l3lab/ntp-mathlib huggingface dataset:
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  

Proof state:
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
⊢ x ∈
      Finset.image (fun s => ↑s.2 * Multiset.prod s.1)
        (Multiset.toFinset (Multiset.powerset (normalizedFactors y)) ×ˢ Finset.univ) ↔
    x ∈ fun x => ∃ c, y = x * c

Next tactic:
simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  

Proof state:
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
⊢ (∃ a, a.1 ≤ normalizedFactors y ∧ ↑a.2 * Multiset.prod a.1 = x) ↔ x ∈ fun x => ∃ c, y = x * c

Next tactic:
constructor
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · 

Proof state:
case mp
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
⊢ (∃ a, a.1 ≤ normalizedFactors y ∧ ↑a.2 * Multiset.prod a.1 = x) → x ∈ fun x => ∃ c, y = x * c

Next tactic:
rintro ⟨s, hs, rfl⟩
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    

Proof state:
case mp.intro.intro
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
s : Multiset M × Mˣ
hs : s.1 ≤ normalizedFactors y
⊢ ↑s.2 * Multiset.prod s.1 ∈ fun x => ∃ c, y = x * c

Next tactic:
show (s.snd : M) * s.fst.prod ∣ y
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    

Proof state:
case mp.intro.intro
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
s : Multiset M × Mˣ
hs : s.1 ≤ normalizedFactors y
⊢ ↑s.2 * Multiset.prod s.1 ∣ y

Next tactic:
rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    

Proof state:
case mp.intro.intro
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
s : Multiset M × Mˣ
hs : s.1 ≤ normalizedFactors y
⊢ Multiset.prod s.1 ∣ Multiset.prod (normalizedFactors y)

Next tactic:
exact Multiset.prod_dvd_prod_of_le hs
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    exact Multiset.prod_dvd_prod_of_le hs
  · 

Proof state:
case mpr
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
⊢ (x ∈ fun x => ∃ c, y = x * c) → ∃ a, a.1 ≤ normalizedFactors y ∧ ↑a.2 * Multiset.prod a.1 = x

Next tactic:
rintro (h : x ∣ y)
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    exact Multiset.prod_dvd_prod_of_le hs
  · rintro (h : x ∣ y)
    

Proof state:
case mpr
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
⊢ ∃ a, a.1 ≤ normalizedFactors y ∧ ↑a.2 * Multiset.prod a.1 = x

Next tactic:
have hx : x ≠ 0 := by
      refine' mt (fun hx => _) hy
      rwa [hx, zero_dvd_iff] at h
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    exact Multiset.prod_dvd_prod_of_le hs
  · rintro (h : x ∣ y)
    have hx : x ≠ 0 := by
      

Proof state:
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
⊢ x ≠ 0

Next tactic:
refine' mt (fun hx => _) hy
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    exact Multiset.prod_dvd_prod_of_le hs
  · rintro (h : x ∣ y)
    have hx : x ≠ 0 := by
      refine' mt (fun hx => _) hy
      

Proof state:
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x = 0
⊢ y = 0

Next tactic:
rwa [hx, zero_dvd_iff] at h
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    exact Multiset.prod_dvd_prod_of_le hs
  · rintro (h : x ∣ y)
    have hx : x ≠ 0 := by
      refine' mt (fun hx => _) hy
      rwa [hx, zero_dvd_iff] at h
    

Proof state:
case mpr
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x ≠ 0
⊢ ∃ a, a.1 ≤ normalizedFactors y ∧ ↑a.2 * Multiset.prod a.1 = x

Next tactic:
obtain ⟨u, hu⟩ := normalizedFactors_prod hx
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    exact Multiset.prod_dvd_prod_of_le hs
  · rintro (h : x ∣ y)
    have hx : x ≠ 0 := by
      refine' mt (fun hx => _) hy
      rwa [hx, zero_dvd_iff] at h
    obtain ⟨u, hu⟩ := normalizedFactors_prod hx
    

