awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : ¬P = 0
span_ne_zero : Ne (α := Ideal K[X]) (Ideal.span {P}) 0 ∧ Ne (α := Ideal K[X]) (Ideal.span {Polynomial.X}) 0
span_ne_zero' : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0 ∧ Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) 0
neg :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))
neg₁ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₂ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₃ : Ideal.IsPrime (asIdeal (idealX K))
⊢ Associates.count (α := Ideal K[X]) (Associates.mk (Ideal.span {Polynomial.X}))
      (Associates.factors (Associates.mk (Ideal.span {P}))) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))
have neg₄ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
Tactic: classical
  rw [Ideal.count_associates_factors_eq span_ne_zero.1
    (Ideal.span_singleton_prime Polynomial.X_ne_zero |>.mpr prime_X) span_ne_zero.2,
    Ideal.count_associates_factors_eq]
  on_goal 1 => convert! (normalized_count_X_eq_of_coe hP).symm
  exacts [Ideal.count_span_normalizedFactors_eq_of_normUnit hP Polynomial.normUnit_X prime_X,
    Ideal.count_span_normalizedFactors_eq_of_normUnit (by simp [hP]) normUnit_X X_prime,
    span_ne_zero'.1, (idealX K).isPrime, span_ne_zero'.2]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : ¬P = 0
span_ne_zero : Ne (α := Ideal K[X]) (Ideal.span {P}) 0 ∧ Ne (α := Ideal K[X]) (Ideal.span {Polynomial.X}) 0
span_ne_zero' : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0 ∧ Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) 0
neg :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))
neg₁ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₂ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₃ : Ideal.IsPrime (asIdeal (idealX K))
neg₄ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
neg₅ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₆ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₇ : Ideal.IsPrime (asIdeal (idealX K))
neg₈ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
neg₉ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₁₀ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₁₁ : Ideal.IsPrime (asIdeal (idealX K))
neg₁₂ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
⊢ Associates.count (α := Ideal K[X]) (Associates.mk (Ideal.span {Polynomial.X}))
      (Associates.factors (Associates.mk (Ideal.span {P}))) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))
have h : Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
    (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
  Multiset.count Polynomial.X (UniqueFactorizationMonoid.normalizedFactors P)
Tactic: classical
  rw [Ideal.count_associates_factors_eq span_ne_zero.1
    (Ideal.span_singleton_prime Polynomial.X_ne_zero |>.mpr prime_X) span_ne_zero.2,
    Ideal.count_associates_factors_eq]
  on_goal 1 => convert! (normalized_count_X_eq_of_coe hP).symm
  exacts [Ideal.count_span_normalizedFactors_eq_of_normUnit hP Polynomial.normUnit_X prime_X,
    Ideal.count_span_normalizedFactors_eq_of_normUnit (by simp [hP]) normUnit_X X_prime,
    span_ne_zero'.1, (idealX K).isPrime, span_ne_zero'.2]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : ¬P = 0
span_ne_zero : Ne (α := Ideal K[X]) (Ideal.span {P}) 0 ∧ Ne (α := Ideal K[X]) (Ideal.span {Polynomial.X}) 0
span_ne_zero' : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0 ∧ Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) 0
neg :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))
neg₁ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₂ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₃ : Ideal.IsPrime (asIdeal (idealX K))
neg₄ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
neg₅ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₆ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₇ : Ideal.IsPrime (asIdeal (idealX K))
neg₈ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
neg₉ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₁₀ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₁₁ : Ideal.IsPrime (asIdeal (idealX K))
neg₁₂ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
h :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count Polynomial.X (UniqueFactorizationMonoid.normalizedFactors P)
⊢ Associates.count (α := Ideal K[X]) (Associates.mk (Ideal.span {Polynomial.X}))
      (Associates.factors (Associates.mk (Ideal.span {P}))) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))
