awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑0 : ℤ_[p]) : R) (1 : R)
rw₁ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 0 1), Nat.choose 0 i • r ^ i) 1
rw₂ : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range 1, Nat.choose 0 i • r ^ i) 1
rw₃ : Nat.choose 0 0 • HPow.hPow (β := ℕ) r 0 = 1
rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) = 1
rw₅ : HSMul.hSMul (α := ℕ) (β := R) 1 1 = 1
⊢ Eq.{u_1 + 1} (α := R) ((mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) 0 : R) (1 : R)
have rw₆ : Eq.{u_1 + 1} (α := R) 1 1
Tactic: rw [← Nat.cast_zero, mahlerSeries_apply_nat hr le_rfl, zero_add, Finset.sum_range_one,
      Nat.choose_self, pow_zero, one_smul]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
n : ℕ
⊢ (F : (a : ℤ_[p]) → R) (↑n : ℤ_[p]) = (r + 1) ^ n
have rw : Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range (n + 1), n.choose i • r ^ i)
  (∑ m ∈ Finset.range (n + 1), r ^ m * HPow.hPow (α := R) 1 (n - m) * (↑(n.choose m) : R))
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, (Commute.one_right _).add_pow]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
n : ℕ
⊢ Eq.{u_1 + 1} (α := R) (∑ i ∈ Finset.range (n + 1), n.choose i • r ^ i)
    (∑ m ∈ Finset.range (n + 1), r ^ m * HPow.hPow (α := R) 1 (n - m) * (↑(n.choose m) : R))
have refine : ∀ (i : ℕ) (hi : i ∈ Finset.range (n + 1)),
  n.choose i • r ^ i = r ^ i * HPow.hPow (α := R) 1 (n - i) * (↑(n.choose i) : R)
Tactic: refine Finset.sum_congr rfl fun i hi ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
n i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ n.choose i • r ^ i = r ^ i * HPow.hPow (α := R) 1 (n - i) * (↑(n.choose i) : R)
have rw : n.choose i • r ^ i = r ^ i * 1 * (↑(n.choose i) : R)
Tactic: rw [one_pow, mul_one, nsmul_eq_mul, Nat.cast_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
n i : ℕ
hi : i ∈ Finset.range (n + 1)
rw : n.choose i • r ^ i = r ^ i * 1 * (↑(n.choose i) : R)
⊢ n.choose i • r ^ i = r ^ i * HPow.hPow (α := R) 1 (n - i) * (↑(n.choose i) : R)
have rw₁ : n.choose i • r ^ i = r ^ i * (↑(n.choose i) : R)
Tactic: rw [one_pow, mul_one, nsmul_eq_mul, Nat.cast_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
n i : ℕ
hi : i ∈ Finset.range (n + 1)
rw : n.choose i • r ^ i = r ^ i * 1 * (↑(n.choose i) : R)
rw₁ : n.choose i • r ^ i = r ^ i * (↑(n.choose i) : R)
⊢ n.choose i • r ^ i = r ^ i * HPow.hPow (α := R) 1 (n - i) * (↑(n.choose i) : R)
have rw₂ : (↑(n.choose i) : R) * r ^ i = r ^ i * (↑(n.choose i) : R)
Tactic: rw [one_pow, mul_one, nsmul_eq_mul, Nat.cast_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
n i : ℕ
hi : i ∈ Finset.range (n + 1)
rw : n.choose i • r ^ i = r ^ i * 1 * (↑(n.choose i) : R)
rw₁ : n.choose i • r ^ i = r ^ i * (↑(n.choose i) : R)
rw₂ : (↑(n.choose i) : R) * r ^ i = r ^ i * (↑(n.choose i) : R)
⊢ n.choose i • r ^ i = r ^ i * HPow.hPow (α := R) 1 (n - i) * (↑(n.choose i) : R)
have rw₃ : r ^ i * (↑(n.choose i) : R) = r ^ i * (↑(n.choose i) : R)
Tactic: rw [one_pow, mul_one, nsmul_eq_mul, Nat.cast_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
⊢ (F : (a : ℤ_[p]) → R) (a + b) =
    HMul.hMul (α := R) (β := R) ((F : (a : ℤ_[p]) → R) a : R) ((F : (a : ℤ_[p]) → R) b : R)
have hF : ∀ (n : ℕ), (F : (a : ℤ_[p]) → R) (↑n : ℤ_[p]) = (r + 1) ^ n
Tactic: have hF (n : ℕ) : F n = (r + 1) ^ n := by
      rw [mahlerSeries_apply_nat hr le_rfl, (Commute.one_right _).add_pow]
