awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ (∀ (x : ℕ+),
      Summable (α := ℝ) fun (y : ↥(↑x : ℕ).divisorsAntidiagonal) ↦
        ‖HPow.hPow (α := 𝕜)
              (↑(↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                          (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                      ℕ × ℕ).2 :
                𝕜)
              k *
            r ^
              HMul.hMul (α := ℕ)
                (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                        (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                    ℕ × ℕ).1
                (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                        (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                    ℕ × ℕ).2‖) ∧
    Summable (α := ℝ) fun (x : ℕ+) ↦
      ∑' (y : ↥(↑x : ℕ).divisorsAntidiagonal),
        ‖HPow.hPow (α := 𝕜)
              (↑(↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                          (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                      ℕ × ℕ).2 :
                𝕜)
              k *
            r ^
              HMul.hMul (α := ℕ)
                (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                        (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                    ℕ × ℕ).1
                (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                        (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                    ℕ × ℕ).2‖
have hf : ∀ (x : ℕ+),
  Summable (α := ℝ) fun (y : ↥(↑x : ℕ).divisorsAntidiagonal) ↦
    ‖HPow.hPow (α := 𝕜)
          (↑(↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                      (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                  ℕ × ℕ).2 :
            𝕜)
          k *
        r ^
          HMul.hMul (α := ℕ)
            (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                    (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                ℕ × ℕ).1
            (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                    (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                ℕ × ℕ).2‖
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Summable (α := ℝ) fun (x : ℕ+) ↦
    ∑' (y : ↥(↑x : ℕ).divisorsAntidiagonal),
      ‖HPow.hPow (α := 𝕜)
            (↑(↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                        (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                    ℕ × ℕ).2 :
              𝕜)
            k *
          r ^
            HMul.hMul (α := ℕ)
              (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                      (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                  ℕ × ℕ).1
              (↑(Sigma.snd (β := fun (n : ℕ+) ↦ ↥(↑n : ℕ).divisorsAntidiagonal)
                      (Sigma.mk (β := fun (x : ℕ+) ↦ ↥(↑x : ℕ).divisorsAntidiagonal) x y)) :
                  ℕ × ℕ).2‖
have hf : Summable (α := ℝ) fun (x : ℕ+) ↦
  ∑ y ∈ (↑x : ℕ).divisorsAntidiagonal.attach,
    norm.{u_1} (E := 𝕜) (↑(↑y : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑y : ℕ × ℕ).1 * (↑y : ℕ × ℕ).2)
Tactic: simp only [norm_mul, norm_pow, tsum_fintype, Finset.univ_eq_attach]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Summable (α := ℝ) fun (x : ℕ+) ↦
