ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mellin_convergent_of_isBigO_scalar
theorem mellin_convergent_of_isBigO_scalar {a b : ℝ} {f : ℝ → ℝ} {s : ℝ}
    (hfc : LocallyIntegrableOn f <| Ioi 0) (hf_top : f =O[atTop] (· ^ (-a)))
    (hs_top : s < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s) :
    IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioi 0) 
Arguments: {a b : ℝ} {f : ℝ → ℝ} {s : ℝ} (hfc : MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume) (hf_top : f =O[Filter.atTop] fun x => x ^ (-a)) (hs_top : s < a) (hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)) (hs_bot : b < s)
Goal: MeasureTheory.IntegrableOn (fun t => t ^ (s - 1) * f t) (Set.Ioi 0) MeasureTheory.volume
-- mellinConvergent_of_isBigO_rpow
theorem mellinConvergent_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ} {f : ℝ → E} {s : ℂ}
    (hfc : LocallyIntegrableOn f <| Ioi 0) (hf_top : f =O[atTop] (· ^ (-a)))
    (hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
    MellinConvergent f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f : ℝ → E} {s : ℂ} (hfc : MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume) (hf_top : f =O[Filter.atTop] fun x => x ^ (-a)) (hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)) (hs_bot : b < s.re)
Goal: MellinConvergent f s
-- isBigO_rpow_top_log_smul
theorem isBigO_rpow_top_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : b < a)
    (hf : f =O[atTop] (· ^ (-a))) :
    (fun t : ℝ => log t • f t) =O[atTop] (· ^ (-b)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : b < a) (hf : f =O[Filter.atTop] fun x => x ^ (-a))
Goal: (fun t => Real.log t • f t) =O[Filter.atTop] fun x => x ^ (-b)
-- isBigO_rpow_zero_log_smul
theorem isBigO_rpow_zero_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : a < b)
    (hf : f =O[𝓝[>] 0] (· ^ (-a))) :
    (fun t : ℝ => log t • f t) =O[𝓝[>] 0] (· ^ (-b)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : a < b) (hf : f =O[𝓝[>] 0] fun x => x ^ (-a))
Goal: (fun t => Real.log t • f t) =O[𝓝[>] 0] fun x => x ^ (-b)
-- mellin_hasDerivAt_of_isBigO_rpow
theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
    {f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a)))
    (hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
    MellinConvergent (fun t => log t • f t) s ∧
      HasDerivAt (mellin f) (mellin (fun t => log t • f t) s) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f : ℝ → E} {s : ℂ} (hfc : MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume) (hf_top : f =O[Filter.atTop] fun x => x ^ (-a)) (hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)) (hs_bot : b < s.re)
Goal: MellinConvergent (fun t => Real.log t • f t) s ∧ HasDerivAt (mellin f) (mellin (fun t => Real.log t • f t) s) s
-- mellin_differentiableAt_of_isBigO_rpow
theorem mellin_differentiableAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
    {f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f <| Ioi 0)
