awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (hypothesis)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
⊢ ∃ (M : Subalgebra A K),
    Submodule.FG.{u_2, u_3} (R := A) (M := K) ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) M) ∧
      x ∈ M
have x_mul_mem : ∀ (b : K),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
Tactic: have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) := by
    intro b hb
    rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI0)]
    rw [mem_inv_iff hJ0] at hx
    simp_rw [mul_assoc, mul_comm b]
    exact fun y hy ↦ hx _ (mul_mem_mul hy hb)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
⊢ ∃ (M : Subalgebra A K),
    Submodule.FG.{u_2, u_3} (R := A) (M := K) ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) M) ∧
      x ∈ M
have neg : ∀ (y : K)
  (hy :
    Membership.mem (γ := Submodule A K)
      ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
        Submodule A K)
      y),
  Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) y
Tactic: refine ⟨AlgHom.range (Polynomial.aeval x : A[X] →ₐ[A] K),
    isNoetherian_submodule.mp (isNoetherian (I : FractionalIdeal A⁰ K)⁻¹) _ fun y hy => ?_,
    ⟨Polynomial.X, Polynomial.aeval_X x⟩⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
y : K
hy :
  Membership.mem (γ := Submodule A K)
    ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
      Submodule A K)
    y
⊢ Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) y
have neg : ∀ (p : A[X])
  (hy :
    Membership.mem (α := K) (γ := Submodule A K)
      ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
        Submodule A K)
      ((Polynomial.aeval x : (a : A[X]) → K) p : K)),
  Membership.mem.{u_3, u_3} (α := K) (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
Tactic: obtain ⟨p, rfl⟩ := (AlgHom.mem_range _).mp hy
Cleared hypotheses: y, hy
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
p : A[X]
hy :
  Membership.mem (α := K) (γ := Submodule A K)
    ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
      Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
⊢ Membership.mem.{u_3, u_3} (α := K) (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
have neg : Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K)
  (∑ i ∈ Finset.range (p.natDegree + 1), p.coeff i • x ^ i)
Tactic: rw [Polynomial.aeval_eq_sum_range]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
p : A[X]
hy :
  Membership.mem (α := K) (γ := Submodule A K)
    ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
      Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
⊢ Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K)
    (∑ i ∈ Finset.range (p.natDegree + 1), p.coeff i • x ^ i)
have neg : ∀ (i : ℕ) (hi : i ∈ Finset.range (p.natDegree + 1)),
  Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x ^ i)
Tactic: refine Submodule.sum_mem _ fun i hi => Submodule.smul_mem _ _ ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
p : A[X]
hy :
  Membership.mem (α := K) (γ := Submodule A K)
    ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
      Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
⊢ Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (HPow.hPow (β := ℕ) x 0)
have neg : Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) 1
Tactic: rw [pow_zero]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
p : A[X]
hy :
  Membership.mem (α := K) (γ := Submodule A K)
    ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
      Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
i : ℕ
ih : Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x ^ i)
⊢ Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x ^ (i + 1))
have neg : Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x * x ^ i)
Tactic: rw [pow_succ']
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
p : A[X]
hy :
  Membership.mem (α := K) (γ := Submodule A K)
    ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
      Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
i : ℕ
⊢ Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x ^ i)
have neg : Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (HPow.hPow (β := ℕ) x 0)
Tactic: induction i with
  | zero => rw [pow_zero]; exact one_mem_inv_coe_ideal hI0
  | succ i ih => rw [pow_succ']; exact x_mul_mem _ ih