    ∑ y ∈ (↑x : ℕ).divisorsAntidiagonal.attach,
      norm.{u_1} (E := 𝕜) (↑(↑y : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑y : ℕ × ℕ).1 * (↑y : ℕ × ℕ).2)
have apply : ∀ (b : ℕ+),
  LE.le (α := ℝ)
    (∑ i ∈ (↑b : ℕ).divisorsAntidiagonal.attach,
      norm.{u_1} (E := 𝕜) (↑(↑i : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑i : ℕ × ℕ).1 * (↑i : ℕ × ℕ).2))
    ((fun (c : ℕ+) ↦ ‖HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) (k + 1) * r ^ (↑c : ℕ)‖) b)
Tactic: apply Summable.of_nonneg_of_le (f := fun c : ℕ+ ↦ ‖(c : 𝕜) ^ (k + 1) * r ^ (c : ℕ)‖)
      (fun b ↦ Finset.sum_nonneg (fun _ _ ↦ mul_nonneg (by simp) (by simp))) (fun b ↦ ?_)
      (by apply (summable_norm_pow_mul_geometric_of_norm_lt_one (k + 1) hr).subtype)
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
⊢ LE.le (α := ℝ)
    (∑ i ∈ (↑b : ℕ).divisorsAntidiagonal.attach,
      norm.{u_1} (E := 𝕜) (↑(↑i : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑i : ℕ × ℕ).1 * (↑i : ℕ × ℕ).2))
    ((fun (c : ℕ+) ↦ ‖HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) (k + 1) * r ^ (↑c : ℕ)‖) b)
have transitivity : LE.le (α := ℝ)
  (∑ i ∈ (↑b : ℕ).divisorsAntidiagonal.attach,
    norm.{u_1} (E := 𝕜) (↑(↑i : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑i : ℕ × ℕ).1 * (↑i : ℕ × ℕ).2))
  (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
Tactic: transitivity ∑ _ ∈ (b : ℕ).divisors, ‖(b : 𝕜)‖ ^ k * ‖r ^ (b : ℕ)‖
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
transitivity :
  LE.le (α := ℝ)
    (∑ i ∈ (↑b : ℕ).divisorsAntidiagonal.attach,
      norm.{u_1} (E := 𝕜) (↑(↑i : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑i : ℕ × ℕ).1 * (↑i : ℕ × ℕ).2))
    (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
⊢ LE.le (α := ℝ)
    (∑ i ∈ (↑b : ℕ).divisorsAntidiagonal.attach,
      norm.{u_1} (E := 𝕜) (↑(↑i : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑i : ℕ × ℕ).1 * (↑i : ℕ × ℕ).2))
    ((fun (c : ℕ+) ↦ ‖HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) (k + 1) * r ^ (↑c : ℕ)‖) b)
have transitivity₁ : LE.le (α := ℝ) (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
  ((fun (c : ℕ+) ↦ ‖HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) (k + 1) * r ^ (↑c : ℕ)‖) b)
Tactic: transitivity ∑ _ ∈ (b : ℕ).divisors, ‖(b : 𝕜)‖ ^ k * ‖r ^ (b : ℕ)‖
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
⊢ LE.le (α := ℝ)
    (∑ i ∈ (↑b : ℕ).divisorsAntidiagonal.attach,
      norm.{u_1} (E := 𝕜) (↑(↑i : ℕ × ℕ).2 : 𝕜) ^ k * ‖r‖ ^ ((↑i : ℕ × ℕ).1 * (↑i : ℕ × ℕ).2))
    (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
have rw : LE.le (α := ℝ) (∑ i ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑((↑b : ℕ) / i) : 𝕜) ^ k * ‖r‖ ^ (i * ((↑b : ℕ) / i)))
  (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
Tactic: rw [(b : ℕ).divisorsAntidiagonal.sum_attach (fun x ↦ ‖(x.2 : 𝕜)‖ ^ _ * _ ^ (x.1 * x.2)),
          sum_divisorsAntidiagonal ((fun x y ↦ ‖(y : 𝕜)‖ ^ k * _ ^ (x * y)))]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
⊢ LE.le (α := ℝ) (∑ i ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑((↑b : ℕ) / i) : 𝕜) ^ k * ‖r‖ ^ (i * ((↑b : ℕ) / i)))
    (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
have h : ∀ (i : ℕ) (hi : i ∈ (↑b : ℕ).divisors), norm.{u_1} (E := 𝕜) (↑((↑b : ℕ) / i) : 𝕜) ≤ norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜)