    (hf_top : f =O[atTop] (· ^ (-a))) (hs_top : s.re < a)
    (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
    DifferentiableAt ℂ (mellin f) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f : ℝ → E} {s : ℂ} (hfc : MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume) (hf_top : f =O[Filter.atTop] fun x => x ^ (-a)) (hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)) (hs_bot : b < s.re)
Goal: DifferentiableAt ℂ (mellin f) s
-- mellinConvergent_of_isBigO_rpow_exp
theorem mellinConvergent_of_isBigO_rpow_exp [NormedSpace ℂ E] {a b : ℝ} (ha : 0 < a) {f : ℝ → E}
    {s : ℂ} (hfc : LocallyIntegrableOn f <| Ioi 0) (hf_top : f =O[atTop] fun t => exp (-a * t))
    (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) : MellinConvergent f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} (ha : 0 < a) {f : ℝ → E} {s : ℂ} (hfc : MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume) (hf_top : f =O[Filter.atTop] fun t => rexp (-a * t)) (hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)) (hs_bot : b < s.re)
Goal: MellinConvergent f s
-- mellin_differentiableAt_of_isBigO_rpow_exp
theorem mellin_differentiableAt_of_isBigO_rpow_exp [NormedSpace ℂ E] {a b : ℝ}
    (ha : 0 < a) {f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f <| Ioi 0)
    (hf_top : f =O[atTop] fun t => exp (-a * t)) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b)))
    (hs_bot : b < s.re) : DifferentiableAt ℂ (mellin f) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} (ha : 0 < a) {f : ℝ → E} {s : ℂ} (hfc : MeasureTheory.LocallyIntegrableOn f (Set.Ioi 0) MeasureTheory.volume) (hf_top : f =O[Filter.atTop] fun t => rexp (-a * t)) (hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)) (hs_bot : b < s.re)
Goal: DifferentiableAt ℂ (mellin f) s
-- hasMellin_one_Ioc
theorem hasMellin_one_Ioc {s : ℂ} (hs : 0 < re s) :
    HasMellin (indicator (Ioc 0 1) (fun _ => 1 : ℝ → ℂ)) s (1 / s) 
Arguments: {s : ℂ} (hs : 0 < s.re)
Goal: HasMellin ((Set.Ioc 0 1).indicator fun x => 1) s (1 / s)
-- hasMellin_cpow_Ioc
theorem hasMellin_cpow_Ioc (a : ℂ) {s : ℂ} (hs : 0 < re s + re a) :
    HasMellin (indicator (Ioc 0 1) (fun t => ↑t ^ a : ℝ → ℂ)) s (1 / (s + a)) 
Arguments: (a : ℂ) {s : ℂ} (hs : 0 < s.re + a.re)
Goal: HasMellin ((Set.Ioc 0 1).indicator fun t => ↑t ^ a) s (1 / (s + a))
-- tendsto_div_of_monotone_of_exists_subseq_tendsto_div
theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
    (hmono : Monotone u)
    (hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
      Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) 
Arguments: (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u) (hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in Filter.atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Filter.Tendsto c Filter.atTop Filter.atTop ∧ Filter.Tendsto (fun n => u (c n) / ↑(c n)) Filter.atTop (𝓝 l))
Goal: Filter.Tendsto (fun n => u n / ↑n) Filter.atTop (𝓝 l)
-- tendsto_div_of_monotone_of_tendsto_div_floor_pow
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) 
