awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
U : Set K⸨X⸩
hU : U ∈ 𝓝 (limit hℱ)
γ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hU₁ :
  {y : K⸨X⸩ |
      ((Valuation.restrict Valued.v :
              Valuation K⸨X⸩
                (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                  (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
            (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
          (y - limit hℱ) <
        (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    U
D : ℤ :=
  -HSub.hSub (α := ℤ)
      (log
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
          (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v)))
      1
hD₀ :
  D =
    -HSub.hSub (α := ℤ)
        (log
          ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v)))
        1
hD :
  exp (-D) <
    (embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
      (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v)
x : K⸨X⸩
hf : ∀ d < D, coeff hℱ d = x.coeff d
n : ℤ
hn : n < D
⊢ (x - limit hℱ).coeff n = 0
have rw : (limit hℱ).coeff n = x.coeff n
Tactic: rw [HahnSeries.coeff_sub, sub_eq_zero, eq_comm]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
⊢ ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
have pos : ∀ (h_neg : 1 < η),
  ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
Tactic: by_cases! h_neg : 1 < η
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
pos :
  ∀ (h_neg : 1 < η),
    ∃ (P : K[X]),
      ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
            (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
          (F - (↑P : K⟦X⟧)) <
        (↑η : WithZero (Multiplicative ℤ))
⊢ ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
have neg : ∀ (h_neg : η ≤ 1),
  ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
Tactic: by_cases! h_neg : 1 < η
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : 1 < η
⊢ ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
have h : ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
      (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
    (F - (↑0 : K⟦X⟧)) <
  (↑η : WithZero (Multiplicative ℤ))
Tactic: use 0
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : η ≤ 1
⊢ ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
have h_neg₁ : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero sorry) : ℤ) (0 : ℤ)
Tactic: rw [← Units.val_le_val, Units.val_one, ← WithZero.coe_one, ← coe_unzero η.ne_zero,
      coe_le_coe, ← Multiplicative.toAdd_le, toAdd_one] at h_neg
Cleared hypotheses: h_neg
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
⊢ ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
have neg : ∀ (d : ℕ) (hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero sorry) : ℤ) (-(↑d : ℤ) : ℤ)),
  ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
Tactic: obtain ⟨d, hd⟩ := Int.exists_eq_neg_ofNat h_neg
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
⊢ ∃ (P : K[X]),
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑η : WithZero (Multiplicative ℤ))
have h : ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
      (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
    (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧)) <
  (↑η : WithZero (Multiplicative ℤ))
Tactic: use F.trunc (d + 1)
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
⊢ LE.le (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⟧)
          (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) :
      WithZero (Multiplicative ℤ))
    ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-HAdd.hAdd (α := ℤ) (↑d : ℤ) 1)) :
        WithZero (Multiplicative ℤ)) :
      WithZero (Multiplicative ℤ))
have apply : ∀ n < d + 1,
  Eq (α := K) ((PowerSeries.coeff n : (a : K⟦X⟧) → K) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧)) : K)
    (0 : K)
Tactic: apply (intValuation_le_iff_coeff_lt_eq_zero K _).mpr
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
⊢ ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
        (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧)) <
    (↑η : WithZero (Multiplicative ℤ))
have this : LE.le (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⟧)
        (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) :
    WithZero (Multiplicative ℤ))
  ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-HAdd.hAdd (α := ℤ) (↑d : ℤ) 1)) :
      WithZero (Multiplicative ℤ)) :
    WithZero (Multiplicative ℤ))
Tactic: have : Valued.v ((ofPowerSeries ℤ K) (F - (trunc (d + 1) F))) ≤
      (Multiplicative.ofAdd (-(d + 1 : ℤ))) := by
      apply (intValuation_le_iff_coeff_lt_eq_zero K _).mpr
      simpa only [map_sub, sub_eq_zero, Polynomial.coeff_coe, coeff_trunc] using
        fun _ h ↦ (if_pos h).symm
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
this :
  LE.le (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⟧)
          (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) :
      WithZero (Multiplicative ℤ))
    ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-HAdd.hAdd (α := ℤ) (↑d : ℤ) 1)) :
        WithZero (Multiplicative ℤ)) :
      WithZero (Multiplicative ℤ))
⊢ ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
        (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧)) <
    (↑η : WithZero (Multiplicative ℤ))
have this₁ : ((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
    ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) ≤
  HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
    ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
      WithZero (Multiplicative ℤ))
Tactic: rw [neg_add, ofAdd_add, ← hd, ofAdd_toAdd, WithZero.coe_mul, coe_unzero,
      ← coe_algebraMap] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
this :
  ((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) ≤
    HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
⊢ ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
        (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧)) <
    (↑η : WithZero (Multiplicative ℤ))
have h : ((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
      (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
    (↑(F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧)) : K⸨X⸩) <
  (↑η : WithZero (Multiplicative ℤ))
Tactic: rw [← valuation_of_algebraMap (K := K⸨X⸩) (PowerSeries.idealX K) (F - F.trunc (d + 1))]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