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 :
  ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0
x : K
hx :
  x ∈
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹
x_mul_mem :
  ∀ (b : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ (x * b)
p : A[X]
hy :
  Membership.mem (α := K) (γ := Submodule A K)
    ((Subalgebra.toSubmodule : (a : Subalgebra A K) → Submodule A K) (AlgHom.range (Polynomial.aeval x)) :
      Submodule A K)
    ((Polynomial.aeval x : (a : A[X]) → K) p : K)
i : ℕ
neg : Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (HPow.hPow (β := ℕ) x 0)
⊢ Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x ^ i)
have neg₁ : ∀ (i : ℕ) (ih : Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x ^ i)),
  Membership.mem (γ := Submodule A K) (↑(↑I : FractionalIdeal A⁰ K)⁻¹ : Submodule A K) (x ^ (i + 1))
Tactic: induction i with
  | zero => rw [pow_zero]; exact one_mem_inv_coe_ideal hI0
  | succ i ih => rw [pow_succ']; exact x_mul_mem _ ih
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
inst_5 : IsDedekindDomain A
I_1 : Ideal A
I : FractionalIdeal (R := A) A⁰ (FractionRing A)
hI : I ≠ ⊥
⊢ I * I⁻¹ = 1
have obtain : ∀ (a : A) (J : Ideal A) (ha : a ≠ 0)
  (hJ :
    I =
      HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A))),
  I * I⁻¹ = 1
Tactic: obtain ⟨a, J, ha, hJ⟩ := exists_eq_spanSingleton_mul (K := FractionRing A) I
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
inst_5 : IsDedekindDomain A
I_1 : Ideal A
I : FractionalIdeal (R := A) A⁰ (FractionRing A)
hI : I ≠ ⊥
a : A
J : Ideal A
ha : a ≠ 0
hJ :
  I =
    HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
      (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
        FractionalIdeal (R := A) A⁰ (FractionRing A))
      ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A))
h₂ :
  I *
      HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a) :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal A⁰ (FractionRing A))⁻¹ : FractionalIdeal (R := A) A⁰ (FractionRing A)) =
    1
⊢ I * I⁻¹ = 1
have rw : ∃ (J : FractionalIdeal (R := A) A⁰ (FractionRing A)), I * J = 1
Tactic: rw [mul_inv_cancel_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
inst_5 : IsDedekindDomain A
I_1 : Ideal A
I : FractionalIdeal (R := A) A⁰ (FractionRing A)
hI : I ≠ ⊥
a : A
J : Ideal A
ha : a ≠ 0
hJ :
  I =
    HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
      (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
        FractionalIdeal (R := A) A⁰ (FractionRing A))
      ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A))
⊢ I * I⁻¹ = 1
have suffices : I *
    HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
      (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a) :
        FractionalIdeal (R := A) A⁰ (FractionRing A))
      ((↑J : FractionalIdeal A⁰ (FractionRing A))⁻¹ : FractionalIdeal (R := A) A⁰ (FractionRing A)) =
  1
Tactic: suffices h₂ : I * (spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ _)⁻¹) = 1 by
      rw [mul_inv_cancel_iff]
      exact ⟨spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ _)⁻¹, h₂⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
inst_5 : IsDedekindDomain A
I_1 : Ideal A
I : FractionalIdeal (R := A) A⁰ (FractionRing A)
hI : I ≠ ⊥
a : A
J : Ideal A
ha : a ≠ 0
hJ :
  I =
    HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
      (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
        FractionalIdeal (R := A) A⁰ (FractionRing A))
      ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A))
⊢ I *
      HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a) :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal A⁰ (FractionRing A))⁻¹ : FractionalIdeal (R := A) A⁰ (FractionRing A)) =
    1
have subst : ∀
  (hI :
    HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A)) ≠
      ⊥),
  HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A)) *
      HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a) :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal A⁰ (FractionRing A))⁻¹ : FractionalIdeal (R := A) A⁰ (FractionRing A)) =
    1
Tactic: subst hJ
Cleared hypotheses: I, hI, hJ
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
inst_5 : IsDedekindDomain A
I : Ideal A
a : A