Arguments: (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u) (c : ℕ → ℝ) (cone : ∀ (k : ℕ), 1 < c k) (clim : Filter.Tendsto c Filter.atTop (𝓝 1)) (hc : ∀ (k : ℕ), Filter.Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) Filter.atTop (𝓝 l))
Goal: Filter.Tendsto (fun n => u n / ↑n) Filter.atTop (𝓝 l)
-- sum_div_pow_sq_le_div_sq
theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
    (∑ i ∈ (range N).filter (j < c ^ ·), (1 : ℝ) / (c ^ i) ^ 2) ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 
Arguments: (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c)
Goal: ∑ i ∈ Finset.filter (fun x => j < c ^ x) (Finset.range N), 1 / (c ^ i) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2
-- mul_pow_le_nat_floor_pow
theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹) * c ^ i ≤ ⌊c ^ i⌋₊ 
Arguments: {c : ℝ} (hc : 1 < c) (i : ℕ)
Goal: (1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊
-- sum_div_nat_floor_pow_sq_le_div_sq
theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
    (∑ i ∈ (range N).filter (j < ⌊c ^ ·⌋₊), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
      c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 
Arguments: (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c)
Goal: ∑ i ∈ Finset.filter (fun x => j < ↑⌊c ^ x⌋₊) (Finset.range N), 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2
-- tendsto_norm_atTop_atTop
theorem tendsto_norm_atTop_atTop : Tendsto (norm : ℝ → ℝ) atTop atTop 
Goal: Filter.Tendsto norm Filter.atTop Filter.atTop
-- summable_of_absolute_convergence_real
theorem summable_of_absolute_convergence_real {f : ℕ → ℝ} :
    (∃ r, Tendsto (fun n ↦ ∑ i ∈ range n, |f i|) atTop (𝓝 r)) → Summable f
  
Arguments: {f : ℕ → ℝ}
Goal: (∃ r, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, |f i|) Filter.atTop (𝓝 r)) → Summable f
-- tendsto_norm_zero'
theorem tendsto_norm_zero' {𝕜 : Type*} [NormedAddCommGroup 𝕜] :
    Tendsto (norm : 𝕜 → ℝ) (𝓝[≠] 0) (𝓝[>] 0) 
Arguments: {𝕜 : Type u_4} [NormedAddCommGroup 𝕜]
Goal: Filter.Tendsto norm (𝓝[≠] 0) (𝓝[>] 0)
-- NormedField.tendsto_norm_inverse_nhdsWithin_0_atTop
theorem tendsto_norm_inverse_nhdsWithin_0_atTop {𝕜 : Type*} [NormedDivisionRing 𝕜] :
    Tendsto (fun x : 𝕜 ↦ ‖x⁻¹‖) (𝓝[≠] 0) atTop 
Arguments: {𝕜 : Type u_4} [NormedDivisionRing 𝕜]
Goal: Filter.Tendsto (fun x => ‖x⁻¹‖) (𝓝[≠] 0) Filter.atTop
-- NormedField.tendsto_norm_zpow_nhdsWithin_0_atTop
theorem tendsto_norm_zpow_nhdsWithin_0_atTop {𝕜 : Type*} [NormedDivisionRing 𝕜] {m : ℤ}
    (hm : m < 0) :
    Tendsto (fun x : 𝕜 ↦ ‖x ^ m‖) (𝓝[≠] 0) atTop 
Arguments: {𝕜 : Type u_4} [NormedDivisionRing 𝕜] {m : ℤ} (hm : m < 0)
Goal: Filter.Tendsto (fun x => ‖x ^ m‖) (𝓝[≠] 0) Filter.atTop
-- NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded
theorem tendsto_zero_smul_of_tendsto_zero_of_bounded {ι 𝕜 𝔸 : Type*} [NormedDivisionRing 𝕜]
    [NormedAddCommGroup 𝔸] [Module 𝕜 𝔸] [BoundedSMul 𝕜 𝔸] {l : Filter ι} {ε : ι → 𝕜} {f : ι → 𝔸}
