awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n j : ℕ
hj : j ∈ range (n + 1)
⊢ HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (HSMul.hSMul (β := ℤ_[p]) j 1) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
have rw : HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
    ((mahlerSeries a : (a : ℤ_[p]) → E) (↑j : ℤ_[p]) : E) =
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
    (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
Tactic: rw [nsmul_one, nsmul_one,
      mahlerSeries_apply_nat ha (Nat.lt_succ_iff.mp <| Finset.mem_range.mp hj), Nat.cast_id]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n j : ℕ
hj : j ∈ range (n + 1)
rw :
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (↑j : ℤ_[p]) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
⊢ HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (HSMul.hSMul (β := ℤ_[p]) j 1) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
have rw₁ : HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
    ((mahlerSeries a : (a : ℤ_[p]) → E) (↑j : ℤ_[p]) : E) =
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
    (∑ x ∈ range (n + 1), Nat.choose (↑j : ℕ) x • a x)
Tactic: rw [nsmul_one, nsmul_one,
      mahlerSeries_apply_nat ha (Nat.lt_succ_iff.mp <| Finset.mem_range.mp hj), Nat.cast_id]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n j : ℕ
hj : j ∈ range (n + 1)
rw :
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (↑j : ℤ_[p]) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
rw₁ :
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (↑j : ℤ_[p]) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), Nat.choose (↑j : ℕ) x • a x)
⊢ HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (HSMul.hSMul (β := ℤ_[p]) j 1) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
have rw₂ : HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ)) (∑ i ∈ range (n + 1), j.choose i • a i) =
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
    (∑ x ∈ range (n + 1), Nat.choose (↑j : ℕ) x • a x)
Tactic: rw [nsmul_one, nsmul_one,
      mahlerSeries_apply_nat ha (Nat.lt_succ_iff.mp <| Finset.mem_range.mp hj), Nat.cast_id]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n j : ℕ
hj : j ∈ range (n + 1)
rw :
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (↑j : ℤ_[p]) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
rw₁ :
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (↑j : ℤ_[p]) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), Nat.choose (↑j : ℕ) x • a x)
rw₂ :
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ)) (∑ i ∈ range (n + 1), j.choose i • a i) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), Nat.choose (↑j : ℕ) x • a x)
⊢ HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      ((mahlerSeries a : (a : ℤ_[p]) → E) (HSMul.hSMul (β := ℤ_[p]) j 1) : E) =
    HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ))
      (∑ x ∈ range (n + 1), (HSMul.hSMul (β := ℕ) j 1).choose x • a x)
have rw₃ : HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ)) (∑ i ∈ range (n + 1), j.choose i • a i) =
  HSMul.hSMul (β := E) (HPow.hPow (α := ℤ) (-1) (n - j) * (↑(n.choose j) : ℤ)) (∑ x ∈ range (n + 1), j.choose x • a x)
Tactic: rw [nsmul_one, nsmul_one,
      mahlerSeries_apply_nat ha (Nat.lt_succ_iff.mp <| Finset.mem_range.mp hj), Nat.cast_id]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n : ℕ
