l3lab/ntp-mathlib huggingface dataset:
theorem normalizedFactors_prod_eq (s : Multiset α) (hs : ∀ a ∈ s, Irreducible a) :
    normalizedFactors s.prod = s.map normalize := by
  induction' s using Multiset.induction with a s ih
  · rw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero]
  · have ia := hs a (Multiset.mem_cons_self a _)
    

Proof state:
case cons
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
s : Multiset α
ih : (∀ a ∈ s, Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (⇑normalize) s
hs : ∀ a_1 ∈ a ::ₘ s, Irreducible a_1
ia : Irreducible a
⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (⇑normalize) (a ::ₘ s)

Next tactic:
have ib := fun b h => hs b (Multiset.mem_cons_of_mem h)
theorem normalizedFactors_prod_eq (s : Multiset α) (hs : ∀ a ∈ s, Irreducible a) :
    normalizedFactors s.prod = s.map normalize := by
  induction' s using Multiset.induction with a s ih
  · rw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero]
  · have ia := hs a (Multiset.mem_cons_self a _)
    have ib := fun b h => hs b (Multiset.mem_cons_of_mem h)
    

Proof state:
case cons
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
s : Multiset α
ih : (∀ a ∈ s, Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (⇑normalize) s
hs : ∀ a_1 ∈ a ::ₘ s, Irreducible a_1
ia : Irreducible a
ib : ∀ b ∈ s, Irreducible b
⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (⇑normalize) (a ::ₘ s)

Next tactic:
obtain rfl | ⟨b, hb⟩ := s.empty_or_exists_mem
theorem normalizedFactors_prod_eq (s : Multiset α) (hs : ∀ a ∈ s, Irreducible a) :
    normalizedFactors s.prod = s.map normalize := by
  induction' s using Multiset.induction with a s ih
  · rw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero]
  · have ia := hs a (Multiset.mem_cons_self a _)
    have ib := fun b h => hs b (Multiset.mem_cons_of_mem h)
    obtain rfl | ⟨b, hb⟩ := s.empty_or_exists_mem
    · 

Proof state:
case cons.inl
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ia : Irreducible a
ih : (∀ a ∈ 0, Irreducible a) → normalizedFactors (Multiset.prod 0) = Multiset.map (⇑normalize) 0
hs : ∀ a_1 ∈ a ::ₘ 0, Irreducible a_1
ib : ∀ b ∈ 0, Irreducible b
⊢ normalizedFactors (Multiset.prod (a ::ₘ 0)) = Multiset.map (⇑normalize) (a ::ₘ 0)

Next tactic:
rw [Multiset.cons_zero, Multiset.prod_singleton, Multiset.map_singleton,
        normalizedFactors_irreducible ia]
theorem normalizedFactors_prod_eq (s : Multiset α) (hs : ∀ a ∈ s, Irreducible a) :
    normalizedFactors s.prod = s.map normalize := by
  induction' s using Multiset.induction with a s ih
  · rw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero]
  · have ia := hs a (Multiset.mem_cons_self a _)
    have ib := fun b h => hs b (Multiset.mem_cons_of_mem h)
    obtain rfl | ⟨b, hb⟩ := s.empty_or_exists_mem
    · rw [Multiset.cons_zero, Multiset.prod_singleton, Multiset.map_singleton,
        normalizedFactors_irreducible ia]
    

Proof state:
case cons.inr.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
s : Multiset α
ih : (∀ a ∈ s, Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (⇑normalize) s
hs : ∀ a_1 ∈ a ::ₘ s, Irreducible a_1
ia : Irreducible a
ib : ∀ b ∈ s, Irreducible b
b : α
hb : b ∈ s
⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (⇑normalize) (a ::ₘ s)

