ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoxIntegral.Prepartition.not_disjoint_imp_le_of_subset_of_mem_splitMany
theorem not_disjoint_imp_le_of_subset_of_mem_splitMany {I J Js : Box ι} {s : Finset (ι × ℝ)}
    (H : ∀ i, {(i, J.lower i), (i, J.upper i)} ⊆ s) (HJs : Js ∈ splitMany I s)
    (Hn : ¬Disjoint (J : WithBot (Box ι)) Js) : Js ≤ J 
Arguments: {ι : Type u_1} {I J Js : BoxIntegral.Box ι} {s : Finset (ι × ℝ)} (H : ∀ (i : ι), {(i, J.lower i), (i, J.upper i)} ⊆ s) (HJs : Js ∈ BoxIntegral.Prepartition.splitMany I s) (Hn : ¬Disjoint ↑J ↑Js)
Goal: Js ≤ J
-- BoxIntegral.Prepartition.eventually_not_disjoint_imp_le_of_mem_splitMany
theorem eventually_not_disjoint_imp_le_of_mem_splitMany (s : Finset (Box ι)) :
    ∀ᶠ t : Finset (ι × ℝ) in atTop, ∀ (I : Box ι), ∀ J ∈ s, ∀ J' ∈ splitMany I t,
      ¬Disjoint (J : WithBot (Box ι)) J' → J' ≤ J 
Arguments: {ι : Type u_1} [Finite ι] (s : Finset (BoxIntegral.Box ι))
Goal: ∀ᶠ (t : Finset (ι × ℝ)) in Filter.atTop,
  ∀ (I J : BoxIntegral.Box ι), J ∈ s → ∀ J' ∈ BoxIntegral.Prepartition.splitMany I t, ¬Disjoint ↑J ↑J' → J' ≤ J
-- BoxIntegral.Prepartition.eventually_splitMany_inf_eq_filter
theorem eventually_splitMany_inf_eq_filter (π : Prepartition I) :
    ∀ᶠ t : Finset (ι × ℝ) in atTop,
      π ⊓ splitMany I t = (splitMany I t).filter fun J => ↑J ⊆ π.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Finite ι] (π : BoxIntegral.Prepartition I)
Goal: ∀ᶠ (t : Finset (ι × ℝ)) in Filter.atTop,
  π ⊓ BoxIntegral.Prepartition.splitMany I t = (BoxIntegral.Prepartition.splitMany I t).filter fun J => ↑J ⊆ π.iUnion
-- BoxIntegral.Prepartition.exists_splitMany_inf_eq_filter_of_finite
theorem exists_splitMany_inf_eq_filter_of_finite (s : Set (Prepartition I)) (hs : s.Finite) :
    ∃ t : Finset (ι × ℝ),
      ∀ π ∈ s, π ⊓ splitMany I t = (splitMany I t).filter fun J => ↑J ⊆ π.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Finite ι] (s : Set (BoxIntegral.Prepartition I)) (hs : s.Finite)
Goal: ∃ t,
  ∀ π ∈ s,
    π ⊓ BoxIntegral.Prepartition.splitMany I t = (BoxIntegral.Prepartition.splitMany I t).filter fun J => ↑J ⊆ π.iUnion
-- BoxIntegral.Prepartition.IsPartition.exists_splitMany_le
theorem IsPartition.exists_splitMany_le {I : Box ι} {π : Prepartition I} (h : IsPartition π) :
    ∃ s, splitMany I s ≤ π 
Arguments: {ι : Type u_1} [Finite ι] {I : BoxIntegral.Box ι} {π : BoxIntegral.Prepartition I} (h : π.IsPartition)
Goal: ∃ s, BoxIntegral.Prepartition.splitMany I s ≤ π
-- BoxIntegral.Prepartition.exists_iUnion_eq_diff
theorem exists_iUnion_eq_diff (π : Prepartition I) :
