awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
this : i.IsPrime
⊢ ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) ⋯ :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi
have refine_2 : ∀ (this : ∃ q ∈ normalizedFactors r, Associated (Submodule.IsPrincipal.generator i) q),
  ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) sorry :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi
Tactic: have := exists_mem_normalizedFactors_of_dvd hr
        (Submodule.IsPrincipal.prime_generator_of_isPrime i
        (prime_of_normalized_factor i hi).ne_zero).irreducible ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
this : i.IsPrime
refine_2 :
  ∀ (this : ∃ q ∈ normalizedFactors r, Associated (Submodule.IsPrincipal.generator i) q),
    ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
      (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
            (Subtype.mk (p := fun (x : Ideal R) ↦
                Membership.mem (γ := Set (Ideal R))
                  {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
                (span {(↑d : R)}) ⋯ :
              { x : Ideal R //
                Membership.mem (γ := Set (Ideal R))
                  {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
          a =
        Subtype.mk (p := fun (x : Ideal R) ↦
          Membership.mem (γ := Set (Ideal R))
            {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
          i hi
⊢ ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) ⋯ :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi
have refine_2₁ : Submodule.IsPrincipal.generator i ∣ r
Tactic: have := exists_mem_normalizedFactors_of_dvd hr
        (Submodule.IsPrincipal.prime_generator_of_isPrime i
        (prime_of_normalized_factor i hi).ne_zero).irreducible ?_
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
this_1 : i.IsPrime
this : ∃ q ∈ normalizedFactors r, Associated (Submodule.IsPrincipal.generator i) q
⊢ ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) ⋯ :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi
have refine_2 : ∀ (a : R) (ha : a ∈ normalizedFactors r) (ha' : Associated (Submodule.IsPrincipal.generator i) a),
  ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) sorry :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi
Tactic: obtain ⟨a, ha, ha'⟩ := this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
this : i.IsPrime
a : R
ha : a ∈ normalizedFactors r
ha' : Associated (Submodule.IsPrincipal.generator i) a
⊢ ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) ⋯ :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi
have h : (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
      (Subtype.mk (p := fun (x : Ideal R) ↦
          Membership.mem (γ := Set (Ideal R))
            {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
          (span {(↑d : R)}) sorry :
        { x : Ideal R //
          Membership.mem (γ := Set (Ideal R))
            {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
    (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha) =
  Subtype.mk (p := fun (x : Ideal R) ↦
    Membership.mem (γ := Set (Ideal R))
      {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
    i hi
Tactic: use ⟨a, ha⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
r : R
hr : r ≠ 0
I : (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
⊢ emultiplicity
      (↑(((Equiv.symm (normalizedFactorsEquivSpanNormalizedFactors hr) :
                (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} :
                    Type u_1) ≃
                  (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) :
              (a : (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1)) →
                (↑{d : R | d ∈ normalizedFactors r} : Type u_1))
            I) :
        R)
      r =
    emultiplicity (↑I : Ideal R) (span {r})
have obtain : ∀ (x : (↑{d : R | d ∈ normalizedFactors r} : Type u_1))
  (hx :
    ((normalizedFactorsEquivSpanNormalizedFactors hr :
            (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
              (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)) :
          (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) →
            (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1))
        x =
      I),
  emultiplicity
      (↑(((Equiv.symm (normalizedFactorsEquivSpanNormalizedFactors hr) :
                (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} :
                    Type u_1) ≃
                  (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) :
              (a : (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1)) →
                (↑{d : R | d ∈ normalizedFactors r} : Type u_1))
            I) :
        R)
      r =
    emultiplicity (↑I : Ideal R) (span {r})
Tactic: obtain ⟨x, hx⟩ := (normalizedFactorsEquivSpanNormalizedFactors hr).surjective I
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
r : R
hr : r ≠ 0
I : (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
x : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)
hx :
  ((normalizedFactorsEquivSpanNormalizedFactors hr :
          (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
