awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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)

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)]
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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)
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)
⊢ count K v (I * I') = count K v I + count K v I'

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]

Premises used: FractionalIdeal.count_well_defined, Ideal.span_singleton_mul_span_singleton, Associates.mk_mul_mk, Associates.count_mul, mul_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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)
⊢ HAdd.hAdd (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
        (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J'))) : ℤ) -
      HAdd.hAdd (α := ℤ)
        (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : 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) +
      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:
ring
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
inst_5 : Decidable (I ≠ 0 ∧ I' ≠ 0)
pos : ∀ (h : I ≠ 0 ∧ I' ≠ 0), count K v (I * I') = count K v I + count K v I'
neg : ∀ (h : ¬(I ≠ 0 ∧ I' ≠ 0)), count K v (I * I') = 0
⊢ count K v (I * I') = if I ≠ 0 ∧ I' ≠ 0 then count K v I + count K v I' else 0

Tactic:
split_ifs with h
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
inst_5 : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 ∧ I' ≠ 0
⊢ count K v (I * I') = count K v I + count K v I'

Tactic:
count_mul K v h.1 h.2

Premises used: FractionalIdeal.count_mul, And.right, And.left
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
inst_5 : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I ≠ 0 ∧ I' ≠ 0
⊢ count K v (I * I') = count K v I + count K v I'

Tactic:
exact count_mul K v h.1 h.2

Premises used: FractionalIdeal.count_mul, And.right, And.left
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
inst_5 : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : ¬(I ≠ 0 ∧ I' ≠ 0)
⊢ I * I' = 0

Tactic:
rw [← mul_ne_zero_iff, not_ne_iff] at h

Premises used: not_ne_iff, mul_ne_zero_iff
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
inst_5 : Decidable (I ≠ 0 ∧ I' ≠ 0)
h : I * I' = 0
neg : count K v 0 = 0
neg₁ neg₂ : Eq (α := ℤ) 0 0
⊢ count K v (I * I') = 0

Tactic:
rw [h, count_zero]

Premises used: FractionalIdeal.count_zero
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
rw : 1 = HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (spanSingleton R⁰ 1⁻¹) (↑1 : FractionalIdeal R⁰ K)
rw₁ : 1 = HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (spanSingleton R⁰ 1⁻¹) (↑⊤ : FractionalIdeal R⁰ K)
rw₂ : 1 = HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (spanSingleton R⁰ 1⁻¹) 1
rw₃ : Eq.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 (spanSingleton R⁰ 1⁻¹)
rw₄ : Eq.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 (spanSingleton R⁰ 1)
rw₅ rw₆ : Eq.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) 1 1
⊢ 1 =
    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) 1)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑1 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)

Tactic:
rw [(algebraMap R K).map_one, Ideal.one_eq_top, coeIdeal_top, mul_one, inv_one,
      spanSingleton_one]

Premises used: FractionalIdeal.coeIdeal_top, Ideal.one_eq_top, Algebra.algebraMap, inv_one, RingHom.map_one, FractionalIdeal.spanSingleton_one, mul_one
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
⊢ 1 =
    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) 1)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑1 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)

Tactic:
have h1 : (1 : FractionalIdeal R⁰ K) =
      spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑(1 : Ideal R) := by
    rw [(algebraMap R K).map_one, Ideal.one_eq_top, coeIdeal_top, mul_one, inv_one,
      spanSingleton_one]
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
h1 :
  1 =
    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) 1)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑1 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
rw :
  HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk 1))) : ℤ)
      (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {1})))) : ℤ) =
    0
rw₁ :
  HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk 1))) : ℤ)
      (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk ⊤))) : ℤ) =
    0
rw₂ :
  HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk ⊤))) : ℤ)
      (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk ⊤))) : ℤ) =
    0
rw₃ rw₄ : Eq (α := ℤ) 0 0
⊢ count K v 1 = 0

Tactic:
rw [count_well_defined K v one_ne_zero h1, Ideal.span_singleton_one, Ideal.one_eq_top, sub_self]

Premises used: Ideal.one_eq_top, FractionalIdeal.count_well_defined, sub_self, Ideal.span_singleton_one, one_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
ι : Type u_3
I : ι → FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hS : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → I i ≠ 0
empty : count K v 1 = ∑ i ∈ ∅, count K v (I i)
empty₁ : count K v 1 = 0
empty₂ empty₃ : Eq (α := ℤ) 0 0
⊢ count K v (∏ i ∈ ∅, I i) = ∑ i ∈ ∅, count K v (I i)

Tactic:
rw [Finset.prod_empty, Finset.sum_empty, count_one]

Premises used: Finset.prod_empty, FractionalIdeal.count_one, Finset.sum_empty
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
ι : Type u_3
I : ι → FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
i : ι
s : Finset ι
hi : i ∉ s
hrec : ∀ (hS : ∀ i ∈ s, I i ≠ 0), count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)
hS : ∀ i_1 ∈ insert i s, I i_1 ≠ 0
⊢ ∀ i ∈ s, I i ≠ 0