    (hε : Tendsto ε l (𝓝 0)) (hf : Filter.IsBoundedUnder (· ≤ ·) l (norm ∘ f)) :
    Tendsto (ε • f) l (𝓝 0) 
Arguments: {ι : Type u_4} {𝕜 : Type u_5} {𝔸 : Type u_6} [NormedDivisionRing 𝕜] [NormedAddCommGroup 𝔸] [Module 𝕜 𝔸] [BoundedSMul 𝕜 𝔸] {l : Filter ι} {ε : ι → 𝕜} {f : ι → 𝔸} (hε : Filter.Tendsto ε l (𝓝 0)) (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f))
Goal: Filter.Tendsto (ε • f) l (𝓝 0)
-- NormedField.continuousAt_zpow
@[simp]
theorem continuousAt_zpow {𝕜 : Type*} [NontriviallyNormedField 𝕜] {m : ℤ} {x : 𝕜} :
    ContinuousAt (fun x ↦ x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m 
Arguments: {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {m : ℤ} {x : 𝕜}
Goal: ContinuousAt (fun x => x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m
-- NormedField.continuousAt_inv
@[simp]
theorem continuousAt_inv {𝕜 : Type*} [NontriviallyNormedField 𝕜] {x : 𝕜} :
    ContinuousAt Inv.inv x ↔ x ≠ 0 
Arguments: {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {x : 𝕜}
Goal: ContinuousAt Inv.inv x ↔ x ≠ 0
-- isLittleO_pow_pow_of_lt_left
theorem isLittleO_pow_pow_of_lt_left {r₁ r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ < r₂) :
    (fun n : ℕ ↦ r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n 
Arguments: {r₁ r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ < r₂)
Goal: (fun n => r₁ ^ n) =o[Filter.atTop] fun n => r₂ ^ n
-- isBigO_pow_pow_of_le_left
theorem isBigO_pow_pow_of_le_left {r₁ r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ ≤ r₂) :
    (fun n : ℕ ↦ r₁ ^ n) =O[atTop] fun n ↦ r₂ ^ n 
Arguments: {r₁ r₂ : ℝ} (h₁ : 0 ≤ r₁) (h₂ : r₁ ≤ r₂)
Goal: (fun n => r₁ ^ n) =O[Filter.atTop] fun n => r₂ ^ n
-- isLittleO_pow_pow_of_abs_lt_left
theorem isLittleO_pow_pow_of_abs_lt_left {r₁ r₂ : ℝ} (h : |r₁| < |r₂|) :
    (fun n : ℕ ↦ r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n 
Arguments: {r₁ r₂ : ℝ} (h : |r₁| < |r₂|)
Goal: (fun n => r₁ ^ n) =o[Filter.atTop] fun n => r₂ ^ n
-- TFAE_exists_lt_isLittleO_pow
theorem TFAE_exists_lt_isLittleO_pow (f : ℕ → ℝ) (R : ℝ) :
    TFAE
      [∃ a ∈ Ioo (-R) R, f =o[atTop] (a ^ ·), ∃ a ∈ Ioo 0 R, f =o[atTop] (a ^ ·),
        ∃ a ∈ Ioo (-R) R, f =O[atTop] (a ^ ·), ∃ a ∈ Ioo 0 R, f =O[atTop] (a ^ ·),
        ∃ a < R, ∃ C : ℝ, (0 < C ∨ 0 < R) ∧ ∀ n, |f n| ≤ C * a ^ n,
        ∃ a ∈ Ioo 0 R, ∃ C > 0, ∀ n, |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ n in atTop, |f n| ≤ a ^ n,
        ∃ a ∈ Ioo 0 R, ∀ᶠ n in atTop, |f n| ≤ a ^ n] 
Arguments: (f : ℕ → ℝ) (R : ℝ)
Goal: [∃ a ∈ Set.Ioo (-R) R, f =o[Filter.atTop] fun x => a ^ x, ∃ a ∈ Set.Ioo 0 R, f =o[Filter.atTop] fun x => a ^ x,
    ∃ a ∈ Set.Ioo (-R) R, f =O[Filter.atTop] fun x => a ^ x, ∃ a ∈ Set.Ioo 0 R, f =O[Filter.atTop] fun x => a ^ x,
    ∃ a < R, ∃ C, (0 < C ∨ 0 < R) ∧ ∀ (n : ℕ), |f n| ≤ C * a ^ n,
    ∃ a ∈ Set.Ioo 0 R, ∃ C > 0, ∀ (n : ℕ), |f n| ≤ C * a ^ n, ∃ a < R, ∀ᶠ (n : ℕ) in Filter.atTop, |f n| ≤ a ^ n,
    ∃ a ∈ Set.Ioo 0 R, ∀ᶠ (n : ℕ) in Filter.atTop, |f n| ≤ a ^ n].TFAE