J : Ideal A
ha : a ≠ 0
hI :
  HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
      (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
        FractionalIdeal (R := A) A⁰ (FractionRing A))
      ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A)) ≠
    ⊥
⊢ HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A)) *
      HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a) :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal A⁰ (FractionRing A))⁻¹ : FractionalIdeal (R := A) A⁰ (FractionRing A)) =
    1
have rw : Ne (α := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a : FractionRing A)
  (0 : FractionRing A)
Tactic: rw [mul_assoc, mul_left_comm (J : FractionalIdeal A⁰ _), coe_ideal_mul_inv, mul_one,
      spanSingleton_mul_spanSingleton, inv_mul_cancel₀, spanSingleton_one]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
inst_5 : IsDedekindDomain A
I : Ideal A
a : A
J : Ideal A
ha : a ≠ 0
hI :
  HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
      (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
        FractionalIdeal (R := A) A⁰ (FractionRing A))
      ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A)) ≠
    ⊥
rw :
  Ne (α := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a : FractionRing A)
    (0 : FractionRing A)
⊢ HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a)⁻¹ :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal (R := A) A⁰ (FractionRing A)) : FractionalIdeal (R := A) A⁰ (FractionRing A)) *
      HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (β := FractionalIdeal (R := A) A⁰ (FractionRing A))
        (spanSingleton A⁰ ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) a) :
          FractionalIdeal (R := A) A⁰ (FractionRing A))
        ((↑J : FractionalIdeal A⁰ (FractionRing A))⁻¹ : FractionalIdeal (R := A) A⁰ (FractionRing A)) =
    1
have hI0 : J ≠ ⊥
Tactic: rw [mul_assoc, mul_left_comm (J : FractionalIdeal A⁰ _), coe_ideal_mul_inv, mul_one,
      spanSingleton_mul_spanSingleton, inv_mul_cancel₀, spanSingleton_one]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K_1 : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K_1
inst_3 : Algebra A K_1
inst_4 : IsFractionRing A K_1
inst_5 : IsDedekindDomain A
I_1 : Ideal A
I : { x : Ideal A // 0 < x }
J K : Ideal A
e :
  (fun (x1 x2 : Ideal A) ↦ x1 ≤ x2) ((fun (x : { x : Ideal A // 0 < x }) (y : Ideal A) ↦ (↑x : Ideal A) * y) I J)
    ((fun (x : { x : Ideal A // 0 < x }) (y : Ideal A) ↦ (↑x : Ideal A) * y) I K)
⊢ J ≤ K
have rwa : LE.le (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑J : FractionalIdeal A⁰ (FractionRing A))
  (↑K : FractionalIdeal A⁰ (FractionRing A))
Tactic: rwa [← FractionalIdeal.coeIdeal_le_coeIdeal (FractionRing A),
      ← mul_le_mul_iff_right₀ (α := FractionalIdeal A⁰ (FractionRing A)) (a := I.1)
        (by simpa [pos_iff_ne_zero] using I.2.ne'),
      ← FractionalIdeal.coeIdeal_mul, ← FractionalIdeal.coeIdeal_mul,
      FractionalIdeal.coeIdeal_le_coeIdeal]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
R : Type u_1
A : Type u_2
K_1 : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K_1
inst_3 : Algebra A K_1
inst_4 : IsFractionRing A K_1
inst_5 : IsDedekindDomain A
I_1 : Ideal A
I : { x : Ideal A // 0 < x }
J K : Ideal A
e :
  (fun (x1 x2 : Ideal A) ↦ x1 ≤ x2) ((fun (x : { x : Ideal A // 0 < x }) (y : Ideal A) ↦ (↑x : Ideal A) * y) I J)
    ((fun (x : { x : Ideal A // 0 < x }) (y : Ideal A) ↦ (↑x : Ideal A) * y) I K)
rwa :
  LE.le (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑J : FractionalIdeal A⁰ (FractionRing A))
    (↑K : FractionalIdeal A⁰ (FractionRing A))
⊢ J ≤ K
have rwa₁ : HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑(↑I : Ideal A) : FractionalIdeal A⁰ (FractionRing A))
    (↑J : FractionalIdeal A⁰ (FractionRing A)) ≤
  HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑(↑I : Ideal A) : FractionalIdeal A⁰ (FractionRing A))
    (↑K : FractionalIdeal A⁰ (FractionRing A))
Tactic: rwa [← FractionalIdeal.coeIdeal_le_coeIdeal (FractionRing A),
      ← mul_le_mul_iff_right₀ (α := FractionalIdeal A⁰ (FractionRing A)) (a := I.1)
        (by simpa [pos_iff_ne_zero] using I.2.ne'),
      ← FractionalIdeal.coeIdeal_mul, ← FractionalIdeal.coeIdeal_mul,
      FractionalIdeal.coeIdeal_le_coeIdeal]