            (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)) :
        (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) →
          (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1))
      x =
    I
⊢ emultiplicity
      (↑(((Equiv.symm (normalizedFactorsEquivSpanNormalizedFactors hr) :
                (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} :
                    Type u_1) ≃
                  (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) :
              (a : (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1)) →
                (↑{d : R | d ∈ normalizedFactors r} : Type u_1))
            I) :
        R)
      r =
    emultiplicity (↑I : Ideal R) (span {r})
have obtain : ∀ (a : R) (ha : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} a)
  (hx :
    ((normalizedFactorsEquivSpanNormalizedFactors hr :
            (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
              (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)) :
          (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) →
            (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1))
        (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha) =
      I),
  emultiplicity
      (↑(((Equiv.symm (normalizedFactorsEquivSpanNormalizedFactors hr) :
                (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} :
                    Type u_1) ≃
                  (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) :
              (a : (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1)) →
                (↑{d : R | d ∈ normalizedFactors r} : Type u_1))
            I) :
        R)
      r =
    emultiplicity (↑I : Ideal R) (span {r})
Tactic: obtain ⟨a, ha⟩ := x
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
r : R
hr : r ≠ 0
I : (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
a : R
ha : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} a
hx :
  ((normalizedFactorsEquivSpanNormalizedFactors hr :
          (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
            (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)) :
        (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) →
          (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1))
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha) =
    I
⊢ emultiplicity
      (↑(((Equiv.symm (normalizedFactorsEquivSpanNormalizedFactors hr) :
                (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} :
                    Type u_1) ≃
                  (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) :
              (a : (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1)) →
                (↑{d : R | d ∈ normalizedFactors r} : Type u_1))
            I) :
        R)
      r =
    emultiplicity (↑I : Ideal R) (span {r})
have rw : emultiplicity
    (↑(((Equiv.symm (normalizedFactorsEquivSpanNormalizedFactors hr) :
              (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1) ≃
                (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) :
            (a : (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1)) →
              (↑{d : R | d ∈ normalizedFactors r} : Type u_1))
          (((normalizedFactorsEquivSpanNormalizedFactors hr :
                (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
                  (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} :
                    Type u_1)) :
              (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) →
                (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1))
            (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha))) :
      R)
    r =
  emultiplicity (α := Ideal R)
    (↑(((normalizedFactorsEquivSpanNormalizedFactors hr :
              (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
                (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} :
                  Type u_1)) :
            (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) →
              (↑{I : Ideal R | I ∈ normalizedFactors (span {r})} : Type u_1))
          (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha)) :
      Ideal R)
    (span {r})
Tactic: rw [hx.symm, Equiv.symm_apply_apply, Subtype.coe_mk,
    emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_emultiplicity hr ha]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
inst_4 : DecidableEq R
r X : R
hr : r ≠ 0
hX : Prime X
⊢ Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
    Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)
have this : emultiplicity (α := Ideal R) (span {X}) (span {r}) = emultiplicity X r
Tactic: have := emultiplicity_eq_emultiplicity_span (R := R) (a := X) (b := r)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
inst_4 : DecidableEq R
r X : R
hr : r ≠ 0
hX : Prime X
this : emultiplicity (α := Ideal R) (span {X}) (span {r}) = emultiplicity X r
⊢ Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
    Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)
have rw : ∀
  (this :
    Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
      Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)),
  Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
    Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)
Tactic: rw [emultiplicity_eq_count_normalizedFactors (Prime.irreducible hX) hr,
    emultiplicity_eq_count_normalizedFactors (Prime.irreducible ?_), normalize_apply,
    normUnit_eq_one, Units.val_one, one_eq_top, mul_top, Nat.cast_inj] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
inst_4 : DecidableEq R
r X : R
hr : r ≠ 0
hX : Prime X
this : emultiplicity (α := Ideal R) (span {X}) (span {r}) = emultiplicity X r
rw :
  ∀
    (this :
      Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
        Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)),
    Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
      Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)
⊢ Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
    Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)
have rw₁ : ∀
  (this :
    emultiplicity (α := Ideal R) (span {X}) (span {r}) =
      (↑(Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)) : ℕ∞)),
  Ne (α := Ideal R) (span {r}) 0
Tactic: rw [emultiplicity_eq_count_normalizedFactors (Prime.irreducible hX) hr,
    emultiplicity_eq_count_normalizedFactors (Prime.irreducible ?_), normalize_apply,
    normUnit_eq_one, Units.val_one, one_eq_top, mul_top, Nat.cast_inj] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
inst_4 : DecidableEq R
r X : R
hr : r ≠ 0
hX : Prime X
this : emultiplicity (α := Ideal R) (span {X}) (span {r}) = emultiplicity X r
rw :
  ∀
    (this :
      Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
        Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)),
    Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
      Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)
rw₁ :
  ∀
    (this :
      emultiplicity (α := Ideal R) (span {X}) (span {r}) =
        (↑(Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)) : ℕ∞)),
    Ne (α := Ideal R) (span {r}) 0
⊢ Multiset.count (α := Ideal R) (span {X}) (normalizedFactors (span {r})) =
    Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)
have rw₂ : ∀
  (this :
    emultiplicity (α := Ideal R) (span {X}) (span {r}) =
      (↑(Multiset.count ((normalize : (a : R) → R) X) (normalizedFactors r)) : ℕ∞)),
  Prime (M := Ideal R) (span {X})
Tactic: rw [emultiplicity_eq_count_normalizedFactors (Prime.irreducible hX) hr,
    emultiplicity_eq_count_normalizedFactors (Prime.irreducible ?_), normalize_apply,
    normUnit_eq_one, Units.val_one, one_eq_top, mul_top, Nat.cast_inj] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
A : Type u_4
inst : CommRing A
p : Ideal A
hpb : p ≠ ⊥
hpm : p.IsMaximal
B : Type u_5
inst_1 : CommRing B
inst_2 : IsDedekindDomain B
inst_3 : Algebra A B
inst_4 : IsDomain A
inst_5 : IsTorsionFree A B
⊢ (↑(primesOverFinset p B) : Set (Ideal B)) = p.primesOver B
have h : ∀ (x : Ideal B), x ∈ (↑(primesOverFinset p B) : Set (Ideal B)) ↔ Membership.mem (γ := Set (Ideal B)) (p.primesOver B) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
A : Type u_4
inst : CommRing A
p : Ideal A
hpb : p ≠ ⊥
hpm : p.IsMaximal
B : Type u_5
inst_1 : CommRing B
inst_2 : IsDedekindDomain B
inst_3 : Algebra A B
inst_4 : IsDomain A
inst_5 : IsTorsionFree A B
P : Ideal B
⊢ Membership.mem (γ := Finset (Ideal B)) (primesOverFinset p B) P ↔
    Membership.mem (γ := Set (Ideal B)) (p.primesOver B) P
have rw : P ∈ (↑(primesOverFinset p B) : Set (Ideal B)) ↔ Membership.mem (γ := Set (Ideal B)) (p.primesOver B) P
Tactic: rw [← Finset.mem_coe, coe_primesOverFinset hpb]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
A : Type u_4
inst : CommRing A
p : Ideal A
hpb : p ≠ ⊥
hpm : p.IsMaximal
B : Type u_5
inst_1 : CommRing B
inst_2 : IsDedekindDomain B
inst_3 : Algebra A B
inst_4 : IsDomain A
inst_5 : IsTorsionFree A B
P : Ideal B
rw : P ∈ (↑(primesOverFinset p B) : Set (Ideal B)) ↔ Membership.mem (γ := Set (Ideal B)) (p.primesOver B) P
⊢ Membership.mem (γ := Finset (Ideal B)) (primesOverFinset p B) P ↔
    Membership.mem (γ := Set (Ideal B)) (p.primesOver B) P
have rw₁ : Membership.mem (γ := Set (Ideal B)) (p.primesOver B) P ↔ Membership.mem (γ := Set (Ideal B)) (p.primesOver B) P
Tactic: rw [← Finset.mem_coe, coe_primesOverFinset hpb]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
A : Type u_4
inst_1 : CommRing A
inst_2 : IsDomain A
inst_3 : IsLocalRing A
inst_4 : IsDedekindDomain A
inst_5 : Algebra R A
inst_6 : FaithfulSMul R A
inst_7 : Module.Finite R A
p : Ideal R
inst_8 : p.IsMaximal
hp0 : p ≠ ⊥
⊢ Eq (α := Finset (Ideal A)) (IsDedekindDomain.primesOverFinset p A) {maximalIdeal A}
have this : IsDomain R
Tactic: have : IsDomain R := .of_faithfulSMul R A
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
R : Type u_1
inst : CommRing R
A : Type u_4
inst_1 : CommRing A
inst_2 : IsDomain A
inst_3 : IsLocalRing A
inst_4 : IsDedekindDomain A
inst_5 : Algebra R A
inst_6 : FaithfulSMul R A
inst_7 : Module.Finite R A
p : Ideal R
inst_8 : p.IsMaximal
hp0 : p ≠ ⊥
this : IsDomain R
⊢ Eq (α := Finset (Ideal A)) (IsDedekindDomain.primesOverFinset p A) {maximalIdeal A}
have rw : (↑(IsDedekindDomain.primesOverFinset p A) : Set (Ideal A)) = {maximalIdeal A}
Tactic: rw [← Finset.coe_eq_singleton, IsDedekindDomain.coe_primesOverFinset hp0,
    IsLocalRing.primesOver_eq A hp0]