have h₁ : Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
    (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)})) =
  Multiset.count X (UniqueFactorizationMonoid.normalizedFactors (↑P : K⟦X⟧))
Tactic: classical
  rw [Ideal.count_associates_factors_eq span_ne_zero.1
    (Ideal.span_singleton_prime Polynomial.X_ne_zero |>.mpr prime_X) span_ne_zero.2,
    Ideal.count_associates_factors_eq]
  on_goal 1 => convert! (normalized_count_X_eq_of_coe hP).symm
  exacts [Ideal.count_span_normalizedFactors_eq_of_normUnit hP Polynomial.normUnit_X prime_X,
    Ideal.count_span_normalizedFactors_eq_of_normUnit (by simp [hP]) normUnit_X X_prime,
    span_ne_zero'.1, (idealX K).isPrime, span_ne_zero'.2]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        X :
      WithZero (Multiplicative ℤ))
    (exp (-1) : WithZero (Multiplicative ℤ))
have rw : Eq (α := WithZero (Multiplicative ℤ))
  (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
        (a : K[X]) → WithZero (Multiplicative ℤ))
      Polynomial.X :
    WithZero (Multiplicative ℤ))
  (exp (-1) : WithZero (Multiplicative ℤ))
Tactic: rw [← Polynomial.coe_X, ← intValuation_eq_of_coe]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K⟮X⟯
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((polynomialValuationX K : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) :
          (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
        P :
      WithZero (Multiplicative ℤ))
    (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
          (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) P) :
      WithZero (Multiplicative ℤ))
have refine : ∀ (p q : K[X]),
  q ≠ 0 →
    Eq (α := WithZero (Multiplicative ℤ))
      (((polynomialValuationX K : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) :
            (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
          (RatFunc.mk p q) :
        WithZero (Multiplicative ℤ))
      (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
            (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
          ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) (RatFunc.mk p q)) :
        WithZero (Multiplicative ℤ))
Tactic: refine RatFunc.induction_on' P ?_
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K⟮X⟯
f g : K[X]
h : g ≠ 0
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((polynomialValuationX K : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) :
          (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
        (RatFunc.mk f g) :
      WithZero (Multiplicative ℤ))
    (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
          (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) (RatFunc.mk f g)) :
      WithZero (Multiplicative ℤ))
have rw : ((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
      (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
    ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩)
      (IsLocalization.mk' K⟮X⟯ f
        (Subtype.mk (p := fun (x : K[X]) ↦ Membership.mem (γ := Submonoid K[X]) (nonZeroDivisors K[X]) x) g sorry))) =
  HDiv.hDiv (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        f :
      WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        g :
      WithZero (Multiplicative ℤ))
Tactic: rw [Polynomial.valuation_of_mk K f h, RatFunc.mk_eq_mk' f h, Eq.comm]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K⟮X⟯
f g : K[X]
h : g ≠ 0
⊢ ((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
        (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩)
        (IsLocalization.mk' K⟮X⟯ f
          (Subtype.mk (p := fun (x : K[X]) ↦ Membership.mem (γ := Submonoid K[X]) (nonZeroDivisors K[X]) x) g ⋯))) =
    HDiv.hDiv (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
            (a : K[X]) → WithZero (Multiplicative ℤ))
          f :
        WithZero (Multiplicative ℤ))
      (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
            (a : K[X]) → WithZero (Multiplicative ℤ))
          g :
        WithZero (Multiplicative ℤ))
have h₁ : Eq (α := K⸨X⸩)
  ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩)
    (IsLocalization.mk' K⟮X⟯ f
      (Subtype.mk (p := fun (x : K[X]) ↦ Membership.mem (γ := Submonoid K[X]) (nonZeroDivisors K[X]) x) g sorry)))
  (IsLocalization.mk' K⸨X⸩ (↑f : K⟦X⟧)
    (Subtype.mk (p := fun (x : K⟦X⟧) ↦ Membership.mem (γ := Submonoid K⟦X⟧) (nonZeroDivisors K⟦X⟧) x) (↑g : K⟦X⟧) sorry))
Tactic: convert!
