l3lab/ntp-mathlib huggingface dataset:
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
  ext ⟨p, hp⟩
  

Proof state:
case intro.intro.a.h.mk
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
p : Associates α
hp : Irreducible p
⊢ (toFinsupp sa) { val := p, property := hp } = (toFinsupp sb) { val := p, property := hp }

Next tactic:
rw [Multiset.toFinsupp_apply, Multiset.toFinsupp_apply, h_count p hp]
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count 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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : factors a ≤ factors b
⊢ count p (factors a) ≤ count p (factors b)

Next tactic:
by_cases ha : a = 0
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by
  by_cases ha : a = 0
  · 

Proof state:
case pos
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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : factors a ≤ factors b
ha : a = 0
⊢ count p (factors a) ≤ count p (factors b)

Next tactic:
simp_all
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by
  by_cases ha : a = 0
  · simp_all
  

Proof state:
case neg
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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : factors a ≤ factors b
ha : ¬a = 0
⊢ count p (factors a) ≤ count p (factors b)

Next tactic:
obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by
  by_cases ha : a = 0
  · simp_all
  obtain ⟨sa, h_sa⟩ := factors_eq_some_iff_ne_zero.mpr ha
  

Proof state:
case neg.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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : factors a ≤ factors b
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
⊢ count p (factors a) ≤ count p (factors b)

Next tactic:
obtain ⟨sb, h_sb⟩ := factors_eq_some_iff_ne_zero.mpr hb
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by
  by_cases ha : a = 0
  · simp_all
  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 neg.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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : factors a ≤ factors b
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h_sb : factors b = some sb
⊢ count p (factors a) ≤ count p (factors b)

Next tactic:
rw [h_sa, h_sb] at h ⊢
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by
  by_cases ha : a = 0
  · simp_all
  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 neg.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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : some sa ≤ some sb
h_sb : factors b = some sb
⊢ count p (some sa) ≤ count p (some sb)

Next tactic:
rw [count_some hp, count_some hp]
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by
  by_cases ha : a = 0
  · simp_all
  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 [count_some hp, count_some hp]; 

Proof state:
case neg.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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : some sa ≤ some sb
h_sb : factors b = some sb
⊢ Multiset.count { val := p, property := hp } sa ≤ Multiset.count { val := p, property := hp } sb

Next tactic:
rw [WithTop.some_le_some] at h
theorem count_le_count_of_factors_le {a b p : Associates α} (hb : b ≠ 0) (hp : Irreducible p)
    (h : a.factors ≤ b.factors) : p.count a.factors ≤ p.count b.factors := by
  by_cases ha : a = 0
  · simp_all
  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 [count_some hp, count_some hp]; rw [WithTop.some_le_some] at h
  

Proof state:
case neg.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 p : Associates α
hb : b ≠ 0
hp : Irreducible p
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : factors a = some sa
sb : Multiset { p // Irreducible p }
h : sa ≤ sb
h_sb : factors b = some sb
⊢ Multiset.count { val := p, property := hp } sa ≤ Multiset.count { val := p, property := hp } sb

Next tactic:
exact Multiset.count_le_of_le _ h
@[simp]
theorem factors_mul (a b : Associates α) : (a * 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
⊢ factors (a * b) = factors a + factors b

Next tactic:
cases subsingleton_or_nontrivial α
@[simp]
theorem factors_mul (a b : Associates α) : (a * b).factors = a.factors + b.factors := by
  cases subsingleton_or_nontrivial α
  · 

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 b : Associates α
h✝ : Subsingleton α
⊢ factors (a * b) = factors a + factors b

Next tactic:
simp [Subsingleton.elim a 0]
@[simp]
theorem factors_mul (a b : Associates α) : (a * b).factors = a.factors + b.factors := by
  cases subsingleton_or_nontrivial α
  · simp [Subsingleton.elim a 0]
  

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 : Associates α
h✝ : Nontrivial α
⊢ factors (a * b) = factors a + factors b

Next tactic:
refine' eq_of_prod_eq_prod (eq_of_factors_eq_factors _)
@[simp]
theorem factors_mul (a b : Associates α) : (a * b).factors = a.factors + b.factors := by
  cases subsingleton_or_nontrivial α
  · simp [Subsingleton.elim a 0]
  refine' eq_of_prod_eq_prod (eq_of_factors_eq_factors _)
  

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 : Associates α
h✝ : Nontrivial α
⊢ factors (FactorSet.prod (factors (a * b))) = factors (FactorSet.prod (factors a + factors b))

