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
pos :
  Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        0 :
      WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑0 : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        P :
      WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑P : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
have pos₁ : Eq (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
  (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      (↑0 : K⟦X⟧) :
    WithZero (Multiplicative ℤ))
Tactic: rw [hP, Valuation.map_zero, Polynomial.coe_zero, Valuation.map_zero]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : P = 0
pos :
  Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        0 :
      WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑0 : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
pos₁ :
  Eq (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑0 : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        P :
      WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑P : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
have pos₂ : Eq (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
  (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      0 :
    WithZero (Multiplicative ℤ))
Tactic: rw [hP, Valuation.map_zero, Polynomial.coe_zero, Valuation.map_zero]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : P = 0
pos :
  Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        0 :
      WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑0 : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
pos₁ :
  Eq (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑0 : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
pos₂ :
  Eq (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        0 :
      WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        P :
      WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑P : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
have pos₃ : Eq (α := WithZero (Multiplicative ℤ)) 0 0
Tactic: rw [hP, Valuation.map_zero, Polynomial.coe_zero, Valuation.map_zero]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : ¬P = 0
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((intValuation (Polynomial.idealX K) : Valuation K[X] (WithZero (Multiplicative ℤ))) :
          (a : K[X]) → WithZero (Multiplicative ℤ))
        P :
      WithZero (Multiplicative ℤ))
    (((intValuation (idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (↑P : K⟦X⟧) :
      WithZero (Multiplicative ℤ))
have neg : Eq (α := WithZero (Multiplicative ℤ))
  (exp
    (-(↑(Associates.count (α := Ideal K[X]) (Associates.mk (asIdeal (Polynomial.idealX K)))
            (Associates.factors (Associates.mk (Ideal.span {P})))) :
        ℤ)))
  (exp
    (-(↑(Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
            (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))) :
        ℤ)))
Tactic: rw [intValuation_if_neg _ hP, intValuation_if_neg _ <| (by simp [hP])]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : ¬P = 0
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (exp
      (-(↑(Associates.count (α := Ideal K[X]) (Associates.mk (asIdeal (Polynomial.idealX K)))
              (Associates.factors (Associates.mk (Ideal.span {P})))) :
          ℤ)))
    (exp
      (-(↑(Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
              (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))) :
          ℤ)))
have neg : 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⟧)})))
Tactic: simp only [idealX_span, exp_neg, inv_inj, exp_inj, Nat.cast_inj]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : ¬P = 0
⊢ 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 span_ne_zero : Ne (α := Ideal K[X]) (Ideal.span {P}) 0 ∧ Ne (α := Ideal K[X]) (Ideal.span {Polynomial.X}) 0
Tactic: have span_ne_zero :
    (Ideal.span {P} : Ideal K[X]) ≠ 0 ∧ (Ideal.span {Polynomial.X} : Ideal K[X]) ≠ 0 := by
    simp only [Ideal.zero_eq_bot, ne_eq, Ideal.span_singleton_eq_bot, hP, Polynomial.X_ne_zero,
      not_false_iff, and_self_iff]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
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
⊢ 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 span_ne_zero' : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0 ∧ Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) 0
Tactic: have span_ne_zero' :
    (Ideal.span {↑P} : Ideal K⟦X⟧) ≠ 0 ∧ ((idealX K).asIdeal : Ideal K⟦X⟧) ≠ 0 := by
    simp only [Ideal.zero_eq_bot, ne_eq, Ideal.span_singleton_eq_bot, coe_eq_zero_iff, hP,
      not_false_eq_true, true_and, (idealX K).3]
-- 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
⊢ 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 neg : 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⟧)}))
Tactic: 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]
-- 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})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
⊢ 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 neg₁ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
Tactic: 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]
-- 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})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₁ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
⊢ 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 neg₂ : Ideal.IsPrime (asIdeal (idealX K))
Tactic: 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]
-- 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})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₁ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₂ : 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 neg₃ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
Tactic: 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]
-- 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
⊢ 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⟧)}))
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: convert! (normalized_count_X_eq_of_coe hP).symm
-- 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
h :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count Polynomial.X (UniqueFactorizationMonoid.normalizedFactors P)
⊢ 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⟧)}))
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: convert! (normalized_count_X_eq_of_coe hP).symm
-- 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
⊢ 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⟧)}))
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: on_goal 1 => convert! (normalized_count_X_eq_of_coe hP).symm
-- 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
h :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count Polynomial.X (UniqueFactorizationMonoid.normalizedFactors P)
⊢ 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⟧)}))
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: on_goal 1 => convert! (normalized_count_X_eq_of_coe hP).symm
-- 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
⊢ 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 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⟧)})))
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⟧)})))
⊢ 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 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⟧)}))
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⟧)}))
⊢ 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 neg₂ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
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
⊢ 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 neg₃ : Ideal.IsPrime (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]