactic:
simp [normalizedFactors, factors]
@[simp]
theorem normalizedFactors_one : normalizedFactors (1 : α) = 0 := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
⊢ normalizedFactors 1 = 0

Next tactic:
cases' subsingleton_or_nontrivial α with h h