l3lab/ntp-mathlib huggingface dataset:
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> 

Proof state:
case coe
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
q : FactorSet α
a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝ = prod q
⊢ ↑a✝ = q

Next tactic:
induction q using WithTop.recTopCoe
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · 

Proof state:
case top.top
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
h : prod ⊤ = prod ⊤
⊢ ⊤ = ⊤

Next tactic:
rfl
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · 

Proof state:
case top.coe
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝ : Multiset { a // Irreducible a }
h : prod ⊤ = prod ↑a✝
⊢ ⊤ = ↑a✝

Next tactic:
rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · 

Proof state:
case coe.top
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝ = prod ⊤
⊢ ↑a✝ = ⊤

Next tactic:
rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · 

Proof state:
case coe.coe
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
⊢ ↑a✝¹ = ↑a✝

Next tactic:
congr 1
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    

Proof state:
case coe.coe.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
⊢ a✝¹ = a✝

Next tactic:
rw [← Multiset.map_eq_map Subtype.coe_injective]
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    rw [← Multiset.map_eq_map Subtype.coe_injective]
    

Proof state:
case coe.coe.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
⊢ map (fun a => ↑a) a✝¹ = map (fun a => ↑a) a✝

Next tactic:
apply unique' _ _ h
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    rw [← Multiset.map_eq_map Subtype.coe_injective]
    apply unique' _ _ h <;>
      · 

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
⊢ ∀ a ∈ map Subtype.val a✝¹, Irreducible a

Next tactic:
intro a ha
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    rw [← Multiset.map_eq_map Subtype.coe_injective]
    apply unique' _ _ h <;>
      · intro a ha
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
a : Associates α
ha : a ∈ map Subtype.val a✝¹
⊢ Irreducible a

Next tactic:
obtain ⟨⟨a', irred⟩, -, rfl⟩ := Multiset.mem_map.mp ha
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    rw [← Multiset.map_eq_map Subtype.coe_injective]
    apply unique' _ _ h <;>
      · intro a ha
        obtain ⟨⟨a', irred⟩, -, rfl⟩ := Multiset.mem_map.mp ha
        

Proof state:
case intro.mk.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
a' : Associates α
irred : Irreducible a'
ha : ↑{ val := a', property := irred } ∈ map Subtype.val a✝¹
⊢ Irreducible ↑{ val := a', property := irred }

Next tactic:
rwa [Subtype.coe_mk]
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    rw [← Multiset.map_eq_map Subtype.coe_injective]
    apply unique' _ _ h <;>
      · 

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
⊢ ∀ a ∈ map Subtype.val a✝, Irreducible a

Next tactic:
intro a ha
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    rw [← Multiset.map_eq_map Subtype.coe_injective]
    apply unique' _ _ h <;>
      · intro a ha
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
a : Associates α
ha : a ∈ map Subtype.val a✝
⊢ Irreducible a

Next tactic:
obtain ⟨⟨a', irred⟩, -, rfl⟩ := Multiset.mem_map.mp ha
theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q := by
  induction p using WithTop.recTopCoe <;> induction q using WithTop.recTopCoe
  · rfl
  · rw [eq_comm, ← FactorSet.prod_eq_zero_iff, ← h, Associates.prod_top]
  · rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top]
  · congr 1
    rw [← Multiset.map_eq_map Subtype.coe_injective]
    apply unique' _ _ h <;>
      · intro a ha
        obtain ⟨⟨a', irred⟩, -, rfl⟩ := Multiset.mem_map.mp ha
        

Proof state:
case intro.mk.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
a' : Associates α
irred : Irreducible a'
ha : ↑{ val := a', property := irred } ∈ map Subtype.val a✝
⊢ Irreducible ↑{ val := a', property := irred }

Next tactic:
rwa [Subtype.coe_mk]
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
⊢ Multiset.prod p ≤ Multiset.prod q → p ≤ q

Next tactic:
classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · exact hp x h
          · exact irreducible_of_factor _ h
        · rw [eqc, Multiset.prod_add]
          congr
          refine' associated_iff_eq.mp (factors_prod fun hc => _).symm
          refine' not_irreducible_zero (hq _ _)
          rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        

Proof state:
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
⊢ Multiset.prod p ≤ Multiset.prod q → p ≤ q

Next tactic:
rintro ⟨c, eqc⟩
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        

Proof state:
case intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
⊢ p ≤ q

Next tactic:
refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · 

Proof state:
case intro.refine'_1
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
x : Associates α
hx : x ∈ p + factors c
⊢ Irreducible x

Next tactic:
obtain h | h := Multiset.mem_add.1 hx
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · 

Proof state:
case intro.refine'_1.inl
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ p
⊢ Irreducible x

Next tactic:
exact hp x h
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · exact hp x h
          · 

Proof state:
case intro.refine'_1.inr
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ factors c
⊢ Irreducible x

Next tactic:
exact irreducible_of_factor _ h
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · exact hp x h
          · exact irreducible_of_factor _ h
        · 

Proof state:
case intro.refine'_2
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
⊢ Multiset.prod q = Multiset.prod (p + factors c)

Next tactic:
rw [eqc, Multiset.prod_add]
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · exact hp x h
          · exact irreducible_of_factor _ h
        · rw [eqc, Multiset.prod_add]
          

