l3lab/ntp-mathlib huggingface dataset:
/-- This returns the multiset of irreducible factors of an associate as a `FactorSet`,
  a multiset of irreducible associates `WithTop`. -/
noncomputable def factors (a : Associates α) : FactorSet α := by
  refine' if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
h : ¬a = 0
⊢ ∀ (a b : α), a ≈ b → HEq (fun x => some (factors' a)) fun x => some (factors' b)

Next tactic:
intro a b hab
/-- This returns the multiset of irreducible factors of an associate as a `FactorSet`,
  a multiset of irreducible associates `WithTop`. -/
noncomputable def factors (a : Associates α) : FactorSet α := by
  refine' if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h
  intro a b hab
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a✝ : Associates α
h : ¬a✝ = 0
a b : α
hab : a ≈ b
⊢ HEq (fun x => some (factors' a)) fun x => some (factors' b)

Next tactic:
apply Function.hfunext
/-- This returns the multiset of irreducible factors of an associate as a `FactorSet`,
  a multiset of irreducible associates `WithTop`. -/
noncomputable def factors (a : Associates α) : FactorSet α := by
  refine' if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h
  intro a b hab
  apply Function.hfunext
  · 

Proof state:
case hα
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a✝ : Associates α
h : ¬a✝ = 0
a b : α
hab : a ≈ b
⊢ (¬⟦a⟧ = 0) = ¬⟦b⟧ = 0

Next tactic:
have : a ~ᵤ 0 ↔ b ~ᵤ 0 := Iff.intro (fun ha0 => hab.symm.trans ha0) fun hb0 => hab.trans hb0
/-- This returns the multiset of irreducible factors of an associate as a `FactorSet`,
  a multiset of irreducible associates `WithTop`. -/
noncomputable def factors (a : Associates α) : FactorSet α := by
  refine' if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h
  intro a b hab
  apply Function.hfunext
  · have : a ~ᵤ 0 ↔ b ~ᵤ 0 := Iff.intro (fun ha0 => hab.symm.trans ha0) fun hb0 => hab.trans hb0
    

Proof state:
case hα
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a✝ : Associates α
h : ¬a✝ = 0
a b : α
hab : a ≈ b
this : a ~ᵤ 0 ↔ b ~ᵤ 0
⊢ (¬⟦a⟧ = 0) = ¬⟦b⟧ = 0

Next tactic:
simp only [associated_zero_iff_eq_zero] at this
/-- This returns the multiset of irreducible factors of an associate as a `FactorSet`,
  a multiset of irreducible associates `WithTop`. -/
noncomputable def factors (a : Associates α) : FactorSet α := by
  refine' if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h
  intro a b hab
  apply Function.hfunext
  · have : a ~ᵤ 0 ↔ b ~ᵤ 0 := Iff.intro (fun ha0 => hab.symm.trans ha0) fun hb0 => hab.trans hb0
    simp only [associated_zero_iff_eq_zero] at this
    

Proof state:
case hα
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a✝ : Associates α
h : ¬a✝ = 0
a b : α
hab : a ≈ b
this : a = 0 ↔ b = 0
⊢ (¬⟦a⟧ = 0) = ¬⟦b⟧ = 0

Next tactic:
simp only [quotient_mk_eq_mk, this, mk_eq_zero]
/-- This returns the multiset of irreducible factors of an associate as a `FactorSet`,
  a multiset of irreducible associates `WithTop`. -/
noncomputable def factors (a : Associates α) : FactorSet α := by
  refine' if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h
  intro a b hab
  apply Function.hfunext
  · have : a ~ᵤ 0 ↔ b ~ᵤ 0 := Iff.intro (fun ha0 => hab.symm.trans ha0) fun hb0 => hab.trans hb0
    simp only [associated_zero_iff_eq_zero] at this
    simp only [quotient_mk_eq_mk, this, mk_eq_zero]
  

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a✝ : Associates α
h : ¬a✝ = 0
a b : α
hab : a ≈ b
⊢ ∀ (a_1 : ¬⟦a⟧ = 0) (a' : ¬⟦b⟧ = 0), HEq a_1 a' → HEq (some (factors' a)) (some (factors' b))

Next tactic:
exact fun ha hb _ => heq_of_eq <| congr_arg some <| factors'_cong hab
@[simp]
theorem factors_mk (a : α) (h : a ≠ 0) : (Associates.mk a).factors = factors' a := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : α
h : a ≠ 0
⊢ factors (Associates.mk a) = ↑(factors' a)

Next tactic:
classical
    apply dif_neg
    apply mt mk_eq_zero.1 h
@[simp]
theorem factors_mk (a : α) (h : a ≠ 0) : (Associates.mk a).factors = factors' a := by
  classical
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : α
h : a ≠ 0
⊢ factors (Associates.mk a) = ↑(factors' a)

Next tactic:
apply dif_neg
@[simp]
theorem factors_mk (a : α) (h : a ≠ 0) : (Associates.mk a).factors = factors' a := by
  classical
    apply dif_neg
    

