awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r})
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)))
rw₁ :
  Eq (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
    (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ))
rw₂ :
  Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)
rw₃ :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) =
    multiplicity v.asIdeal (Ideal.span {r})
rw₄ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℕ∞)
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)) : WithZero (Multiplicative ℤ))
have rw₅ : (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞) =
  emultiplicity v.asIdeal (Ideal.span {r})
Tactic: rw [v.intValuation_if_neg hr, exp_inj, neg_inj, Int.natCast_inj, ← ENat.coe_inj,
    ← FiniteMultiplicity.emultiplicity_eq_multiplicity hfm,
    UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors (irreducible v) hsr,
    normalize_eq, Ideal.count_associates_factors_eq hsr v.isPrime v.ne_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r})
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)))
rw₁ :
  Eq (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
    (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ))
rw₂ :
  Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)
rw₃ :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) =
    multiplicity v.asIdeal (Ideal.span {r})
rw₄ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℕ∞)
rw₅ :
  (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞) =
    emultiplicity v.asIdeal (Ideal.span {r})
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)) : WithZero (Multiplicative ℤ))
have rw₆ : Eq (α := ℕ∞) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
  (↑(Multiset.count (α := Ideal R) ((normalize : (a : Ideal R) → Ideal R) v.asIdeal : Ideal R)
        (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}) : Multiset (Ideal R))) :
    ℕ∞)
Tactic: rw [v.intValuation_if_neg hr, exp_inj, neg_inj, Int.natCast_inj, ← ENat.coe_inj,
    ← FiniteMultiplicity.emultiplicity_eq_multiplicity hfm,
    UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors (irreducible v) hsr,
    normalize_eq, Ideal.count_associates_factors_eq hsr v.isPrime v.ne_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r})
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)))
rw₁ :
  Eq (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
    (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ))
rw₂ :
  Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)
rw₃ :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) =
    multiplicity v.asIdeal (Ideal.span {r})
rw₄ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℕ∞)
rw₅ :
  (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞) =
    emultiplicity v.asIdeal (Ideal.span {r})
rw₆ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(Multiset.count (α := Ideal R) ((normalize : (a : Ideal R) → Ideal R) v.asIdeal : Ideal R)
          (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}) : Multiset (Ideal R))) :
      ℕ∞)
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)) : WithZero (Multiplicative ℤ))
have rw₇ : Eq (α := ℕ∞) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
  (↑(Multiset.count v.asIdeal (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}))) : ℕ∞)
Tactic: rw [v.intValuation_if_neg hr, exp_inj, neg_inj, Int.natCast_inj, ← ENat.coe_inj,
    ← FiniteMultiplicity.emultiplicity_eq_multiplicity hfm,
    UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors (irreducible v) hsr,
    normalize_eq, Ideal.count_associates_factors_eq hsr v.isPrime v.ne_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : r ≠ 0
hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r})
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)))
rw₁ :
  Eq (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
    (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ))
rw₂ :
  Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)
rw₃ :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) =
    multiplicity v.asIdeal (Ideal.span {r})
rw₄ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(multiplicity v.asIdeal (Ideal.span {r})) : ℕ∞)
rw₅ :
  (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞) =
    emultiplicity v.asIdeal (Ideal.span {r})
rw₆ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(Multiset.count (α := Ideal R) ((normalize : (a : Ideal R) → Ideal R) v.asIdeal : Ideal R)
          (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}) : Multiset (Ideal R))) :
      ℕ∞)
rw₇ :
  Eq (α := ℕ∞)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℕ∞)
    (↑(Multiset.count v.asIdeal (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}))) : ℕ∞)
⊢ Eq (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
    (exp (-(↑(multiplicity v.asIdeal (Ideal.span {r})) : ℤ)) : WithZero (Multiplicative ℤ))
have rw₈ : Eq (α := ℕ∞) (↑(Multiset.count v.asIdeal (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}))) : ℕ∞)
  (↑(Multiset.count v.asIdeal (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {r}))) : ℕ∞)
Tactic: rw [v.intValuation_if_neg hr, exp_inj, neg_inj, Int.natCast_inj, ← ENat.coe_inj,
    ← FiniteMultiplicity.emultiplicity_eq_multiplicity hfm,
    UniqueFactorizationMonoid.emultiplicity_eq_count_normalizedFactors (irreducible v) hsr,
    normalize_eq, Ideal.count_associates_factors_eq hsr v.isPrime v.ne_bot]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
hx : x ≠ 0
⊢ Ne (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (0 : WithZero (Multiplicative ℤ))
have rw : Ne (α := WithZero (Multiplicative ℤ))
  (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ))) 0
Tactic: rw [v.intValuation_if_neg hx]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : ↥(nonZeroDivisors R)
⊢ LT.lt (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) (↑x : R) : WithZero (Multiplicative ℤ))
have rw : LT.lt (α := WithZero (Multiplicative ℤ)) 0
  (exp
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {(↑x : R)})))) : ℤ)))
Tactic: rw [v.intValuation_if_neg (nonZeroDivisors.coe_ne_zero x)]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
have inl : LE.le (α := WithZero (Multiplicative ℤ))
  ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) 0 : WithZero (Multiplicative ℤ))
