awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
K : Type u_2
inst : CommRing R
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDomain R
inst_5 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x : K
hx : x ≠ 0
eq_J :
  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (spanSingleton R⁰ x)
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ K) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ K)
⊢ ∃ (z : (FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)ˣ),
    (∃ (x : K), spanSingleton R⁰ x = (↑z : FractionalIdeal R⁰ K)) ∧
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ K) * (↑z : FractionalIdeal R⁰ K) = (↑(↑J : Ideal R) : FractionalIdeal R⁰ K)
have mpr : HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑(↑I : Ideal R) : FractionalIdeal R⁰ K)
    (↑(Units.mk0 (spanSingleton R⁰ x) sorry : (FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)ˣ) : FractionalIdeal R⁰ K) =
  (↑(↑J : Ideal R) : FractionalIdeal R⁰ K)
Tactic: refine ⟨Units.mk0 _ (spanSingleton_ne_zero_iff.mpr hx), ⟨x, rfl⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
⊢ Eq (α := ClassGroup R) ((mk0 : (a : ↥(Ideal R)⁰) → ClassGroup R) I : ClassGroup R)
      ((mk0 : (a : ↥(Ideal R)⁰) → ClassGroup R) J : ClassGroup R) ↔
    ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
have refine_1 : (∃ (x : FractionRing R) (_ : x ≠ 0),
    HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R)) (spanSingleton R⁰ x)
        (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
      (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))) →
  ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
Tactic: refine (ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring (FractionRing R)).trans ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
refine_1 :
  (∃ (x : FractionRing R) (_ : x ≠ 0),
      HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R)) (spanSingleton R⁰ x)
          (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
        (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))) →
    ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
⊢ Eq (α := ClassGroup R) ((mk0 : (a : ↥(Ideal R)⁰) → ClassGroup R) I : ClassGroup R)
      ((mk0 : (a : ↥(Ideal R)⁰) → ClassGroup R) J : ClassGroup R) ↔
    ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
have refine_2 : (∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)) →
  ∃ (x : FractionRing R) (_ : x ≠ 0),
    HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R)) (spanSingleton R⁰ x)
        (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
      (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
Tactic: refine (ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring (FractionRing R)).trans ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
z : FractionRing R
hz : z ≠ 0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R)) (spanSingleton R⁰ z)
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
⊢ ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
have refine_1 : ∀ (x y : R) (hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y)
  (hz :
    Ne (α := FractionRing R)
      (mk' (FractionRing R) x
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
      0)
  (h :
    HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
        (spanSingleton R⁰
          (mk' (FractionRing R) x
            (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
        (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
      (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))),
  ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
Tactic: obtain ⟨x, ⟨y, hy⟩, rfl⟩ := IsLocalization.exists_mk'_eq R⁰ z
Cleared hypotheses: z, hz, h
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y
hz :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
      (spanSingleton R⁰
        (mk' (FractionRing R) x
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
⊢ ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
have refine_1 : x ≠ 0
Tactic: refine ⟨x, y, ?_, mem_nonZeroDivisors_iff_ne_zero.mp hy, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y
hz :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
      (spanSingleton R⁰
        (mk' (FractionRing R) x
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
refine_1 : x ≠ 0
⊢ ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
have refine_1₁ : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
Tactic: refine ⟨x, y, ?_, mem_nonZeroDivisors_iff_ne_zero.mp hy, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y
hz :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
      (spanSingleton R⁰
        (mk' (FractionRing R) x
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
hx : x = 0
⊢ False
have refine_1 : Eq (α := FractionRing R)
  (mk' (FractionRing R) x (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
  0
Tactic: apply hz
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y
hz :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
      (spanSingleton R⁰
        (mk' (FractionRing R) x
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
hx : x = 0
⊢ Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
have refine_1 : Eq (α := FractionRing R)
  (mk' (FractionRing R) 0 (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
  0
Tactic: rw [hx, IsFractionRing.mk'_eq_div, map_zero, zero_div]