Proof state:
case hnc
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : α
h : a ≠ 0
⊢ ¬Associates.mk a = 0

Next tactic:
apply mt mk_eq_zero.1 h
@[simp]
theorem factors_prod (a : Associates α) : a.factors.prod = a :=
  Quotient.inductionOn a fun b =>
    Decidable.byCases (fun (this : Associates.mk b = 0) => by 

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
b : α
this : Associates.mk b = 0
⊢ FactorSet.prod (factors ⟦b⟧) = ⟦b⟧

Next tactic:
simp [quotient_mk_eq_mk, this]
@[simp]
theorem factors_prod (a : Associates α) : a.factors.prod = a :=
  Quotient.inductionOn a fun b =>
    Decidable.byCases (fun (this : Associates.mk b = 0) => by simp [quotient_mk_eq_mk, this])
      fun (_ : Associates.mk b ≠ 0) => by
      

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
b : α
x✝ : Associates.mk b ≠ 0
⊢ FactorSet.prod (factors ⟦b⟧) = ⟦b⟧

Next tactic:
have : b ≠ 0 := by simp_all
@[simp]
theorem factors_prod (a : Associates α) : a.factors.prod = a :=
  Quotient.inductionOn a fun b =>
    Decidable.byCases (fun (this : Associates.mk b = 0) => by simp [quotient_mk_eq_mk, this])
      fun (_ : Associates.mk b ≠ 0) => by
      have : b ≠ 0 := by 

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
b : α
x✝ : Associates.mk b ≠ 0
⊢ b ≠ 0

Next tactic:
simp_all
@[simp]
theorem factors_prod (a : Associates α) : a.factors.prod = a :=
  Quotient.inductionOn a fun b =>
    Decidable.byCases (fun (this : Associates.mk b = 0) => by simp [quotient_mk_eq_mk, this])
      fun (_ : Associates.mk b ≠ 0) => by
      have : b ≠ 0 := by simp_all
      

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
b : α
x✝ : Associates.mk b ≠ 0
this : b ≠ 0
⊢ FactorSet.prod (factors ⟦b⟧) = ⟦b⟧

Next tactic:
simp [this, quotient_mk_eq_mk, prod_mk,
          mk_eq_mk_iff_associated.2 (UniqueFactorizationMonoid.factors_prod this)]
@[nontriviality]
theorem factors_subsingleton [Subsingleton α] {a : Associates α} : a.factors = Option.none := by
  

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Subsingleton α
a : Associates α
⊢ factors a = none

Next tactic:
convert @factors_0 _ _ _ _ _
theorem factors_eq_none_iff_zero {a : Associates α} : a.factors = Option.none ↔ a = 0 := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
⊢ factors a = none ↔ a = 0

Next tactic:
nontriviality α
theorem factors_eq_none_iff_zero {a : Associates α} : a.factors = Option.none ↔ a = 0 := by
  nontriviality α
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
✝ : Nontrivial α
⊢ factors a = none ↔ a = 0

Next tactic:
exact
    ⟨fun h => by rwa [← factors_prod a, FactorSet.prod_eq_zero_iff], fun h => h.symm ▸ factors_0⟩
theorem factors_eq_none_iff_zero {a : Associates α} : a.factors = Option.none ↔ a = 0 := by
  nontriviality α
  exact
    ⟨fun h => by 

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
✝ : Nontrivial α
h : factors a = none
⊢ a = 0

Next tactic:
rwa [← factors_prod a, FactorSet.prod_eq_zero_iff]
theorem factors_eq_some_iff_ne_zero {a : Associates α} :
    (∃ s : Multiset { p : Associates α // Irreducible p }, a.factors = some s) ↔ a ≠ 0 := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a : Associates α
⊢ (∃ s, factors a = some s) ↔ a ≠ 0

Next tactic:
rw [← Option.isSome_iff_exists, ← Option.ne_none_iff_isSome, Ne.def, Ne.def,
    factors_eq_none_iff_zero]
theorem eq_of_factors_eq_factors {a b : Associates α} (h : a.factors = b.factors) : a = b := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
h : factors a = factors b
⊢ a = b

Next tactic:
have : a.factors.prod = b.factors.prod := by rw [h]
theorem eq_of_factors_eq_factors {a b : Associates α} (h : a.factors = b.factors) : a = b := by
  have : a.factors.prod = b.factors.prod := by 

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
h : factors a = factors b
⊢ FactorSet.prod (factors a) = FactorSet.prod (factors b)

Next tactic:
rw [h]
theorem eq_of_factors_eq_factors {a b : Associates α} (h : a.factors = b.factors) : a = b := by
  have : a.factors.prod = b.factors.prod := by rw [h]
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
h : factors a = factors b
this : FactorSet.prod (factors a) = FactorSet.prod (factors b)
⊢ a = b

Next tactic:
rwa [factors_prod, factors_prod] at this
theorem eq_of_prod_eq_prod [Nontrivial α] {a b : FactorSet α} (h : a.prod = b.prod) : a = b := by
  