Next tactic:
rw [prod_add, factors_prod, factors_prod, factors_prod]
theorem factors_mono : ∀ {a b : Associates α}, a ≤ b → a.factors ≤ b.factors
  | s, t, ⟨d, eq⟩ => 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)
s t : Associates α
d : Associates α
eq : t = s * d
⊢ factors s ≤ factors t

Next tactic:
rw [eq, factors_mul]
theorem factors_mono : ∀ {a b : Associates α}, a ≤ b → a.factors ≤ b.factors
  | s, t, ⟨d, eq⟩ => by rw [eq, factors_mul]; 

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)
s t : Associates α
d : Associates α
eq : t = s * d
⊢ factors s ≤ factors s + factors d

Next tactic:
exact le_add_of_nonneg_right bot_le
theorem factors_le {a b : Associates α} : a.factors ≤ b.factors ↔ a ≤ b :=
  Iff.intro
    (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 b : Associates α
h : factors a ≤ factors b
⊢ a ≤ b

Next tactic:
have : a.factors.prod ≤ b.factors.prod := prod_mono h
theorem factors_le {a b : Associates α} : a.factors ≤ b.factors ↔ a ≤ b :=
  Iff.intro
    (fun h => by
      have : a.factors.prod ≤ b.factors.prod := prod_mono 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 prod_le [Nontrivial α] {a b : FactorSet α} : 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
⊢ FactorSet.prod a ≤ FactorSet.prod b ↔ a ≤ b

Next tactic:
classical
    exact
      Iff.intro
        (fun h => by
          have : a.prod.factors ≤ b.prod.factors := factors_mono h
          rwa [prod_factors, prod_factors] at this)
        prod_mono
theorem prod_le [Nontrivial α] {a b : FactorSet α} : 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
⊢ FactorSet.prod a ≤ FactorSet.prod b ↔ a ≤ b

Next tactic:
exact
      Iff.intro
        (fun h => by
          have : a.prod.factors ≤ b.prod.factors := factors_mono h
          rwa [prod_factors, prod_factors] at this)
        prod_mono
theorem prod_le [Nontrivial α] {a b : FactorSet α} : a.prod ≤ b.prod ↔ a ≤ b := by
  classical
    exact
      Iff.intro
        (fun h => 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:
have : a.prod.factors ≤ b.prod.factors := factors_mono h
theorem prod_le [Nontrivial α] {a b : FactorSet α} : a.prod ≤ b.prod ↔ a ≤ b := by
  classical
    exact
      Iff.intro
        (fun h => by
          have : a.prod.factors ≤ b.prod.factors := factors_mono 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 sup_mul_inf (a b : Associates α) : (a ⊔ b) * (a ⊓ b) = a * b :=
  show (a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod = 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
⊢ FactorSet.prod (factors a ⊔ factors b) * FactorSet.prod (factors a ⊓ factors b) = a * b

Next tactic:
nontriviality α
theorem sup_mul_inf (a b : Associates α) : (a ⊔ b) * (a ⊓ b) = a * b :=
  show (a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod = a * b 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 b : Associates α
✝ : Nontrivial α
⊢ FactorSet.prod (factors a ⊔ factors b) * FactorSet.prod (factors a ⊓ factors b) = a * b

Next tactic:
refine' eq_of_factors_eq_factors _
theorem sup_mul_inf (a b : Associates α) : (a ⊔ b) * (a ⊓ b) = a * b :=
  show (a.factors ⊔ b.factors).prod * (a.factors ⊓ b.factors).prod = a * b by
    nontriviality α
    refine' eq_of_factors_eq_factors _
    

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
✝ : Nontrivial α
⊢ factors (FactorSet.prod (factors a ⊔ factors b) * FactorSet.prod (factors a ⊓ factors b)) = factors (a * b)

Next tactic:
rw [← prod_add, prod_factors, factors_mul, FactorSet.sup_add_inf_eq_add]
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ 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 p : Associates α
hp : Irreducible p
hm : p ∈ factors a
⊢ p ∣ a

Next tactic:
by_cases ha0 : a = 0
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · 

Proof state:
case pos
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 p : Associates α
hp : Irreducible p
hm : p ∈ factors a
ha0 : a = 0
⊢ p ∣ a

Next tactic:
rw [ha0]
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    

Proof state:
case pos
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 p : Associates α
hp : Irreducible p
hm : p ∈ factors a
ha0 : a = 0
⊢ p ∣ 0

Next tactic:
exact dvd_zero p
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    exact dvd_zero p
  