    @valuation_of_mk' K⟦X⟧ _ _ K⸨X⸩ _ _ _ (PowerSeries.idealX K) f
      ⟨g, mem_nonZeroDivisors_iff_ne_zero.2 <| (by simp [h])⟩
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K⟮X⟯
f g : K[X]
h : g ≠ 0
h₁ :
  Eq (α := K⸨X⸩)
    ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩)
      (IsLocalization.mk' K⟮X⟯ f
        (Subtype.mk (p := fun (x : K[X]) ↦ Membership.mem (γ := Submonoid K[X]) (nonZeroDivisors K[X]) x) g ⋯)))
    (IsLocalization.mk' K⸨X⸩ (↑f : K⟦X⟧)
      (Subtype.mk (p := fun (x : K⟦X⟧) ↦ Membership.mem (γ := Submonoid K⟦X⟧) (nonZeroDivisors K⟦X⟧) x) (↑g : K⟦X⟧) ⋯))
⊢ ((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
        (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩)
        (IsLocalization.mk' K⟮X⟯ f
          (Subtype.mk (p := fun (x : K[X]) ↦ Membership.mem (γ := Submonoid K[X]) (nonZeroDivisors K[X]) x) g ⋯))) =
    HDiv.hDiv (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
            (a : K[X]) → WithZero (Multiplicative ℤ))
          f :
        WithZero (Multiplicative ℤ))
      (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
            (a : K[X]) → WithZero (Multiplicative ℤ))
          g :
        WithZero (Multiplicative ℤ))
have h₂ : Eq (α := WithZero (Multiplicative ℤ))
  (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
        (a : K[X]) → WithZero (Multiplicative ℤ))
      f :
    WithZero (Multiplicative ℤ))
  (((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
        (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      (↑f : K⟦X⟧) :
    WithZero (Multiplicative ℤ))
Tactic: convert!
    @valuation_of_mk' K⟦X⟧ _ _ K⸨X⸩ _ _ _ (PowerSeries.idealX K) f
      ⟨g, mem_nonZeroDivisors_iff_ne_zero.2 <| (by simp [h])⟩
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K⟮X⟯
f g : K[X]
h : g ≠ 0
h₁ :
  Eq (α := K⸨X⸩)
    ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩)
      (IsLocalization.mk' K⟮X⟯ f
        (Subtype.mk (p := fun (x : K[X]) ↦ Membership.mem (γ := Submonoid K[X]) (nonZeroDivisors K[X]) x) g ⋯)))
    (IsLocalization.mk' K⸨X⸩ (↑f : K⟦X⟧)
      (Subtype.mk (p := fun (x : K⟦X⟧) ↦ Membership.mem (γ := Submonoid K⟦X⟧) (nonZeroDivisors K⟦X⟧) x) (↑g : K⟦X⟧) ⋯))
h₂ :
  Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        f :
      WithZero (Multiplicative ℤ))
    (((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑f : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
⊢ ((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
        (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩)
        (IsLocalization.mk' K⟮X⟯ f
          (Subtype.mk (p := fun (x : K[X]) ↦ Membership.mem (γ := Submonoid K[X]) (nonZeroDivisors K[X]) x) g ⋯))) =
    HDiv.hDiv (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
      (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
            (a : K[X]) → WithZero (Multiplicative ℤ))
          f :
        WithZero (Multiplicative ℤ))
      (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
            (a : K[X]) → WithZero (Multiplicative ℤ))
          g :
        WithZero (Multiplicative ℤ))
have h₃ : Eq (α := WithZero (Multiplicative ℤ))
  (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
        (a : K[X]) → WithZero (Multiplicative ℤ))
      g :
    WithZero (Multiplicative ℤ))
  (((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
        (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      (↑(Subtype.mk (p := fun (x : K⟦X⟧) ↦ Membership.mem (γ := Submonoid K⟦X⟧) (nonZeroDivisors K⟦X⟧) x) (↑g : K⟦X⟧)
            sorry) :
        K⟦X⟧) :
    WithZero (Multiplicative ℤ))
Tactic: convert!