-- isLittleO_pow_const_const_pow_of_one_lt
theorem isLittleO_pow_const_const_pow_of_one_lt {R : Type*} [NormedRing R] (k : ℕ) {r : ℝ}
    (hr : 1 < r) : (fun n ↦ (n : R) ^ k : ℕ → R) =o[atTop] fun n ↦ r ^ n 
Arguments: {R : Type u_4} [NormedRing R] (k : ℕ) {r : ℝ} (hr : 1 < r)
Goal: (fun n => ↑n ^ k) =o[Filter.atTop] fun n => r ^ n
-- isLittleO_coe_const_pow_of_one_lt
theorem isLittleO_coe_const_pow_of_one_lt {R : Type*} [NormedRing R] {r : ℝ} (hr : 1 < r) :
    ((↑) : ℕ → R) =o[atTop] fun n ↦ r ^ n 
Arguments: {R : Type u_4} [NormedRing R] {r : ℝ} (hr : 1 < r)
Goal: Nat.cast =o[Filter.atTop] fun n => r ^ n
-- isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt
theorem isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt {R : Type*} [NormedRing R] (k : ℕ)
    {r₁ : R} {r₂ : ℝ} (h : ‖r₁‖ < r₂) :
    (fun n ↦ (n : R) ^ k * r₁ ^ n : ℕ → R) =o[atTop] fun n ↦ r₂ ^ n 
Arguments: {R : Type u_4} [NormedRing R] (k : ℕ) {r₁ : R} {r₂ : ℝ} (h : ‖r₁‖ < r₂)
Goal: (fun n => ↑n ^ k * r₁ ^ n) =o[Filter.atTop] fun n => r₂ ^ n
-- tendsto_pow_const_div_const_pow_of_one_lt
theorem tendsto_pow_const_div_const_pow_of_one_lt (k : ℕ) {r : ℝ} (hr : 1 < r) :
    Tendsto (fun n ↦ (n : ℝ) ^ k / r ^ n : ℕ → ℝ) atTop (𝓝 0) 
Arguments: (k : ℕ) {r : ℝ} (hr : 1 < r)
Goal: Filter.Tendsto (fun n => ↑n ^ k / r ^ n) Filter.atTop (𝓝 0)
-- tendsto_pow_const_mul_const_pow_of_abs_lt_one
theorem tendsto_pow_const_mul_const_pow_of_abs_lt_one (k : ℕ) {r : ℝ} (hr : |r| < 1) :
    Tendsto (fun n ↦ (n : ℝ) ^ k * r ^ n : ℕ → ℝ) atTop (𝓝 0) 
Arguments: (k : ℕ) {r : ℝ} (hr : |r| < 1)
Goal: Filter.Tendsto (fun n => ↑n ^ k * r ^ n) Filter.atTop (𝓝 0)
-- tendsto_pow_const_mul_const_pow_of_lt_one
theorem tendsto_pow_const_mul_const_pow_of_lt_one (k : ℕ) {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1) :
    Tendsto (fun n ↦ (n : ℝ) ^ k * r ^ n : ℕ → ℝ) atTop (𝓝 0) 
Arguments: (k : ℕ) {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1)
Goal: Filter.Tendsto (fun n => ↑n ^ k * r ^ n) Filter.atTop (𝓝 0)
-- tendsto_self_mul_const_pow_of_abs_lt_one
theorem tendsto_self_mul_const_pow_of_abs_lt_one {r : ℝ} (hr : |r| < 1) :
    Tendsto (fun n ↦ n * r ^ n : ℕ → ℝ) atTop (𝓝 0) 
Arguments: {r : ℝ} (hr : |r| < 1)
Goal: Filter.Tendsto (fun n => ↑n * r ^ n) Filter.atTop (𝓝 0)
-- tendsto_self_mul_const_pow_of_lt_one
theorem tendsto_self_mul_const_pow_of_lt_one {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1) :
    Tendsto (fun n ↦ n * r ^ n : ℕ → ℝ) atTop (𝓝 0) 
Arguments: {r : ℝ} (hr : 0 ≤ r) (h'r : r < 1)
Goal: Filter.Tendsto (fun n => ↑n * r ^ n) Filter.atTop (𝓝 0)
-- tendsto_pow_atTop_nhds_zero_of_norm_lt_one
theorem tendsto_pow_atTop_nhds_zero_of_norm_lt_one {R : Type*} [NormedRing R] {x : R}
    (h : ‖x‖ < 1) :
    Tendsto (fun n : ℕ ↦ x ^ n) atTop (𝓝 0) 
Arguments: {R : Type u_4} [NormedRing R] {x : R} (h : ‖x‖ < 1)
Goal: Filter.Tendsto (fun n => x ^ n) Filter.atTop (𝓝 0)
-- tendsto_pow_atTop_nhds_zero_of_abs_lt_one
theorem tendsto_pow_atTop_nhds_zero_of_abs_lt_one {r : ℝ} (h : |r| < 1) :