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y
hz :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
      (spanSingleton R⁰
        (mk' (FractionRing R) x
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
hx : x = 0
refine_1 :
  Eq (α := FractionRing R)
    (mk' (FractionRing R) 0
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
⊢ Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
have refine_1₁ : HDiv.hDiv (α := FractionRing R) (β := FractionRing R)
    ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) 0 : FractionRing R)
    ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
        (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy) : R) :
      FractionRing R) =
  0
Tactic: rw [hx, IsFractionRing.mk'_eq_div, map_zero, zero_div]
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y
hz :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
      (spanSingleton R⁰
        (mk' (FractionRing R) x
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
hx : x = 0
refine_1 :
  Eq (α := FractionRing R)
    (mk' (FractionRing R) 0
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
refine_1₁ :
  HDiv.hDiv (α := FractionRing R) (β := FractionRing R)
      ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) 0 : FractionRing R)
      ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
          (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy) : R) :
        FractionRing R) =
    0
⊢ Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
have refine_1₂ : HDiv.hDiv (α := FractionRing R) (β := FractionRing R) (0 : FractionRing R)
    ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
        (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy) : R) :
      FractionRing R) =
  0
Tactic: rw [hx, IsFractionRing.mk'_eq_div, map_zero, zero_div]
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) y
hz :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
h :
  HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
      (spanSingleton R⁰
        (mk' (FractionRing R) x
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy)))
      (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
hx : x = 0
refine_1 :
  Eq (α := FractionRing R)
    (mk' (FractionRing R) 0
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
refine_1₁ :
  HDiv.hDiv (α := FractionRing R) (β := FractionRing R)
      ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) 0 : FractionRing R)
      ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
          (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy) : R) :
        FractionRing R) =
    0
refine_1₂ :
  HDiv.hDiv (α := FractionRing R) (β := FractionRing R) (0 : FractionRing R)
      ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
          (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy) : R) :
        FractionRing R) =
    0
⊢ Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
have refine_1₃ : Eq.{u_1 + 1} (α := FractionRing R) 0 0
Tactic: rw [hx, IsFractionRing.mk'_eq_div, map_zero, zero_div]
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hx : x ≠ 0
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
⊢ ∃ (x : FractionRing R) (_ : x ≠ 0),
    HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R)) (spanSingleton R⁰ x)
        (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
      (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
have hy' : Membership.mem (γ := Submonoid R) R⁰ y
Tactic: have hy' : y ∈ R⁰ := mem_nonZeroDivisors_iff_ne_zero.mpr hy
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hx : x ≠ 0
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
hy' : Membership.mem (γ := Submonoid R) R⁰ y
⊢ ∃ (x : FractionRing R) (_ : x ≠ 0),
    HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R)) (spanSingleton R⁰ x)
        (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
      (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
have refine_2 : Ne (α := FractionRing R)
  (mk' (FractionRing R) x
    (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
  0
Tactic: refine ⟨IsLocalization.mk' _ x ⟨y, hy'⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hx : x ≠ 0
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
hy' : Membership.mem (γ := Submonoid R) R⁰ y
refine_2 :
  Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
    0
⊢ ∃ (x : FractionRing R) (_ : x ≠ 0),
    HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R)) (spanSingleton R⁰ x)
        (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
      (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
have refine_2₁ : HMul.hMul (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
    (spanSingleton R⁰
      (mk' (FractionRing R) x
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy')))
    (↑(↑I : Ideal R) : FractionalIdeal R⁰ (FractionRing R)) =
  (↑(↑J : Ideal R) : FractionalIdeal R⁰ (FractionRing R))
Tactic: refine ⟨IsLocalization.mk' _ x ⟨y, hy'⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hx : x ≠ 0
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
hy' : Membership.mem (γ := Submonoid R) R⁰ y
⊢ Ne (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
    0
have refine_2 : ∀
  (hx :
    Eq (α := FractionRing R)
      (mk' (FractionRing R) x
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
