awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
M : Submonoid A
hM : M ≤ A⁰
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization M Aₘ
h :
  ∀ (y : ↥M),
    IsUnit (M := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (↑y : A) : FractionRing A)
this_1 : Algebra.{u_2, u_1} Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)
this_2 : IsScalarTower A Aₘ (FractionRing A)
this : IsFractionRing.{u_2, u_1} Aₘ (FractionRing A)
refine_1 : IsDomain Aₘ
refine_2 : IsNoetherianRing Aₘ
⊢ IsDedekindDomain Aₘ
have refine_3 : Ring.DimensionLEOne Aₘ
Tactic: refine (isDedekindDomain_iff _ (FractionRing A)).mpr ⟨?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
M : Submonoid A
hM : M ≤ A⁰
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization M Aₘ
h :
  ∀ (y : ↥M),
    IsUnit (M := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (↑y : A) : FractionRing A)
this_1 : Algebra.{u_2, u_1} Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)
this_2 : IsScalarTower A Aₘ (FractionRing A)
this : IsFractionRing.{u_2, u_1} Aₘ (FractionRing A)
refine_1 : IsDomain Aₘ
refine_2 : IsNoetherianRing Aₘ
refine_3 : Ring.DimensionLEOne Aₘ
⊢ IsDedekindDomain Aₘ
have refine_4 : ∀ {x : FractionRing A}, IsIntegral Aₘ x → ∃ (y : Aₘ), (algebraMap Aₘ (FractionRing A) : (a : Aₘ) → FractionRing A) y = x
Tactic: refine (isDedekindDomain_iff _ (FractionRing A)).mpr ⟨?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
M : Submonoid A
hM : M ≤ A⁰
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization M Aₘ
h :
  ∀ (y : ↥M),
    IsUnit (M := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (↑y : A) : FractionRing A)
this_1 : Algebra.{u_2, u_1} Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)
this_2 : IsScalarTower A Aₘ (FractionRing A)
this : IsFractionRing.{u_2, u_1} Aₘ (FractionRing A)
x : FractionRing A
hx : IsIntegral Aₘ x
⊢ ∃ (y : Aₘ), (algebraMap Aₘ (FractionRing A) : (a : Aₘ) → FractionRing A) y = x
have refine_4 : ∀ (y : A) (y_mem : y ∈ M) (hy : IsIntegral A (Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x)),
  ∃ (y : Aₘ), (algebraMap Aₘ (FractionRing A) : (a : Aₘ) → FractionRing A) y = x
Tactic: obtain ⟨⟨y, y_mem⟩, hy⟩ := hx.exists_multiple_integral_of_isLocalization M _
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
M : Submonoid A
hM : M ≤ A⁰
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization M Aₘ
h :
  ∀ (y : ↥M),
    IsUnit (M := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (↑y : A) : FractionRing A)
this_1 : Algebra.{u_2, u_1} Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)
this_2 : IsScalarTower A Aₘ (FractionRing A)
this : IsFractionRing.{u_2, u_1} Aₘ (FractionRing A)
x : FractionRing A
hx : IsIntegral Aₘ x
y : A
y_mem : y ∈ M
hy : IsIntegral A (Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x)
⊢ ∃ (y : Aₘ), (algebraMap Aₘ (FractionRing A) : (a : Aₘ) → FractionRing A) y = x
have refine_4 : ∀ (z : A)
  (hz :
    (algebraMap A (FractionRing A) : (a : A) → FractionRing A) z = Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x),
  ∃ (y : Aₘ), (algebraMap Aₘ (FractionRing A) : (a : Aₘ) → FractionRing A) y = x
Tactic: obtain ⟨z, hz⟩ := (isIntegrallyClosed_iff _).mp IsDedekindRing.toIsIntegralClosure hy
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
M : Submonoid A
hM : M ≤ A⁰
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization M Aₘ
h :
  ∀ (y : ↥M),
    IsUnit (M := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (↑y : A) : FractionRing A)
this_1 : Algebra.{u_2, u_1} Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)
this_2 : IsScalarTower A Aₘ (FractionRing A)
this : IsFractionRing.{u_2, u_1} Aₘ (FractionRing A)
x : FractionRing A
hx : IsIntegral Aₘ x
y : A
y_mem : y ∈ M
hy : IsIntegral A (Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x)