Proof state:
case neg
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 p : Associates α
hp : Irreducible p
hm : p ∈ factors a
ha0 : ¬a = 0
⊢ p ∣ a

Next tactic:
obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    exact dvd_zero p
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
  

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
hm : p ∈ factors a
ha0 : ¬a = 0
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ a

Next tactic:
rw [← Associates.factors_prod a]
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    exact dvd_zero p
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
  rw [← Associates.factors_prod a]
  

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
hm : p ∈ factors a
ha0 : ¬a = 0
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ FactorSet.prod (factors a)

Next tactic:
rw [← ha', factors_mk a0 nza] at hm ⊢
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    exact dvd_zero p
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
  rw [← Associates.factors_prod a]
  rw [← ha', factors_mk a0 nza] at hm ⊢
  

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
ha0 : ¬a = 0
a0 : α
hm : p ∈ ↑(factors' a0)
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ FactorSet.prod ↑(factors' a0)

Next tactic:
rw [prod_coe]
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    exact dvd_zero p
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
  rw [← Associates.factors_prod a]
  rw [← ha', factors_mk a0 nza] at hm ⊢
  rw [prod_coe]
  

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
ha0 : ¬a = 0
a0 : α
hm : p ∈ ↑(factors' a0)
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ Multiset.prod (map Subtype.val (factors' a0))

Next tactic:
apply Multiset.dvd_prod
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    exact dvd_zero p
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
  rw [← Associates.factors_prod a]
  rw [← ha', factors_mk a0 nza] at hm ⊢
  rw [prod_coe]
  apply Multiset.dvd_prod; 

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
ha0 : ¬a = 0
a0 : α
hm : p ∈ ↑(factors' a0)
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∈ map Subtype.val (factors' a0)

Next tactic:
apply Multiset.mem_map.mpr
theorem dvd_of_mem_factors {a p : Associates α} {hp : Irreducible p} (hm : p ∈ factors a) :
    p ∣ a := by
  by_cases ha0 : a = 0
  · rw [ha0]
    exact dvd_zero p
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
  rw [← Associates.factors_prod a]
  rw [← ha', factors_mk a0 nza] at hm ⊢
  rw [prod_coe]
  apply Multiset.dvd_prod; apply Multiset.mem_map.mpr
  

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
ha0 : ¬a = 0
a0 : α
hm : p ∈ ↑(factors' a0)
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ ∃ a ∈ factors' a0, ↑a = p

Next tactic:
exact ⟨⟨p, hp⟩, mem_factorSet_some.mp hm, rfl⟩
theorem dvd_of_mem_factors' {a : α} {p : Associates α} {hp : Irreducible p} {hz : a ≠ 0}
    (h_mem : Subtype.mk p hp ∈ factors' a) : p ∣ Associates.mk 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
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem : { val := p, property := hp } ∈ factors' a
⊢ p ∣ Associates.mk a

Next tactic:
haveI := Classical.decEq (Associates α)
theorem dvd_of_mem_factors' {a : α} {p : Associates α} {hp : Irreducible p} {hz : a ≠ 0}
    (h_mem : Subtype.mk p hp ∈ factors' a) : p ∣ Associates.mk a := by
  haveI := Classical.decEq (Associates 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 : α
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem : { val := p, property := hp } ∈ factors' a
this : DecidableEq (Associates α)
⊢ p ∣ Associates.mk a

Next tactic:
apply dvd_of_mem_factors (hp := hp)
theorem dvd_of_mem_factors' {a : α} {p : Associates α} {hp : Irreducible p} {hz : a ≠ 0}
    (h_mem : Subtype.mk p hp ∈ factors' a) : p ∣ Associates.mk a := by
  haveI := Classical.decEq (Associates α)
  apply dvd_of_mem_factors (hp := 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 : α
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem : { val := p, property := hp } ∈ factors' a
this : DecidableEq (Associates α)
⊢ p ∈ factors (Associates.mk a)

Next tactic:
rw [factors_mk _ hz]
theorem dvd_of_mem_factors' {a : α} {p : Associates α} {hp : Irreducible p} {hz : a ≠ 0}
    (h_mem : Subtype.mk p hp ∈ factors' a) : p ∣ Associates.mk a := by
  haveI := Classical.decEq (Associates α)
  apply dvd_of_mem_factors (hp := hp)
  rw [factors_mk _ hz]
  

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
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem : { val := p, property := hp } ∈ factors' a
this : DecidableEq (Associates α)
⊢ p ∈ ↑(factors' a)

Next tactic:
apply mem_factorSet_some.2 h_mem