l3lab/ntp-mathlib huggingface dataset:
theorem count_ne_zero_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    (Associates.mk p).count (Associates.mk a).factors ≠ 0 ↔ p ∣ a := by
  nontriviality α
  rw [← Associates.mk_le_mk_iff_dvd_iff]
  refine'
    ⟨fun h =>
      Associates.le_of_count_ne_zero (Associates.mk_ne_zero.mpr ha0)
        ((Associates.irreducible_mk p).mpr hp) h,
      fun 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)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
inst✝ : Nontrivial α
h : Associates.mk p ≤ Associates.mk a
⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0

Next tactic:
rw [← pow_one (Associates.mk p),
    Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero.mpr ha0)
      ((Associates.irreducible_mk p).mpr hp)] at h
theorem count_ne_zero_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    (Associates.mk p).count (Associates.mk a).factors ≠ 0 ↔ p ∣ a := by
  nontriviality α
  rw [← Associates.mk_le_mk_iff_dvd_iff]
  refine'
    ⟨fun h =>
      Associates.le_of_count_ne_zero (Associates.mk_ne_zero.mpr ha0)
        ((Associates.irreducible_mk p).mpr hp) h,
      fun h => _⟩
  rw [← pow_one (Associates.mk p),
    Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero.mpr ha0)
      ((Associates.irreducible_mk p).mpr hp)] at 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)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
inst✝ : Nontrivial α
h : 1 ≤ count (Associates.mk p) (factors (Associates.mk a))
⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0

Next tactic:
exact (zero_lt_one.trans_le h).ne'
theorem count_self [Nontrivial α] {p : Associates α} (hp : Irreducible p) : p.count p.factors = 1 :=
  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
p : Associates α
hp : Irreducible p
⊢ count p (factors p) = 1

Next tactic:
simp [factors_self hp, Associates.count_some hp]
theorem count_eq_zero_of_ne {p q : Associates α} (hp : Irreducible p) (hq : Irreducible q)
    (h : p ≠ q) : p.count q.factors = 0 :=
  not_ne_iff.mp fun h' =>
    h <|
      associated_iff_eq.mp <|
        hp.associated_of_dvd hq <| 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)
p q : Associates α
hp : Irreducible p
hq : Irreducible q
h : p ≠ q
h' : count p (factors q) ≠ 0
⊢ p ∣ q

Next tactic:
nontriviality α
theorem count_eq_zero_of_ne {p q : Associates α} (hp : Irreducible p) (hq : Irreducible q)
    (h : p ≠ q) : p.count q.factors = 0 :=
  not_ne_iff.mp fun h' =>
    h <|
      associated_iff_eq.mp <|
        hp.associated_of_dvd hq <| 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)
p q : Associates α
hp : Irreducible p
hq : Irreducible q
h : p ≠ q
h' : count p (factors q) ≠ 0
✝ : Nontrivial α
⊢ p ∣ q

Next tactic:
exact le_of_count_ne_zero hq.ne_zero hp h'
theorem count_mul {a : Associates α} (ha : a ≠ 0) {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) : count p (factors (a * b)) = count p a.factors + count p 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 : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)

Next tactic:
obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha
theorem count_mul {a : Associates α} (ha : a ≠ 0) {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) : count p (factors (a * b)) = count p a.factors + count p b.factors := by
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha
  

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 : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)

Next tactic:
obtain ⟨b0, nzb, hb'⟩ := exists_non_zero_rep hb
theorem count_mul {a : Associates α} (ha : a ≠ 0) {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) : count p (factors (a * b)) = count p a.factors + count p b.factors := by
  obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha
  obtain ⟨b0, nzb, hb'⟩ := exists_non_zero_rep hb
  

Proof state:
case intro.intro.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 : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
b0 : α
nzb : b0 ≠ 0
hb' : Associates.mk b0 = b
⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)

Next tactic:
rw [factors_mul, ← ha', ← hb', factors_mk a0 nza, factors_mk b0 nzb, ← FactorSet.coe_add, ←
    WithTop.some_eq_coe, ← WithTop.some_eq_coe, ← WithTop.some_eq_coe, count_some hp,
    Multiset.count_add, count_some hp, count_some hp]
theorem count_of_coprime {a : Associates α} (ha : a ≠ 0) {b : Associates α} (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {p : Associates α} (hp : Irreducible p) :
    count p a.factors = 0 ∨ count p b.factors = 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
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
⊢ count p (factors a) = 0 ∨ count p (factors b) = 0

