l3lab/ntp-mathlib huggingface dataset:
/-- The first half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the prime
    factors of `I*S` are in bijection with those of the minimal polynomial of the generator of `S`
    over `R`, taken `mod I`.-/
noncomputable def normalizedFactorsMapEquivNormalizedFactorsMinPolyMk (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    {J : Ideal S | J ∈ normalizedFactors (I.map (algebraMap R S))} ≃
      {d : (R ⧸ I)[X] |
        d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} := by
  -- Porting note: Lean needs to be reminded about this so it does not time out
  have : IsPrincipalIdealRing (R ⧸ I)[X] := inferInstance
  let f : S ⧸ map (algebraMap R S) I ≃+*
    (R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} := by
    refine (quotAdjoinEquivQuotMap hx ?_).symm.trans
      (((Algebra.adjoin.powerBasis'
        hx').quotientEquivQuotientMinpolyMap I).toRingEquiv.trans (quotEquivOfEq ?_))
    · exact NoZeroSMulDivisors.algebraMap_injective (Algebra.adjoin R {x}) S
    · rw [Algebra.adjoin.powerBasis'_minpoly_gen hx']
  refine (normalizedFactorsEquivOfQuotEquiv f ?_ ?_).trans ?_
  · rwa [Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S),
      ← Ne.def]
  · by_contra h
    

Proof state:
case refine_2
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
this : IsPrincipalIdealRing (R ⧸ I)[X]
f : S ⧸ Ideal.map (algebraMap R S) I ≃+* (R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=
  RingEquiv.trans
    (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S))))
    (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))
      (quotEquivOfEq
        (_ :
          span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =
            span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))
h : span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} = ⊥
⊢ False

Next tactic:
exact (show Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) ≠ 0 from
      Polynomial.map_monic_ne_zero (minpoly.monic hx')) (span_singleton_eq_bot.mp h)
/-- The first half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the prime
    factors of `I*S` are in bijection with those of the minimal polynomial of the generator of `S`
    over `R`, taken `mod I`.-/
noncomputable def normalizedFactorsMapEquivNormalizedFactorsMinPolyMk (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    {J : Ideal S | J ∈ normalizedFactors (I.map (algebraMap R S))} ≃
      {d : (R ⧸ I)[X] |
        d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} := by
  -- Porting note: Lean needs to be reminded about this so it does not time out
  have : IsPrincipalIdealRing (R ⧸ I)[X] := inferInstance
  let f : S ⧸ map (algebraMap R S) I ≃+*
    (R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} := by
    refine (quotAdjoinEquivQuotMap hx ?_).symm.trans
      (((Algebra.adjoin.powerBasis'
        hx').quotientEquivQuotientMinpolyMap I).toRingEquiv.trans (quotEquivOfEq ?_))
    · exact NoZeroSMulDivisors.algebraMap_injective (Algebra.adjoin R {x}) S
    · rw [Algebra.adjoin.powerBasis'_minpoly_gen hx']
  refine (normalizedFactorsEquivOfQuotEquiv f ?_ ?_).trans ?_
  · rwa [Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S),
      ← Ne.def]
  · by_contra h
    exact (show Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) ≠ 0 from
      Polynomial.map_monic_ne_zero (minpoly.monic hx')) (span_singleton_eq_bot.mp h)
  · 

Proof state:
case refine_3
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
this : IsPrincipalIdealRing (R ⧸ I)[X]
f : S ⧸ Ideal.map (algebraMap R S) I ≃+* (R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=
  RingEquiv.trans
    (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S))))
    (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))
      (quotEquivOfEq
        (_ :
          span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =
            span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))
⊢ ↑{M | M ∈ normalizedFactors (span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})} ≃
    ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}

Next tactic:
refine (normalizedFactorsEquivSpanNormalizedFactors ?_).symm
/-- The first half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the prime
    factors of `I*S` are in bijection with those of the minimal polynomial of the generator of `S`
    over `R`, taken `mod I`.-/
noncomputable def normalizedFactorsMapEquivNormalizedFactorsMinPolyMk (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    {J : Ideal S | J ∈ normalizedFactors (I.map (algebraMap R S))} ≃
      {d : (R ⧸ I)[X] |
        d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} := by
  -- Porting note: Lean needs to be reminded about this so it does not time out
  have : IsPrincipalIdealRing (R ⧸ I)[X] := inferInstance
  let f : S ⧸ map (algebraMap R S) I ≃+*
    (R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} := by
    refine (quotAdjoinEquivQuotMap hx ?_).symm.trans
      (((Algebra.adjoin.powerBasis'
        hx').quotientEquivQuotientMinpolyMap I).toRingEquiv.trans (quotEquivOfEq ?_))
    · exact NoZeroSMulDivisors.algebraMap_injective (Algebra.adjoin R {x}) S
    · rw [Algebra.adjoin.powerBasis'_minpoly_gen hx']
  refine (normalizedFactorsEquivOfQuotEquiv f ?_ ?_).trans ?_
  · rwa [Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S),
      ← Ne.def]
  · by_contra h
    exact (show Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) ≠ 0 from
      Polynomial.map_monic_ne_zero (minpoly.monic hx')) (span_singleton_eq_bot.mp h)
  · refine (normalizedFactorsEquivSpanNormalizedFactors ?_).symm
    