    ∃ π' : Prepartition I, π'.iUnion = ↑I \ π.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Finite ι] (π : BoxIntegral.Prepartition I)
Goal: ∃ π', π'.iUnion = ↑I \ π.iUnion
-- BoxIntegral.Prepartition.iUnion_compl
@[simp]
theorem iUnion_compl (π : Prepartition I) : π.compl.iUnion = ↑I \ π.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Finite ι] (π : BoxIntegral.Prepartition I)
Goal: π.compl.iUnion = ↑I \ π.iUnion
-- BoxIntegral.Prepartition.compl_congr
theorem compl_congr {π₁ π₂ : Prepartition I} (h : π₁.iUnion = π₂.iUnion) : π₁.compl = π₂.compl 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Finite ι] {π₁ π₂ : BoxIntegral.Prepartition I} (h : π₁.iUnion = π₂.iUnion)
Goal: π₁.compl = π₂.compl
-- BoxIntegral.Prepartition.IsPartition.compl_eq_bot
theorem IsPartition.compl_eq_bot {π : Prepartition I} (h : IsPartition π) : π.compl = ⊥ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Finite ι] {π : BoxIntegral.Prepartition I} (h : π.IsPartition)
Goal: π.compl = ⊥
-- BoxIntegral.Prepartition.compl_top
@[simp]
theorem compl_top : (⊤ : Prepartition I).compl = ⊥ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Finite ι]
Goal: ⊤.compl = ⊥
-- tendsto_inverse_atTop_nhds_zero_nat
theorem tendsto_inverse_atTop_nhds_zero_nat : Tendsto (fun n : ℕ ↦ (n : ℝ)⁻¹) atTop (𝓝 0) 
Goal: Filter.Tendsto (fun n => (↑n)⁻¹) Filter.atTop (𝓝 0)
-- tendsto_const_div_atTop_nhds_zero_nat
theorem tendsto_const_div_atTop_nhds_zero_nat (C : ℝ) :
    Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) 
Arguments: (C : ℝ)
Goal: Filter.Tendsto (fun n => C / ↑n) Filter.atTop (𝓝 0)
-- tendsto_one_div_atTop_nhds_zero_nat
theorem tendsto_one_div_atTop_nhds_zero_nat : Tendsto (fun n : ℕ ↦ 1/(n : ℝ)) atTop (𝓝 0) 
Goal: Filter.Tendsto (fun n => 1 / ↑n) Filter.atTop (𝓝 0)
-- NNReal.tendsto_inverse_atTop_nhds_zero_nat
theorem NNReal.tendsto_inverse_atTop_nhds_zero_nat :
    Tendsto (fun n : ℕ ↦ (n : ℝ≥0)⁻¹) atTop (𝓝 0) 
Goal: Filter.Tendsto (fun n => (↑n)⁻¹) Filter.atTop (𝓝 0)
-- NNReal.tendsto_const_div_atTop_nhds_zero_nat
theorem NNReal.tendsto_const_div_atTop_nhds_zero_nat (C : ℝ≥0) :
    Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) 
Arguments: (C : ℝ≥0)
Goal: Filter.Tendsto (fun n => C / ↑n) Filter.atTop (𝓝 0)
-- EReal.tendsto_const_div_atTop_nhds_zero_nat
theorem EReal.tendsto_const_div_atTop_nhds_zero_nat {C : EReal} (h : C ≠ ⊥) (h' : C ≠ ⊤) :
    Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) 
Arguments: {C : EReal} (h : C ≠ ⊥) (h' : C ≠ ⊤)
Goal: Filter.Tendsto (fun n => C / ↑n) Filter.atTop (𝓝 0)
-- tendsto_one_div_add_atTop_nhds_zero_nat
theorem tendsto_one_div_add_atTop_nhds_zero_nat :