Proof state:
case mpr.intro
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x ≠ 0
u : Mˣ
hu : Multiset.prod (normalizedFactors x) * ↑u = x
⊢ ∃ a, a.1 ≤ normalizedFactors y ∧ ↑a.2 * Multiset.prod a.1 = x

Next tactic:
refine' ⟨⟨normalizedFactors x, u⟩, _, (mul_comm _ _).trans hu⟩
/-- If `y` is a nonzero element of a unique factorization monoid with finitely
many units (e.g. `ℤ`, `Ideal (ring_of_integers K)`), it has finitely many divisors. -/
noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
    [UniqueFactorizationMonoid M] [Fintype Mˣ] (y : M) (hy : y ≠ 0) : Fintype { x // x ∣ y } := by
  haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
  haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
  haveI := Classical.decEq M
  haveI := Classical.decEq (Associates M)
  -- We'll show `λ (u : Mˣ) (f ⊆ factors y) → u * Π f` is injective
  -- and has image exactly the divisors of `y`.
  refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _
  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
    Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
  constructor
  · rintro ⟨s, hs, rfl⟩
    show (s.snd : M) * s.fst.prod ∣ y
    rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (normalizedFactors_prod hy).dvd_iff_dvd_right]
    exact Multiset.prod_dvd_prod_of_le hs
  · rintro (h : x ∣ y)
    have hx : x ≠ 0 := by
      refine' mt (fun hx => _) hy
      rwa [hx, zero_dvd_iff] at h
    obtain ⟨u, hu⟩ := normalizedFactors_prod hx
    refine' ⟨⟨normalizedFactors x, u⟩, _, (mul_comm _ _).trans hu⟩
    

Proof state:
case mpr.intro
α : Type u_1
M : Type u_2
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : NormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x ≠ 0
u : Mˣ
hu : Multiset.prod (normalizedFactors x) * ↑u = x
⊢ (normalizedFactors x, u).1 ≤ normalizedFactors y

Next tactic:
exact (dvd_iff_normalizedFactors_le_normalizedFactors hx hy).mp h
theorem factorization_eq_count {n p : α} :
    factorization n p = Multiset.count p (normalizedFactors n) := by 

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
n p : α
⊢ (factorization n) p = Multiset.count p (normalizedFactors n)

Next tactic:
simp [factorization]
@[simp]
theorem factorization_zero : factorization (0 : α) = 0 := by 

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
⊢ factorization 0 = 0

Next tactic:
simp [factorization]
@[simp]
theorem factorization_one : factorization (1 : α) = 0 := by 

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
⊢ factorization 1 = 0

Next tactic:
simp [factorization]
/-- The support of `factorization n` is exactly the Finset of normalized factors -/
@[simp]
theorem support_factorization {n : α} :
    (factorization n).support = (normalizedFactors n).toFinset := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
n : α
⊢ (factorization n).support = Multiset.toFinset (normalizedFactors n)

Next tactic:
simp [factorization, Multiset.toFinsupp_support]
/-- For nonzero `a` and `b`, the power of `p` in `a * b` is the sum of the powers in `a` and `b` -/
@[simp]
theorem factorization_mul {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) :
    factorization (a * b) = factorization a + factorization b := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
a b : α
ha : a ≠ 0
hb : b ≠ 0
⊢ factorization (a * b) = factorization a + factorization b

Next tactic:
simp [factorization, normalizedFactors_mul ha hb]
/-- For any `p`, the power of `p` in `x^n` is `n` times the power in `x` -/
theorem factorization_pow {x : α} {n : ℕ} : factorization (x ^ n) = n • factorization x := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
x : α
n : ℕ
⊢ factorization (x ^ n) = n • factorization x

Next tactic:
ext
/-- For any `p`, the power of `p` in `x^n` is `n` times the power in `x` -/
theorem factorization_pow {x : α} {n : ℕ} : factorization (x ^ n) = n • factorization x := by
  ext
  

Proof state:
case h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
x : α
n : ℕ
a✝ : α
⊢ (factorization (x ^ n)) a✝ = (n • factorization x) a✝

Next tactic:
simp [factorization]