l3lab/ntp-mathlib huggingface dataset:
/-- The canonical morphism of rings from `R<x> ⧸ (I*R<x>)` to `S ⧸ (I*S)` is an isomorphism
    when `I` and `(conductor R x) ∩ R` are coprime. -/
noncomputable def quotAdjoinEquivQuotMap (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤)
    (h_alg : Function.Injective (algebraMap R<x> S)) :
    R<x> ⧸ I.map (algebraMap R R<x>) ≃+* S ⧸ I.map (algebraMap R S) := by
  let f : R<x> ⧸ I.map (algebraMap R R<x>) →+* S ⧸ I.map (algebraMap R S) :=
    (Ideal.Quotient.lift (I.map (algebraMap R R<x>))
      ((Ideal.Quotient.mk (I.map (algebraMap R S))).comp (algebraMap R<x> S)) (fun r hr => by
      have : algebraMap R S = (algebraMap R<x> S).comp (algebraMap R R<x>) := by ext; rfl
      rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, this, ← Ideal.map_map]
      exact Ideal.mem_map_of_mem _ hr))
  refine RingEquiv.ofBijective f ⟨?_, ?_⟩
  · --the kernel of the map is clearly `(I * S) ∩ R<x>`. To get injectivity, we need to show that
    --this is contained in `I * R<x>`, which is the content of the previous lemma.
    refine RingHom.lift_injective_of_ker_le_ideal _ _ fun u hu => ?_
    rwa [RingHom.mem_ker, RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, ← Ideal.mem_comap,
      comap_map_eq_map_adjoin_of_coprime_conductor hx h_alg] at hu
  · -- Surjectivity follows from the surjectivity of the canonical map `R<x> → S ⧸ (I * S)`,
    -- which in turn follows from the fact that `I * S + (conductor R x) = S`.
    refine Ideal.Quotient.lift_surjective_of_surjective _ _ fun y => ?_
    obtain ⟨z, hz⟩ := Ideal.Quotient.mk_surjective y
    have : z ∈ conductor R x ⊔ I.map (algebraMap R S) := by
      suffices conductor R x ⊔ I.map (algebraMap R S) = ⊤ by simp only [this, Submodule.mem_top]
      rw [Ideal.eq_top_iff_one] at hx ⊢
      replace hx := Ideal.mem_map_of_mem (algebraMap R S) hx
      

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
h_alg : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S)
f : ↥(adjoin R {x}) ⧸ Ideal.map (algebraMap R ↥(adjoin R {x})) I →+* S ⧸ Ideal.map (algebraMap R S) I :=
  Ideal.Quotient.lift (Ideal.map (algebraMap R ↥(adjoin R {x})) I)
    (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap (↥(adjoin R {x})) S))
    (_ :
      ∀ r ∈ Ideal.map (algebraMap R ↥(adjoin R {x})) I,
        (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap (↥(adjoin R {x})) S)) r = 0)
y : S ⧸ Ideal.map (algebraMap R S) I
z : S
hz : (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y
hx : (algebraMap R S) 1 ∈ Ideal.map (algebraMap R S) (comap (algebraMap R S) (conductor R x) ⊔ I)
⊢ 1 ∈ conductor R x ⊔ Ideal.map (algebraMap R S) I

