l3lab/ntp-mathlib huggingface dataset:
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
    exact coe_ideal_le_self_mul_inv K I
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
hJ0 : ¬J = ⊥
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
by_cases hJ1 : J = ⊤
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
    exact coe_ideal_le_self_mul_inv K I
  by_cases hJ1 : J = ⊤
  · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
hJ0 : ¬J = ⊥
hJ1 : J = ⊤
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
rw [← hJ, hJ1, coeIdeal_top]
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
    exact coe_ideal_le_self_mul_inv K I
  by_cases hJ1 : J = ⊤
  · rw [← hJ, hJ1, coeIdeal_top]
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
hJ0 : ¬J = ⊥
hJ1 : ¬J = ⊤
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
obtain ⟨x, hx, hx1⟩ :
    ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
    exists_not_mem_one_of_ne_bot hNF hJ0 hJ1
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
    exact coe_ideal_le_self_mul_inv K I
  by_cases hJ1 : J = ⊤
  · rw [← hJ, hJ1, coeIdeal_top]
  obtain ⟨x, hx, hx1⟩ :
    ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
    exists_not_mem_one_of_ne_bot hNF hJ0 hJ1
  

Proof state:
case neg.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
hJ0 : ¬J = ⊥
hJ1 : ¬J = ⊤
x : K
hx : x ∈ (↑J)⁻¹
hx1 : x ∉ 1
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
contrapose! hx1 with h_abs
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
    exact coe_ideal_le_self_mul_inv K I
  by_cases hJ1 : J = ⊤
  · rw [← hJ, hJ1, coeIdeal_top]
  obtain ⟨x, hx, hx1⟩ :
    ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
    exists_not_mem_one_of_ne_bot hNF hJ0 hJ1
  contrapose! hx1 with h_abs
  

Proof state:
case neg.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
hJ0 : ¬J = ⊥
hJ1 : ¬J = ⊤
x : K
hx : x ∈ (↑J)⁻¹
h_abs : ↑I * (↑I)⁻¹ ≠ 1
⊢ x ∈ 1

Next tactic:
rw [hJ] at hx
theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
    (hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- Handle a few trivial cases.
  by_cases hI1 : I = ⊤
  · rw [hI1, coeIdeal_top, one_mul, inv_one]
  by_cases hNF : IsField A
  · letI := hNF.toField; rcases hI1 (I.eq_bot_or_top.resolve_left hI0) with ⟨⟩
  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:
  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.
  obtain ⟨J, hJ⟩ : ∃ J : Ideal A, (J : FractionalIdeal A⁰ K) = I * (I : FractionalIdeal A⁰ K)⁻¹ :=
    le_one_iff_exists_coeIdeal.mp mul_one_div_le_one
  by_cases hJ0 : J = ⊥
  · subst hJ0
    refine' absurd _ hI0
    rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ]
    exact coe_ideal_le_self_mul_inv K I
  by_cases hJ1 : J = ⊤
  · rw [← hJ, hJ1, coeIdeal_top]
  obtain ⟨x, hx, hx1⟩ :
    ∃ x : K, x ∈ (J : FractionalIdeal A⁰ K)⁻¹ ∧ x ∉ (1 : FractionalIdeal A⁰ K) :=
    exists_not_mem_one_of_ne_bot hNF hJ0 hJ1
  contrapose! hx1 with h_abs
  rw [hJ] at hx
  

Proof state:
case neg.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hI1 : ¬I = ⊤
hNF : ¬IsField A
J : Ideal A
hJ : ↑J = ↑I * (↑I)⁻¹
hJ0 : ¬J = ⊥
hJ1 : ¬J = ⊤
x : K
hx : x ∈ (↑I * (↑I)⁻¹)⁻¹
h_abs : ↑I * (↑I)⁻¹ ≠ 1
⊢ x ∈ 1

Next tactic:
exact hI hx
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
⊢ ↑I * (↑I)⁻¹ = 1

