l3lab/ntp-mathlib huggingface dataset:
/-- `toGCDMonoid` constructs a GCD monoid out of a unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toGCDMonoid (α : Type*) [CancelCommMonoidWithZero α]
    [UniqueFactorizationMonoid α] [DecidableEq (Associates α)] [DecidableEq α] : GCDMonoid α where
  gcd a b := Quot.out (Associates.mk a ⊓ Associates.mk b : Associates α)
  lcm a b := Quot.out (Associates.mk a ⊔ Associates.mk b : Associates α)
  gcd_dvd_left a b := by
    rw [← mk_dvd_mk, (Associates.mk a ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le]
    exact inf_le_left
  gcd_dvd_right a b := by
    rw [← mk_dvd_mk, (Associates.mk a ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le]
    exact inf_le_right
  dvd_gcd {a b c} hac hab := by
    rw [← mk_dvd_mk, (Associates.mk c ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le, le_inf_iff,
      mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff]
    exact ⟨hac, hab⟩
  lcm_zero_left a := by
    have : Associates.mk (0 : α) = ⊤ := rfl
    dsimp
    rw [this, top_sup_eq, ← this, ← associated_zero_iff_eq_zero, ← mk_eq_mk_iff_associated, ←
      associated_iff_eq, Associates.quot_out]
  lcm_zero_right a := by
    

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
a : α
⊢ (fun a b => Quot.out (Associates.mk a ⊔ Associates.mk b)) a 0 = 0

Next tactic:
have : Associates.mk (0 : α) = ⊤ := rfl
/-- `toGCDMonoid` constructs a GCD monoid out of a unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toGCDMonoid (α : Type*) [CancelCommMonoidWithZero α]
    [UniqueFactorizationMonoid α] [DecidableEq (Associates α)] [DecidableEq α] : GCDMonoid α where
  gcd a b := Quot.out (Associates.mk a ⊓ Associates.mk b : Associates α)
  lcm a b := Quot.out (Associates.mk a ⊔ Associates.mk b : Associates α)
  gcd_dvd_left a b := by
    rw [← mk_dvd_mk, (Associates.mk a ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le]
    exact inf_le_left
  gcd_dvd_right a b := by
    rw [← mk_dvd_mk, (Associates.mk a ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le]
    exact inf_le_right
  dvd_gcd {a b c} hac hab := by
    rw [← mk_dvd_mk, (Associates.mk c ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le, le_inf_iff,
      mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff]
    exact ⟨hac, hab⟩
  lcm_zero_left a := by
    have : Associates.mk (0 : α) = ⊤ := rfl
    dsimp
    rw [this, top_sup_eq, ← this, ← associated_zero_iff_eq_zero, ← mk_eq_mk_iff_associated, ←
      associated_iff_eq, Associates.quot_out]
  lcm_zero_right a := by
    have : Associates.mk (0 : α) = ⊤ := rfl
    

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
a : α
this : Associates.mk 0 = ⊤
⊢ (fun a b => Quot.out (Associates.mk a ⊔ Associates.mk b)) a 0 = 0

Next tactic:
dsimp
/-- `toGCDMonoid` constructs a GCD monoid out of a unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toGCDMonoid (α : Type*) [CancelCommMonoidWithZero α]
    [UniqueFactorizationMonoid α] [DecidableEq (Associates α)] [DecidableEq α] : GCDMonoid α where
  gcd a b := Quot.out (Associates.mk a ⊓ Associates.mk b : Associates α)
  lcm a b := Quot.out (Associates.mk a ⊔ Associates.mk b : Associates α)
  gcd_dvd_left a b := by
    rw [← mk_dvd_mk, (Associates.mk a ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le]
    exact inf_le_left
  gcd_dvd_right a b := by
    rw [← mk_dvd_mk, (Associates.mk a ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le]
    exact inf_le_right
  dvd_gcd {a b c} hac hab := by
    rw [← mk_dvd_mk, (Associates.mk c ⊓ Associates.mk b).quot_out, congr_fun₂ dvd_eq_le, le_inf_iff,
      mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff]
    exact ⟨hac, hab⟩
  lcm_zero_left a := by
    have : Associates.mk (0 : α) = ⊤ := rfl
    dsimp
    rw [this, top_sup_eq, ← this, ← associated_zero_iff_eq_zero, ← mk_eq_mk_iff_associated, ←
      associated_iff_eq, Associates.quot_out]
  lcm_zero_right a := by
    have : Associates.mk (0 : α) = ⊤ := rfl
    dsimp
    

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
a : α
this : Associates.mk 0 = ⊤
⊢ Quot.out (Associates.mk a ⊔ Associates.mk 0) = 0

Next tactic:
rw [this, sup_top_eq, ← this, ← associated_zero_iff_eq_zero, ← mk_eq_mk_iff_associated, ←
      associated_iff_eq, Associates.quot_out]
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b c : α
hac : a ∣ c
hab : a ∣ b
⊢ a ∣ Associates.out (Associates.mk c ⊓ Associates.mk b)

Next tactic:
rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff]
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b c : α
hac : a ∣ c
hab : a ∣ b
⊢ a ∣ c ∧ a ∣ b

Next tactic:
exact ⟨hac, hab⟩
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by simp
    lcm_zero_right := fun a => show (Associates.mk a ⊔ ⊤).out = 0 by simp
    gcd_mul_lcm := fun a b => by
      