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b : FactorSet α
h : FactorSet.prod a = FactorSet.prod b
⊢ a = b

Next tactic:
classical
    have : a.prod.factors = b.prod.factors := by rw [h]
    rwa [prod_factors, prod_factors] at this
theorem eq_of_prod_eq_prod [Nontrivial α] {a b : FactorSet α} (h : a.prod = b.prod) : a = b := by
  classical
    

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b : FactorSet α
h : FactorSet.prod a = FactorSet.prod b
⊢ a = b

Next tactic:
have : a.prod.factors = b.prod.factors := by rw [h]
theorem eq_of_prod_eq_prod [Nontrivial α] {a b : FactorSet α} (h : a.prod = b.prod) : a = b := by
  classical
    have : a.prod.factors = b.prod.factors := by 

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b : FactorSet α
h : FactorSet.prod a = FactorSet.prod b
⊢ factors (FactorSet.prod a) = factors (FactorSet.prod b)

Next tactic:
rw [h]
theorem eq_of_prod_eq_prod [Nontrivial α] {a b : FactorSet α} (h : a.prod = b.prod) : a = b := by
  classical
    have : a.prod.factors = b.prod.factors := by rw [h]
    

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
inst✝ : Nontrivial α
a b : FactorSet α
h : FactorSet.prod a = FactorSet.prod b
this : factors (FactorSet.prod a) = factors (FactorSet.prod b)
⊢ a = b

Next tactic:
rwa [prod_factors, prod_factors] at this
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (p : Associates α), Irreducible p → count p (factors a) = count p (factors b)
⊢ factors a = factors b

Next tactic:
obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  

Proof state:
case intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (p : Associates α), Irreducible p → count p (factors a) = count p (factors b)
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
⊢ factors a = factors b

Next tactic:
obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
  

Proof state:
case intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (p : Associates α), Irreducible p → count p (factors a) = count p (factors b)
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h_sb : factors b = some sb
⊢ factors a = factors b

Next tactic:
rw [h_sa, h_sb] at h ⊢
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
  rw [h_sa, h_sb] at h ⊢
  

Proof state:
case intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → count p (some sa) = count p (some sb)
h_sb : factors b = some sb
⊢ some sa = some sb

Next tactic:
rw [Option.some_inj]
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
  rw [h_sa, h_sb] at h ⊢
  rw [Option.some_inj]
  

Proof state:
case intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → count p (some sa) = count p (some sb)
h_sb : factors b = some sb
⊢ sa = sb

Next tactic:
have h_count : ∀ (p : Associates α) (hp : Irreducible p),
      sa.count ⟨p, hp⟩ = sb.count ⟨p, hp⟩ := by
    intro p hp
    rw [← count_some, ← count_some, h p hp]
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
  rw [h_sa, h_sb] at h ⊢
  rw [Option.some_inj]
  have h_count : ∀ (p : Associates α) (hp : Irreducible p),
      sa.count ⟨p, hp⟩ = sb.count ⟨p, hp⟩ := by
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → count p (some sa) = count p (some sb)
h_sb : factors b = some sb
⊢ ∀ (p : Associates α) (hp : Irreducible p),
    Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb

Next tactic:
intro p hp
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
  rw [h_sa, h_sb] at h ⊢
  rw [Option.some_inj]
  have h_count : ∀ (p : Associates α) (hp : Irreducible p),
      sa.count ⟨p, hp⟩ = sb.count ⟨p, hp⟩ := by
    intro p hp
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → count p (some sa) = count p (some sb)
h_sb : factors b = some sb
p : Associates α
hp : Irreducible p
⊢ Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb

Next tactic:
rw [← count_some, ← count_some, h p hp]
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
  rw [h_sa, h_sb] at h ⊢
  rw [Option.some_inj]
  have h_count : ∀ (p : Associates α) (hp : Irreducible p),
      sa.count ⟨p, hp⟩ = sb.count ⟨p, hp⟩ := by
    intro p hp
    rw [← count_some, ← count_some, h p hp]
  

Proof state:
case intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → count p (some sa) = count p (some sb)
h_sb : factors b = some sb
h_count :
  ∀ (p : Associates α) (hp : Irreducible p),
    Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb
⊢ sa = sb

Next tactic:
apply Multiset.toFinsupp.injective
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
    a.factors = b.factors := by
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
  rw [h_sa, h_sb] at h ⊢
  rw [Option.some_inj]
  have h_count : ∀ (p : Associates α) (hp : Irreducible p),
      sa.count ⟨p, hp⟩ = sb.count ⟨p, hp⟩ := by
    intro p hp
    rw [← count_some, ← count_some, h p hp]
  apply Multiset.toFinsupp.injective
  

Proof state:
case intro.intro.a
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → count p (some sa) = count p (some sb)
h_sb : factors b = some sb
h_count :
  ∀ (p : Associates α) (hp : Irreducible p),
    Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb
⊢ toFinsupp sa = toFinsupp sb

Next tactic:
ext ⟨p, hp⟩