Tactic: gcongr with i hi
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
h :
  ∀ (i : ℕ) (hi : i ∈ (↑b : ℕ).divisors),
    norm.{u_1} (E := 𝕜) (↑((↑b : ℕ) / i) : 𝕜) ≤ norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜)
⊢ LE.le (α := ℝ) (∑ i ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑((↑b : ℕ) / i) : 𝕜) ^ k * ‖r‖ ^ (i * ((↑b : ℕ) / i)))
    (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
have h₁ : ∀ (i : ℕ) (hi : i ∈ (↑b : ℕ).divisors), ‖r‖ ^ (i * ((↑b : ℕ) / i)) ≤ ‖r ^ (↑b : ℕ)‖
Tactic: gcongr with i hi
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
i : ℕ
hi : i ∈ (↑b : ℕ).divisors
⊢ ‖r‖ ^ (i * ((↑b : ℕ) / i)) ≤ ‖r ^ (↑b : ℕ)‖
have h : ‖r‖ ^ (i * ((↑b : ℕ) / i)) ≤ ‖r‖ ^ (↑b : ℕ)
Tactic: rw [norm_pow, mul_comm, Nat.div_mul_cancel (dvd_of_mem_divisors hi)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
i : ℕ
hi : i ∈ (↑b : ℕ).divisors
h : ‖r‖ ^ (i * ((↑b : ℕ) / i)) ≤ ‖r‖ ^ (↑b : ℕ)
⊢ ‖r‖ ^ (i * ((↑b : ℕ) / i)) ≤ ‖r ^ (↑b : ℕ)‖
have h₁ : ‖r‖ ^ ((↑b : ℕ) / i * i) ≤ ‖r‖ ^ (↑b : ℕ)
Tactic: rw [norm_pow, mul_comm, Nat.div_mul_cancel (dvd_of_mem_divisors hi)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
i : ℕ
hi : i ∈ (↑b : ℕ).divisors
h : ‖r‖ ^ (i * ((↑b : ℕ) / i)) ≤ ‖r‖ ^ (↑b : ℕ)
h₁ : ‖r‖ ^ ((↑b : ℕ) / i * i) ≤ ‖r‖ ^ (↑b : ℕ)
⊢ ‖r‖ ^ (i * ((↑b : ℕ) / i)) ≤ ‖r ^ (↑b : ℕ)‖
have h₂ : ‖r‖ ^ (↑b : ℕ) ≤ ‖r‖ ^ (↑b : ℕ)
Tactic: rw [norm_pow, mul_comm, Nat.div_mul_cancel (dvd_of_mem_divisors hi)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
⊢ LE.le (α := ℝ) (∑ x ∈ (↑b : ℕ).divisors, norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r ^ (↑b : ℕ)‖)
    ((fun (c : ℕ+) ↦ ‖HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) (k + 1) * r ^ (↑c : ℕ)‖) b)
have simp : (↑(↑b : ℕ).divisors.card : ℝ) * norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r‖ ^ (↑b : ℕ) ≤
  norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) * norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r‖ ^ (↑b : ℕ)
Tactic: simp only [norm_pow, Finset.sum_const, nsmul_eq_mul, ← mul_assoc, add_comm k 1, pow_add,
        pow_one, norm_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
b : ℕ+
⊢ (↑(↑b : ℕ).divisors.card : ℝ) * norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r‖ ^ (↑b : ℕ) ≤
    norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) * norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜) ^ k * ‖r‖ ^ (↑b : ℕ)
have hbc : (↑(↑b : ℕ).divisors.card : ℝ) ≤ norm.{u_1} (E := 𝕜) (↑(↑b : ℕ) : 𝕜)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑' (d : ℕ+) (c : ℕ+), HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) k * r ^ ((↑d : ℕ) * (↑c : ℕ)))
    (∑' (e : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑e : ℕ)) : 𝕜) * r ^ (↑e : ℕ))
have suffices : Eq.{u_1 + 1} (α := 𝕜) (∑' (c : ℕ+ × ℕ+), HPow.hPow (α := 𝕜) (↑(↑c.2 : ℕ) : 𝕜) k * r ^ ((↑c.1 : ℕ) * (↑c.2 : ℕ)))
  (∑' (e : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑e : ℕ)) : 𝕜) * r ^ (↑e : ℕ))
Tactic: suffices ∑' c : ℕ+ × ℕ+, c.2 ^ k * r ^ (c.1 * c.2 : ℕ) =
    ∑' e : ℕ+, σ k e * r ^ (e : ℕ) by rwa [← (summable_prod_mul_pow k hr).tsum_prod]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑' (c : ℕ+ × ℕ+), HPow.hPow (α := 𝕜) (↑(↑c.2 : ℕ) : 𝕜) k * r ^ ((↑c.1 : ℕ) * (↑c.2 : ℕ)))
    (∑' (e : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑e : ℕ)) : 𝕜) * r ^ (↑e : ℕ))
