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
pos :
  (if 0 * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0 * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
pos₁ :
  (if Eq.{u_1 + 1} (α := R) 0 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have pos₂ : 0 =
  HMul.hMul (α := WithZero (Multiplicative ℤ)) 0
    (if y = 0 then 0
    else
      exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
Tactic: rw [hx, zero_mul, if_pos rfl, zero_mul]
-- 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
pos :
  (if 0 * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0 * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
pos₁ :
  (if Eq.{u_1 + 1} (α := R) 0 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
pos₂ :
  0 =
    HMul.hMul (α := WithZero (Multiplicative ℤ)) 0
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have pos₃ : Eq (α := WithZero (Multiplicative ℤ)) 0 0
Tactic: rw [hx, zero_mul, if_pos rfl, zero_mul]
-- 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
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have pos : ∀ (hy : y = 0),
  (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
Tactic: by_cases hy : y = 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 y : R
hx : ¬x = 0
pos :
  ∀ (hy : y = 0),
    (if x * y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
              ℤ))) =
      HMul.hMul (α := WithZero (Multiplicative ℤ))
        (if x = 0 then 0
        else
          exp
            (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) :
                ℤ)))
        (if y = 0 then 0
        else
          exp
            (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) :
                ℤ)))
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have neg : ∀ (hy : ¬y = 0),
  (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
Tactic: by_cases hy : y = 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 y : R
hx : ¬x = 0
hy : y = 0
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have pos : (if x * 0 = 0 then 0
  else
    exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * 0})))) :
          ℤ))) =
  HMul.hMul (α := WithZero (Multiplicative ℤ))
    (if x = 0 then 0
    else
      exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
    (if Eq.{u_1 + 1} (α := R) 0 0 then 0
    else
      exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
Tactic: rw [hy, mul_zero, if_pos rfl, mul_zero]
-- 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
pos :
  (if x * 0 = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * 0})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have pos₁ : (if Eq.{u_1 + 1} (α := R) 0 0 then 0
  else
    exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ))) =
  HMul.hMul (α := WithZero (Multiplicative ℤ))
    (if x = 0 then 0
    else
      exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
    (if Eq.{u_1 + 1} (α := R) 0 0 then 0
    else
      exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
Tactic: rw [hy, mul_zero, if_pos rfl, mul_zero]
-- 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
pos :
  (if x * 0 = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * 0})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
pos₁ :
  (if Eq.{u_1 + 1} (α := R) 0 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have pos₂ : 0 =
  HMul.hMul (α := WithZero (Multiplicative ℤ))
    (if x = 0 then 0
    else
      exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
    0
Tactic: rw [hy, mul_zero, if_pos rfl, mul_zero]
-- 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
pos :
  (if x * 0 = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * 0})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
pos₁ :
  (if Eq.{u_1 + 1} (α := R) 0 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if Eq.{u_1 + 1} (α := R) 0 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {0})))) : ℤ)))
pos₂ :
  0 =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      0
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have pos₃ : Eq (α := WithZero (Multiplicative ℤ)) 0 0
Tactic: rw [hy, mul_zero, if_pos rfl, mul_zero]
-- 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
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have neg : (if x * y = 0 then 0
  else
    exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
          ℤ))) =
  HMul.hMul (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
    (if y = 0 then 0
    else
      exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
Tactic: rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← exp_add,
        ← Ideal.span_singleton_mul_span_singleton, ← Associates.mk_mul_mk, ← neg_add,
        Associates.count_mul (Associates.mk_ne_zero'.mpr hx) (Associates.mk_ne_zero'.mpr hy)
          v.associates_irreducible,
        Nat.cast_add]
-- 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
neg :
  (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have neg₁ : (if x * y = 0 then 0
  else
    exp
      (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
          ℤ))) =
  HMul.hMul (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
Tactic: rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← exp_add,
        ← Ideal.span_singleton_mul_span_singleton, ← Associates.mk_mul_mk, ← neg_add,
        Associates.count_mul (Associates.mk_ne_zero'.mpr hx) (Associates.mk_ne_zero'.mpr hy)
          v.associates_irreducible,
        Nat.cast_add]
-- 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
neg :
  (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
neg₁ :
  (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
⊢ (if x * y = 0 then 0
    else
      exp
        (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) :
            ℤ))) =
    HMul.hMul (α := WithZero (Multiplicative ℤ))
      (if x = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
      (if y = 0 then 0
      else
        exp
          (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
have neg₂ : exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x * y})))) : ℤ)) =
  HMul.hMul (α := WithZero (Multiplicative ℤ))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {x})))) : ℤ)))
    (exp (-(↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {y})))) : ℤ)))
Tactic: rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← exp_add,
        ← Ideal.span_singleton_mul_span_singleton, ← Associates.mk_mul_mk, ← neg_add,
        Associates.count_mul (Associates.mk_ne_zero'.mpr hx) (Associates.mk_ne_zero'.mpr hy)
          v.associates_irreducible,
        Nat.cast_add]