    Tendsto (fun n : ℕ ↦ r ^ n) atTop (𝓝 0) 
Arguments: {r : ℝ} (h : |r| < 1)
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝 0)
-- hasSum_geometric_of_norm_lt_one
theorem hasSum_geometric_of_norm_lt_one (h : ‖ξ‖ < 1) : HasSum (fun n : ℕ ↦ ξ ^ n) (1 - ξ)⁻¹ 
Arguments: {K : Type u_4} [NormedDivisionRing K] {ξ : K} (h : ‖ξ‖ < 1)
Goal: HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹
-- summable_geometric_of_norm_lt_one
theorem summable_geometric_of_norm_lt_one (h : ‖ξ‖ < 1) : Summable fun n : ℕ ↦ ξ ^ n 
Arguments: {K : Type u_4} [NormedDivisionRing K] {ξ : K} (h : ‖ξ‖ < 1)
Goal: Summable fun n => ξ ^ n
-- tsum_geometric_of_norm_lt_one
theorem tsum_geometric_of_norm_lt_one (h : ‖ξ‖ < 1) : ∑' n : ℕ, ξ ^ n = (1 - ξ)⁻¹ 
Arguments: {K : Type u_4} [NormedDivisionRing K] {ξ : K} (h : ‖ξ‖ < 1)
Goal: ∑' (n : ℕ), ξ ^ n = (1 - ξ)⁻¹
-- hasSum_geometric_of_abs_lt_one
theorem hasSum_geometric_of_abs_lt_one {r : ℝ} (h : |r| < 1) :
    HasSum (fun n : ℕ ↦ r ^ n) (1 - r)⁻¹ 
Arguments: {r : ℝ} (h : |r| < 1)
Goal: HasSum (fun n => r ^ n) (1 - r)⁻¹
-- summable_geometric_of_abs_lt_one
theorem summable_geometric_of_abs_lt_one {r : ℝ} (h : |r| < 1) : Summable fun n : ℕ ↦ r ^ n 
Arguments: {r : ℝ} (h : |r| < 1)
Goal: Summable fun n => r ^ n
-- tsum_geometric_of_abs_lt_one
theorem tsum_geometric_of_abs_lt_one {r : ℝ} (h : |r| < 1) : ∑' n : ℕ, r ^ n = (1 - r)⁻¹ 
Arguments: {r : ℝ} (h : |r| < 1)
Goal: ∑' (n : ℕ), r ^ n = (1 - r)⁻¹
-- summable_geometric_iff_norm_lt_one
@[simp]
theorem summable_geometric_iff_norm_lt_one : (Summable fun n : ℕ ↦ ξ ^ n) ↔ ‖ξ‖ < 1 
Arguments: {K : Type u_4} [NormedDivisionRing K] {ξ : K}
Goal: (Summable fun n => ξ ^ n) ↔ ‖ξ‖ < 1
-- summable_norm_pow_mul_geometric_of_norm_lt_one
theorem summable_norm_pow_mul_geometric_of_norm_lt_one {R : Type*} [NormedRing R] (k : ℕ) {r : R}
    (hr : ‖r‖ < 1) : Summable fun n : ℕ ↦ ‖((n : R) ^ k * r ^ n : R)‖ 
Arguments: {R : Type u_4} [NormedRing R] (k : ℕ) {r : R} (hr : ‖r‖ < 1)
Goal: Summable fun n => ‖↑n ^ k * r ^ n‖
-- summable_pow_mul_geometric_of_norm_lt_one
theorem summable_pow_mul_geometric_of_norm_lt_one {R : Type*} [NormedRing R] [CompleteSpace R]
    (k : ℕ) {r : R} (hr : ‖r‖ < 1) : Summable (fun n ↦ (n : R) ^ k * r ^ n : ℕ → R) 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (k : ℕ) {r : R} (hr : ‖r‖ < 1)
Goal: Summable fun n => ↑n ^ k * r ^ n
-- hasSum_coe_mul_geometric_of_norm_lt_one
theorem hasSum_coe_mul_geometric_of_norm_lt_one {𝕜 : Type*} [NormedDivisionRing 𝕜] [CompleteSpace 𝕜]
    {r : 𝕜} (hr : ‖r‖ < 1) : HasSum (fun n ↦ n * r ^ n : ℕ → 𝕜) (r / (1 - r) ^ 2) 
Arguments: {𝕜 : Type u_4} [NormedDivisionRing 𝕜] [CompleteSpace 𝕜] {r : 𝕜} (hr : ‖r‖ < 1)
Goal: HasSum (fun n => ↑n * r ^ n) (r / (1 - r) ^ 2)
-- tsum_coe_mul_geometric_of_norm_lt_one
theorem tsum_coe_mul_geometric_of_norm_lt_one {𝕜 : Type*} [NormedDivisionRing 𝕜] [CompleteSpace 𝕜]
    {r : 𝕜} (hr : ‖r‖ < 1) : (∑' n : ℕ, n * r ^ n : 𝕜) = r / (1 - r) ^ 2 