⊢ Eq.{u_1 + 1} (α := E) (Δ_[1]^[n] (fun (k : ℕ) ↦ ∑ j ∈ range (n + 1), k.choose j • a j) 0 : E)
    (∑ j ∈ range (n + 1), Δ_[1]^[n] (fun (k : ℕ) ↦ k.choose j • a j) 0 : E)
have simp : Eq.{u_1 + 1} (α := E)
  (∑ x ∈ range (n + 1),
    ∑ x_1 ∈ range (n + 1),
      (HPow.hPow (α := ℤ) (-1) (n - x) * (↑(n.choose x) : ℤ)) • (0 + HSMul.hSMul (β := ℕ) x 1).choose x_1 • a x_1)
  (∑ x ∈ range (n + 1),
    ∑ k ∈ range (n + 1),
      (HPow.hPow (α := ℤ) (-1) (n - k) * (↑(n.choose k) : ℤ)) • (0 + HSMul.hSMul (β := ℕ) k 1).choose x • a x)
Tactic: simp only [fwdDiff_iter_eq_sum_shift, smul_sum]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n : ℕ
⊢ Eq.{u_1 + 1} (α := E)
    (∑ x ∈ range (n + 1),
      ∑ x_1 ∈ range (n + 1),
        (HPow.hPow (α := ℤ) (-1) (n - x) * (↑(n.choose x) : ℤ)) • (0 + HSMul.hSMul (β := ℕ) x 1).choose x_1 • a x_1)
    (∑ x ∈ range (n + 1),
      ∑ k ∈ range (n + 1),
        (HPow.hPow (α := ℤ) (-1) (n - k) * (↑(n.choose k) : ℤ)) • (0 + HSMul.hSMul (β := ℕ) k 1).choose x • a x)
have rw : Eq.{u_1 + 1} (α := E)
  (∑ y ∈ range (n + 1),
    ∑ x ∈ range (n + 1),
      (HPow.hPow (α := ℤ) (-1) (n - x) * (↑(n.choose x) : ℤ)) • (0 + HSMul.hSMul (β := ℕ) x 1).choose y • a y)
  (∑ x ∈ range (n + 1),
    ∑ k ∈ range (n + 1),
      (HPow.hPow (α := ℤ) (-1) (n - k) * (↑(n.choose k) : ℤ)) • (0 + HSMul.hSMul (β := ℕ) k 1).choose x • a x)
Tactic: rw [sum_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
f : C(ℤ_[p], E)
⊢ HasSum (fun (n : ℕ) ↦ mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n) f
have this : HasSum.{u_1, 0} (α := C(ℤ_[p], E)) (fun (n : ℕ) ↦ mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n)
  (mahlerSeries fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0)
Tactic: have : HasSum (fun n ↦ mahlerTerm (Δ_[1]^[n] f 0) n)
      (mahlerSeries (Δ_[1]^[·] f 0) : C(ℤ_[p], E)) :=
    hasSum_mahlerSeries (fwdDiff_tendsto_zero f)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
f : C(ℤ_[p], E)
this :
  HasSum.{u_1, 0} (α := C(ℤ_[p], E)) (fun (n : ℕ) ↦ mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n)
    (mahlerSeries fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0)
⊢ HasSum (fun (n : ℕ) ↦ mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n) f
have h : f = mahlerSeries fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0
Tactic: convert! this using 1
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
f : C(ℤ_[p], E)
this :
  HasSum.{u_1, 0} (α := C(ℤ_[p], E)) (fun (n : ℕ) ↦ mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n)
    (mahlerSeries fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0)
⊢ f = mahlerSeries fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0
have h : ∀ (n : ℕ),
  Eq (α := E) (Function.comp (β := ℤ_[p]) (⇑f : (a : ℤ_[p]) → E) Nat.cast n)
    (Function.comp (β := ℤ_[p]) (⇑(mahlerSeries fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0) : (a : ℤ_[p]) → E)
      Nat.cast n)
Tactic: refine ContinuousMap.coe_injective (denseRange_natCast.equalizer
    (map_continuous f) (map_continuous _) (funext fun n ↦ ?_))
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
f g : C(ℤ_[p], E)
⊢ { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑(f + g) : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ } =
    HAdd.hAdd (α := ℕ →C₀ E)
      { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ :
        ℕ →C₀ E }
      { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑g : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ :
        ℕ →C₀ E }
have h : ∀ (x : ℕ),
  Eq (α := E)
    (({ toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑(f + g) : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry } :
          (a : ℕ) → E)
        x :
      E)
    ((HAdd.hAdd (α := ℕ →C₀ E)
            { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry :
              ℕ →C₀ E }
            { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑g : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry :
              ℕ →C₀ E } :
          (a : ℕ) → E)
        x :
      E)
Tactic: ext x
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
r : ℤ_[p]
f : C(ℤ_[p], E)
⊢ { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑(r • f) : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ } =
    HSMul.hSMul (α := ℤ_[p]) (β := ℕ →C₀ E) ((RingHom.id ℤ_[p] : (a : ℤ_[p]) → ℤ_[p]) r : ℤ_[p])
      { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯, zero_at_infty' := ⋯ :
        ℕ →C₀ E }
have h : ∀ (n : ℕ),
  Eq (α := E)
    (({ toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑(r • f) : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry } :
          (a : ℕ) → E)
        n :
      E)
    ((HSMul.hSMul (α := ℤ_[p]) (β := ℕ →C₀ E) ((RingHom.id ℤ_[p] : (a : ℤ_[p]) → ℤ_[p]) r : ℤ_[p])
            { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry :
              ℕ →C₀ E } :
          (a : ℕ) → E)
        n :
      E)
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ norm (E := ℕ →C₀ E)
      (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯,
                  zero_at_infty' := ⋯ : ℕ →C₀ E },
              map_add' := ⋯, map_smul' := ⋯, invFun := fun (a : ℕ →C₀ E) ↦ mahlerSeries (⇑a : (a : ℕ) → E),
              left_inv := ⋯, right_inv := ⋯ : C(ℤ_[p], E) ≃ₗ[ℤ_[p]] ℕ →C₀ E } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          f :
        ℕ →C₀ E) =
    ‖f‖
have simp : norm.{u_1} (E := BoundedContinuousFunction ℕ E)
    (ZeroAtInftyContinuousMap.toBCF
      (({
            toFun := fun (f : C(ℤ_[p], E)) ↦
              { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry :
                ℕ →C₀ E },
            map_add' := sorry, map_smul' := sorry : LinearMap (R := ℤ_[p]) (RingHom.id ℤ_[p]) C(ℤ_[p], E) (ℕ →C₀ E) } :
          (a : C(ℤ_[p], E)) → ℕ →C₀ E)
        f)) =
  ‖f‖
Tactic: simp only [LinearEquiv.coe_mk, ← ZeroAtInftyContinuousMap.norm_toBCF_eq_norm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ norm.{u_1} (E := BoundedContinuousFunction ℕ E)
      (ZeroAtInftyContinuousMap.toBCF
        (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯,
                  zero_at_infty' := ⋯ : ℕ →C₀ E },
              map_add' := ⋯, map_smul' := ⋯ : LinearMap (R := ℤ_[p]) (RingHom.id ℤ_[p]) C(ℤ_[p], E) (ℕ →C₀ E) } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          f)) =
    ‖f‖
have a₁ : norm.{u_1} (E := BoundedContinuousFunction ℕ E)
    (ZeroAtInftyContinuousMap.toBCF
      (({
            toFun := fun (f : C(ℤ_[p], E)) ↦
              { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry :
                ℕ →C₀ E },
            map_add' := sorry, map_smul' := sorry : LinearMap (R := ℤ_[p]) (RingHom.id ℤ_[p]) C(ℤ_[p], E) (ℕ →C₀ E) } :
          (a : C(ℤ_[p], E)) → ℕ →C₀ E)
        f)) ≤
  ‖f‖
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
a₁ :
  norm.{u_1} (E := BoundedContinuousFunction ℕ E)