Tactic:
have hS' : ∀ i ∈ s, I i ≠ 0 := fun j hj => hS j (Finset.mem_insert_of_mem hj)
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
ι : Type u_3
I : ι → FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
i : ι
s : Finset ι
hi : i ∉ s
hrec : ∀ (hS : ∀ i ∈ s, I i ≠ 0), count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)
hS : ∀ i_1 ∈ insert i s, I i_1 ≠ 0
hS' : ∀ i ∈ s, I i ≠ 0
⊢ Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (∏ i ∈ s, I i) 0

Tactic:
have hS0 : ∏ i ∈ s, I i ≠ 0 := Finset.prod_ne_zero_iff.mpr hS'
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
ι : Type u_3
I : ι → FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
i : ι
s : Finset ι
hi : i ∉ s
hrec : ∀ (hS : ∀ i ∈ s, I i ≠ 0), count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)
hS : ∀ i_1 ∈ insert i s, I i_1 ≠ 0
hS' : ∀ i ∈ s, I i ≠ 0
hS0 : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (∏ i ∈ s, I i) 0
⊢ I i ≠ 0

Tactic:
have hi0 : I i ≠ 0 := hS i (Finset.mem_insert_self i s)
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
ι : Type u_3
I : ι → FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
i : ι
s : Finset ι
hi : i ∉ s
hrec : ∀ (hS : ∀ i ∈ s, I i ≠ 0), count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)
hS : ∀ i_1 ∈ Insert.insert i s, I i_1 ≠ 0
hS' : ∀ i ∈ s, I i ≠ 0
hS0 : Ne.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (∏ i ∈ s, I i) 0
hi0 : I i ≠ 0
insert : count K v (I i * ∏ x ∈ s, I x) = ∑ i ∈ Insert.insert i s, count K v (I i)
insert₁ : count K v (I i * ∏ x ∈ s, I x) = count K v (I i) + ∑ x ∈ s, count K v (I x)
insert₂ : count K v (I i) + count K v (∏ i ∈ s, I i) = count K v (I i) + ∑ x ∈ s, count K v (I x)
insert₃ insert₄ : count K v (I i) + ∑ i ∈ s, count K v (I i) = count K v (I i) + ∑ x ∈ s, count K v (I x)
⊢ count K v (∏ i ∈ Insert.insert i s, I i) = ∑ i ∈ Insert.insert i s, count K v (I i)

Tactic:
rw [Finset.prod_insert hi, Finset.sum_insert hi, count_mul K v hi0 hS0, hrec hS']

Premises used: Finset.prod_insert, FractionalIdeal.count_mul, Finset.sum_insert
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
ι : Type u_3
s : Finset ι
I : ι → FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hS : ∀ i ∈ s, I i ≠ 0
empty :
  ∀ (hS : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → I i ≠ 0), count K v (∏ i ∈ ∅, I i) = ∑ i ∈ ∅, count K v (I i)
insert :
  ∀ (i : ι) (s : Finset ι) (hi : i ∉ s)
    (hrec : ∀ (hS : ∀ i ∈ s, I i ≠ 0), count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i))
    (hS : ∀ i_1 ∈ Insert.insert i s, I i_1 ≠ 0),
    count K v (∏ i ∈ Insert.insert i s, I i) = ∑ i ∈ Insert.insert i s, count K v (I i)
⊢ count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)

Tactic:
induction s using Finset.induction with
  | empty => rw [Finset.prod_empty, Finset.sum_empty, count_one]
  | insert i s hi hrec =>
    have hS' : ∀ i ∈ s, I i ≠ 0 := fun j hj => hS j (Finset.mem_insert_of_mem hj)
    have hS0 : ∏ i ∈ s, I i ≠ 0 := Finset.prod_ne_zero_iff.mpr hS'
    have hi0 : I i ≠ 0 := hS i (Finset.mem_insert_self i s)
    rw [Finset.prod_insert hi, Finset.sum_insert hi, count_mul K v hi0 hS0, hrec hS']

Premises used: Finset.prod_empty, Insert.insert, FractionalIdeal.count_one, Finset.sum_empty
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
ι : Type u_3
s : Finset ι
I : ι → FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hS : ∀ i ∈ s, I i ≠ 0
empty :
  ∀ (hS : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → I i ≠ 0), count K v (∏ i ∈ ∅, I i) = ∑ i ∈ ∅, count K v (I i)
insert :
  ∀ (i : ι) (s : Finset ι) (hi : i ∉ s)
    (hrec : ∀ (hS : ∀ i ∈ s, I i ≠ 0), count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i))
    (hS : ∀ i_1 ∈ Insert.insert i s, I i_1 ≠ 0),
    count K v (∏ i ∈ Insert.insert i s, I i) = ∑ i ∈ Insert.insert i s, count K v (I i)
⊢ count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i)