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b : α
⊢ (fun a b => Associates.out (Associates.mk a ⊓ Associates.mk b)) a b *
      (fun a b => Associates.out (Associates.mk a ⊔ Associates.mk b)) a b ~ᵤ
    a * b

Next tactic:
rw [← out_mul, mul_comm, sup_mul_inf, mk_mul_mk, out_mk]
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by simp
    lcm_zero_right := fun a => show (Associates.mk a ⊔ ⊤).out = 0 by simp
    gcd_mul_lcm := fun a b => by
      rw [← out_mul, mul_comm, sup_mul_inf, mk_mul_mk, out_mk]
      

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b : α
⊢ normalize (a * b) ~ᵤ a * b

Next tactic:
exact normalize_associated (a * b)
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by 

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a : α
⊢ Associates.out (⊤ ⊔ Associates.mk a) = 0

Next tactic:
simp
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by simp
    lcm_zero_right := fun a => show (Associates.mk a ⊔ ⊤).out = 0 by 

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a : α
⊢ Associates.out (Associates.mk a ⊔ ⊤) = 0

Next tactic:
simp
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by simp
    lcm_zero_right := fun a => show (Associates.mk a ⊔ ⊤).out = 0 by simp
    gcd_mul_lcm := fun a b => by
      rw [← out_mul, mul_comm, sup_mul_inf, mk_mul_mk, out_mk]
      exact normalize_associated (a * b)
    normalize_gcd := fun a b => by 

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b : α
⊢ normalize (gcd a b) = gcd a b

Next tactic:
congr
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by simp
    lcm_zero_right := fun a => show (Associates.mk a ⊔ ⊤).out = 0 by simp
    gcd_mul_lcm := fun a b => by
      rw [← out_mul, mul_comm, sup_mul_inf, mk_mul_mk, out_mk]
      exact normalize_associated (a * b)
    normalize_gcd := fun a b => by congr; 

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b : α
⊢ normalize (gcd a b) = gcd a b

Next tactic:
apply normalize_out _
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by simp
    lcm_zero_right := fun a => show (Associates.mk a ⊔ ⊤).out = 0 by simp
    gcd_mul_lcm := fun a b => by
      rw [← out_mul, mul_comm, sup_mul_inf, mk_mul_mk, out_mk]
      exact normalize_associated (a * b)
    normalize_gcd := fun a b => by congr; apply normalize_out _
    normalize_lcm := fun a b => by 

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b : α
⊢ normalize (lcm a b) = lcm a b

Next tactic:
congr
/-- `toNormalizedGCDMonoid` constructs a GCD monoid out of a normalization on a
  unique factorization domain. -/
noncomputable def UniqueFactorizationMonoid.toNormalizedGCDMonoid (α : Type*)
    [CancelCommMonoidWithZero α] [UniqueFactorizationMonoid α] [NormalizationMonoid α]
    [DecidableEq (Associates α)] [DecidableEq α] : NormalizedGCDMonoid α :=
  { ‹NormalizationMonoid α› with
    gcd := fun a b => (Associates.mk a ⊓ Associates.mk b).out
    lcm := fun a b => (Associates.mk a ⊔ Associates.mk b).out
    gcd_dvd_left := fun a b => (out_dvd_iff a (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_left
    gcd_dvd_right := fun a b =>
      (out_dvd_iff b (Associates.mk a ⊓ Associates.mk b)).2 <| inf_le_right
    dvd_gcd := fun {a} {b} {c} hac hab =>
      show a ∣ (Associates.mk c ⊓ Associates.mk b).out by
        rw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff];
          exact ⟨hac, hab⟩
    lcm_zero_left := fun a => show (⊤ ⊔ Associates.mk a).out = 0 by simp
    lcm_zero_right := fun a => show (Associates.mk a ⊔ ⊤).out = 0 by simp
    gcd_mul_lcm := fun a b => by
      rw [← out_mul, mul_comm, sup_mul_inf, mk_mul_mk, out_mk]
      exact normalize_associated (a * b)
    normalize_gcd := fun a b => by congr; apply normalize_out _
    normalize_lcm := fun a b => by congr; 

Proof state:
α✝ : Type u_1
α : Type u_2
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : NormalizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : DecidableEq α
src✝ : NormalizationMonoid α := inst✝²
a b : α
⊢ normalize (lcm a b) = lcm a b

Next tactic:
apply normalize_out _
/-- 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
  

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
⊢ Fintype { x // x ∣ y }

Next tactic:
haveI : Nontrivial M := ⟨⟨y, 0, 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⟩⟩
  

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 : Nontrivial M
⊢ Fintype { x // x ∣ y }

Next tactic:
haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
/-- 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
  

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✝ : Nontrivial M
this : NormalizationMonoid M
⊢ Fintype { x // x ∣ y }

Next tactic:
haveI := Classical.decEq M
/-- 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
  

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✝¹ : Nontrivial M
this✝ : NormalizationMonoid M
this : DecidableEq M
⊢ Fintype { x // x ∣ y }

Next tactic:
haveI := Classical.decEq (Associates M)
/-- 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`.
  

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)
⊢ Fintype { x // x ∣ y }

Next tactic:
refine'
    Fintype.ofFinset
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => _