awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have pos : ∀ (hJ0 : J = ⊥),
  HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
Tactic: by_cases hJ0 : J = ⊥
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
pos :
  ∀ (hJ0 : J = ⊥),
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹ =
      1
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have neg : ∀ (hJ0 : ¬J = ⊥),
  HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
Tactic: by_cases hJ0 : J = ⊥
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
hJ0 : J = ⊥
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have hJ₁ : (↑⊥ : FractionalIdeal A⁰ K) =
  HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
Tactic: subst hJ0
Cleared hypotheses: J, hJ, hJ0
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
hJ :
  (↑⊥ : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have pos : I = ⊥
Tactic: refine absurd ?_ hI0
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
hJ :
  (↑⊥ : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
⊢ I = ⊥
have pos : (↑I : FractionalIdeal A⁰ K) ≤
  HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
Tactic: rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
hJ0 : ¬J = ⊥
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have pos : ∀ (hJ1 : J = ⊤),
  HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
Tactic: by_cases hJ1 : J = ⊤
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
hJ0 : ¬J = ⊥
pos :
  ∀ (hJ1 : J = ⊤),
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹ =
      1
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have neg : ∀ (hJ1 : ¬J = ⊤),
  HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
Tactic: by_cases hJ1 : J = ⊤
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
hJ0 : ¬J = ⊥
hJ1 : J = ⊤
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have pos : Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑J : FractionalIdeal A⁰ K) 1
Tactic: rw [← hJ, hJ1, coeIdeal_top]
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
hJ0 : ¬J = ⊥
hJ1 : J = ⊤
pos : Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑J : FractionalIdeal A⁰ K) 1
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have pos₁ : Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑⊤ : FractionalIdeal A⁰ K) 1
Tactic: rw [← hJ, hJ1, coeIdeal_top]
-- 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 ≠ ⊥
hI :
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
J : Ideal A
hJ :
  (↑J : FractionalIdeal A⁰ K) =
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹
hJ0 : ¬J = ⊥
hJ1 : J = ⊤
pos : Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑J : FractionalIdeal A⁰ K) 1
pos₁ : Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑⊤ : FractionalIdeal A⁰ K) 1
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have pos₂ : Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 1
Tactic: rw [← hJ, hJ1, coeIdeal_top]
-- 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 ≠ ⊥
⊢ HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ =
    1
have apply : (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
      (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
  1
Tactic: apply mul_inv_cancel_of_le_one hI0
-- 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 ≠ ⊥
⊢ (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
have pos : ∀
  (hJ0 :
    HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹ =
      0),
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
Tactic: by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
-- 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 ≠ ⊥
pos :
  ∀
    (hJ0 :
      HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
          (↑I : FractionalIdeal A⁰ K)⁻¹ =
        0),
    (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
          (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
      1
⊢ (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
have neg : ∀
  (hJ0 :
    ¬HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
          (↑I : FractionalIdeal A⁰ K)⁻¹ =
        0),
  (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
Tactic: by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
-- 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
⊢ (HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤
    1
have pos : LE.le.{u_3} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 0 1
Tactic: rw [hJ0, inv_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
this : Membership.mem (γ := Subalgebra A K) (integralClosure A K) x
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
have rwa : ∀ (this : Membership.mem (γ := Subalgebra A K) ⊥ x), Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
Tactic: rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
Cleared hypotheses: this
-- 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
this : Membership.mem (γ := Subalgebra A K) (integralClosure A K) x
rwa :
  ∀ (this : Membership.mem (γ := Subalgebra A K) ⊥ x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
have rwa₁ : ∀ (this : Membership.mem (γ := Set K) (Set.range.{u_3, u_2 + 1} (ι := A) (⇑(algebraMap A K) : (a : A) → K)) x),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
Tactic: rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
Cleared hypotheses: this
-- 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
this : Membership.mem (γ := Subalgebra A K) (integralClosure A K) x
rwa :
  ∀ (this : Membership.mem (γ := Subalgebra A K) ⊥ x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
rwa₁ :
  ∀ (this : Membership.mem (γ := Set K) (Set.range.{u_3, u_2 + 1} (ι := A) (⇑(algebraMap A K) : (a : A) → K)) x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
have rwa₂ : ∀ (this : ∃ (y : A), (algebraMap A K : (a : A) → K) y = x),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
Tactic: rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
Cleared hypotheses: this
-- 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
this : Membership.mem (γ := Subalgebra A K) (integralClosure A K) x
rwa :
  ∀ (this : Membership.mem (γ := Subalgebra A K) ⊥ x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
rwa₁ :
  ∀ (this : Membership.mem (γ := Set K) (Set.range.{u_3, u_2 + 1} (ι := A) (⇑(algebraMap A K) : (a : A) → K)) x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
rwa₂ :
  ∀ (this : ∃ (y : A), (algebraMap A K : (a : A) → K) y = x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
have rwa₃ : ∀ (this : ∃ (y : A), (algebraMap A K : (a : A) → K) y = x)
  (this : Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
Tactic: rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
Cleared hypotheses: this
-- 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
this : Membership.mem (γ := Subalgebra A K) (integralClosure A K) x
rwa :
  ∀ (this : Membership.mem (γ := Subalgebra A K) ⊥ x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
rwa₁ :
  ∀ (this : Membership.mem (γ := Set K) (Set.range.{u_3, u_2 + 1} (ι := A) (⇑(algebraMap A K) : (a : A) → K)) x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
rwa₂ :
  ∀ (this : ∃ (y : A), (algebraMap A K : (a : A) → K) y = x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
rwa₃ :
  ∀ (this : ∃ (y : A), (algebraMap A K : (a : A) → K) y = x)
    (this : Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
have S : (this : ∃ (y : A), (algebraMap A K : (a : A) → K) y = x) → Submonoid A
Tactic: rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
Cleared hypotheses: this
-- 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
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 x
have neg : Membership.mem (γ := Subalgebra A K) (integralClosure A K) x
Tactic: suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
-- 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
⊢ Membership.mem (γ := Subalgebra A K) (integralClosure A K) x
have neg : ∃ (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
Tactic: rw [mem_integralClosure_iff_mem_fg]
-- 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
b : K
hb : 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)
have rw : ∀ (y : K),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) y →
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (x * b * y)
Tactic: rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI0)]
-- 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
b : K
hb : Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b
⊢ ∀ (y : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) y →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (x * b * y)
have rw : ∀
  (hx :
    ∀
      y ∈
        HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
          (↑I : FractionalIdeal A⁰ K)⁻¹,
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (x * y))
  (b : K) (hb : Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b) (y : K),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) y →
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (x * b * y)
Tactic: rw [mem_inv_iff hJ0] at hx
Cleared hypotheses: hx, b, 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 :
  ∀
    y ∈
      HMul.hMul (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)
        (↑I : FractionalIdeal A⁰ K)⁻¹,
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (x * y)
b : K
hb : Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ b
⊢ ∀ (y : K),
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) y →
      Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (x * b * y)
have simp_rw : ∀ (y : K),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) y →
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (x * (y * b))
Tactic: simp_rw [mul_assoc, mul_comm b]