    @valuation_of_mk' K⟦X⟧ _ _ K⸨X⸩ _ _ _ (PowerSeries.idealX K) f
      ⟨g, mem_nonZeroDivisors_iff_ne_zero.2 <| (by simp [h])⟩
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
s : ℕ
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
        (HPow.hPow (α := K⟦ℤ⟧) (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X : K⟦ℤ⟧) s) :
      WithZero (Multiplicative ℤ))
    (exp (-(↑s : ℤ)) : WithZero (Multiplicative ℤ))
have rw : HPow.hPow (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
        (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X) :
      WithZero (Multiplicative ℤ))
    s =
  exp (-(↑s : ℤ))
Tactic: rw [map_pow, valuation_def, ← LaurentSeries.coe_algebraMap,
    valuation_of_algebraMap, intValuation_X, ← exp_nsmul, smul_neg, nsmul_one]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
s : ℕ
rw :
  HPow.hPow (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
          (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X) :
        WithZero (Multiplicative ℤ))
      s =
    exp (-(↑s : ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
        (HPow.hPow (α := K⟦ℤ⟧) (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X : K⟦ℤ⟧) s) :
      WithZero (Multiplicative ℤ))
    (exp (-(↑s : ℤ)) : WithZero (Multiplicative ℤ))
have rw₁ : HPow.hPow (α := WithZero (Multiplicative ℤ))
    (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
          (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
        (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X) :
      WithZero (Multiplicative ℤ))
    s =
  exp (-(↑s : ℤ))
Tactic: rw [map_pow, valuation_def, ← LaurentSeries.coe_algebraMap,
    valuation_of_algebraMap, intValuation_X, ← exp_nsmul, smul_neg, nsmul_one]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
s : ℕ
rw :
  HPow.hPow (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
          (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X) :
        WithZero (Multiplicative ℤ))
      s =
    exp (-(↑s : ℤ))
rw₁ :
  HPow.hPow (α := WithZero (Multiplicative ℤ))
      (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
            (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
          (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X) :
        WithZero (Multiplicative ℤ))
      s =
    exp (-(↑s : ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
        (HPow.hPow (α := K⟦ℤ⟧) (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X : K⟦ℤ⟧) s) :
      WithZero (Multiplicative ℤ))
    (exp (-(↑s : ℤ)) : WithZero (Multiplicative ℤ))
have rw₂ : HPow.hPow (α := WithZero (Multiplicative ℤ))
    (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
          (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
        ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) PowerSeries.X) :
      WithZero (Multiplicative ℤ))
    s =
  exp (-(↑s : ℤ))
Tactic: rw [map_pow, valuation_def, ← LaurentSeries.coe_algebraMap,
    valuation_of_algebraMap, intValuation_X, ← exp_nsmul, smul_neg, nsmul_one]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
s : ℕ
rw :
  HPow.hPow (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
          (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X) :
        WithZero (Multiplicative ℤ))
      s =
    exp (-(↑s : ℤ))
rw₁ :
  HPow.hPow (α := WithZero (Multiplicative ℤ))
      (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
            (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
          (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X) :
        WithZero (Multiplicative ℤ))
      s =
    exp (-(↑s : ℤ))
rw₂ :
  HPow.hPow (α := WithZero (Multiplicative ℤ))
      (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
            (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
          ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) PowerSeries.X) :
        WithZero (Multiplicative ℤ))
      s =
    exp (-(↑s : ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
        (HPow.hPow (α := K⟦ℤ⟧) (((ofPowerSeries ℤ K : K⟦X⟧ →+* K⟦ℤ⟧) : (a : K⟦X⟧) → K⟦ℤ⟧) PowerSeries.X : K⟦ℤ⟧) s) :
      WithZero (Multiplicative ℤ))
    (exp (-(↑s : ℤ)) : WithZero (Multiplicative ℤ))
have rw₃ : HPow.hPow (α := WithZero (Multiplicative ℤ))
    (((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        PowerSeries.X :
      WithZero (Multiplicative ℤ))
    s =
  exp (-(↑s : ℤ))
Tactic: rw [map_pow, valuation_def, ← LaurentSeries.coe_algebraMap,
    valuation_of_algebraMap, intValuation_X, ← exp_nsmul, smul_neg, nsmul_one]