Next tactic:
rw [or_iff_not_imp_left, ← Ne.def]
theorem count_of_coprime {a : Associates α} (ha : a ≠ 0) {b : Associates α} (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {p : Associates α} (hp : Irreducible p) :
    count p a.factors = 0 ∨ count p b.factors = 0 := by
  rw [or_iff_not_imp_left, ← Ne.def]
  

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
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
⊢ count p (factors a) ≠ 0 → count p (factors b) = 0

Next tactic:
intro hca
theorem count_of_coprime {a : Associates α} (ha : a ≠ 0) {b : Associates α} (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {p : Associates α} (hp : Irreducible p) :
    count p a.factors = 0 ∨ count p b.factors = 0 := by
  rw [or_iff_not_imp_left, ← Ne.def]
  intro hca
  

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
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
hca : count p (factors a) ≠ 0
⊢ count p (factors b) = 0

Next tactic:
contrapose! hab with hcb
theorem count_of_coprime {a : Associates α} (ha : a ≠ 0) {b : Associates α} (hb : b ≠ 0)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {p : Associates α} (hp : Irreducible p) :
    count p a.factors = 0 ∨ count p b.factors = 0 := by
  rw [or_iff_not_imp_left, ← Ne.def]
  intro hca
  contrapose! hab with hcb
  

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
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hca : count p (factors a) ≠ 0
hcb : count p (factors b) ≠ 0
⊢ ∃ d, d ∣ a ∧ d ∣ b ∧ Prime d

Next tactic:
exact
    ⟨p, le_of_count_ne_zero ha hp hca, le_of_count_ne_zero hb hp hcb, irreducible_iff_prime.mp hp⟩
theorem count_mul_of_coprime {a : Associates α} {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p a.factors = 0 ∨ count p a.factors = count p (a * 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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
⊢ count p (factors a) = 0 ∨ count p (factors a) = count p (factors (a * b))

Next tactic:
by_cases ha : a = 0
theorem count_mul_of_coprime {a : Associates α} {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p a.factors = 0 ∨ count p a.factors = count p (a * 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 : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : a = 0
⊢ count p (factors a) = 0 ∨ count p (factors a) = count p (factors (a * b))

Next tactic:
simp [ha]
theorem count_mul_of_coprime {a : Associates α} {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p a.factors = 0 ∨ count p a.factors = count p (a * b).factors := by
  by_cases ha : a = 0
  · simp [ha]
  

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 : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
⊢ count p (factors a) = 0 ∨ count p (factors a) = count p (factors (a * b))

Next tactic:
cases' count_of_coprime ha hb hab hp with hz hb0
theorem count_mul_of_coprime {a : Associates α} {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p a.factors = 0 ∨ count p a.factors = count p (a * b).factors := by
  by_cases ha : a = 0
  · simp [ha]
  cases' count_of_coprime ha hb hab hp with hz hb0; · 

Proof state:
case neg.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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hz : count p (factors a) = 0
⊢ count p (factors a) = 0 ∨ count p (factors a) = count p (factors (a * b))

Next tactic:
tauto
theorem count_mul_of_coprime {a : Associates α} {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p a.factors = 0 ∨ count p a.factors = count p (a * b).factors := by
  by_cases ha : a = 0
  · simp [ha]
  cases' count_of_coprime ha hb hab hp with hz hb0; · tauto
  

Proof state:
case neg.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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb0 : count p (factors b) = 0
⊢ count p (factors a) = 0 ∨ count p (factors a) = count p (factors (a * b))

Next tactic:
apply Or.intro_right
theorem count_mul_of_coprime {a : Associates α} {b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p a.factors = 0 ∨ count p a.factors = count p (a * b).factors := by
  by_cases ha : a = 0
  · simp [ha]
  cases' count_of_coprime ha hb hab hp with hz hb0; · tauto
  apply Or.intro_right
  

Proof state:
case neg.inr.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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb0 : count p (factors b) = 0
⊢ count p (factors a) = count p (factors (a * b))

