l3lab/ntp-mathlib huggingface dataset:
theorem prod_finsetApprox_ne_zero : algebraMap R S (∏ m in finsetApprox bS adm, m) ≠ 0 := by
  refine' mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) _
  

Proof state:
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁶ : EuclideanDomain R
inst✝¹⁵ : CommRing S
inst✝¹⁴ : IsDomain S
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra R K
inst✝¹⁰ : IsFractionRing R K
inst✝⁹ : Algebra K L
inst✝⁸ : FiniteDimensional K L
inst✝⁷ : IsSeparable K L
algRL : Algebra R L
inst✝⁶ : IsScalarTower R K L
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝¹ : Infinite R
inst✝ : DecidableEq R
⊢ ¬∏ m in finsetApprox bS adm, m = 0

Next tactic:
simp only [Finset.prod_eq_zero_iff, not_exists]
theorem prod_finsetApprox_ne_zero : algebraMap R S (∏ m in finsetApprox bS adm, m) ≠ 0 := by
  refine' mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) _
  simp only [Finset.prod_eq_zero_iff, not_exists]
  

Proof state:
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁶ : EuclideanDomain R
inst✝¹⁵ : CommRing S
inst✝¹⁴ : IsDomain S
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra R K
inst✝¹⁰ : IsFractionRing R K
inst✝⁹ : Algebra K L
inst✝⁸ : FiniteDimensional K L
inst✝⁷ : IsSeparable K L
algRL : Algebra R L
inst✝⁶ : IsScalarTower R K L
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝¹ : Infinite R
inst✝ : DecidableEq R
⊢ ∀ (x : R), ¬(x ∈ finsetApprox bS adm ∧ x = 0)

Next tactic:
rintro x ⟨hx, rfl⟩
theorem prod_finsetApprox_ne_zero : algebraMap R S (∏ m in finsetApprox bS adm, m) ≠ 0 := by
  refine' mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) _
  simp only [Finset.prod_eq_zero_iff, not_exists]
  rintro x ⟨hx, rfl⟩
  

Proof state:
case intro
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁶ : EuclideanDomain R
inst✝¹⁵ : CommRing S
inst✝¹⁴ : IsDomain S
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra R K
inst✝¹⁰ : IsFractionRing R K
inst✝⁹ : Algebra K L
inst✝⁸ : FiniteDimensional K L
inst✝⁷ : IsSeparable K L
algRL : Algebra R L
inst✝⁶ : IsScalarTower R K L
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝¹ : Infinite R
inst✝ : DecidableEq R
hx : 0 ∈ finsetApprox bS adm
⊢ False

Next tactic:
exact finsetApprox.zero_not_mem bS adm hx
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  

Proof state:
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
⊢ ∃ J, mk0 I = mk0 J ∧ (algebraMap R S) (∏ m in finsetApprox bS adm, m) ∈ ↑J

Next tactic:
set M := ∏ m in finsetApprox bS adm, m
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  

Proof state:
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
⊢ ∃ J, mk0 I = mk0 J ∧ (algebraMap R S) M ∈ ↑J

Next tactic:
have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  

Proof state:
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
⊢ ∃ J, mk0 I = mk0 J ∧ (algebraMap R S) M ∈ ↑J

Next tactic:
obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  

Proof state:
case intro.intro.intro
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
⊢ ∃ J, mk0 I = mk0 J ∧ (algebraMap R S) M ∈ ↑J

Next tactic:
suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    refine' ⟨⟨J, _⟩, _, _⟩
    · rw [mem_nonZeroDivisors_iff_ne_zero]
      rintro rfl
      rw [Ideal.zero_eq_bot, Ideal.mul_bot] at hJ
      exact hM (Ideal.span_singleton_eq_bot.mp (I.2 _ hJ))
    · rw [ClassGroup.mk0_eq_mk0_iff]
      exact ⟨algebraMap _ _ M, b, hM, b_ne_zero, hJ⟩
    rw [← SetLike.mem_coe, ← Set.singleton_subset_iff, ← Ideal.span_le, ← Ideal.dvd_iff_le]
    refine' (mul_dvd_mul_iff_left _).mp _
    swap; · exact mt Ideal.span_singleton_eq_bot.mp b_ne_zero
    rw [Subtype.coe_mk, Ideal.dvd_iff_le, ← hJ, mul_comm]
    apply Ideal.mul_mono le_rfl
    rw [Ideal.span_le, Set.singleton_subset_iff]
    exact b_mem
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    

Proof state:
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
this : Ideal.span {b} ∣ Ideal.span {(algebraMap R S) M} * ↑I
⊢ ∃ J, mk0 I = mk0 J ∧ (algebraMap R S) M ∈ ↑J

Next tactic:
obtain ⟨J, hJ⟩ := this
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    

Proof state:
case intro
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
J : Ideal S
hJ : Ideal.span {(algebraMap R S) M} * ↑I = Ideal.span {b} * J
⊢ ∃ J, mk0 I = mk0 J ∧ (algebraMap R S) M ∈ ↑J

