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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : x + y = 0
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
have pos : 0 ≤ max (v.intValuationDef x) (v.intValuationDef y)
Tactic: rw [intValuationDef, if_pos hxy]
-- 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 y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
have neg : LE.le (α := WithZero (Multiplicative ℤ))
    (exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))) : ℤ)))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ))) ∨
  LE.le (α := WithZero (Multiplicative ℤ))
    (exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))) : ℤ)))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
Tactic: rw [v.intValuationDef_if_neg hxy, v.intValuationDef_if_neg hx, v.intValuationDef_if_neg hy,
          le_max_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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      (exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))) :
            ℤ)))
      (exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ))) ∨
    LE.le (α := WithZero (Multiplicative ℤ))
      (exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))) :
            ℤ)))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have neg : min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) ≤
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
Tactic: simp only [exp_le_exp, neg_le_neg_iff, Nat.cast_le, ← min_le_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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
⊢ x ∈ v.asIdeal ^ nmin
have rw : nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))
Tactic: rw [← Associates.le_singleton_iff x nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr 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 y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
rw : nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))
⊢ x ∈ v.asIdeal ^ nmin
have rw₁ : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
Tactic: rw [← Associates.le_singleton_iff x nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hx) _]
-- 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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
have h_dvd_x : x ∈ v.asIdeal ^ nmin
Tactic: have h_dvd_x : x ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff x nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hx) _]
          · exact min_le_left _ _
          exact 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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
⊢ y ∈ v.asIdeal ^ nmin
have rw : nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))
Tactic: rw [← Associates.le_singleton_iff y nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]
-- 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 y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
rw : nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))
⊢ y ∈ v.asIdeal ^ nmin
have rw₁ : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
Tactic: rw [← Associates.le_singleton_iff y nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]
-- 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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
have h_dvd_y : y ∈ v.asIdeal ^ nmin
Tactic: have h_dvd_y : y ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff y nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]
          · exact min_le_right _ _
          exact 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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
h_dvd_y : y ∈ v.asIdeal ^ nmin
⊢ HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})
have rw : x + y ∈ v.asIdeal ^ nmin
Tactic: rw [Associates.le_singleton_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
x y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
h_dvd_y : y ∈ v.asIdeal ^ nmin
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
have h_dvd_xy : HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})
Tactic: have h_dvd_xy : Associates.mk v.asIdeal ^ nmin ≤ Associates.mk (Ideal.span {x + y}) := by
          rw [Associates.le_singleton_iff]
          exact Ideal.add_mem (v.asIdeal ^ nmin) h_dvd_x h_dvd_y
-- 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 y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
h_dvd_y : y ∈ v.asIdeal ^ nmin
h_dvd_xy : HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
have neg : ∀
  (h_dvd_xy :
    nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))),
  nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
Tactic: rw [Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hxy) _] at h_dvd_xy
Cleared hypotheses: h_dvd_xy
-- 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 y : R
hx : ¬x = 0
hy : ¬y = 0
hxy : ¬x + y = 0
nmin : ℕ :=
  min (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x}))))
    (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y}))))
h_dvd_x : x ∈ v.asIdeal ^ nmin
h_dvd_y : y ∈ v.asIdeal ^ nmin
h_dvd_xy : HPow.hPow (α := Associates (Ideal R)) (Associates.mk v.asIdeal) nmin ≤ Associates.mk (Ideal.span {x + y})
neg :
  ∀
    (h_dvd_xy :
      nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))),
    nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
⊢ nmin ≤ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x + y})))
have rw : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
Tactic: rw [Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hxy) _] at h_dvd_xy
-- 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 y : R
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
have pos : ∀ (hx : x = 0), v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
Tactic: classical
  by_cases hx : x = 0
  · rw [hx, zero_add]
    order
  · by_cases hy : y = 0
    · rw [hy, add_zero]
      order
    · by_cases hxy : x + y = 0
      · rw [intValuationDef, if_pos hxy]; exact zero_le'
      · rw [v.intValuationDef_if_neg hxy, v.intValuationDef_if_neg hx, v.intValuationDef_if_neg hy,
          le_max_iff]
        simp only [exp_le_exp, neg_le_neg_iff, Nat.cast_le, ← min_le_iff]
        set nmin :=
          min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)
            ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors)
        have h_dvd_x : x ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff x nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hx) _]
          · exact min_le_left _ _
          exact v.associates_irreducible
        have h_dvd_y : y ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff y nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]
          · exact min_le_right _ _
          exact v.associates_irreducible
        have h_dvd_xy : Associates.mk v.asIdeal ^ nmin ≤ Associates.mk (Ideal.span {x + y}) := by
          rw [Associates.le_singleton_iff]
          exact Ideal.add_mem (v.asIdeal ^ nmin) h_dvd_x h_dvd_y
        rw [Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hxy) _] at h_dvd_xy
        · exact h_dvd_xy
        exact 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
x y : R
pos : ∀ (hx : x = 0), v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
have neg : ∀ (hx : ¬x = 0), v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
Tactic: classical
  by_cases hx : x = 0
  · rw [hx, zero_add]
    order
  · by_cases hy : y = 0
    · rw [hy, add_zero]
      order
    · by_cases hxy : x + y = 0
      · rw [intValuationDef, if_pos hxy]; exact zero_le'
      · rw [v.intValuationDef_if_neg hxy, v.intValuationDef_if_neg hx, v.intValuationDef_if_neg hy,
          le_max_iff]
        simp only [exp_le_exp, neg_le_neg_iff, Nat.cast_le, ← min_le_iff]
        set nmin :=
          min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)
            ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors)
        have h_dvd_x : x ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff x nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hx) _]
          · exact min_le_left _ _
          exact v.associates_irreducible
        have h_dvd_y : y ∈ v.asIdeal ^ nmin := by
          rw [← Associates.le_singleton_iff y nmin _,
            Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hy) _]
          · exact min_le_right _ _
          exact v.associates_irreducible
        have h_dvd_xy : Associates.mk v.asIdeal ^ nmin ≤ Associates.mk (Ideal.span {x + y}) := by
          rw [Associates.le_singleton_iff]
          exact Ideal.add_mem (v.asIdeal ^ nmin) h_dvd_x h_dvd_y
        rw [Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hxy) _] at h_dvd_xy
        · exact h_dvd_xy
        exact v.associates_irreducible
-- 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
⊢ 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 hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
Tactic: have hsr : Ideal.span {r} ≠ 0 := Submodule.span_singleton_eq_bot.mp.mt hr
-- 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
hsr : Ne (α := Ideal R) (Ideal.span {r}) 0
⊢ 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 hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r})
Tactic: have hfm : FiniteMultiplicity v.asIdeal (Ideal.span {r}) :=
    FiniteMultiplicity.of_prime_left v.prime hsr
-- 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})
⊢ 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 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)))
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)))
⊢ 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 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))
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))
⊢ 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 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)
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)
⊢ 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 rw₃ : Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {r}))) =
  multiplicity 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})
⊢ 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 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∞)
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]