z : A
hz : (algebraMap A (FractionRing A) : (a : A) → FractionRing A) z = Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x
⊢ ∃ (y : Aₘ), (algebraMap Aₘ (FractionRing A) : (a : Aₘ) → FractionRing A) y = x
have refine_4 : (algebraMap A (FractionRing A) : (a : A) → FractionRing A) z =
  HMul.hMul (α := FractionRing A)
    ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (↑(Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem) : A) :
      FractionRing A)
    x
Tactic: refine ⟨IsLocalization.mk' Aₘ z ⟨y, y_mem⟩, (IsLocalization.lift_mk'_spec _ _ _ _).mpr ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
M : Submonoid A
hM : M ≤ A⁰
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization M Aₘ
h :
  ∀ (y : ↥M),
    IsUnit (M := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (↑y : A) : FractionRing A)
this_1 : Algebra.{u_2, u_1} Aₘ (FractionRing A) := RingHom.toAlgebra (lift h)
this_2 : IsScalarTower A Aₘ (FractionRing A)
this : IsFractionRing.{u_2, u_1} Aₘ (FractionRing A)
x : FractionRing A
hx : IsIntegral Aₘ x
y : A
y_mem : y ∈ M
hy : IsIntegral A (Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x)
z : A
hz : (algebraMap A (FractionRing A) : (a : A) → FractionRing A) z = Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x
⊢ (algebraMap A (FractionRing A) : (a : A) → FractionRing A) z =
    HMul.hMul (α := FractionRing A)
      ((algebraMap A (FractionRing A) : (a : A) → FractionRing A)
          (↑(Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem) : A) :
        FractionRing A)
      x
have refine_4 : Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem • x = (↑(Subtype.mk (p := fun (x : A) ↦ x ∈ M) y y_mem) : A) • x
Tactic: rw [hz, ← Algebra.smul_def]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
P : Ideal A
hP : P ≠ ⊥
pP : P.IsPrime
Aₘ : Type u_2
inst_2 : CommRing Aₘ
inst_3 : Algebra A Aₘ
inst_4 : IsLocalization.AtPrime Aₘ P
h : IsField Aₘ
this : Field Aₘ := h.toField
⊢ False
have obtain : ∀ (x : A) (x_mem : x ∈ P) (x_ne : x ≠ 0), False
Tactic: obtain ⟨x, x_mem, x_ne⟩ := P.ne_bot_iff.mp hP
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (hypothesis)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
P : Ideal A
hP : P ≠ ⊥
pP : P.IsPrime
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization.AtPrime Aₘ P
this_1 : IsNoetherianRing Aₘ := ⋯
this : IsLocalRing Aₘ := ⋯
⊢ IsDiscreteValuationRing Aₘ
have hnf : ¬IsField Aₘ
Tactic: have hnf := IsLocalization.AtPrime.not_isField A hP Aₘ
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
inst_2 : IsDedekindDomain A
P : Ideal A
hP : P ≠ ⊥
pP : P.IsPrime
Aₘ : Type u_2
inst_3 : CommRing Aₘ
inst_4 : IsDomain Aₘ
inst_5 : Algebra A Aₘ
inst_6 : IsLocalization.AtPrime Aₘ P
letI letI₁ letI₂ letI₃ : IsDiscreteValuationRing Aₘ
⊢ IsDiscreteValuationRing Aₘ
have letI₄ : ∀ (hnf : ¬IsField Aₘ), IsDiscreteValuationRing Aₘ
Tactic: classical
  letI : IsNoetherianRing Aₘ :=
    IsLocalization.isNoetherianRing P.primeCompl _ IsDedekindRing.toIsNoetherian
  letI : IsLocalRing Aₘ := IsLocalization.AtPrime.isLocalRing Aₘ P
  have hnf := IsLocalization.AtPrime.not_isField A hP Aₘ
  exact
    ((IsDiscreteValuationRing.TFAE Aₘ hnf).out 0 2).mpr
      (IsLocalization.AtPrime.isDedekindDomain A P _)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (goal)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
h : IsDedekindDomainDvr A
p : Ideal A
hp : p ≠ ⊥
hpp : p.IsPrime
⊢ p.IsMaximal
have rcases : ∀ (q : Ideal A) (hq : q.IsMaximal) (hpq : p ≤ q), p.IsMaximal
Tactic: rcases p.exists_le_maximal (Ideal.IsPrime.ne_top hpp) with ⟨q, hq, hpq⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (hypothesis)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
h : IsDedekindDomainDvr A
p : Ideal A
hp : p ≠ ⊥
hpp : p.IsPrime
q : Ideal A