Next tactic:
apply mul_inv_cancel_of_le_one hI0
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
⊢ (↑I * (↑I)⁻¹)⁻¹ ≤ 1

Next tactic:
by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ↑I * (↑I)⁻¹ = 0
⊢ (↑I * (↑I)⁻¹)⁻¹ ≤ 1

Next tactic:
rw [hJ0, inv_zero']
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; 

Proof state:
case pos
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ↑I * (↑I)⁻¹ = 0
⊢ 0 ≤ 1

Next tactic:
exact zero_le _
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
⊢ (↑I * (↑I)⁻¹)⁻¹ ≤ 1

Next tactic:
intro x hx
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹
⊢ x ∈ (fun a => ↑a) 1

Next tactic:
suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  suffices x ∈ integralClosure A K by
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹
this : x ∈ integralClosure A K
⊢ x ∈ (fun a => ↑a) 1

Next tactic:
rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
  -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
  -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹
⊢ x ∈ integralClosure A K

Next tactic:
rw [mem_integralClosure_iff_mem_fg]
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
  -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
  -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
  rw [mem_integralClosure_iff_mem_fg]
  

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹
⊢ ∃ M, Submodule.FG (Subalgebra.toSubmodule M) ∧ x ∈ M

Next 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)]
    dsimp only at hx
    rw [val_eq_coe, mem_coe, mem_inv_iff hJ0] at hx
    simp only [mul_assoc, mul_comm b] at hx ⊢
    intro y hy
    exact hx _ (mul_mem_mul hy hb)
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
  -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
  -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
  rw [mem_integralClosure_iff_mem_fg]
  have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) := by
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹
⊢ ∀ b ∈ (↑I)⁻¹, x * b ∈ (↑I)⁻¹

Next tactic:
intro b hb
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
  -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
  -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
  rw [mem_integralClosure_iff_mem_fg]
  have x_mul_mem : ∀ b ∈ (I⁻¹ : FractionalIdeal A⁰ K), x * b ∈ (I⁻¹ : FractionalIdeal A⁰ K) := by
    intro b hb
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹
b : K
hb : b ∈ (↑I)⁻¹
⊢ x * b ∈ (↑I)⁻¹

Next tactic:
rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI0)]
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
  -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
  -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
  rw [mem_integralClosure_iff_mem_fg]
  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)]
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹
b : K
hb : b ∈ (↑I)⁻¹
⊢ ∀ y ∈ ↑I, x * b * y ∈ 1

Next tactic:
dsimp only at hx
/-- Nonzero integral ideals in a Dedekind domain are invertible.

We will use this to show that nonzero fractional ideals are invertible,
and finally conclude that fractional ideals in a Dedekind domain form a group with zero.
-/
theorem coe_ideal_mul_inv [h : IsDedekindDomain A] (I : Ideal A) (hI0 : I ≠ ⊥) :
    I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 := by
  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.
  apply mul_inv_cancel_of_le_one hI0
  by_cases hJ0 : I * (I : FractionalIdeal A⁰ K)⁻¹ = 0
  · rw [hJ0, inv_zero']; exact zero_le _
  intro x hx
  -- In particular, we'll show all `x ∈ J⁻¹` are integral.
  suffices x ∈ integralClosure A K by
    rwa [IsIntegrallyClosed.integralClosure_eq_bot, Algebra.mem_bot, Set.mem_range,
      ← mem_one_iff] at this
  -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.
  -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.
  rw [mem_integralClosure_iff_mem_fg]
  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)]
    dsimp only at hx
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hJ0 : ¬↑I * (↑I)⁻¹ = 0
x : K
hx : x ∈ ↑(↑I * (↑I)⁻¹)⁻¹
b : K
hb : b ∈ (↑I)⁻¹
⊢ ∀ y ∈ ↑I, x * b * y ∈ 1

Next tactic:
rw [val_eq_coe, mem_coe, mem_inv_iff hJ0] at hx