Next tactic:
rw [count_mul ha hb hp, hb0, add_zero]
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p 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
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
⊢ count p (factors (a * b)) = count p (factors a) ∨ count p (factors (a * b)) = count p (factors b)

Next tactic:
by_cases ha : a = 0
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p 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
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : a = 0
⊢ count p (factors (a * b)) = count p (factors a) ∨ count p (factors (a * b)) = count p (factors b)

Next tactic:
simp [ha]
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  

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
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
⊢ count p (factors (a * b)) = count p (factors a) ∨ count p (factors (a * b)) = count p (factors b)

Next tactic:
by_cases hb : b = 0
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 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
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : b = 0
⊢ count p (factors (a * b)) = count p (factors a) ∨ count p (factors (a * b)) = count p (factors b)

Next tactic:
simp [hb]
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  

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
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
⊢ count p (factors (a * b)) = count p (factors a) ∨ count p (factors (a * b)) = count p (factors b)

Next tactic:
rw [count_mul ha hb hp]
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  rw [count_mul ha hb hp]
  

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
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
⊢ count p (factors a) + count p (factors b) = count p (factors a) ∨
    count p (factors a) + count p (factors b) = count p (factors b)

Next tactic:
cases' count_of_coprime ha hb hab hp with ha0 hb0
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  rw [count_mul ha hb hp]
  cases' count_of_coprime ha hb hab hp with ha0 hb0
  · 

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
ha0 : count p (factors a) = 0
⊢ count p (factors a) + count p (factors b) = count p (factors a) ∨
    count p (factors a) + count p (factors b) = count p (factors b)

Next tactic:
apply Or.intro_right
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  rw [count_mul ha hb hp]
  cases' count_of_coprime ha hb hab hp with ha0 hb0
  · apply Or.intro_right
    

Proof state:
case neg.inl.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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
ha0 : count p (factors a) = 0
⊢ count p (factors a) + count p (factors b) = count p (factors b)

Next tactic:
rw [ha0, zero_add]
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  rw [count_mul ha hb hp]
  cases' count_of_coprime ha hb hab hp with ha0 hb0
  · apply Or.intro_right
    rw [ha0, zero_add]
  · 

Proof state:
case neg.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 p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
hb0 : count p (factors b) = 0
⊢ count p (factors a) + count p (factors b) = count p (factors a) ∨
    count p (factors a) + count p (factors b) = count p (factors b)

Next tactic:
apply Or.intro_left
theorem count_mul_of_coprime' {a b : Associates α} {p : Associates α} (hp : Irreducible p)
    (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) :
    count p (a * b).factors = count p a.factors ∨ count p (a * b).factors = count p b.factors := by
  by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  rw [count_mul ha hb hp]
  cases' count_of_coprime ha hb hab hp with ha0 hb0
  · apply Or.intro_right
    rw [ha0, zero_add]
  · apply Or.intro_left
    

Proof state:
case neg.inr.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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
hb0 : count p (factors b) = 0
⊢ count p (factors a) + count p (factors b) = count p (factors a)

Next tactic:
rw [hb0, add_zero]
theorem dvd_count_of_dvd_count_mul {a b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ}
    (habk : k ∣ count p (a * b).factors) : k ∣ count p a.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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ count p (factors (a * b))
⊢ k ∣ count p (factors a)

Next tactic:
by_cases ha : a = 0
theorem dvd_count_of_dvd_count_mul {a b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ}
    (habk : k ∣ count p (a * b).factors) : k ∣ count p a.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 : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ count p (factors (a * b))
ha : a = 0
⊢ k ∣ count p (factors a)

Next tactic:
simpa [*] using habk
theorem dvd_count_of_dvd_count_mul {a b : Associates α} (hb : b ≠ 0) {p : Associates α}
    (hp : Irreducible p) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {k : ℕ}
    (habk : k ∣ count p (a * b).factors) : k ∣ count p a.factors := by
  by_cases ha : a = 0
  · simpa [*] using habk
  

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 : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ count p (factors (a * b))
ha : ¬a = 0
⊢ k ∣ count p (factors a)

Next tactic:
cases' count_of_coprime ha hb hab hp with hz h