Next tactic:
rw [Ideal.map_sup, RingHom.map_one] at hx
/-- The canonical morphism of rings from `R<x> ⧸ (I*R<x>)` to `S ⧸ (I*S)` is an isomorphism
    when `I` and `(conductor R x) ∩ R` are coprime. -/
noncomputable def quotAdjoinEquivQuotMap (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤)
    (h_alg : Function.Injective (algebraMap R<x> S)) :
    R<x> ⧸ I.map (algebraMap R R<x>) ≃+* S ⧸ I.map (algebraMap R S) := by
  let f : R<x> ⧸ I.map (algebraMap R R<x>) →+* S ⧸ I.map (algebraMap R S) :=
    (Ideal.Quotient.lift (I.map (algebraMap R R<x>))
      ((Ideal.Quotient.mk (I.map (algebraMap R S))).comp (algebraMap R<x> S)) (fun r hr => by
      have : algebraMap R S = (algebraMap R<x> S).comp (algebraMap R R<x>) := by ext; rfl
      rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, this, ← Ideal.map_map]
      exact Ideal.mem_map_of_mem _ hr))
  refine RingEquiv.ofBijective f ⟨?_, ?_⟩
  · --the kernel of the map is clearly `(I * S) ∩ R<x>`. To get injectivity, we need to show that
    --this is contained in `I * R<x>`, which is the content of the previous lemma.
    refine RingHom.lift_injective_of_ker_le_ideal _ _ fun u hu => ?_
    rwa [RingHom.mem_ker, RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, ← Ideal.mem_comap,
      comap_map_eq_map_adjoin_of_coprime_conductor hx h_alg] at hu
  · -- Surjectivity follows from the surjectivity of the canonical map `R<x> → S ⧸ (I * S)`,
    -- which in turn follows from the fact that `I * S + (conductor R x) = S`.
    refine Ideal.Quotient.lift_surjective_of_surjective _ _ fun y => ?_
    obtain ⟨z, hz⟩ := Ideal.Quotient.mk_surjective y
    have : z ∈ conductor R x ⊔ I.map (algebraMap R S) := by
      suffices conductor R x ⊔ I.map (algebraMap R S) = ⊤ by simp only [this, Submodule.mem_top]
      rw [Ideal.eq_top_iff_one] at hx ⊢
      replace hx := Ideal.mem_map_of_mem (algebraMap R S) hx
      rw [Ideal.map_sup, RingHom.map_one] at hx
      

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
h_alg : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S)
f : ↥(adjoin R {x}) ⧸ Ideal.map (algebraMap R ↥(adjoin R {x})) I →+* S ⧸ Ideal.map (algebraMap R S) I :=
  Ideal.Quotient.lift (Ideal.map (algebraMap R ↥(adjoin R {x})) I)
    (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap (↥(adjoin R {x})) S))
    (_ :
      ∀ r ∈ Ideal.map (algebraMap R ↥(adjoin R {x})) I,
        (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap (↥(adjoin R {x})) S)) r = 0)
y : S ⧸ Ideal.map (algebraMap R S) I
z : S
hz : (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y
hx : 1 ∈ Ideal.map (algebraMap R S) (comap (algebraMap R S) (conductor R x)) ⊔ Ideal.map (algebraMap R S) I
⊢ 1 ∈ conductor R x ⊔ Ideal.map (algebraMap R S) I

Next tactic:
exact (sup_le_sup
        (show ((conductor R x).comap (algebraMap R S)).map (algebraMap R S) ≤ conductor R x
          from Ideal.map_comap_le)
          (le_refl (I.map (algebraMap R S)))) hx
/-- The canonical morphism of rings from `R<x> ⧸ (I*R<x>)` to `S ⧸ (I*S)` is an isomorphism
    when `I` and `(conductor R x) ∩ R` are coprime. -/
noncomputable def quotAdjoinEquivQuotMap (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤)
    (h_alg : Function.Injective (algebraMap R<x> S)) :
    R<x> ⧸ I.map (algebraMap R R<x>) ≃+* S ⧸ I.map (algebraMap R S) := by
  let f : R<x> ⧸ I.map (algebraMap R R<x>) →+* S ⧸ I.map (algebraMap R S) :=
    (Ideal.Quotient.lift (I.map (algebraMap R R<x>))
      ((Ideal.Quotient.mk (I.map (algebraMap R S))).comp (algebraMap R<x> S)) (fun r hr => by
      have : algebraMap R S = (algebraMap R<x> S).comp (algebraMap R R<x>) := by ext; rfl
      rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, this, ← Ideal.map_map]
      exact Ideal.mem_map_of_mem _ hr))
  refine RingEquiv.ofBijective f ⟨?_, ?_⟩
  · --the kernel of the map is clearly `(I * S) ∩ R<x>`. To get injectivity, we need to show that
    --this is contained in `I * R<x>`, which is the content of the previous lemma.
    refine RingHom.lift_injective_of_ker_le_ideal _ _ fun u hu => ?_
    rwa [RingHom.mem_ker, RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem, ← Ideal.mem_comap,
      comap_map_eq_map_adjoin_of_coprime_conductor hx h_alg] at hu
  · -- Surjectivity follows from the surjectivity of the canonical map `R<x> → S ⧸ (I * S)`,
    -- which in turn follows from the fact that `I * S + (conductor R x) = S`.
    refine Ideal.Quotient.lift_surjective_of_surjective _ _ fun y => ?_
    obtain ⟨z, hz⟩ := Ideal.Quotient.mk_surjective y
    have : z ∈ conductor R x ⊔ I.map (algebraMap R S) := by
      suffices conductor R x ⊔ I.map (algebraMap R S) = ⊤ by simp only [this, Submodule.mem_top]
      rw [Ideal.eq_top_iff_one] at hx ⊢
      replace hx := Ideal.mem_map_of_mem (algebraMap R S) hx
      rw [Ideal.map_sup, RingHom.map_one] at hx
      exact (sup_le_sup
        (show ((conductor R x).comap (algebraMap R S)).map (algebraMap R S) ≤ conductor R x
          from Ideal.map_comap_le)
          (le_refl (I.map (algebraMap R S)))) hx
    

