awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
I : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
k : K
hk : I = spanSingleton R⁰ k
n : R := choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯
d : ↥R⁰ :=
  choose (p := fun (x : ↥R⁰) ↦ mk' K (choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯) x = k) ⋯
hnd : mk' K n d = k
hn0 : n ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑v.asIdeal : FractionalIdeal R⁰ K)
        (HSub.hSub (α := ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal)
                (Associates.factors
                  (Associates.mk
                    (Ideal.span {choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯})))) :
            ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {(↑d : R)})))) :
            ℤ)) =
    I
have rw : spanSingleton R⁰ (mk' K n d) = I
Tactic: rw [finprod_heightOneSpectrum_factorization_principal_fraction hn0 d, hk, hnd]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
I : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
k : K
hk : I = spanSingleton R⁰ k
n : R := choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯
d : ↥R⁰ :=
  choose (p := fun (x : ↥R⁰) ↦ mk' K (choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯) x = k) ⋯
hnd : mk' K n d = k
hn0 : n ≠ 0
rw : spanSingleton R⁰ (mk' K n d) = I
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑v.asIdeal : FractionalIdeal R⁰ K)
        (HSub.hSub (α := ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal)
                (Associates.factors
                  (Associates.mk
                    (Ideal.span {choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯})))) :
            ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {(↑d : R)})))) :
            ℤ)) =
    I
have rw₁ : Eq (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (spanSingleton R⁰ (mk' K n d)) (spanSingleton R⁰ k)
Tactic: rw [finprod_heightOneSpectrum_factorization_principal_fraction hn0 d, hk, hnd]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
I : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
k : K
hk : I = spanSingleton R⁰ k
n : R := choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯
d : ↥R⁰ :=
  choose (p := fun (x : ↥R⁰) ↦ mk' K (choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯) x = k) ⋯
hnd : mk' K n d = k
hn0 : n ≠ 0
rw : spanSingleton R⁰ (mk' K n d) = I
rw₁ : Eq (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (spanSingleton R⁰ (mk' K n d)) (spanSingleton R⁰ k)
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑v.asIdeal : FractionalIdeal R⁰ K)
        (HSub.hSub (α := ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal)
                (Associates.factors
                  (Associates.mk
                    (Ideal.span {choose (p := fun (x : R) ↦ ∃ (y : ↥(R⁰ : Submonoid R)), mk' K x y = k) ⋯})))) :
            ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {(↑d : R)})))) :
            ℤ)) =
    I
have rw₂ : Eq (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (spanSingleton R⁰ k) (spanSingleton R⁰ k)
Tactic: rw [finprod_heightOneSpectrum_factorization_principal_fraction hn0 d, hk, hnd]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
⊢ count K v (I * I') = count K v I + count K v I'
have hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
Tactic: have hv : Irreducible (Associates.mk v.asIdeal) := by apply v.associates_irreducible
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ count K v (I * I') = count K v I + count K v I'
have obtain : ∀ (a : R) (J : Ideal R) (ha : a ≠ 0)
  (haJ :
    I =
      HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)),
  count K v (I * I') = count K v I + count K v I'
Tactic: obtain ⟨a, J, ha, haJ⟩ := exists_eq_spanSingleton_mul I
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
⊢ Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
have rw : ¬a = 0
Tactic: rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
⊢ count K v (I * I') = count K v I + count K v I'
have ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
Tactic: have ha_ne_zero : Associates.mk (Ideal.span {a} : Ideal R) ≠ 0 := by
    rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]; exact ha
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
⊢ count K v (I * I') = count K v I + count K v I'
have hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
Tactic: have hJ_ne_zero : Associates.mk J ≠ 0 := Associates.mk_ne_zero.mpr (ideal_factor_ne_zero hI haJ)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
⊢ count K v (I * I') = count K v I + count K v I'
have obtain : ∀ (a' : R) (J' : Ideal R) (ha' : a' ≠ 0)
  (haJ' :
    I' =
      HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a')⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
        ((↑J' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)),
  count K v (I * I') = count K v I + count K v I'
Tactic: obtain ⟨a', J', ha', haJ'⟩ := exists_eq_spanSingleton_mul I'
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
a' : R
J' : Ideal R
ha' : a' ≠ 0
haJ' :
  I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a')⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
⊢ Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a'})) 0
have rw : ¬a' = 0
Tactic: rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
a' : R
J' : Ideal R
ha' : a' ≠ 0
haJ' :
  I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a')⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
⊢ count K v (I * I') = count K v I + count K v I'
have ha'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a'})) 0