      refine Finset.sum_congr rfl fun i hi ↦ ?_
      rw [one_pow, mul_one, nsmul_eq_mul, Nat.cast_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
a b : ℤ_[p]
F : C(ℤ_[p], R) := mahlerSeries fun (x : ℕ) ↦ r ^ x
hF : ∀ (n : ℕ), (F : (a : ℤ_[p]) → R) (↑n : ℤ_[p]) = (r + 1) ^ n
⊢ (F : (a : ℤ_[p]) → R) (a + b) =
    HMul.hMul (α := R) (β := R) ((F : (a : ℤ_[p]) → R) a : R) ((F : (a : ℤ_[p]) → R) b : R)
have refine : ∀ (x : ℕ × ℕ) (m n : ℕ),
  Eq (α := R)
    (Function.comp (β := ℤ_[p] × ℤ_[p])
      (Function.comp (β := ℤ_[p]) (⇑F : (a : ℤ_[p]) → R) fun (p_1 : ℤ_[p] × ℤ_[p]) ↦ p_1.1 + p_1.2)
      (Prod.map Nat.cast Nat.cast) (m, n))
    (Function.comp (β := ℤ_[p] × ℤ_[p]) ((fun (p : R × R) ↦ p.1 * p.2) ∘ (⇑(F.prodMap F) : (a : ℤ_[p] × ℤ_[p]) → R × R))
      (Prod.map Nat.cast Nat.cast) (m, n))
Tactic: refine congr_fun ((denseRange_natCast.prodMap denseRange_natCast).equalizer
      ((map_continuous F).comp continuous_add)
      (continuous_mul.comp (map_continuous <| F.prodMap F)) (funext fun ⟨m, n⟩ ↦ ?_)) (a, b)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
⊢ (addChar_of_value_at_one r hr : (a : ℤ_[p]) → R) 1 = 1 + r
have change : (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
Tactic: change mahlerSeries (r ^ ·) ↑(1 : ℕ) = _
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₂ : Nat.choose 1 0 • HPow.hPow (β := ℕ) r 0 + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₂ : Nat.choose 1 0 • HPow.hPow (β := ℕ) r 0 + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₃ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₂ : Nat.choose 1 0 • HPow.hPow (β := ℕ) r 0 + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₃ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₂ : Nat.choose 1 0 • HPow.hPow (β := ℕ) r 0 + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₃ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₅ : HPow.hPow (β := ℕ) r 0 + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₂ : Nat.choose 1 0 • HPow.hPow (β := ℕ) r 0 + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₃ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
rw₅ : HPow.hPow (β := ℕ) r 0 + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₆ : HPow.hPow (β := ℕ) r 0 + HPow.hPow (β := ℕ) r 1 = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₂ : Nat.choose 1 0 • HPow.hPow (β := ℕ) r 0 + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₃ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
rw₅ : HPow.hPow (β := ℕ) r 0 + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
rw₆ : HPow.hPow (β := ℕ) r 0 + HPow.hPow (β := ℕ) r 1 = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₇ : 1 + HPow.hPow (β := ℕ) r 1 = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.AddChar (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : NormedRing R
inst_2 : Algebra ℤ_[p] R
inst_3 : IsBoundedSMul ℤ_[p] R
inst_4 : IsUltrametricDist R
inst_5 : CompleteSpace R
r : R
hr : Tendsto.{0, u_1} (β := R) (fun (x : ℕ) ↦ r ^ x) atTop (𝓝 0)
rw : ∑ i ∈ Finset.range (HAdd.hAdd (α := ℕ) 1 1), Nat.choose 1 i • r ^ i = 1 + r
rw₁ : ∑ x ∈ Finset.range 1, Nat.choose 1 x • r ^ x + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₂ : Nat.choose 1 0 • HPow.hPow (β := ℕ) r 0 + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₃ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + Nat.choose 1 1 • HPow.hPow (β := ℕ) r 1 = 1 + r