Next tactic:
haveI := nontrivial_of_ne b 0 (ib b hb).ne_zero
theorem normalizedFactors_prod_eq (s : Multiset α) (hs : ∀ a ∈ s, Irreducible a) :
    normalizedFactors s.prod = s.map normalize := by
  induction' s using Multiset.induction with a s ih
  · rw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero]
  · have ia := hs a (Multiset.mem_cons_self a _)
    have ib := fun b h => hs b (Multiset.mem_cons_of_mem h)
    obtain rfl | ⟨b, hb⟩ := s.empty_or_exists_mem
    · rw [Multiset.cons_zero, Multiset.prod_singleton, Multiset.map_singleton,
        normalizedFactors_irreducible ia]
    haveI := nontrivial_of_ne b 0 (ib b hb).ne_zero
    

Proof state:
case cons.inr.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
s : Multiset α
ih : (∀ a ∈ s, Irreducible a) → normalizedFactors (Multiset.prod s) = Multiset.map (⇑normalize) s
hs : ∀ a_1 ∈ a ::ₘ s, Irreducible a_1
ia : Irreducible a
ib : ∀ b ∈ s, Irreducible b
b : α
hb : b ∈ s
this : Nontrivial α
⊢ normalizedFactors (Multiset.prod (a ::ₘ s)) = Multiset.map (⇑normalize) (a ::ₘ s)

Next tactic:
rw [Multiset.prod_cons, Multiset.map_cons,
      normalizedFactors_mul ia.ne_zero (Multiset.prod_ne_zero fun h => (ib 0 h).ne_zero rfl),
      normalizedFactors_irreducible ia, ih ib, Multiset.singleton_add]
theorem dvd_iff_normalizedFactors_le_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ∣ y ↔ normalizedFactors x ≤ normalizedFactors y := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ x ∣ y ↔ normalizedFactors x ≤ normalizedFactors y

Next tactic:
constructor
theorem dvd_iff_normalizedFactors_le_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ∣ y ↔ normalizedFactors x ≤ normalizedFactors y := by
  constructor
  · 

Proof state:
case mp
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ x ∣ y → normalizedFactors x ≤ normalizedFactors y

Next tactic:
rintro ⟨c, rfl⟩
theorem dvd_iff_normalizedFactors_le_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ∣ y ↔ normalizedFactors x ≤ normalizedFactors y := by
  constructor
  · rintro ⟨c, rfl⟩
    

Proof state:
case mp.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
c : α
hy : x * c ≠ 0
⊢ normalizedFactors x ≤ normalizedFactors (x * c)

Next tactic:
simp [hx, right_ne_zero_of_mul hy]
theorem dvd_iff_normalizedFactors_le_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ∣ y ↔ normalizedFactors x ≤ normalizedFactors y := by
  constructor
  · rintro ⟨c, rfl⟩
    simp [hx, right_ne_zero_of_mul hy]
  · 

Proof state:
case mpr
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ normalizedFactors x ≤ normalizedFactors y → x ∣ y

Next tactic:
rw [← (normalizedFactors_prod hx).dvd_iff_dvd_left, ←
      (normalizedFactors_prod hy).dvd_iff_dvd_right]
theorem dvd_iff_normalizedFactors_le_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ∣ y ↔ normalizedFactors x ≤ normalizedFactors y := by
  constructor
  · rintro ⟨c, rfl⟩
    simp [hx, right_ne_zero_of_mul hy]
  · rw [← (normalizedFactors_prod hx).dvd_iff_dvd_left, ←
      (normalizedFactors_prod hy).dvd_iff_dvd_right]
    

Proof state:
case mpr
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ normalizedFactors x ≤ normalizedFactors y → Multiset.prod (normalizedFactors x) ∣ Multiset.prod (normalizedFactors y)

Next tactic:
apply Multiset.prod_dvd_prod_of_le
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y

Next tactic:
refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by 

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : normalizedFactors x = normalizedFactors y
⊢ Multiset.prod (normalizedFactors x) ~ᵤ Multiset.prod (normalizedFactors y)

Next tactic:
rw [h]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ normalizedFactors x = normalizedFactors y

Next tactic:
apply le_antisymm
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> 

Proof state:
case a
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ normalizedFactors x ≤ normalizedFactors y

Next tactic:
rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> 

Proof state:
case a
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ normalizedFactors y ≤ normalizedFactors x