      0),
  x = 0
Tactic: contrapose hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
hy' : Membership.mem (γ := Submonoid R) R⁰ y
hx :
  Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
    0
⊢ x = 0
have refine_2 : ∀
  (hx :
    (algebraMap R (FractionRing R) : (a : R) → FractionRing R) x =
      HMul.hMul (α := FractionRing R) (β := FractionRing R) (0 : FractionRing R)
        ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
            (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy') : R) :
          FractionRing R)),
  x = 0
Tactic: rwa [mk'_eq_iff_eq_mul, zero_mul, ← (algebraMap R (FractionRing R)).map_zero,
        (IsFractionRing.injective R (FractionRing R)).eq_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
hy' : Membership.mem (γ := Submonoid R) R⁰ y
hx :
  Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
    0
refine_2 :
  ∀
    (hx :
      (algebraMap R (FractionRing R) : (a : R) → FractionRing R) x =
        HMul.hMul (α := FractionRing R) (β := FractionRing R) (0 : FractionRing R)
          ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
              (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy') : R) :
            FractionRing R)),
    x = 0
⊢ x = 0
have refine_2₁ : ∀
  (hx :
    Eq (α := FractionRing R) ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) x : FractionRing R)
      (0 : FractionRing R)),
  x = 0
Tactic: rwa [mk'_eq_iff_eq_mul, zero_mul, ← (algebraMap R (FractionRing R)).map_zero,
        (IsFractionRing.injective R (FractionRing R)).eq_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
hy' : Membership.mem (γ := Submonoid R) R⁰ y
hx :
  Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
    0
refine_2 :
  ∀
    (hx :
      (algebraMap R (FractionRing R) : (a : R) → FractionRing R) x =
        HMul.hMul (α := FractionRing R) (β := FractionRing R) (0 : FractionRing R)
          ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
              (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy') : R) :
            FractionRing R)),
    x = 0
refine_2₁ :
  ∀
    (hx :
      Eq (α := FractionRing R) ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) x : FractionRing R)
        (0 : FractionRing R)),
    x = 0
⊢ x = 0
have refine_2₂ : ∀
  (hx :
    Eq (α := FractionRing R) ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) x : FractionRing R)
      ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) 0 : FractionRing R)),
  x = 0
Tactic: rwa [mk'_eq_iff_eq_mul, zero_mul, ← (algebraMap R (FractionRing R)).map_zero,
        (IsFractionRing.injective R (FractionRing R)).eq_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsDedekindDomain R
I J : ↥(Ideal R)⁰
x y : R
hy : y ≠ 0
h : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
hy' : Membership.mem (γ := Submonoid R) R⁰ y
hx :
  Eq (α := FractionRing R)
    (mk' (FractionRing R) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy'))
    0
refine_2 :
  ∀
    (hx :
      (algebraMap R (FractionRing R) : (a : R) → FractionRing R) x =
        HMul.hMul (α := FractionRing R) (β := FractionRing R) (0 : FractionRing R)
          ((algebraMap R (FractionRing R) : (a : R) → FractionRing R)
              (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy') : R) :
            FractionRing R)),
    x = 0
refine_2₁ :
  ∀
    (hx :
      Eq (α := FractionRing R) ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) x : FractionRing R)
        (0 : FractionRing R)),
    x = 0
refine_2₂ :
  ∀
    (hx :
      Eq (α := FractionRing R) ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) x : FractionRing R)
        ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) 0 : FractionRing R)),
    x = 0
⊢ x = 0
have refine_2₃ : ∀ (hx : x = 0), x = 0
Tactic: rwa [mk'_eq_iff_eq_mul, zero_mul, ← (algebraMap R (FractionRing R)).map_zero,
        (IsFractionRing.injective R (FractionRing R)).eq_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
I : FractionalIdeal (R := R) R⁰ (FractionRing R)
⊢ IsUnit (M := FractionalIdeal (R := R) R⁰ (FractionRing R)) (↑I.num : FractionalIdeal R⁰ (FractionRing R)) ↔ IsUnit I
have hden0 : Ne (α := FractionRing R) ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) (↑I.den : R) : FractionRing R)
  (0 : FractionRing R)
Tactic: have hden0 : algebraMap R (FractionRing R) I.den ≠ 0 :=
    IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors I.den.prop
-- Lean 4 proof step from Mathlib.RingTheory.ClassGroup (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
I : FractionalIdeal (R := R) R⁰ (FractionRing R)
hden0 :
  Ne (α := FractionRing R) ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) (↑I.den : R) : FractionRing R)
    (0 : FractionRing R)
u : (FractionRing R)ˣ := Units.mk0 ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) (↑I.den : R)) hden0
⊢ IsUnit (M := FractionalIdeal (R := R) R⁰ (FractionRing R)) (↑I.num : FractionalIdeal R⁰ (FractionRing R)) ↔ IsUnit I
have hdenUnit : IsUnit (M := FractionalIdeal (R := R) R⁰ (FractionRing R))
  (spanSingleton R⁰ ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) (↑I.den : R)) :
    FractionalIdeal (R := R) R⁰ (FractionRing R))
Tactic: have hdenUnit : IsUnit (spanSingleton R⁰ (algebraMap R (FractionRing R) I.den)) :=
    ⟨toPrincipalIdeal R (FractionRing R) u, by simp [u]⟩