l3lab/ntp-mathlib huggingface dataset:
/-- 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 _
      · rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
          mul_comm, mul_comm p _, mul_assoc]
        

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
⊢ ↑u * Multiset.prod (p ::ₘ s) = ↑u * (p * Multiset.prod s)

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
  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 _
      · rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
          mul_comm, mul_comm p _, mul_assoc]
        simp
      

Proof state:
case intro.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
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 (p * ↑u * Multiset.prod (Multiset.erase s q))

Next tactic:
apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_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 _
      · 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 _)
      

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
⊢ ¬IsUnit p

Next tactic:
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
      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
    

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
hs0 : s = 0
⊢ a ~ᵤ p ∧ p ::ₘ s = {p}

Next tactic:
simp only [mul_one, Multiset.prod_cons, Multiset.prod_zero, hs0] at *
/-- 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 _
      · 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
    simp only [mul_one, Multiset.prod_cons, Multiset.prod_zero, hs0] at *
    

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
u : αˣ
a✝ : 1 ~ᵤ a → (∀ b ∈ 0, Prime b) → ∃ p, a ~ᵤ p ∧ 0 = {p}
hu : p * ↑u = a
hs : ∀ b ∈ p ::ₘ 0, Prime b
hs0 : True
⊢ a ~ᵤ p ∧ p ::ₘ 0 = {p}

Next tactic:
exact ⟨Associated.symm ⟨u, hu⟩, rfl⟩
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
⊢ WellFounded DvdNotUnit

Next tactic:
classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h]
        erw [WithTop.coe_lt_coe]
        have cne0 : c ≠ 0 := by
          refine' mt (fun con => _) h
          rw [b_eq, con, mul_zero]
        calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)

        · convert (Classical.choose_spec (pf c cne0)).2.symm
          rw [con, Multiset.prod_zero]
        · intro x hadd
          rw [Multiset.mem_add] at hadd
          cases' hadd with h h <;> apply (Classical.choose_spec (pf _ _)).1 _ h <;> assumption
        · rw [Multiset.prod_add]
          trans a * c
          · apply Associated.mul_mul <;> apply (Classical.choose_spec (pf _ _)).2 <;> assumption
          · rw [← b_eq]
            apply (Classical.choose_spec (pf _ _)).2.symm; assumption
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
⊢ WellFounded DvdNotUnit

Next tactic:
refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · 

Proof state:
case refine'_1
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
⊢ α → ℕ∞

Next tactic:
intro a
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        

Proof state:
case refine'_1
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a : α
⊢ ℕ∞

Next tactic:
by_cases h : a = 0
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · 

Proof state:
case pos
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a : α
h : a = 0
⊢ ℕ∞

Next tactic:
exact ⊤
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        

Proof state:
case neg
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a : α
h : ¬a = 0
⊢ ℕ∞

Next tactic:
exact ↑(Multiset.card (Classical.choose (pf a h)))
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      

Proof state:
case refine'_2
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
⊢ ∀ {a b : α},
    DvdNotUnit a b →
      (if h : a = 0 then ⊤
        else ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)))) <
        if h : b = 0 then ⊤
        else ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      

Proof state:
case refine'_2.intro.intro.intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
⊢ (if h : a = 0 then ⊤ else ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a)))) <
    if h : b = 0 then ⊤ else ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
rw [dif_neg ane0]
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      

Proof state:
case refine'_2.intro.intro.intro
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
⊢ ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) <
    if h : b = 0 then ⊤ else ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
by_cases h : b = 0
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · 

Proof state:
case pos
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : b = 0
⊢ ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) <
    if h : b = 0 then ⊤ else ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
simp [h, lt_top_iff_ne_top]
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · 

Proof state:
case neg
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) <
    if h : b = 0 then ⊤ else ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
rw [dif_neg h]
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h]
        

Proof state:
case neg
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) <
    ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
erw [WithTop.coe_lt_coe]
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h]
        erw [WithTop.coe_lt_coe]
        

Proof state:
case neg
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) <
    ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
have cne0 : c ≠ 0 := by
          refine' mt (fun con => _) h
          rw [b_eq, con, mul_zero]
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h]
        erw [WithTop.coe_lt_coe]
        have cne0 : c ≠ 0 := by
          

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ c ≠ 0

Next tactic:
refine' mt (fun con => _) h
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h]
        erw [WithTop.coe_lt_coe]
        have cne0 : c ≠ 0 := by
          refine' mt (fun con => _) h
          

Proof state:
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
con : c = 0
⊢ b = 0

Next tactic:
rw [b_eq, con, mul_zero]
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h]
        erw [WithTop.coe_lt_coe]
        have cne0 : c ≠ 0 := by
          refine' mt (fun con => _) h
          rw [b_eq, con, mul_zero]
        

Proof state:
case neg
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a))) <
    ↑(Multiset.card (Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ b)))

Next tactic:
calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)
theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
  ⟨by
    classical
      refine'
        RelHomClass.wellFounded
          (RelHom.mk _ _ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h]
        erw [WithTop.coe_lt_coe]
        have cne0 : c ≠ 0 := by
          refine' mt (fun con => _) h
          rw [b_eq, con, mul_zero]
        calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)

        · 

Proof state:
case neg.calc_1
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
con : Classical.choose (_ : ∃ f, (∀ b ∈ f, Prime b) ∧ Multiset.prod f ~ᵤ c) = 0
⊢ c ~ᵤ 1

Next tactic:
convert (Classical.choose_spec (pf c cne0)).2.symm