rw₄ : HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 0) + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
rw₅ : HPow.hPow (β := ℕ) r 0 + HSMul.hSMul (α := ℕ) 1 (HPow.hPow (β := ℕ) r 1) = 1 + r
rw₆ : HPow.hPow (β := ℕ) r 0 + HPow.hPow (β := ℕ) r 1 = 1 + r
rw₇ : 1 + HPow.hPow (β := ℕ) r 1 = 1 + r
⊢ (mahlerSeries fun (x : ℕ) ↦ r ^ x : (a : ℤ_[p]) → R) (↑1 : ℤ_[p]) = 1 + r
have rw₈ : 1 + r = 1 + r
Tactic: rw [mahlerSeries_apply_nat hr le_rfl, Finset.sum_range_succ, Finset.sum_range_one,
    Nat.choose_zero_right, Nat.choose_self, one_smul, one_smul, pow_zero, pow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ (Valued.v : (a : PadicAlgCl p) → ℝ≥0) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p])) =
    HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)
have a : (↑((Valued.v : (a : PadicAlgCl p) → ℝ≥0)
        ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p]))) :
    ℝ) =
  (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ (↑((Valued.v : (a : PadicAlgCl p) → ℝ≥0)
          ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p]))) :
      ℝ) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
have a : Norm.norm (E := PadicAlgCl p) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p])) =
  (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
Tactic: rw [valuation_coe, norm_extends, Padic.norm_p, one_div, NNReal.coe_inv,
    NNReal.coe_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a :
  Norm.norm (E := PadicAlgCl p) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p])) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
⊢ (↑((Valued.v : (a : PadicAlgCl p) → ℝ≥0)
          ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p]))) :
      ℝ) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
have a₁ : Norm.norm (E := ℚ_[p]) (↑p : ℚ_[p]) = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
Tactic: rw [valuation_coe, norm_extends, Padic.norm_p, one_div, NNReal.coe_inv,
    NNReal.coe_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a :
  Norm.norm (E := PadicAlgCl p) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p])) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₁ : Norm.norm (E := ℚ_[p]) (↑p : ℚ_[p]) = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
⊢ (↑((Valued.v : (a : PadicAlgCl p) → ℝ≥0)
          ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p]))) :
      ℝ) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
have a₂ : (↑p : ℝ)⁻¹ = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
Tactic: rw [valuation_coe, norm_extends, Padic.norm_p, one_div, NNReal.coe_inv,
    NNReal.coe_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a :
  Norm.norm (E := PadicAlgCl p) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p])) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₁ : Norm.norm (E := ℚ_[p]) (↑p : ℚ_[p]) = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₂ : (↑p : ℝ)⁻¹ = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
⊢ (↑((Valued.v : (a : PadicAlgCl p) → ℝ≥0)
          ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p]))) :
      ℝ) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
have a₃ : (↑p : ℝ)⁻¹ = (↑(↑p : ℝ≥0)⁻¹ : ℝ)
Tactic: rw [valuation_coe, norm_extends, Padic.norm_p, one_div, NNReal.coe_inv,
    NNReal.coe_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a :
