l3lab/ntp-mathlib huggingface dataset:
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
⊢ p ∈ p ::ₘ f

Next tactic:
simp
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
⊢ p ∣ Multiset.prod (p ::ₘ f)

Next tactic:
simp
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    

Proof state:
case cons
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
b : α
hbg : b ∈ g
hb : p ~ᵤ b
⊢ Multiset.Rel Associated (p ::ₘ f) g

Next tactic:
haveI := Classical.decEq α
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    haveI := Classical.decEq α
    

Proof state:
case cons
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ Multiset.Rel Associated (p ::ₘ f) g

Next tactic:
rw [← Multiset.cons_erase hbg]
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    haveI := Classical.decEq α
    rw [← Multiset.cons_erase hbg]
    

Proof state:
case cons
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ Multiset.Rel Associated (p ::ₘ f) (b ::ₘ Multiset.erase g b)

Next tactic:
exact
      Multiset.Rel.cons hb
        (ih (fun q hq => hf _ (by simp [hq]))
          (fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq))
          (Associated.of_mul_left
            (by rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]) hb
            (hf p (by simp)).ne_zero))
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    haveI := Classical.decEq α
    rw [← Multiset.cons_erase hbg]
    exact
      Multiset.Rel.cons hb
        (ih (fun q hq => hf _ (by 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
q : α
hq : q ∈ f
⊢ q ∈ p ::ₘ f

Next tactic:
simp [hq]
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    haveI := Classical.decEq α
    rw [← Multiset.cons_erase hbg]
    exact
      Multiset.Rel.cons hb
        (ih (fun q hq => hf _ (by simp [hq]))
          (fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq))
          (Associated.of_mul_left
            (by 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ p * Multiset.prod f ~ᵤ b * Multiset.prod (Multiset.erase g b)

Next tactic:
rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]
theorem prime_factors_unique [CancelCommMonoidWithZero α] :
    ∀ {f g : Multiset α},
      (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g := by
  classical
  intro f
  induction' f using Multiset.induction_on with p f ih
  · intros g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_not_mem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  · intros g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    haveI := Classical.decEq α
    rw [← Multiset.cons_erase hbg]
    exact
      Multiset.Rel.cons hb
        (ih (fun q hq => hf _ (by simp [hq]))
          (fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq))
          (Associated.of_mul_left
            (by rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]) hb
            (hf p (by 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
f : Multiset α
ih :
  ∀ {g : Multiset α},
    (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → Multiset.prod f ~ᵤ Multiset.prod g → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : Multiset.prod (p ::ₘ f) ~ᵤ Multiset.prod g
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ p ∈ p ::ₘ f

Next tactic:
simp
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
⊢ ∃ p, a ~ᵤ p ∧ f = {p}

Next tactic:
haveI := Classical.decEq α
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
⊢ ∃ p, a ~ᵤ p ∧ f = {p}

Next tactic:
refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · 

Proof state:
case refine_1
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
⊢ (fun g => Multiset.prod g ~ᵤ a → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) 0

Next tactic:
intro h
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; 

Proof state:
case refine_1
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
h : Multiset.prod 0 ~ᵤ a
⊢ (∀ b ∈ 0, Prime b) → ∃ p, a ~ᵤ p ∧ 0 = {p}

Next tactic:
exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · 

Proof state:
case refine_2
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
⊢ ∀ ⦃a_1 : α⦄ {s : Multiset α},
    (fun g => Multiset.prod g ~ᵤ a → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) s →
      (fun g => Multiset.prod g ~ᵤ a → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) (a_1 ::ₘ s)

Next tactic:
rintro p s _ ⟨u, hu⟩ hs
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    

Proof state:
case refine_2.intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
⊢ ∃ p_1, a ~ᵤ p_1 ∧ p ::ₘ s = {p_1}

Next tactic:
use p
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    use p
    

Proof state:
case h
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
⊢ a ~ᵤ p ∧ p ::ₘ s = {p}

Next tactic:
have hs0 : s = 0 := by
      by_contra hs0
      obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
      apply (hs q (by simp [hq])).2.1
      refine' (ha.isUnit_or_isUnit (_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left _
      · rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
          mul_comm, mul_comm p _, mul_assoc]
        simp
      apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _)
      apply (hs p (Multiset.mem_cons_self _ _)).2.1
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    use p
    have hs0 : s = 0 := by
      

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
⊢ s = 0

Next tactic:
by_contra hs0
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    use p
    have hs0 : s = 0 := by
      by_contra hs0
      

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
⊢ False

Next tactic:
obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    use p
    have hs0 : s = 0 := by
      by_contra hs0
      obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
      

Proof state:
case intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ False

Next tactic:
apply (hs q (by simp [hq])).2.1
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    use p
    have hs0 : s = 0 := by
      by_contra hs0
      obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
      apply (hs q (by 

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ q ∈ p ::ₘ s

Next tactic:
simp [hq]
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    use p
    have hs0 : s = 0 := by
      by_contra hs0
      obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
      apply (hs q (by simp [hq])).2.1
      

Proof state:
case intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ IsUnit q

Next tactic:
refine' (ha.isUnit_or_isUnit (_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left _
/-- If an irreducible has a prime factorization,
  then it is an associate of one of its prime factors. -/
theorem prime_factors_irreducible [CancelCommMonoidWithZero α] {a : α} {f : Multiset α}
    (ha : Irreducible a) (pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a) : ∃ p, a ~ᵤ p ∧ f = {p} := by
  haveI := Classical.decEq α
  refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1
  · intro h; exact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim
  · rintro p s _ ⟨u, hu⟩ hs
    use p
    have hs0 : s = 0 := by
      by_contra hs0
      obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
      apply (hs q (by simp [hq])).2.1
      refine' (ha.isUnit_or_isUnit (_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left _
      · 

Proof state:
case intro.refine'_1
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : Multiset.prod s ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : Multiset.prod (p ::ₘ s) * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ a = p * ↑u * Multiset.prod (Multiset.erase s q) * q

Next tactic:
rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
          mul_comm, mul_comm p _, mul_assoc]