have simp : Eq.{u_1 + 1} (α := 𝕜)
  (∑' (b : ℕ+) (c : ↥(↑b : ℕ).divisorsAntidiagonal),
    HPow.hPow (α := 𝕜) (↑(↑c : ℕ × ℕ).2 : 𝕜) k * r ^ ((↑c : ℕ × ℕ).1 * (↑c : ℕ × ℕ).2))
  (∑' (e : ℕ+), (∑ x ∈ (↑e : ℕ).divisors, HPow.hPow (α := 𝕜) (↑((↑e : ℕ) / x) : 𝕜) k) * r ^ (↑e : ℕ))
Tactic: simp only [← sigmaAntidiagonalEquivProd.tsum_eq, sigmaAntidiagonalEquivProd,
    divisorsAntidiagonalFactors, PNat.mk_coe, Equiv.coe_fn_mk, sigma_eq_sum_div, cast_sum,
    cast_pow, Summable.tsum_sigma (summable_divisorsAntidiagonal_aux k hr)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
⊢ Eq.{u_1 + 1} (α := 𝕜)
    (∑' (b : ℕ+) (c : ↥(↑b : ℕ).divisorsAntidiagonal),
      HPow.hPow (α := 𝕜) (↑(↑c : ℕ × ℕ).2 : 𝕜) k * r ^ ((↑c : ℕ × ℕ).1 * (↑c : ℕ × ℕ).2))
    (∑' (e : ℕ+), (∑ x ∈ (↑e : ℕ).divisors, HPow.hPow (α := 𝕜) (↑((↑e : ℕ) / x) : 𝕜) k) * r ^ (↑e : ℕ))
have refine : ∀ (n : ℕ+),
  ∑' (c : ↥(↑n : ℕ).divisorsAntidiagonal),
      HPow.hPow (α := 𝕜) (↑(↑c : ℕ × ℕ).2 : 𝕜) k * r ^ ((↑c : ℕ × ℕ).1 * (↑c : ℕ × ℕ).2) =
    (∑ x ∈ (↑n : ℕ).divisors, HPow.hPow (α := 𝕜) (↑((↑n : ℕ) / x) : 𝕜) k) * r ^ (↑n : ℕ)
Tactic: refine tsum_congr fun n ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
k : ℕ
r : 𝕜
hr : ‖r‖ < 1
n : ℕ+
i : ℕ
hi : i ∈ (↑n : ℕ).divisors
⊢ HPow.hPow (α := 𝕜) (↑((↑n : ℕ) / i) : 𝕜) k * r ^ (i * ((↑n : ℕ) / i)) =
    HPow.hPow (α := 𝕜) (↑((↑n : ℕ) / i) : 𝕜) k * r ^ (↑n : ℕ)
have rw : HPow.hPow (α := 𝕜) (↑((↑n : ℕ) / i) : 𝕜) k * r ^ (↑n : ℕ) = HPow.hPow (α := 𝕜) (↑((↑n : ℕ) / i) : 𝕜) k * r ^ (↑n : ℕ)
Tactic: rw [Nat.mul_div_cancel' (dvd_of_mem_divisors hi)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+), HPow.hPow (α := 𝕜) (↑(↑n : ℕ) : 𝕜) k * r ^ (↑n : ℕ) / (1 - r ^ (↑n : ℕ)))
    (∑' (n : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑n : ℕ)) : 𝕜) * r ^ (↑n : ℕ))
have this : ∀ (m : ℕ) [NeZero m], ∑' (n : ℕ), (r ^ m) ^ n = (1 - r ^ m)⁻¹
Tactic: have (m : ℕ) [NeZero m] := tsum_geometric_of_norm_lt_one (ξ := r ^ m)
    (by simpa using pow_lt_one₀ (by simp) hr (NeZero.ne _))
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
this : ∀ (m : ℕ) [NeZero m], ∑' (n : ℕ), (r ^ m) ^ n = (1 - r ^ m)⁻¹
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+), HPow.hPow (α := 𝕜) (↑(↑n : ℕ) : 𝕜) k * r ^ (↑n : ℕ) / (1 - r ^ (↑n : ℕ)))
    (∑' (n : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑n : ℕ)) : 𝕜) * r ^ (↑n : ℕ))
have simp : Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+) (n_1 : ℕ+), HPow.hPow (α := 𝕜) (↑(↑n : ℕ) : 𝕜) k * (r ^ (↑n : ℕ)) ^ (↑n_1 : ℕ))
  (∑' (n : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑n : ℕ)) : 𝕜) * r ^ (↑n : ℕ))
Tactic: simp only [div_eq_mul_inv, ← this, ← tsum_mul_left, mul_assoc, ← _root_.pow_succ',
    ← fun (n : ℕ) ↦ tsum_pnat_eq_tsum_succ (f := fun m ↦ n ^ k * (r ^ n) ^ m)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
this : ∀ (m : ℕ) [NeZero m], ∑' (n : ℕ), (r ^ m) ^ n = (1 - r ^ m)⁻¹
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+) (n_1 : ℕ+), HPow.hPow (α := 𝕜) (↑(↑n : ℕ) : 𝕜) k * (r ^ (↑n : ℕ)) ^ (↑n_1 : ℕ))
    (∑' (n : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑n : ℕ)) : 𝕜) * r ^ (↑n : ℕ))
have h00 : Eq.{u_1 + 1} (α := 𝕜) (∑' (d : ℕ+) (c : ℕ+), HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) k * r ^ ((↑d : ℕ) * (↑c : ℕ)))
  (∑' (e : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑e : ℕ)) : 𝕜) * r ^ (↑e : ℕ))