Proof state:
case refine_3
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
this : IsPrincipalIdealRing (R ⧸ I)[X]
f : S ⧸ Ideal.map (algebraMap R S) I ≃+* (R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} :=
  RingEquiv.trans
    (RingEquiv.symm (quotAdjoinEquivQuotMap hx (_ : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S))))
    (RingEquiv.trans (AlgEquiv.toRingEquiv (PowerBasis.quotientEquivQuotientMinpolyMap (adjoin.powerBasis' hx') I))
      (quotEquivOfEq
        (_ :
          span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =
            span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)})))
⊢ Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) ≠ 0

Next tactic:
exact Polynomial.map_monic_ne_zero (minpoly.monic hx')
/-- The second half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the
    bijection `FactorsEquiv'` defined in the first half preserves multiplicities. -/
theorem multiplicity_factors_map_eq_multiplicity (hI : IsMaximal I) (hI' : I ≠ ⊥)
    (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) {J : Ideal S}
    (hJ : J ∈ normalizedFactors (I.map (algebraMap R S))) :
    multiplicity J (I.map (algebraMap R S)) =
      multiplicity (↑(normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx' ⟨J, hJ⟩))
        (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) := by
  

Proof state:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ multiplicity J (Ideal.map (algebraMap R S) I) =
    multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))
      (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))

Next tactic:
rw [normalizedFactorsMapEquivNormalizedFactorsMinPolyMk, Equiv.coe_trans, Function.comp_apply,
    multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity,
    normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity]
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  

Proof state:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
⊢ normalizedFactors (Ideal.map (algebraMap R S) I) =
    Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
      (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))))

Next tactic:
ext J
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  

Proof state:
case a
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count J
      (Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
        (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))

Next tactic:
by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S))
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S));
  

Proof state:
case pos
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count J
      (Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
        (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))
case neg
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count J
      (Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
        (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))

Next tactic:
swap
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S));
  swap
  · 

Proof state:
case neg
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count J
      (Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
        (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))

Next tactic:
rw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S));
  swap
  · rw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]
    

Proof state:
case neg
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ ¬∃ a ∈ Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))),
      ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm a) = J

Next tactic:
simp only [Multiset.mem_attach, true_and_iff, not_exists]
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S));
  swap
  · rw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]
    simp only [Multiset.mem_attach, true_and_iff, not_exists]
    

Proof state:
case neg
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∉ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ ∀ (x_1 : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}),
    ¬↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm x_1) = J

Next tactic:
rintro J' rfl
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S));
  swap
  · rw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]
    simp only [Multiset.mem_attach, true_and_iff, not_exists]
    rintro J' rfl
    

Proof state:
case neg
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J' : ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}
hJ :
  ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J') ∉
    normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ False

Next tactic:
exact
      hJ ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J').prop
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S));
  swap
  · rw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]
    simp only [Multiset.mem_attach, true_and_iff, not_exists]
    rintro J' rfl
    exact
      hJ ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J').prop
  -- Then we just have to compare the multiplicities, which we already proved are equal.
  

Proof state:
case pos
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count J
      (Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
        (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))

Next tactic:
have := multiplicity_factors_map_eq_multiplicity hI hI' hx hx' hJ
/-- The **Kummer-Dedekind Theorem**. -/
theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
    (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
    normalizedFactors (I.map (algebraMap R S)) =
      Multiset.map
        (fun f =>
          ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f : Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach := by
  ext J
  -- WLOG, assume J is a normalized factor
  by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S));
  swap
  · rw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]
    simp only [Multiset.mem_attach, true_and_iff, not_exists]
    rintro J' rfl
    exact
      hJ ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J').prop
  -- Then we just have to compare the multiplicities, which we already proved are equal.
  have := multiplicity_factors_map_eq_multiplicity hI hI' hx hx' hJ
  

Proof state:
case pos
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
x : S
I : Ideal R
inst✝⁴ : IsDomain R
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : IsMaximal I
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
J : Ideal S
hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)
this :
  multiplicity J (Ideal.map (algebraMap R S) I) =
    multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') { val := J, property := hJ }))
      (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
⊢ Multiset.count J (normalizedFactors (Ideal.map (algebraMap R S) I)) =
    Multiset.count J
      (Multiset.map (fun f => ↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm f))
        (Multiset.attach (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))))

Next tactic:
rw [multiplicity_eq_count_normalizedFactors, multiplicity_eq_count_normalizedFactors,
    UniqueFactorizationMonoid.normalize_normalized_factor _ hJ,
    UniqueFactorizationMonoid.normalize_normalized_factor, PartENat.natCast_inj] at this