Next tactic:
rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
  

Proof state:
case a
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ x ∣ y
case a.hx
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ x ≠ 0
case a.hy
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ y ≠ 0
case a
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ y ∣ x
case a.hx
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ y ≠ 0
case a.hy
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ x ≠ 0

Next tactic:
all_goals simp [*, h.dvd, h.symm.dvd]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
  all_goals 

Proof state:
case a
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ x ∣ y

Next tactic:
simp [*, h.dvd, h.symm.dvd]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
  all_goals 

Proof state:
case a.hx
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ x ≠ 0

Next tactic:
simp [*, h.dvd, h.symm.dvd]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
  all_goals 

Proof state:
case a.hy
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ y ≠ 0

Next tactic:
simp [*, h.dvd, h.symm.dvd]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
  all_goals 

Proof state:
case a
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ y ∣ x

Next tactic:
simp [*, h.dvd, h.symm.dvd]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
  all_goals 

Proof state:
case a.hx
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ y ≠ 0

Next tactic:
simp [*, h.dvd, h.symm.dvd]
theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
    x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by
  refine'
    ⟨fun h => _, fun h =>
      (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
  apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
  all_goals 

Proof state:
case a.hy
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : x ~ᵤ y
⊢ x ≠ 0

Next tactic:
simp [*, h.dvd, h.symm.dvd]
theorem normalizedFactors_of_irreducible_pow {p : α} (hp : Irreducible p) (k : ℕ) :
    normalizedFactors (p ^ k) = Multiset.replicate k (normalize p) := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
p : α
hp : Irreducible p
k : ℕ
⊢ normalizedFactors (p ^ k) = Multiset.replicate k (normalize p)

Next tactic:
rw [normalizedFactors_pow, normalizedFactors_irreducible hp, Multiset.nsmul_singleton]
theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a) : p ∣ a := 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
H : p ∈ normalizedFactors a
⊢ p ∣ a

Next tactic:
by_cases hcases : a = 0
theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a) : p ∣ a := by
  by_cases hcases : a = 0
  · 

Proof state:
case pos
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
H : p ∈ normalizedFactors a
hcases : a = 0
⊢ p ∣ a

Next tactic:
rw [hcases]
theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a) : p ∣ a := by
  by_cases hcases : a = 0
  · rw [hcases]
    

Proof state:
case pos
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
H : p ∈ normalizedFactors a
hcases : a = 0
⊢ p ∣ 0

Next tactic:
exact dvd_zero p
theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a) : p ∣ a := by
  by_cases hcases : a = 0
  · rw [hcases]
    exact dvd_zero p
  · 

Proof state:
case neg
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
H : p ∈ normalizedFactors a
hcases : ¬a = 0
⊢ p ∣ a

Next tactic:
exact dvd_trans (Multiset.dvd_prod H) (Associated.dvd (normalizedFactors_prod hcases))
theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
    p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
  

Proof state:
α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : DecidableEq α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Unique αˣ
p x : α
hx : x ≠ 0
⊢ p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x

Next tactic:
constructor
theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
    p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
  constructor
  · 

Proof state:
case mp
α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : DecidableEq α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Unique αˣ
p x : α
hx : x ≠ 0
⊢ p ∈ normalizedFactors x → Prime p ∧ p ∣ x

Next tactic:
intro h
theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
    p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
  constructor
  · intro h
    

Proof state:
case mp
α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : DecidableEq α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Unique αˣ
p x : α
hx : x ≠ 0
h : p ∈ normalizedFactors x
⊢ Prime p ∧ p ∣ x

Next tactic:
exact ⟨prime_of_normalized_factor p h, dvd_of_mem_normalizedFactors h⟩
theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
    p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
  constructor
  · intro h
    exact ⟨prime_of_normalized_factor p h, dvd_of_mem_normalizedFactors h⟩
  · 

Proof state:
case mpr
α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : DecidableEq α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Unique αˣ
p x : α
hx : x ≠ 0
⊢ Prime p ∧ p ∣ x → p ∈ normalizedFactors x