Proof state:
case refine_2.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S)
f : ↥(adjoin R {x}) ⧸ Ideal.map (algebraMap R ↥(adjoin R {x})) I →+* S ⧸ Ideal.map (algebraMap R S) I :=
  Ideal.Quotient.lift (Ideal.map (algebraMap R ↥(adjoin R {x})) I)
    (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap (↥(adjoin R {x})) S))
    (_ :
      ∀ r ∈ Ideal.map (algebraMap R ↥(adjoin R {x})) I,
        (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap (↥(adjoin R {x})) S)) r = 0)
y : S ⧸ Ideal.map (algebraMap R S) I
z : S
hz : (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) z = y
this : z ∈ conductor R x ⊔ Ideal.map (algebraMap R S) I
⊢ ∃ a, (RingHom.comp (Ideal.Quotient.mk (Ideal.map (algebraMap R S) I)) (algebraMap (↥(adjoin R {x})) S)) a = y

Next tactic:
rw [← Ideal.mem_quotient_iff_mem_sup, hz, Ideal.mem_map_iff_of_surjective] at this
/-- 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
  

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 | J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)} ≃
    ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}

Next tactic:
have : IsPrincipalIdealRing (R ⧸ I)[X] := inferInstance
/-- 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
  

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
this : IsPrincipalIdealRing (R ⧸ I)[X]
⊢ ↑{J | J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)} ≃
    ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}

Next tactic:
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']
/-- 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
    

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
this : IsPrincipalIdealRing (R ⧸ I)[X]
⊢ S ⧸ Ideal.map (algebraMap R S) I ≃+* (R ⧸ I)[X] ⧸ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)}

Next tactic:
refine (quotAdjoinEquivQuotMap hx ?_).symm.trans
      (((Algebra.adjoin.powerBasis'
        hx').quotientEquivQuotientMinpolyMap I).toRingEquiv.trans (quotEquivOfEq ?_))
/-- 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 ?_))
    · 

Proof state:
case refine_1
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]
⊢ Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S)

Next tactic:
exact NoZeroSMulDivisors.algebraMap_injective (Algebra.adjoin R {x}) S
/-- 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
    · 

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]
⊢ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R (adjoin.powerBasis' hx').gen)} =
    span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)}

Next tactic:
rw [Algebra.adjoin.powerBasis'_minpoly_gen hx']
/-- 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']
  

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
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)})))
⊢ ↑{J | J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)} ≃
    ↑{d | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))}

Next tactic:
refine (normalizedFactorsEquivOfQuotEquiv f ?_ ?_).trans ?_
/-- 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 ?_
  · 

Proof state:
case refine_1
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)})))
⊢ Ideal.map (algebraMap R S) I ≠ ⊥

Next tactic:
rwa [Ne.def, map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S),
      ← Ne.def]
/-- 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]
  · 

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)})))
⊢ span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)} ≠ ⊥

Next tactic:
by_contra h