awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.ClassGroup
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 : (FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)ˣ
hx : (↑(↑I : Ideal R) : FractionalIdeal R⁰ K) * (↑X : FractionalIdeal R⁰ K) = (↑(↑J : Ideal R) : FractionalIdeal R⁰ K)
hX : spanSingleton R⁰ 0 = (↑X : FractionalIdeal R⁰ K)
⊢ False

Tactic:
simp [X.ne_zero.symm] at hX

Premises used: Units.ne_zero, Ne.symm
-- Module: Mathlib.RingTheory.ClassGroup
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 : (FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)ˣ
hx : (↑(↑I : Ideal R) : FractionalIdeal R⁰ K) * (↑X : FractionalIdeal R⁰ K) = (↑(↑J : Ideal R) : FractionalIdeal R⁰ K)
x : K
hX : spanSingleton R⁰ x = (↑X : FractionalIdeal R⁰ K)
⊢ 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)

Tactic:
simpa only [hX, mul_comm] using hx

Premises used: mul_comm
-- Module: Mathlib.RingTheory.ClassGroup
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)

Tactic:
⟨Units.mk0 _ (spanSingleton_ne_zero_iff.mpr hx), ⟨x, rfl⟩, ?_⟩

Premises used: Iff.mpr, FractionalIdeal.spanSingleton_ne_zero_iff, Units.mk0, And.intro, Exists.intro, rfl
-- Module: Mathlib.RingTheory.ClassGroup
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)
mpr :
  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑(↑I : Ideal R) : FractionalIdeal R⁰ K)
      (↑(Units.mk0 (spanSingleton R⁰ x) ⋯ : (FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)ˣ) : 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)

Tactic:
refine ⟨Units.mk0 _ (spanSingleton_ne_zero_iff.mpr hx), ⟨x, rfl⟩, ?_⟩
-- Module: Mathlib.RingTheory.ClassGroup
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)
⊢ HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑(↑I : Ideal R) : FractionalIdeal R⁰ K)
      (↑(Units.mk0 (spanSingleton R⁰ x) ⋯ : (FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)ˣ) : FractionalIdeal R⁰ K) =
    (↑(↑J : Ideal R) : FractionalIdeal R⁰ K)

Tactic:
simpa only [mul_comm] using eq_J

Premises used: mul_comm
-- Module: Mathlib.RingTheory.ClassGroup
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)

Tactic:
(ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring (FractionRing R)).trans ⟨?_, ?_⟩

Premises used: FractionRing, ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring, Iff.intro, Iff.trans
-- Module: Mathlib.RingTheory.ClassGroup
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)
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))
⊢ 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)

Tactic:
refine (ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring (FractionRing R)).trans ⟨?_, ?_⟩

Premises used: FractionRing, ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring, Iff.intro, Iff.trans
-- Module: Mathlib.RingTheory.ClassGroup
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))
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)
⊢ ∃ (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

Premises used: nonZeroDivisors, rfl, IsLocalization.exists_mk'_eq
-- Module: Mathlib.RingTheory.ClassGroup
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)

Tactic:
⟨x, y, ?_, mem_nonZeroDivisors_iff_ne_zero.mp hy, ?_⟩

Premises used: Exists.intro, Iff.mp, mem_nonZeroDivisors_iff_ne_zero
-- Module: Mathlib.RingTheory.ClassGroup
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
refine_1₁ : Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)
⊢ ∃ (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)

Tactic:
refine ⟨x, y, ?_, mem_nonZeroDivisors_iff_ne_zero.mp hy, ?_⟩
-- Module: Mathlib.RingTheory.ClassGroup
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) x
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (R⁰ : Submonoid R) x) y hy))
    0
⊢ False

Tactic:
apply hz
-- Module: Mathlib.RingTheory.ClassGroup
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
refine_1₃ refine_1₄ : Eq.{u_1 + 1} (α := FractionRing R) 0 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

Tactic:
rw [hx, IsFractionRing.mk'_eq_div, map_zero, zero_div]

Premises used: IsFractionRing.mk'_eq_div, zero_div, map_zero
-- Module: Mathlib.RingTheory.ClassGroup
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))
⊢ Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)

Tactic:
(FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal _ hy).mp h

Premises used: FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal, Iff.mp
-- Module: Mathlib.RingTheory.ClassGroup
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))
⊢ Ideal.span {x} * (↑I : Ideal R) = Ideal.span {y} * (↑J : Ideal R)

Tactic:
exact (FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal _ hy).mp h

Premises used: FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal, Iff.mp
-- Module: Mathlib.RingTheory.ClassGroup
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)
⊢ Membership.mem (γ := Submonoid R) R⁰ y

Tactic:
have hy' : y ∈ R⁰ := mem_nonZeroDivisors_iff_ne_zero.mpr hy
-- Module: Mathlib.RingTheory.ClassGroup
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))

Tactic:
⟨IsLocalization.mk' _ x ⟨y, hy'⟩, ?_, ?_⟩

Premises used: Subtype.mk, IsLocalization.mk', Exists.intro
-- Module: Mathlib.RingTheory.ClassGroup
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
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))
⊢ ∃ (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 ⟨IsLocalization.mk' _ x ⟨y, hy'⟩, ?_, ?_⟩
-- Module: Mathlib.RingTheory.ClassGroup
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 :
  ∀
    (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
⊢ 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:
contrapose hx
-- Module: Mathlib.RingTheory.ClassGroup
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
refine_2₃ refine_2₄ refine_2₅ refine_2₆ : ∀ (hx : x = 0), 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

Premises used: Algebra.algebraMap, IsFractionRing.injective, MulZeroClass.zero_mul, Function.Injective.eq_iff, FractionRing, IsLocalization.mk'_eq_iff_eq_mul, RingHom.map_zero
-- Module: Mathlib.RingTheory.ClassGroup
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
⊢ 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:
(FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal _ hy').mpr h

Premises used: Iff.mpr, FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal
-- Module: Mathlib.RingTheory.ClassGroup
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
⊢ 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:
exact (FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal _ hy').mpr h

Premises used: Iff.mpr, FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal
-- Module: Mathlib.RingTheory.ClassGroup
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
I : FractionalIdeal (R := R) R⁰ (FractionRing R)
⊢ 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
-- Module: Mathlib.RingTheory.ClassGroup
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
⊢ Eq (α := FractionalIdeal (R := R) R⁰ (FractionRing R))
    ((↑((toPrincipalIdeal R (FractionRing R) : (a : (FractionRing R)ˣ) → (FractionalIdeal R⁰ (FractionRing R))ˣ) u) :
        FractionalIdeal (R := R) R⁰ (FractionRing R)) :
      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:
simp [u]
-- Module: Mathlib.RingTheory.ClassGroup
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))
    (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]⟩
-- Module: Mathlib.RingTheory.ClassGroup
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
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))
obtain :
  ∀ (c : (FractionalIdeal (R := R) R⁰ (FractionRing R))ˣ)
    (hc :
      (↑c : FractionalIdeal R⁰ (FractionRing R)) =
        spanSingleton R⁰ ((algebraMap R (FractionRing R) : (a : R) → FractionRing R) (↑I.den : R))),
    IsUnit (M := FractionalIdeal (R := R) R⁰ (FractionRing R)) (↑I.num : FractionalIdeal R⁰ (FractionRing R)) ↔ IsUnit I
⊢ IsUnit (M := FractionalIdeal (R := R) R⁰ (FractionRing R)) (↑I.num : FractionalIdeal R⁰ (FractionRing R)) ↔ IsUnit I

Tactic:
obtain ⟨c, hc⟩ := hdenUnit