Arguments: {𝕜 : Type u_4} [NormedDivisionRing 𝕜] [CompleteSpace 𝕜] {r : 𝕜} (hr : ‖r‖ < 1)
Goal: ∑' (n : ℕ), ↑n * r ^ n = r / (1 - r) ^ 2
-- SeminormedAddCommGroup.cauchySeq_of_le_geometric
nonrec theorem SeminormedAddCommGroup.cauchySeq_of_le_geometric {C : ℝ} {r : ℝ} (hr : r < 1)
    {u : ℕ → α} (h : ∀ n, ‖u n - u (n + 1)‖ ≤ C * r ^ n) : CauchySeq u 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {C r : ℝ} (hr : r < 1) {u : ℕ → α} (h : ∀ (n : ℕ), ‖u n - u (n + 1)‖ ≤ C * r ^ n)
Goal: CauchySeq u
-- dist_partial_sum_le_of_le_geometric
theorem dist_partial_sum_le_of_le_geometric (hf : ∀ n, ‖f n‖ ≤ C * r ^ n) (n : ℕ) :
    dist (∑ i ∈ range n, f i) (∑ i ∈ range (n + 1), f i) ≤ C * r ^ n 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {r C : ℝ} {f : ℕ → α} (hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n) (n : ℕ)
Goal: dist (∑ i ∈ Finset.range n, f i) (∑ i ∈ Finset.range (n + 1), f i) ≤ C * r ^ n
-- cauchySeq_finset_of_geometric_bound
theorem cauchySeq_finset_of_geometric_bound (hr : r < 1) (hf : ∀ n, ‖f n‖ ≤ C * r ^ n) :
    CauchySeq fun s : Finset ℕ ↦ ∑ x ∈ s, f x 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {r C : ℝ} {f : ℕ → α} (hr : r < 1) (hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n)
Goal: CauchySeq fun s => ∑ x ∈ s, f x
-- norm_sub_le_of_geometric_bound_of_hasSum
theorem norm_sub_le_of_geometric_bound_of_hasSum (hr : r < 1) (hf : ∀ n, ‖f n‖ ≤ C * r ^ n) {a : α}
    (ha : HasSum f a) (n : ℕ) : ‖(∑ x ∈ Finset.range n, f x) - a‖ ≤ C * r ^ n / (1 - r) 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {r C : ℝ} {f : ℕ → α} (hr : r < 1) (hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n) {a : α} (ha : HasSum f a) (n : ℕ)
Goal: ‖∑ x ∈ Finset.range n, f x - a‖ ≤ C * r ^ n / (1 - r)
-- dist_partial_sum
@[simp]
theorem dist_partial_sum (u : ℕ → α) (n : ℕ) :
    dist (∑ k ∈ range (n + 1), u k) (∑ k ∈ range n, u k) = ‖u n‖ 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] (u : ℕ → α) (n : ℕ)
Goal: dist (∑ k ∈ Finset.range (n + 1), u k) (∑ k ∈ Finset.range n, u k) = ‖u n‖
-- dist_partial_sum'
@[simp]
theorem dist_partial_sum' (u : ℕ → α) (n : ℕ) :
    dist (∑ k ∈ range n, u k) (∑ k ∈ range (n + 1), u k) = ‖u n‖ 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] (u : ℕ → α) (n : ℕ)
Goal: dist (∑ k ∈ Finset.range n, u k) (∑ k ∈ Finset.range (n + 1), u k) = ‖u n‖
-- cauchy_series_of_le_geometric
theorem cauchy_series_of_le_geometric {C : ℝ} {u : ℕ → α} {r : ℝ} (hr : r < 1)
    (h : ∀ n, ‖u n‖ ≤ C * r ^ n) : CauchySeq fun n ↦ ∑ k ∈ range n, u k 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {C : ℝ} {u : ℕ → α} {r : ℝ} (hr : r < 1) (h : ∀ (n : ℕ), ‖u n‖ ≤ C * r ^ n)
Goal: CauchySeq fun n => ∑ k ∈ Finset.range n, u k
-- NormedAddCommGroup.cauchy_series_of_le_geometric'
theorem NormedAddCommGroup.cauchy_series_of_le_geometric' {C : ℝ} {u : ℕ → α} {r : ℝ} (hr : r < 1)
    (h : ∀ n, ‖u n‖ ≤ C * r ^ n) : CauchySeq fun n ↦ ∑ k ∈ range (n + 1), u k 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {C : ℝ} {u : ℕ → α} {r : ℝ} (hr : r < 1) (h : ∀ (n : ℕ), ‖u n‖ ≤ C * r ^ n)