hq : q.IsMaximal
hpq : p ≤ q
f : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) } ≃o
  { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  OrderIso.symm (IsLocalization.orderIsoOfPrime q.primeCompl (Localization.AtPrime q))
P : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) p ⋯)
Q : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) q ⋯)
⊢ p.IsMaximal
have hinj : Function.Injective (α := A) (β := Localization.AtPrime q)
  (⇑(algebraMap A (Localization.AtPrime q)) : (a : A) → Localization.AtPrime q)
Tactic: have hinj : Function.Injective (algebraMap A (Localization.AtPrime q)) :=
      IsLocalization.injective (Localization.AtPrime q) q.primeCompl_le_nonZeroDivisors
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (hypothesis)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
h : IsDedekindDomainDvr A
p : Ideal A
hp : p ≠ ⊥
hpp : p.IsPrime
q : Ideal A
hq : q.IsMaximal
hpq : p ≤ q
f : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) } ≃o
  { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  OrderIso.symm (IsLocalization.orderIsoOfPrime q.primeCompl (Localization.AtPrime q))
P : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) p ⋯)
Q : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) q ⋯)
hinj :
  Function.Injective (α := A) (β := Localization.AtPrime q)
    (⇑(algebraMap A (Localization.AtPrime q)) : (a : A) → Localization.AtPrime q)
⊢ p.IsMaximal
have hp1 : (↑P : Ideal (Localization.AtPrime q)) ≠ ⊥
Tactic: have hp1 : P.1 ≠ ⊥ := fun x => hp ((p.map_eq_bot_iff_of_injective hinj).mp x)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (hypothesis)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
h : IsDedekindDomainDvr A
p : Ideal A
hp : p ≠ ⊥
hpp : p.IsPrime
q : Ideal A
hq : q.IsMaximal
hpq : p ≤ q
f : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) } ≃o
  { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  OrderIso.symm (IsLocalization.orderIsoOfPrime q.primeCompl (Localization.AtPrime q))
P : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) p ⋯)
Q : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) q ⋯)
hinj :
  Function.Injective (α := A) (β := Localization.AtPrime q)
    (⇑(algebraMap A (Localization.AtPrime q)) : (a : A) → Localization.AtPrime q)
hp1 : (↑P : Ideal (Localization.AtPrime q)) ≠ ⊥
⊢ p.IsMaximal
have hq1 : (↑Q : Ideal (Localization.AtPrime q)) ≠ ⊥
Tactic: have hq1 : Q.1 ≠ ⊥ :=
      fun x => (ne_bot_of_le_ne_bot hp hpq) ((q.map_eq_bot_iff_of_injective hinj).mp x)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (hypothesis)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
h : IsDedekindDomainDvr A
p : Ideal A
hp : p ≠ ⊥
hpp : p.IsPrime
q : Ideal A
hq : q.IsMaximal
hpq : p ≤ q
f : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) } ≃o
  { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  OrderIso.symm (IsLocalization.orderIsoOfPrime q.primeCompl (Localization.AtPrime q))
P : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) p ⋯)
Q : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) q ⋯)
hinj :
  Function.Injective (α := A) (β := Localization.AtPrime q)
    (⇑(algebraMap A (Localization.AtPrime q)) : (a : A) → Localization.AtPrime q)
hp1 : (↑P : Ideal (Localization.AtPrime q)) ≠ ⊥
hq1 : (↑Q : Ideal (Localization.AtPrime q)) ≠ ⊥
⊢ p.IsMaximal
have left : IsPrincipalIdealRing.{u_1} (Localization.AtPrime q)
Tactic: rcases (IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime (Localization.AtPrime q)).mp
      (h.is_dvr_at_nonzero_prime q (ne_bot_of_le_ne_bot hp hpq) hq.isPrime) with ⟨_, huq⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Dvr (hypothesis)
Goal:
A : Type u_1
inst : CommRing A
inst_1 : IsDomain A
h : IsDedekindDomainDvr A
p : Ideal A
hp : p ≠ ⊥
hpp : p.IsPrime
q : Ideal A
hq : q.IsMaximal