Next tactic:
refine' ⟨⟨J, _⟩, _, _⟩
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    refine' ⟨⟨J, _⟩, _, _⟩
    · 

Proof state:
case intro.refine'_1
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
J : Ideal S
hJ : Ideal.span {(algebraMap R S) M} * ↑I = Ideal.span {b} * J
⊢ J ∈ (Ideal S)⁰

Next tactic:
rw [mem_nonZeroDivisors_iff_ne_zero]
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    refine' ⟨⟨J, _⟩, _, _⟩
    · rw [mem_nonZeroDivisors_iff_ne_zero]
      

Proof state:
case intro.refine'_1
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
J : Ideal S
hJ : Ideal.span {(algebraMap R S) M} * ↑I = Ideal.span {b} * J
⊢ J ≠ 0

Next tactic:
rintro rfl
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    refine' ⟨⟨J, _⟩, _, _⟩
    · rw [mem_nonZeroDivisors_iff_ne_zero]
      rintro rfl
      

Proof state:
case intro.refine'_1
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
hJ : Ideal.span {(algebraMap R S) M} * ↑I = Ideal.span {b} * 0
⊢ False

Next tactic:
rw [Ideal.zero_eq_bot, Ideal.mul_bot] at hJ
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    refine' ⟨⟨J, _⟩, _, _⟩
    · rw [mem_nonZeroDivisors_iff_ne_zero]
      rintro rfl
      rw [Ideal.zero_eq_bot, Ideal.mul_bot] at hJ
      

Proof state:
case intro.refine'_1
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
hJ : Ideal.span {(algebraMap R S) M} * ↑I = ⊥
⊢ False

Next tactic:
exact hM (Ideal.span_singleton_eq_bot.mp (I.2 _ hJ))
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    refine' ⟨⟨J, _⟩, _, _⟩
    · rw [mem_nonZeroDivisors_iff_ne_zero]
      rintro rfl
      rw [Ideal.zero_eq_bot, Ideal.mul_bot] at hJ
      exact hM (Ideal.span_singleton_eq_bot.mp (I.2 _ hJ))
    · 

Proof state:
case intro.refine'_2
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
J : Ideal S
hJ : Ideal.span {(algebraMap R S) M} * ↑I = Ideal.span {b} * J
⊢ mk0 I = mk0 { val := J, property := (_ : J ∈ (Ideal S)⁰) }

Next tactic:
rw [ClassGroup.mk0_eq_mk0_iff]
/-- Each class in the class group contains an ideal `J`
such that `M := Π m ∈ finsetApprox` is in `J`. -/
theorem exists_mk0_eq_mk0 [IsDedekindDomain S] (h : Algebra.IsAlgebraic R L) (I : (Ideal S)⁰) :
    ∃ J : (Ideal S)⁰,
      ClassGroup.mk0 I = ClassGroup.mk0 J ∧
        algebraMap _ _ (∏ m in finsetApprox bS adm, m) ∈ (J : Ideal S) := by
  set M := ∏ m in finsetApprox bS adm, m
  have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
  obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
  suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
    obtain ⟨J, hJ⟩ := this
    refine' ⟨⟨J, _⟩, _, _⟩
    · rw [mem_nonZeroDivisors_iff_ne_zero]
      rintro rfl
      rw [Ideal.zero_eq_bot, Ideal.mul_bot] at hJ
      exact hM (Ideal.span_singleton_eq_bot.mp (I.2 _ hJ))
    · rw [ClassGroup.mk0_eq_mk0_iff]
      

Proof state:
case intro.refine'_2
R : Type u_1
S : Type u_2
K : Type u_3
L : Type u_4
inst✝¹⁷ : EuclideanDomain R
inst✝¹⁶ : CommRing S
inst✝¹⁵ : IsDomain S
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra R K
inst✝¹¹ : IsFractionRing R K
inst✝¹⁰ : Algebra K L
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : IsSeparable K L
algRL : Algebra R L
inst✝⁷ : IsScalarTower R K L
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
bS : Basis ι R S
adm : AbsoluteValue.IsAdmissible abv
inst✝² : Infinite R
inst✝¹ : DecidableEq R
inst✝ : IsDedekindDomain S
h : Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
M : R := ∏ m in finsetApprox bS adm, m
hM : (algebraMap R S) M ≠ 0
b : S
b_mem : b ∈ ↑I
b_ne_zero : b ≠ 0
b_min : ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0
J : Ideal S
hJ : Ideal.span {(algebraMap R S) M} * ↑I = Ideal.span {b} * J
⊢ ∃ x y,
    ∃ (_ : x ≠ 0) (_ : y ≠ 0), Ideal.span {x} * ↑I = Ideal.span {y} * ↑{ val := J, property := (_ : J ∈ (Ideal S)⁰) }

Next tactic:
exact ⟨algebraMap _ _ M, b, hM, b_ne_zero, hJ⟩