Goal: CauchySeq fun n => ∑ k ∈ Finset.range (n + 1), u k
-- NormedAddCommGroup.cauchy_series_of_le_geometric''
theorem NormedAddCommGroup.cauchy_series_of_le_geometric'' {C : ℝ} {u : ℕ → α} {N : ℕ} {r : ℝ}
    (hr₀ : 0 < r) (hr₁ : r < 1) (h : ∀ n ≥ N, ‖u n‖ ≤ C * r ^ n) :
    CauchySeq fun n ↦ ∑ k ∈ range (n + 1), u k 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {C : ℝ} {u : ℕ → α} {N : ℕ} {r : ℝ} (hr₀ : 0 < r) (hr₁ : r < 1) (h : ∀ n ≥ N, ‖u n‖ ≤ C * r ^ n)
Goal: CauchySeq fun n => ∑ k ∈ Finset.range (n + 1), u k
-- exists_norm_le_of_cauchySeq
lemma exists_norm_le_of_cauchySeq (h : CauchySeq fun n ↦ ∑ k ∈ range n, f k) :
    ∃ C, ∀ n, ‖f n‖ ≤ C 
Arguments: {α : Type u_1} [SeminormedAddCommGroup α] {f : ℕ → α} (h : CauchySeq fun n => ∑ k ∈ Finset.range n, f k)
Goal: ∃ C, ∀ (n : ℕ), ‖f n‖ ≤ C
-- NormedRing.summable_geometric_of_norm_lt_one
theorem NormedRing.summable_geometric_of_norm_lt_one (x : R) (h : ‖x‖ < 1) :
    Summable fun n : ℕ ↦ x ^ n 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1)
Goal: Summable fun n => x ^ n
-- NormedRing.tsum_geometric_of_norm_lt_one
theorem NormedRing.tsum_geometric_of_norm_lt_one (x : R) (h : ‖x‖ < 1) :
    ‖∑' n : ℕ, x ^ n‖ ≤ ‖(1 : R)‖ - 1 + (1 - ‖x‖)⁻¹ 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1)
Goal: ‖∑' (n : ℕ), x ^ n‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹
-- geom_series_mul_neg
theorem geom_series_mul_neg (x : R) (h : ‖x‖ < 1) : (∑' i : ℕ, x ^ i) * (1 - x) = 1 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1)
Goal: (∑' (i : ℕ), x ^ i) * (1 - x) = 1
-- mul_neg_geom_series
theorem mul_neg_geom_series (x : R) (h : ‖x‖ < 1) : ((1 - x) * ∑' i : ℕ, x ^ i) = 1 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1)
Goal: (1 - x) * ∑' (i : ℕ), x ^ i = 1
-- geom_series_succ
theorem geom_series_succ (x : R) (h : ‖x‖ < 1) : ∑' i : ℕ, x ^ (i + 1) = ∑' i : ℕ, x ^ i - 1 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1)
Goal: ∑' (i : ℕ), x ^ (i + 1) = ∑' (i : ℕ), x ^ i - 1
-- geom_series_mul_shift
theorem geom_series_mul_shift (x : R) (h : ‖x‖ < 1) :
    x * ∑' i : ℕ, x ^ i = ∑' i : ℕ, x ^ (i + 1) 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1)
Goal: x * ∑' (i : ℕ), x ^ i = ∑' (i : ℕ), x ^ (i + 1)
-- geom_series_mul_one_add
theorem geom_series_mul_one_add (x : R) (h : ‖x‖ < 1) :
    (1 + x) * ∑' i : ℕ, x ^ i = 2 * ∑' i : ℕ, x ^ i - 1 
Arguments: {R : Type u_4} [NormedRing R] [CompleteSpace R] (x : R) (h : ‖x‖ < 1)
Goal: (1 + x) * ∑' (i : ℕ), x ^ i = 2 * ∑' (i : ℕ), x ^ i - 1
-- summable_of_ratio_norm_eventually_le
theorem summable_of_ratio_norm_eventually_le {α : Type*} [SeminormedAddCommGroup α]
    [CompleteSpace α] {f : ℕ → α} {r : ℝ} (hr₁ : r < 1)
    (h : ∀ᶠ n in atTop, ‖f (n + 1)‖ ≤ r * ‖f n‖) : Summable f 
Arguments: {α : Type u_4} [SeminormedAddCommGroup α] [CompleteSpace α] {f : ℕ → α} {r : ℝ} (hr₁ : r < 1) (h : ∀ᶠ (n : ℕ) in Filter.atTop, ‖f (n + 1)‖ ≤ r * ‖f n‖)
Goal: Summable f