  (1 : WithZero (Multiplicative ℤ))
Tactic: obtain rfl | hx := eq_or_ne x 0
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
inl :
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) 0 : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
have inr : ∀ (hx : x ≠ 0),
  LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
Tactic: obtain rfl | hx := eq_or_ne x 0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
hx : x ≠ 0
⊢ LE.le (α := WithZero (Multiplicative ℤ))
    ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
have inr : LE.le (α := ℤ) 0
  (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)
Tactic: rw [v.intValuation_if_neg hx, ← exp_zero, exp_le_exp, Right.neg_nonpos_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
have pos : ∀ (hr : r = 0),
  LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
Tactic: by_cases hr : r = 0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
pos :
  ∀ (hr : r = 0),
    LT.lt (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ∣ Ideal.span {r}
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
have neg : ∀ (hr : ¬r = 0),
  LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
Tactic: by_cases hr : r = 0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : ¬r = 0
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
have neg : Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ≠ 0 ↔
  v.asIdeal ∣ Ideal.span {r}
Tactic: rw [v.intValuation_if_neg hr, ← exp_zero, exp_lt_exp,
      neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
hr : ¬r = 0
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) ≠ 0 ↔
    v.asIdeal ∣ Ideal.span {r}
have h : Ne (α := Ideal R) (Ideal.span {r}) 0
Tactic: have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
      rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
      exact hr
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
have pos : ∀ (hr : r = 0),
  LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
Tactic: classical
  by_cases hr : r = 0
  · simp [hr]
  · rw [v.intValuation_if_neg hr, ← exp_zero, exp_lt_exp,
      neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
    have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
      rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
      exact hr
    exact Associates.count_ne_zero_iff_dvd h v.irreducible
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
pos :
  ∀ (hr : r = 0),
    LT.lt (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ∣ Ideal.span {r}
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
have neg : ∀ (hr : ¬r = 0),
  LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ∣ Ideal.span {r}
Tactic: classical
  by_cases hr : r = 0
  · simp [hr]
  · rw [v.intValuation_if_neg hr, ← exp_zero, exp_lt_exp,
      neg_lt_zero, ← Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
    have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
      rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
      exact hr
    exact Associates.count_ne_zero_iff_dvd h v.irreducible
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    r ∈ v.asIdeal
have rw : v.asIdeal ∣ Ideal.span {r} ↔ r ∈ v.asIdeal
Tactic: rw [intValuation_lt_one_iff_dvd, Ideal.dvd_span_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
rw : v.asIdeal ∣ Ideal.span {r} ↔ r ∈ v.asIdeal
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ)) ↔
    r ∈ v.asIdeal
have rw₁ : r ∈ v.asIdeal ↔ r ∈ v.asIdeal
Tactic: rw [intValuation_lt_one_iff_dvd, Ideal.dvd_span_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
have pos : ∀ (hr : r = 0),
  LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
Tactic: by_cases hr : r = 0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
pos :
  ∀ (hr : r = 0),
    LE.le (α := WithZero (Multiplicative ℤ))
        ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
        (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
      v.asIdeal ^ n ∣ Ideal.span {r}
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
have neg : ∀ (hr : ¬r = 0),
  LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
Tactic: by_cases hr : r = 0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hr : ¬r = 0
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
have neg : LE.le (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
    (exp (-(↑n : ℤ))) ↔
  v.asIdeal ^ n ∣ Ideal.span {r}
Tactic: rw [v.intValuation_if_neg hr, exp_le_exp, neg_le_neg_iff, Int.ofNat_le,
      Ideal.dvd_span_singleton, ← Associates.le_singleton_iff,
      Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr) v.associates_irreducible]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hr : ¬r = 0
neg :
  LE.le (α := WithZero (Multiplicative ℤ))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
      (exp (-(↑n : ℤ))) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
have neg₁ : LE.le (α := ℤ)
    (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
    (-(↑n : ℤ)) ↔
  v.asIdeal ^ n ∣ Ideal.span {r}
Tactic: rw [v.intValuation_if_neg hr, exp_le_exp, neg_le_neg_iff, Int.ofNat_le,
      Ideal.dvd_span_singleton, ← Associates.le_singleton_iff,
      Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr) v.associates_irreducible]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.AdicValuation (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
r : R
n : ℕ
hr : ¬r = 0
neg :
  LE.le (α := WithZero (Multiplicative ℤ))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ)))
      (exp (-(↑n : ℤ))) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
neg₁ :
  LE.le (α := ℤ)
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ))
      (-(↑n : ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      ((v.intValuation : (a : R) → WithZero (Multiplicative ℤ)) r : WithZero (Multiplicative ℤ))
      (exp (-(↑n : ℤ)) : WithZero (Multiplicative ℤ)) ↔
    v.asIdeal ^ n ∣ Ideal.span {r}
have neg₂ : LE.le (α := ℤ) (↑n : ℤ)
    (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r})))) : ℤ) ↔
  v.asIdeal ^ n ∣ Ideal.span {r}
Tactic: rw [v.intValuation_if_neg hr, exp_le_exp, neg_le_neg_iff, Int.ofNat_le,
      Ideal.dvd_span_singleton, ← Associates.le_singleton_iff,
      Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr) v.associates_irreducible]