Tactic: have ha'_ne_zero : Associates.mk (Ideal.span {a'} : Ideal R) ≠ 0 := by
    rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]; exact ha'
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
a' : R
J' : Ideal R
ha' : a' ≠ 0
haJ' :
  I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a')⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a'})) 0
⊢ count K v (I * I') = count K v I + count K v I'
have hJ'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J') 0
Tactic: have hJ'_ne_zero : Associates.mk J' ≠ 0 :=
    Associates.mk_ne_zero.mpr (ideal_factor_ne_zero hI' haJ')
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
a' : R
J' : Ideal R
ha' : a' ≠ 0
haJ' :
  I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a')⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a'})) 0
hJ'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J') 0
⊢ I * I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (a * a'))⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑(J * J') : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
have rw : HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) *
    I' =
  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (a * a'))⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    ((↑(J * J') : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
Tactic: rw [haJ, haJ', mul_assoc, mul_comm (J : FractionalIdeal R⁰ K), mul_assoc, ← mul_assoc,
      spanSingleton_mul_spanSingleton, coeIdeal_mul, map_mul, mul_inv,
      mul_comm (J : FractionalIdeal R⁰ K)]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
a' : R
J' : Ideal R
ha' : a' ≠ 0
haJ' :
  I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a')⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a'})) 0
hJ'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J') 0
⊢ count K v (I * I') = count K v I + count K v I'
have h_prod : I * I' =
  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (a * a'))⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    ((↑(J * J') : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
Tactic: have h_prod : I * I' = spanSingleton R⁰ ((algebraMap R K) (a * a'))⁻¹ * ↑(J * J') := by
    rw [haJ, haJ', mul_assoc, mul_comm (J : FractionalIdeal R⁰ K), mul_assoc, ← mul_assoc,
      spanSingleton_mul_spanSingleton, coeIdeal_mul, map_mul, mul_inv,
      mul_comm (J : FractionalIdeal R⁰ K)]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
a : R
J : Ideal R
ha : a ≠ 0
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a})) 0
hJ_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J) 0
a' : R
J' : Ideal R
ha' : a' ≠ 0
haJ' :
  I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a')⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk (Ideal.span {a'})) 0
hJ'_ne_zero : Ne (α := Associates (Ideal R)) (Associates.mk J') 0
h_prod :
  I * I' =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) (a * a'))⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑(J * J') : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
⊢ count K v (I * I') = count K v I + count K v I'
have rw : HSub.hSub (α := ℤ)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) +
          Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J'))) :
      ℤ)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) +
          Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a'})))) :
      ℤ) =
  HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
      (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) +
    HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J'))) : ℤ)
      (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a'})))) : ℤ)
Tactic: rw [count_well_defined K v hI haJ, count_well_defined K v hI' haJ',
    count_well_defined K v (mul_ne_zero hI hI') h_prod, ← Associates.mk_mul_mk,
    Associates.count_mul hJ_ne_zero hJ'_ne_zero hv, ← Ideal.span_singleton_mul_span_singleton,
    ← Associates.mk_mul_mk, Associates.count_mul ha_ne_zero ha'_ne_zero hv]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
v : HeightOneSpectrum R
I I' : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
hI' : I' ≠ 0
⊢ count K v (I * I') = count K v I + count K v I'
have have : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
Tactic: classical
  have hv : Irreducible (Associates.mk v.asIdeal) := by apply v.associates_irreducible
  obtain ⟨a, J, ha, haJ⟩ := exists_eq_spanSingleton_mul I
  have ha_ne_zero : Associates.mk (Ideal.span {a} : Ideal R) ≠ 0 := by
    rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]; exact ha
  have hJ_ne_zero : Associates.mk J ≠ 0 := Associates.mk_ne_zero.mpr (ideal_factor_ne_zero hI haJ)
  obtain ⟨a', J', ha', haJ'⟩ := exists_eq_spanSingleton_mul I'
  have ha'_ne_zero : Associates.mk (Ideal.span {a'} : Ideal R) ≠ 0 := by
    rw [ne_eq, Associates.mk_eq_zero, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]; exact ha'
  have hJ'_ne_zero : Associates.mk J' ≠ 0 :=
    Associates.mk_ne_zero.mpr (ideal_factor_ne_zero hI' haJ')
  have h_prod : I * I' = spanSingleton R⁰ ((algebraMap R K) (a * a'))⁻¹ * ↑(J * J') := by
    rw [haJ, haJ', mul_assoc, mul_comm (J : FractionalIdeal R⁰ K), mul_assoc, ← mul_assoc,
      spanSingleton_mul_spanSingleton, coeIdeal_mul, map_mul, mul_inv,
      mul_comm (J : FractionalIdeal R⁰ K)]
  rw [count_well_defined K v hI haJ, count_well_defined K v hI' haJ',
    count_well_defined K v (mul_ne_zero hI hI') h_prod, ← Associates.mk_mul_mk,
    Associates.count_mul hJ_ne_zero hJ'_ne_zero hv, ← Ideal.span_singleton_mul_span_singleton,
    ← Associates.mk_mul_mk, Associates.count_mul ha_ne_zero ha'_ne_zero hv]
  push_cast
  ring