      (ZeroAtInftyContinuousMap.toBCF
        (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯,
                  zero_at_infty' := ⋯ : ℕ →C₀ E },
              map_add' := ⋯, map_smul' := ⋯ : LinearMap (R := ℤ_[p]) (RingHom.id ℤ_[p]) C(ℤ_[p], E) (ℕ →C₀ E) } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          f)) ≤
    ‖f‖
⊢ norm.{u_1} (E := BoundedContinuousFunction ℕ E)
      (ZeroAtInftyContinuousMap.toBCF
        (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯,
                  zero_at_infty' := ⋯ : ℕ →C₀ E },
              map_add' := ⋯, map_smul' := ⋯ : LinearMap (R := ℤ_[p]) (RingHom.id ℤ_[p]) C(ℤ_[p], E) (ℕ →C₀ E) } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          f)) =
    ‖f‖
have a₂ : ‖f‖ ≤
  norm.{u_1} (E := BoundedContinuousFunction ℕ E)
    (ZeroAtInftyContinuousMap.toBCF
      (({
            toFun := fun (f : C(ℤ_[p], E)) ↦
              { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry },
            map_add' := sorry, map_smul' := sorry } :
          (a : C(ℤ_[p], E)) → ℕ →C₀ E)
        f))
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ ‖f‖ ≤
    norm.{u_1} (E := BoundedContinuousFunction ℕ E)
      (ZeroAtInftyContinuousMap.toBCF
        (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯,
                  zero_at_infty' := ⋯ },
              map_add' := ⋯, map_smul' := ⋯ } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          f))
have a₁ : norm.{u_1} (E := C(ℤ_[p], E)) (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f : (a : ℤ_[p]) → E) 0) b) ≤
  norm.{u_1} (E := BoundedContinuousFunction ℕ E)
    (ZeroAtInftyContinuousMap.toBCF
      (({
            toFun := fun (f : C(ℤ_[p], E)) ↦
              { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry, zero_at_infty' := sorry },
            map_add' := sorry, map_smul' := sorry } :
          (a : C(ℤ_[p], E)) → ℕ →C₀ E)
        (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f : (a : ℤ_[p]) → E) 0) b)))
Tactic: rw [← (hasSum_mahler f).tsum_eq]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
⊢ norm.{u_1} (E := C(ℤ_[p], E)) (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f : (a : ℤ_[p]) → E) 0) b) ≤
    norm.{u_1} (E := BoundedContinuousFunction ℕ E)
      (ZeroAtInftyContinuousMap.toBCF
        (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯,
                  zero_at_infty' := ⋯ },
              map_add' := ⋯, map_smul' := ⋯ } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f : (a : ℤ_[p]) → E) 0) b)))
have a₁ : ∀ (n : ℕ),
  norm (E := C(ℤ_[p], E)) (mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n : C(ℤ_[p], E)) ≤
    norm.{u_1} (E := BoundedContinuousFunction ℕ E)
      (ZeroAtInftyContinuousMap.toBCF
        (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry,
                  zero_at_infty' := sorry },
              map_add' := sorry, map_smul' := sorry } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f : (a : ℤ_[p]) → E) 0) b)))
Tactic: refine (norm_tsum_le _).trans (ciSup_le fun n ↦ ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.MahlerBasis (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : Module ℤ_[p] E
inst_2 : IsBoundedSMul ℤ_[p] E
a_1 : E
n_1 : ℕ
x : ℤ_[p]
inst_3 : IsUltrametricDist E
inst_4 : CompleteSpace E
a : ℕ → E
f : C(ℤ_[p], E)
n : ℕ
⊢ norm (E := C(ℤ_[p], E)) (mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n : C(ℤ_[p], E)) ≤