Tactic:
classical
  induction s using Finset.induction with
  | empty => rw [Finset.prod_empty, Finset.sum_empty, count_one]
  | insert i s hi hrec =>
    have hS' : ∀ i ∈ s, I i ≠ 0 := fun j hj => hS j (Finset.mem_insert_of_mem hj)
    have hS0 : ∏ i ∈ s, I i ≠ 0 := Finset.prod_ne_zero_iff.mpr hS'
    have hi0 : I i ≠ 0 := hS i (Finset.mem_insert_self i s)
    rw [Finset.prod_insert hi, Finset.sum_insert hi, count_mul K v hi0 hS0, hrec hS']

Premises used: Finset.prod_empty, Insert.insert, FractionalIdeal.count_one, Finset.sum_empty
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
zero : count K v 1 = (↑0 : ℤ) * count K v I
zero₁ : count K v 1 = 0 * count K v I
zero₂ : count K v 1 = 0
zero₃ zero₄ : Eq (α := ℤ) 0 0
⊢ count K v (HPow.hPow (β := ℕ) I 0) = (↑0 : ℤ) * count K v I

Tactic:
rw [pow_zero, ofNat_zero, zero_mul, count_one]

Premises used: MulZeroClass.zero_mul, pow_zero, Int.ofNat_zero, FractionalIdeal.count_one
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
n : ℕ
h : count K v (I ^ n) = (↑n : ℤ) * count K v I
succ : (if I ^ n ≠ 0 ∧ I ≠ 0 then count K v (I ^ n) + count K v I else 0) = (↑(n + 1) : ℤ) * count K v I
⊢ count K v (I ^ (n + 1)) = (↑(n + 1) : ℤ) * count K v I

Tactic:
rw [pow_succ, count_mul']

Premises used: pow_succ, FractionalIdeal.count_mul'
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
n : ℕ
h : count K v (I ^ n) = (↑n : ℤ) * count K v I
succ : (if I ^ n ≠ 0 ∧ I ≠ 0 then count K v (I ^ n) + count K v I else 0) = (↑(n + 1) : ℤ) * count K v I
⊢ count K v (I ^ (n + 1)) = (↑(n + 1) : ℤ) * count K v I

Tactic:
classical rw [pow_succ, count_mul']

Premises used: pow_succ, FractionalIdeal.count_mul'
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
n : ℕ
h : count K v (I ^ n) = (↑n : ℤ) * count K v I
pos : ∀ (hI : I = 0), (if I ^ n ≠ 0 ∧ I ≠ 0 then count K v (I ^ n) + count K v I else 0) = (↑(n + 1) : ℤ) * count K v I
neg : ∀ (hI : ¬I = 0), (if I ^ n ≠ 0 ∧ I ≠ 0 then count K v (I ^ n) + count K v I else 0) = (↑(n + 1) : ℤ) * count K v I
⊢ (if I ^ n ≠ 0 ∧ I ≠ 0 then count K v (I ^ n) + count K v I else 0) = (↑(n + 1) : ℤ) * count K v I

Tactic:
by_cases hI : I = 0
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
n : ℕ
h : count K v (I ^ n) = (↑n : ℤ) * count K v I
hI : I = 0
order order₁ order₂ order₃ order₄ order₅ order₆ order₇ order₈ order₉ order₁₀ order₁₁ : I ^ n ≠ 0 ∧ I ≠ 0 → False
⊢ ¬(I ^ n ≠ 0 ∧ I ≠ 0)

Tactic:
order
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
n : ℕ
h : count K v (I ^ n) = (↑n : ℤ) * count K v I
hI : I = 0
⊢ ¬(I ^ n ≠ 0 ∧ I ≠ 0)

Tactic:
have h_neg : ¬(I ^ n ≠ 0 ∧ I ≠ 0) := by order
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
n : ℕ
h : count K v (I ^ n) = (↑n : ℤ) * count K v I
hI : I = 0
h_neg : ¬(I ^ n ≠ 0 ∧ I ≠ 0)
pos : 0 = (↑(n + 1) : ℤ) * count K v I
pos₁ : 0 = (↑(n + 1) : ℤ) * count K v 0
pos₂ : 0 = HMul.hMul (α := ℤ) (↑(n + 1) : ℤ) 0
pos₃ pos₄ : Eq (α := ℤ) 0 0
⊢ (if I ^ n ≠ 0 ∧ I ≠ 0 then count K v (I ^ n) + count K v I else 0) = (↑(n + 1) : ℤ) * count K v I

Tactic:
rw [if_neg h_neg, hI, count_zero, mul_zero]

Premises used: MulZeroClass.mul_zero, FractionalIdeal.count_zero, if_neg