Next tactic:
rintro ⟨hprime, hdvd⟩
theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
    p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
  constructor
  · intro h
    exact ⟨prime_of_normalized_factor p h, dvd_of_mem_normalizedFactors h⟩
  · rintro ⟨hprime, hdvd⟩
    

Proof state:
case mpr.intro
α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : DecidableEq α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Unique αˣ
p x : α
hx : x ≠ 0
hprime : Prime p
hdvd : p ∣ x
⊢ p ∈ normalizedFactors x

Next tactic:
obtain ⟨q, hqmem, hqeq⟩ := exists_mem_normalizedFactors_of_dvd hx hprime.irreducible hdvd
theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
    p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
  constructor
  · intro h
    exact ⟨prime_of_normalized_factor p h, dvd_of_mem_normalizedFactors h⟩
  · rintro ⟨hprime, hdvd⟩
    obtain ⟨q, hqmem, hqeq⟩ := exists_mem_normalizedFactors_of_dvd hx hprime.irreducible hdvd
    

Proof state:
case mpr.intro.intro.intro
α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : DecidableEq α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Unique αˣ
p x : α
hx : x ≠ 0
hprime : Prime p
hdvd : p ∣ x
q : α
hqmem : q ∈ normalizedFactors x
hqeq : p ~ᵤ q
⊢ p ∈ normalizedFactors x

Next tactic:
rw [associated_iff_eq] at hqeq
theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
    p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
  constructor
  · intro h
    exact ⟨prime_of_normalized_factor p h, dvd_of_mem_normalizedFactors h⟩
  · rintro ⟨hprime, hdvd⟩
    obtain ⟨q, hqmem, hqeq⟩ := exists_mem_normalizedFactors_of_dvd hx hprime.irreducible hdvd
    rw [associated_iff_eq] at hqeq
    

Proof state:
case mpr.intro.intro.intro
α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : DecidableEq α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Unique αˣ
p x : α
hx : x ≠ 0
hprime : Prime p
hdvd : p ∣ x
q : α
hqmem : q ∈ normalizedFactors x
hqeq : p = q
⊢ p ∈ normalizedFactors x

Next tactic:
exact hqeq ▸ hqmem
theorem exists_associated_prime_pow_of_unique_normalized_factor {p r : α}
    (h : ∀ {m}, m ∈ normalizedFactors r → m = p) (hr : r ≠ 0) : ∃ i : ℕ, Associated (p ^ i) r := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
p r : α
h : ∀ {m : α}, m ∈ normalizedFactors r → m = p
hr : r ≠ 0
⊢ ∃ i, p ^ i ~ᵤ r

Next tactic:
use Multiset.card.toFun (normalizedFactors r)
theorem exists_associated_prime_pow_of_unique_normalized_factor {p r : α}
    (h : ∀ {m}, m ∈ normalizedFactors r → m = p) (hr : r ≠ 0) : ∃ i : ℕ, Associated (p ^ i) r := by
  use Multiset.card.toFun (normalizedFactors r)
  

Proof state:
case h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
p r : α
h : ∀ {m : α}, m ∈ normalizedFactors r → m = p
hr : r ≠ 0
⊢ p ^ ZeroHom.toFun (↑Multiset.card) (normalizedFactors r) ~ᵤ r

Next tactic:
have := UniqueFactorizationMonoid.normalizedFactors_prod hr
theorem exists_associated_prime_pow_of_unique_normalized_factor {p r : α}
    (h : ∀ {m}, m ∈ normalizedFactors r → m = p) (hr : r ≠ 0) : ∃ i : ℕ, Associated (p ^ i) r := by
  use Multiset.card.toFun (normalizedFactors r)
  have := UniqueFactorizationMonoid.normalizedFactors_prod hr
  

Proof state:
case h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : DecidableEq α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
p r : α
h : ∀ {m : α}, m ∈ normalizedFactors r → m = p
hr : r ≠ 0
this : Multiset.prod (normalizedFactors r) ~ᵤ r
⊢ p ^ ZeroHom.toFun (↑Multiset.card) (normalizedFactors r) ~ᵤ r

Next tactic:
rwa [Multiset.eq_replicate_of_mem fun b => h, Multiset.prod_replicate] at this