hpq : p ≤ q
f : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) } ≃o
  { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  OrderIso.symm (IsLocalization.orderIsoOfPrime q.primeCompl (Localization.AtPrime q))
P : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) p ⋯)
Q : { p : Ideal (Localization.AtPrime q) // p.IsPrime } :=
  (f :
      (a : { p : Ideal A // p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A) }) →
        { p : Ideal (Localization.AtPrime q) // p.IsPrime })
    (Subtype.mk (p := fun (p : Ideal A) ↦ p.IsPrime ∧ Disjoint (↑q.primeCompl : Set A) (↑p : Set A)) q ⋯)
hinj :
  Function.Injective (α := A) (β := Localization.AtPrime q)
    (⇑(algebraMap A (Localization.AtPrime q)) : (a : A) → Localization.AtPrime q)
hp1 : (↑P : Ideal (Localization.AtPrime q)) ≠ ⊥
hq1 : (↑Q : Ideal (Localization.AtPrime q)) ≠ ⊥
left : IsPrincipalIdealRing.{u_1} (Localization.AtPrime q)
⊢ p.IsMaximal
have huq : ∃! P : Ideal (Localization.AtPrime q), P ≠ ⊥ ∧ P.IsPrime
Tactic: rcases (IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime (Localization.AtPrime q)).mp
      (h.is_dvr_at_nonzero_prime q (ne_bot_of_le_ne_bot hp hpq) hq.isPrime) with ⟨_, huq⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
⊢ v.maxPowDividing I = v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
have rw : v.asIdeal ^ count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) =
  v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
Tactic: rw [maxPowDividing, factors_mk _ hI, count_some (irreducible_mk.mpr v.irreducible),
    ← Multiset.count_map_eq_count' _ _ Subtype.val_injective, map_subtype_coe_factors',
    factors_eq_normalizedFactors, ← Multiset.count_map_eq_count' _ _ (mk_injective (M := Ideal R))]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
rw :
  v.asIdeal ^ count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
⊢ v.maxPowDividing I = v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
have rw₁ : v.asIdeal ^
    count (Associates.mk v.asIdeal) (↑(factors' I) : WithTop (Multiset { a : Associates (Ideal R) // Irreducible a })) =
  v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
Tactic: rw [maxPowDividing, factors_mk _ hI, count_some (irreducible_mk.mpr v.irreducible),
    ← Multiset.count_map_eq_count' _ _ Subtype.val_injective, map_subtype_coe_factors',
    factors_eq_normalizedFactors, ← Multiset.count_map_eq_count' _ _ (mk_injective (M := Ideal R))]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
rw :
  v.asIdeal ^ count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₁ :
  v.asIdeal ^
      count (Associates.mk v.asIdeal)
        (↑(factors' I) : WithTop (Multiset { a : Associates (Ideal R) // Irreducible a })) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
⊢ v.maxPowDividing I = v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
have rw₂ : v.asIdeal ^
    Multiset.count (Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) sorry)
      (factors' I) =
  v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
Tactic: rw [maxPowDividing, factors_mk _ hI, count_some (irreducible_mk.mpr v.irreducible),
    ← Multiset.count_map_eq_count' _ _ Subtype.val_injective, map_subtype_coe_factors',
    factors_eq_normalizedFactors, ← Multiset.count_map_eq_count' _ _ (mk_injective (M := Ideal R))]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
rw :
  v.asIdeal ^ count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₁ :
  v.asIdeal ^
      count (Associates.mk v.asIdeal)
        (↑(factors' I) : WithTop (Multiset { a : Associates (Ideal R) // Irreducible a })) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₂ :
  v.asIdeal ^
      Multiset.count (Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯)
        (factors' I) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