    norm.{u_1} (E := BoundedContinuousFunction ℕ E)
      (ZeroAtInftyContinuousMap.toBCF
        (({
              toFun := fun (f : C(ℤ_[p], E)) ↦
                { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := ⋯,
                  zero_at_infty' := ⋯ },
              map_add' := ⋯, map_smul' := ⋯ } :
            (a : C(ℤ_[p], E)) → ℕ →C₀ E)
          (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f : (a : ℤ_[p]) → E) 0) b)))
have a₁ : norm (E := C(ℤ_[p], E)) (mahlerTerm (Δ_[1]^[n] (⇑f : (a : ℤ_[p]) → E) 0) n : C(ℤ_[p], E)) =
  norm.{u_1} (E := E)
    ((ZeroAtInftyContinuousMap.toBCF
            (({
                  toFun := fun (f : C(ℤ_[p], E)) ↦
                    { toFun := fun (x : ℕ) ↦ Δ_[1]^[x] (⇑f : (a : ℤ_[p]) → E) 0, continuous_toFun := sorry,
                      zero_at_infty' := sorry },
                  map_add' := sorry, map_smul' := sorry } :
                (a : C(ℤ_[p], E)) → ℕ →C₀ E)
              (∑' (b : ℕ), mahlerTerm (Δ_[1]^[b] (⇑f : (a : ℤ_[p]) → E) 0) b)) :
          (a : ℕ) → E)
        n :
      E)
Tactic: refine le_trans (le_of_eq ?_) (BoundedContinuousFunction.norm_coe_le_norm _ n)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
⊢ (↑x : ℚ_[p]) = 0 ↔ x = 0
have rw : (↑x : ℚ_[p]) = (↑0 : ℚ_[p]) ↔ x = 0
Tactic: rw [← coe_zero, Subtype.coe_inj]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
rw : (↑x : ℚ_[p]) = (↑0 : ℚ_[p]) ↔ x = 0
⊢ (↑x : ℚ_[p]) = 0 ↔ x = 0
have rw₁ : x = 0 ↔ x = 0
Tactic: rw [← coe_zero, Subtype.coe_inj]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ IsOpenEmbedding (X := { x : ℚ_[p] // ‖x‖ ≤ 1 }) (Subtype.val (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1))
have refine : IsOpen (X := ℚ_[p]) {y : ℚ_[p] | ‖y‖ ≤ 1}
Tactic: refine (?_ : IsOpen {y : ℚ_[p] | ‖y‖ ≤ 1}).isOpenEmbedding_subtypeVal
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
m n : ℕ
h : Eq (α := ℤ_[p]) (↑m : ℤ_[p]) (↑n : ℤ_[p])
⊢ Eq (α := ℚ_[p]) (↑m : ℚ_[p]) (↑n : ℚ_[p])
have h₁ : Eq (α := ℚ_[p]) (↑(↑m : ℤ_[p]) : ℚ_[p]) (↑(↑n : ℤ_[p]) : ℚ_[p])
Tactic: rw [Subtype.ext_iff] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
seq : ℕ → ℤ
h : IsCauSeq (padicNorm p) fun (n : ℕ) ↦ (↑(seq n) : ℚ)
⊢ LE.le (α := ℝ)
    (↑(if hf : Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h ≈ 0 then 0
        else
          padicNorm p
            ((↑(Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h) : ℕ → ℚ)
              (PadicSeq.stationaryPoint hf))) :
      ℝ)
    1
have pos : ∀ (hne : Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h ≈ 0),
  LE.le (α := ℝ) (↑0 : ℝ) 1
Tactic: split_ifs with hne
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
seq : ℕ → ℤ
h : IsCauSeq (padicNorm p) fun (n : ℕ) ↦ (↑(seq n) : ℚ)
pos :
  ∀ (hne : Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h ≈ 0),
    LE.le (α := ℝ) (↑0 : ℝ) 1
⊢ LE.le (α := ℝ)
    (↑(if hf : Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h ≈ 0 then 0
        else
          padicNorm p
            ((↑(Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h) : ℕ → ℚ)
              (PadicSeq.stationaryPoint hf))) :
      ℝ)
    1
have neg : ∀ (hne : ¬Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h ≈ 0),
  LE.le (α := ℝ)
    (↑(padicNorm p
          ((↑(Subtype.mk (p := fun (f : ℕ → ℚ) ↦ IsCauSeq (padicNorm p) f) (fun (n : ℕ) ↦ (↑(seq n) : ℚ)) h) : ℕ → ℚ)
            (PadicSeq.stationaryPoint hne))) :
      ℝ)
    1
Tactic: split_ifs with hne