  Norm.norm (E := PadicAlgCl p) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p])) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₁ : Norm.norm (E := ℚ_[p]) (↑p : ℚ_[p]) = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₂ : (↑p : ℝ)⁻¹ = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₃ : (↑p : ℝ)⁻¹ = (↑(↑p : ℝ≥0)⁻¹ : ℝ)
⊢ (↑((Valued.v : (a : PadicAlgCl p) → ℝ≥0)
          ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p]))) :
      ℝ) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
have a₄ : (↑p : ℝ)⁻¹ = (↑(↑p : ℝ≥0) : ℝ)⁻¹
Tactic: rw [valuation_coe, norm_extends, Padic.norm_p, one_div, NNReal.coe_inv,
    NNReal.coe_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a :
  Norm.norm (E := PadicAlgCl p) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p])) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₁ : Norm.norm (E := ℚ_[p]) (↑p : ℚ_[p]) = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₂ : (↑p : ℝ)⁻¹ = (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
a₃ : (↑p : ℝ)⁻¹ = (↑(↑p : ℝ≥0)⁻¹ : ℝ)
a₄ : (↑p : ℝ)⁻¹ = (↑(↑p : ℝ≥0) : ℝ)⁻¹
⊢ (↑((Valued.v : (a : PadicAlgCl p) → ℝ≥0)
          ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) (↑p : ℚ_[p]))) :
      ℝ) =
    (↑(HDiv.hDiv (α := ℝ≥0) 1 (↑p : ℝ≥0)) : ℝ)
have a₅ : Eq (α := ℝ) (↑p : ℝ)⁻¹ (↑p : ℝ)⁻¹
Tactic: rw [valuation_coe, norm_extends, Padic.norm_p, one_div, NNReal.coe_inv,
    NNReal.coe_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ∃ (x : PadicAlgCl p),
    Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) x : ℝ≥0) (0 : ℝ≥0) ∧
      Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) x : ℝ≥0) (1 : ℝ≥0)
have h : Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) (↑p : PadicAlgCl p)) 0 ∧
  Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) (↑p : PadicAlgCl p)) 1
Tactic: use p
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) (↑p : PadicAlgCl p)) 0 ∧
    Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) (↑p : PadicAlgCl p)) 1
have hp₁ : Nat.Prime p
Tactic: have hp : Nat.Prime p := hp.1
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
hp_1 : Fact (Nat.Prime p)
hp : Nat.Prime p
⊢ Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) (↑p : PadicAlgCl p)) 0 ∧
    Ne (α := ℝ≥0) ((Valued.v : (a : PadicAlgCl p) → ℝ≥0) (↑p : PadicAlgCl p)) 1
have h : ¬p = 0 ∧ ¬p = 1
Tactic: simp only [valuation_p, one_div, ne_eq, inv_eq_zero, Nat.cast_eq_zero, inv_eq_one,
      Nat.cast_eq_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ∃ (x : PadicAlgCl p), 1 < ‖x‖
have choose : ∀ (x : ℚ_[p]) (hx : 1 < ‖x‖), ∃ (x : PadicAlgCl p), 1 < ‖x‖
Tactic: choose x hx using NontriviallyNormedField.non_trivial (α := ℚ_[p])
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
hx : 1 < ‖x‖
⊢ ∃ (x : PadicAlgCl p), 1 < ‖x‖
have h : 1 < Norm.norm (E := PadicAlgCl p) ((algebraMap ℚ_[p] (PadicAlgCl p) : (a : ℚ_[p]) → PadicAlgCl p) x : PadicAlgCl p)
Tactic: use x
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.Complex (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ Eq (α := UniformSpace.Completion (PadicAlgCl p)) (↑(↑n : PadicAlgCl p) : UniformSpace.Completion (PadicAlgCl p))
    (↑n : ℂ_[p])
have rw : Eq (α := UniformSpace.Completion (PadicAlgCl p)) (↑(↑n : PadicAlgCl p) : UniformSpace.Completion (PadicAlgCl p))
  ((algebraMap (PadicAlgCl p) ℂ_[p] : (a : PadicAlgCl p) → ℂ_[p]) (↑n : PadicAlgCl p))
Tactic: rw [← map_natCast (algebraMap (PadicAlgCl p) ℂ_[p]) n, coe_eq]