this :
  ((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) ≤
    HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
⊢ ((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
        (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
      (↑(F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧)) : K⸨X⸩) <
    (↑η : WithZero (Multiplicative ℤ))
have h : HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
    ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
      WithZero (Multiplicative ℤ)) <
  (↑η : WithZero (Multiplicative ℤ))
Tactic: apply lt_of_le_of_lt this
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
this :
  ((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) ≤
    HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
⊢ HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ)) <
    (↑η : WithZero (Multiplicative ℤ))
have h : HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
    ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
      WithZero (Multiplicative ℤ)) <
  (↑η : WithZero (Multiplicative ℤ)) * 1
Tactic: rw [← mul_one (η : ℤᵐ⁰), mul_assoc, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
this :
  ((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) ≤
    HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
⊢ HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ)) <
    (↑η : WithZero (Multiplicative ℤ)) * 1
have h : 0 < (↑η : WithZero (Multiplicative ℤ))
Tactic: gcongr
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
this :
  ((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) ≤
    HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
h : 0 < (↑η : WithZero (Multiplicative ℤ))
⊢ HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ)) <
    (↑η : WithZero (Multiplicative ℤ)) * 1
have h₁ : LT.lt (α := WithZero (Multiplicative ℤ))
  ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
    WithZero (Multiplicative ℤ))
  (1 : WithZero (Multiplicative ℤ))
Tactic: gcongr
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : LE.le (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (0 : ℤ)
d : ℕ
hd : Eq (α := ℤ) ((Multiplicative.toAdd : (a : Multiplicative ℤ) → ℤ) (unzero ⋯) : ℤ) (-(↑d : ℤ) : ℤ)
this :
  ((Valued.v : Valuation K⟦ℤ⟧ (WithZero (Multiplicative ℤ))) : (a : K⟦ℤ⟧) → WithZero (Multiplicative ℤ))
      ((algebraMap K⟦X⟧ K⸨X⸩ : (a : K⟦X⟧) → K⸨X⸩) (F - (↑((trunc (d + 1) : (a : K⟦X⟧) → K[X]) F) : K⟦X⟧))) ≤
    HMul.hMul (β := WithZero (Multiplicative ℤ)) (↑η : WithZero (Multiplicative ℤ))
      ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
        WithZero (Multiplicative ℤ))
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
    ((↑((Multiplicative.ofAdd : (a : ℤ) → Multiplicative ℤ) (-1)) : WithZero (Multiplicative ℤ)) :
      WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
have h : LT.lt (α := ℤ) 0 1
Tactic: rw [← WithZero.coe_one, coe_lt_coe, ofAdd_neg, Right.inv_lt_one_iff, ← ofAdd_zero,
      Multiplicative.ofAdd_lt]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
γ : (WithZero (Multiplicative ℤ))ˣ
⊢ ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
have hF : let F := f.powerSeriesPart;
F = f.powerSeriesPart
Tactic: set F := f.powerSeriesPart with hF
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
γ : (WithZero (Multiplicative ℤ))ˣ
F : K⟦X⟧ := f.powerSeriesPart
hF : F = f.powerSeriesPart
⊢ ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
have pos : ∀ (ord_nonpos : HahnSeries.order f < 0),
  ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
Tactic: by_cases! ord_nonpos : f.order < 0
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
γ : (WithZero (Multiplicative ℤ))ˣ
F : K⟦X⟧ := f.powerSeriesPart
hF : F = f.powerSeriesPart
pos :
  ∀ (ord_nonpos : HahnSeries.order f < 0),
    ∃ (Q : K⟮X⟯),
      ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
          (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
        (↑γ : WithZero (Multiplicative ℤ))
⊢ ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
have neg : ∀ (ord_nonpos : 0 ≤ HahnSeries.order f),
  ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
Tactic: by_cases! ord_nonpos : f.order < 0
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
γ : (WithZero (Multiplicative ℤ))ˣ
F : K⟦X⟧ := f.powerSeriesPart
hF : F = f.powerSeriesPart
ord_nonpos : HahnSeries.order f < 0
⊢ ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
have hη : let η := Units.mk0 (exp (HahnSeries.order f)) sorry;
η = Units.mk0 (exp (HahnSeries.order f)) sorry
Tactic: set η : ℤᵐ⁰ˣ := Units.mk0 (exp f.order) coe_ne_zero
      with hη
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
γ : (WithZero (Multiplicative ℤ))ˣ
F : K⟦X⟧ := f.powerSeriesPart
hF : F = f.powerSeriesPart
ord_nonpos : HahnSeries.order f < 0
η : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (HahnSeries.order f)) ⋯
hη : η = Units.mk0 (exp (HahnSeries.order f)) ⋯
⊢ ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
have pos : ∀ (P : K[X])
  (hP :
    ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
          (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
        (F - (↑P : K⟦X⟧)) <
      (↑(η * γ) : WithZero (Multiplicative ℤ))),
  ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
Tactic: obtain ⟨P, hP⟩ := exists_Polynomial_intValuation_lt F (η * γ)
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
γ : (WithZero (Multiplicative ℤ))ˣ
F : K⟦X⟧ := f.powerSeriesPart
hF : F = f.powerSeriesPart
ord_nonpos : HahnSeries.order f < 0
η : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (HahnSeries.order f)) ⋯
hη : η = Units.mk0 (exp (HahnSeries.order f)) ⋯
P : K[X]
hP :
  ((intValuation (PowerSeries.idealX K) : Valuation K⟦X⟧ (WithZero (Multiplicative ℤ))) :
        (a : K⟦X⟧) → WithZero (Multiplicative ℤ))
      (F - (↑P : K⟦X⟧)) <
    (↑(η * γ) : WithZero (Multiplicative ℤ))
⊢ ∃ (Q : K⟮X⟯),
    ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        (HSub.hSub (β := K⸨X⸩) f ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) Q : K⸨X⸩)) <
      (↑γ : WithZero (Multiplicative ℤ))
have h : ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
    (HSub.hSub (β := K⸨X⸩) f
      ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) (HPow.hPow (α := K⟮X⟯) RatFunc.X (HahnSeries.order f) * (↑P : K⟮X⟯)) :
        K⸨X⸩)) <
  (↑γ : WithZero (Multiplicative ℤ))
Tactic: use RatFunc.X ^ f.order * (P : K⟮X⟯)