⊢ v.maxPowDividing I = v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
have rw₃ : v.asIdeal ^
    Multiset.count
      (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) sorry) :
        Associates (Ideal R))
      (Multiset.map (Subtype.val (p := fun (a : Associates (Ideal R)) ↦ Irreducible a)) (factors' I)) =
  v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
Tactic: rw [maxPowDividing, factors_mk _ hI, count_some (irreducible_mk.mpr v.irreducible),
    ← Multiset.count_map_eq_count' _ _ Subtype.val_injective, map_subtype_coe_factors',
    factors_eq_normalizedFactors, ← Multiset.count_map_eq_count' _ _ (mk_injective (M := Ideal R))]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
rw :
  v.asIdeal ^ count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₁ :
  v.asIdeal ^
      count (Associates.mk v.asIdeal)
        (↑(factors' I) : WithTop (Multiset { a : Associates (Ideal R) // Irreducible a })) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₂ :
  v.asIdeal ^
      Multiset.count (Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯)
        (factors' I) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₃ :
  v.asIdeal ^
      Multiset.count
        (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯) :
          Associates (Ideal R))
        (Multiset.map (Subtype.val (p := fun (a : Associates (Ideal R)) ↦ Irreducible a)) (factors' I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
⊢ v.maxPowDividing I = v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
have rw₄ : v.asIdeal ^
    Multiset.count
      (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) sorry) :
        Associates (Ideal R))
      (Multiset.map Associates.mk (UniqueFactorizationMonoid.factors I)) =
  v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
Tactic: rw [maxPowDividing, factors_mk _ hI, count_some (irreducible_mk.mpr v.irreducible),
    ← Multiset.count_map_eq_count' _ _ Subtype.val_injective, map_subtype_coe_factors',
    factors_eq_normalizedFactors, ← Multiset.count_map_eq_count' _ _ (mk_injective (M := Ideal R))]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
rw :
  v.asIdeal ^ count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₁ :
  v.asIdeal ^
      count (Associates.mk v.asIdeal)
        (↑(factors' I) : WithTop (Multiset { a : Associates (Ideal R) // Irreducible a })) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₂ :
  v.asIdeal ^
      Multiset.count (Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯)
        (factors' I) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₃ :
  v.asIdeal ^
      Multiset.count
        (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯) :
          Associates (Ideal R))
        (Multiset.map (Subtype.val (p := fun (a : Associates (Ideal R)) ↦ Irreducible a)) (factors' I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₄ :
  v.asIdeal ^
      Multiset.count
        (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯) :
          Associates (Ideal R))
        (Multiset.map Associates.mk (UniqueFactorizationMonoid.factors I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
⊢ v.maxPowDividing I = v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
have rw₅ : v.asIdeal ^
    Multiset.count
      (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) sorry) :
        Associates (Ideal R))
      (Multiset.map Associates.mk (normalizedFactors I)) =
  v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
Tactic: rw [maxPowDividing, factors_mk _ hI, count_some (irreducible_mk.mpr v.irreducible),
    ← Multiset.count_map_eq_count' _ _ Subtype.val_injective, map_subtype_coe_factors',
    factors_eq_normalizedFactors, ← Multiset.count_map_eq_count' _ _ (mk_injective (M := Ideal R))]