Proof state:
case intro.refine'_2
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
⊢ Multiset.prod p * c = Multiset.prod p * Multiset.prod (factors c)

Next tactic:
congr
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · exact hp x h
          · exact irreducible_of_factor _ h
        · rw [eqc, Multiset.prod_add]
          congr
          

Proof state:
case intro.refine'_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
⊢ c = Multiset.prod (factors c)

Next tactic:
refine' associated_iff_eq.mp (factors_prod fun hc => _).symm
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · exact hp x h
          · exact irreducible_of_factor _ h
        · rw [eqc, Multiset.prod_add]
          congr
          refine' associated_iff_eq.mp (factors_prod fun hc => _).symm
          

Proof state:
case intro.refine'_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
hc : c = 0
⊢ False

Next tactic:
refine' not_irreducible_zero (hq _ _)
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
    (hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q :=
  Iff.intro
    (by
      classical
        rintro ⟨c, eqc⟩
        refine' Multiset.le_iff_exists_add.2 ⟨factors c, unique' hq (fun x hx => _) _⟩
        · obtain h | h := Multiset.mem_add.1 hx
          · exact hp x h
          · exact irreducible_of_factor _ h
        · rw [eqc, Multiset.prod_add]
          congr
          refine' associated_iff_eq.mp (factors_prod fun hc => _).symm
          refine' not_irreducible_zero (hq _ _)
          

Proof state:
case intro.refine'_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : Multiset.prod q = Multiset.prod p * c
hc : c = 0
⊢ 0 ∈ q

Next tactic:
rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
@[simp]
theorem map_subtype_coe_factors' {a : α} :
    (factors' a).map (↑) = (factors a).map Associates.mk := 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
⊢ map Subtype.val (factors' a) = map Associates.mk (factors a)

Next tactic:
simp [factors', Multiset.map_pmap, Multiset.pmap_eq_map]
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' 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 : α
h : a ~ᵤ b
⊢ factors' a = factors' b

Next tactic:
obtain rfl | hb := eq_or_ne b 0
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · 

Proof state:
case inl
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 ~ᵤ 0
⊢ factors' a = factors' 0

Next tactic:
rw [associated_zero_iff_eq_zero] at h
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · rw [associated_zero_iff_eq_zero] at h
    

Proof state:
case inl
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' a = factors' 0

Next tactic:
rw [h]
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · rw [associated_zero_iff_eq_zero] at h
    rw [h]
  

Proof state:
case inr
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 : α
h : a ~ᵤ b
hb : b ≠ 0
⊢ factors' a = factors' b

Next tactic:
have ha : a ≠ 0 := by
    contrapose! hb with ha
    rw [← associated_zero_iff_eq_zero, ← ha]
    exact h.symm
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · rw [associated_zero_iff_eq_zero] at h
    rw [h]
  have ha : 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 b : α
h : a ~ᵤ b
hb : b ≠ 0
⊢ a ≠ 0

Next tactic:
contrapose! hb with ha
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · rw [associated_zero_iff_eq_zero] at h
    rw [h]
  have ha : a ≠ 0 := by
    contrapose! hb with ha
    

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 : α
h : a ~ᵤ b
ha : a = 0
⊢ b = 0

Next tactic:
rw [← associated_zero_iff_eq_zero, ← ha]
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · rw [associated_zero_iff_eq_zero] at h
    rw [h]
  have ha : a ≠ 0 := by
    contrapose! hb with ha
    rw [← associated_zero_iff_eq_zero, ← ha]
    

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 : α
h : a ~ᵤ b
ha : a = 0
⊢ b ~ᵤ a

Next tactic:
exact h.symm
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · rw [associated_zero_iff_eq_zero] at h
    rw [h]
  have ha : a ≠ 0 := by
    contrapose! hb with ha
    rw [← associated_zero_iff_eq_zero, ← ha]
    exact h.symm
  

Proof state:
case inr
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 : α
h : a ~ᵤ b
hb : b ≠ 0
ha : a ≠ 0
⊢ factors' a = factors' b

Next tactic:
rw [← Multiset.map_eq_map Subtype.coe_injective, map_subtype_coe_factors',
    map_subtype_coe_factors', ← rel_associated_iff_map_eq_map]
theorem factors'_cong {a b : α} (h : a ~ᵤ b) : factors' a = factors' b := by
  obtain rfl | hb := eq_or_ne b 0
  · rw [associated_zero_iff_eq_zero] at h
    rw [h]
  have ha : a ≠ 0 := by
    contrapose! hb with ha
    rw [← associated_zero_iff_eq_zero, ← ha]
    exact h.symm
  rw [← Multiset.map_eq_map Subtype.coe_injective, map_subtype_coe_factors',
    map_subtype_coe_factors', ← rel_associated_iff_map_eq_map]
  

Proof state:
case inr
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 : α
h : a ~ᵤ b
hb : b ≠ 0
ha : a ≠ 0
⊢ Rel Associated (factors a) (factors b)

Next tactic:
exact
    factors_unique irreducible_of_factor irreducible_of_factor
      ((factors_prod ha).trans <| h.trans <| (factors_prod hb).symm)
/-- 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
  

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
⊢ FactorSet α

Next tactic:
refine' if h : a = 0 then ⊤ else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h