Tactic: have h00 := tsum_prod_pow_eq_tsum_sigma k hr
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
this : ∀ (m : ℕ) [NeZero m], ∑' (n : ℕ), (r ^ m) ^ n = (1 - r ^ m)⁻¹
h00 :
  Eq.{u_1 + 1} (α := 𝕜) (∑' (d : ℕ+) (c : ℕ+), HPow.hPow (α := 𝕜) (↑(↑c : ℕ) : 𝕜) k * r ^ ((↑d : ℕ) * (↑c : ℕ)))
    (∑' (e : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑e : ℕ)) : 𝕜) * r ^ (↑e : ℕ))
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+) (n_1 : ℕ+), HPow.hPow (α := 𝕜) (↑(↑n : ℕ) : 𝕜) k * (r ^ (↑n : ℕ)) ^ (↑n_1 : ℕ))
    (∑' (n : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑n : ℕ)) : 𝕜) * r ^ (↑n : ℕ))
have h00₁ : Eq.{u_1 + 1} (α := 𝕜) (∑' (b : ℕ+) (c : ℕ+), HPow.hPow (α := 𝕜) (↑(↑b : ℕ) : 𝕜) k * r ^ ((↑c : ℕ) * (↑b : ℕ)))
  (∑' (e : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑e : ℕ)) : 𝕜) * r ^ (↑e : ℕ))
Tactic: rw [Summable.tsum_comm (by apply (summable_prod_mul_pow k hr).prod_symm)] at h00
Cleared hypotheses: h00
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
inst_1 : CompleteSpace 𝕜
inst_2 : NormSMulClass ℤ 𝕜
r : 𝕜
hr : ‖r‖ < 1
k : ℕ
this : ∀ (m : ℕ) [NeZero m], ∑' (n : ℕ), (r ^ m) ^ n = (1 - r ^ m)⁻¹
h00 :
  Eq.{u_1 + 1} (α := 𝕜) (∑' (b : ℕ+) (c : ℕ+), HPow.hPow (α := 𝕜) (↑(↑b : ℕ) : 𝕜) k * r ^ ((↑c : ℕ) * (↑b : ℕ)))
    (∑' (e : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑e : ℕ)) : 𝕜) * r ^ (↑e : ℕ))
⊢ Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+) (n_1 : ℕ+), HPow.hPow (α := 𝕜) (↑(↑n : ℕ) : 𝕜) k * (r ^ (↑n : ℕ)) ^ (↑n_1 : ℕ))
    (∑' (n : ℕ+), (↑((σ k : (a : ℕ) → ℕ) (↑n : ℕ)) : 𝕜) * r ^ (↑n : ℕ))
have rw : Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+) (n_1 : ℕ+), HPow.hPow (α := 𝕜) (↑(↑n : ℕ) : 𝕜) k * (r ^ (↑n : ℕ)) ^ (↑n_1 : ℕ))
  (∑' (b : ℕ+) (c : ℕ+), HPow.hPow (α := 𝕜) (↑(↑b : ℕ) : 𝕜) k * r ^ ((↑c : ℕ) * (↑b : ℕ)))
Tactic: rw [← h00]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
m : ℕ+
⊢ ‖r ^ (↑m : ℕ)‖ < 1
have rwa : ‖r‖ ^ (↑m : ℕ) < 1
Tactic: rwa [norm_pow, pow_lt_one_iff_of_nonneg (norm_nonneg _) (NeZero.ne _)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
⊢ Tendsto.{0, u_1} (β := 𝕜) (fun (m : ℕ+) ↦ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ))) atTop (𝓝 0)
have h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
Tactic: have h1 (m : ℕ+) : ‖r ^ (m : ℕ)‖ < 1 := by
    rwa [norm_pow, pow_lt_one_iff_of_nonneg (norm_nonneg _) (NeZero.ne _)]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
m : ℕ+
⊢ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
have this : ∑' (n : ℕ), (r ^ (↑m : ℕ)) ^ n = (1 - r ^ (↑m : ℕ))⁻¹
Tactic: have := tsum_geometric_of_norm_lt_one (h1 m)
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
m : ℕ+
this : ∑' (n : ℕ), (r ^ (↑m : ℕ)) ^ n = (1 - r ^ (↑m : ℕ))⁻¹
⊢ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
have this₁ : HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) = (1 - r ^ (↑m : ℕ))⁻¹
Tactic: rw [← tsum_zero_pnat_eq_tsum_nat (summable_geometric_of_norm_lt_one (h1 m))] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
m : ℕ+
this : HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) = (1 - r ^ (↑m : ℕ))⁻¹
⊢ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
have simp_rw : ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) =
  HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) - 1
Tactic: simp_rw [← this, pow_zero, add_sub_cancel_left, mul_comm, pow_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
m : ℕ+
this : HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) = (1 - r ^ (↑m : ℕ))⁻¹
simp_rw :
  ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) =
    HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) - 1
⊢ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
have simp_rw₁ : ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = HAdd.hAdd (α := 𝕜) 1 (∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ)) - 1
Tactic: simp_rw [← this, pow_zero, add_sub_cancel_left, mul_comm, pow_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
m : ℕ+
this : HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) = (1 - r ^ (↑m : ℕ))⁻¹
simp_rw :
  ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) =
    HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) - 1
simp_rw₁ : ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = HAdd.hAdd (α := 𝕜) 1 (∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ)) - 1
⊢ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
have simp_rw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ))) (∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ))
Tactic: simp_rw [← this, pow_zero, add_sub_cancel_left, mul_comm, pow_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
m : ℕ+
this : HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) = (1 - r ^ (↑m : ℕ))⁻¹
simp_rw :
  ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) =
    HPow.hPow (β := ℕ) (r ^ (↑m : ℕ)) 0 + ∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ) - 1
simp_rw₁ : ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = HAdd.hAdd (α := 𝕜) 1 (∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ)) - 1
simp_rw₂ : Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ))) (∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ))
⊢ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
have simp_rw₃ : Eq.{u_1 + 1} (α := 𝕜) (∑' (n : ℕ+), r ^ ((↑m : ℕ) * (↑n : ℕ))) (∑' (n : ℕ+), (r ^ (↑m : ℕ)) ^ (↑n : ℕ))
Tactic: simp_rw [← this, pow_zero, add_sub_cancel_left, mul_comm, pow_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (hypothesis)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
⊢ Tendsto.{0, u_1} (β := 𝕜) (fun (m : ℕ+) ↦ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ))) atTop (𝓝 0)
have h2 : ∀ (m : ℕ+), ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
Tactic: have h2 (m : ℕ+) : ∑' n : ℕ+, r ^ (n * m : ℕ) = (1 - r ^ (m : ℕ))⁻¹ - 1 := by
    have := tsum_geometric_of_norm_lt_one (h1 m)
    rw [← tsum_zero_pnat_eq_tsum_nat (summable_geometric_of_norm_lt_one (h1 m))] at this
    simp_rw [← this, pow_zero, add_sub_cancel_left, mul_comm, pow_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.TsumDivisorsAntidiagonal (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
r : 𝕜
hr : ‖r‖ < 1
h1 : ∀ (m : ℕ+), ‖r ^ (↑m : ℕ)‖ < 1
h2 : ∀ (m : ℕ+), ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ)) = (1 - r ^ (↑m : ℕ))⁻¹ - 1
⊢ Tendsto.{0, u_1} (β := 𝕜) (fun (m : ℕ+) ↦ ∑' (n : ℕ+), r ^ ((↑n : ℕ) * (↑m : ℕ))) atTop (𝓝 0)
have rw : Tendsto.{0, u_1} (β := 𝕜) (fun (x : ℕ+) ↦ r ^ (↑x : ℕ)) atTop (𝓝 0)
Tactic: rw [funext h2, (by simp : 𝓝 (0 : 𝕜) = 𝓝 ((1 - 0)⁻¹ - 1)), tendsto_sub_const_iff,
    tendsto_inv_iff₀ (by simp), tendsto_const_sub_iff]