    Tendsto (fun n : ℕ ↦ 1 / ((n : ℝ) + 1)) atTop (𝓝 0) 
Goal: Filter.Tendsto (fun n => 1 / (↑n + 1)) Filter.atTop (𝓝 0)
-- NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat
theorem NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat (𝕜 : Type*) [Semiring 𝕜]
    [Algebra ℝ≥0 𝕜] [TopologicalSpace 𝕜] [ContinuousSMul ℝ≥0 𝕜] :
    Tendsto (algebraMap ℝ≥0 𝕜 ∘ fun n : ℕ ↦ (n : ℝ≥0)⁻¹) atTop (𝓝 0) 
Arguments: (𝕜 : Type u_4) [Semiring 𝕜] [Algebra ℝ≥0 𝕜] [TopologicalSpace 𝕜] [ContinuousSMul ℝ≥0 𝕜]
Goal: Filter.Tendsto (⇑(algebraMap ℝ≥0 𝕜) ∘ fun n => (↑n)⁻¹) Filter.atTop (𝓝 0)
-- tendsto_algebraMap_inverse_atTop_nhds_zero_nat
theorem tendsto_algebraMap_inverse_atTop_nhds_zero_nat (𝕜 : Type*) [Semiring 𝕜] [Algebra ℝ 𝕜]
    [TopologicalSpace 𝕜] [ContinuousSMul ℝ 𝕜] :
    Tendsto (algebraMap ℝ 𝕜 ∘ fun n : ℕ ↦ (n : ℝ)⁻¹) atTop (𝓝 0) 
Arguments: (𝕜 : Type u_4) [Semiring 𝕜] [Algebra ℝ 𝕜] [TopologicalSpace 𝕜] [ContinuousSMul ℝ 𝕜]
Goal: Filter.Tendsto (⇑(algebraMap ℝ 𝕜) ∘ fun n => (↑n)⁻¹) Filter.atTop (𝓝 0)
-- tendsto_natCast_div_add_atTop
theorem tendsto_natCast_div_add_atTop {𝕜 : Type*} [DivisionRing 𝕜] [TopologicalSpace 𝕜]
    [CharZero 𝕜] [Algebra ℝ 𝕜] [ContinuousSMul ℝ 𝕜] [TopologicalDivisionRing 𝕜] (x : 𝕜) :
    Tendsto (fun n : ℕ ↦ (n : 𝕜) / (n + x)) atTop (𝓝 1) 
Arguments: {𝕜 : Type u_4} [DivisionRing 𝕜] [TopologicalSpace 𝕜] [CharZero 𝕜] [Algebra ℝ 𝕜] [ContinuousSMul ℝ 𝕜] [TopologicalDivisionRing 𝕜] (x : 𝕜)
Goal: Filter.Tendsto (fun n => ↑n / (↑n + x)) Filter.atTop (𝓝 1)
-- tendsto_add_one_pow_atTop_atTop_of_pos
theorem tendsto_add_one_pow_atTop_atTop_of_pos [LinearOrderedSemiring α] [Archimedean α] {r : α}
    (h : 0 < r) : Tendsto (fun n : ℕ ↦ (r + 1) ^ n) atTop atTop 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] [Archimedean α] {r : α} (h : 0 < r)
Goal: Filter.Tendsto (fun n => (r + 1) ^ n) Filter.atTop Filter.atTop
-- tendsto_pow_atTop_atTop_of_one_lt
theorem tendsto_pow_atTop_atTop_of_one_lt [LinearOrderedRing α] [Archimedean α] {r : α}
    (h : 1 < r) : Tendsto (fun n : ℕ ↦ r ^ n) atTop atTop 
Arguments: {α : Type u_1} [LinearOrderedRing α] [Archimedean α] {r : α} (h : 1 < r)
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop Filter.atTop
-- Nat.tendsto_pow_atTop_atTop_of_one_lt
theorem Nat.tendsto_pow_atTop_atTop_of_one_lt {m : ℕ} (h : 1 < m) :
    Tendsto (fun n : ℕ ↦ m ^ n) atTop atTop 
Arguments: {m : ℕ} (h : 1 < m)
Goal: Filter.Tendsto (fun n => m ^ n) Filter.atTop Filter.atTop
-- tendsto_pow_atTop_nhds_zero_of_lt_one
theorem tendsto_pow_atTop_nhds_zero_of_lt_one {𝕜 : Type*} [LinearOrderedField 𝕜] [Archimedean 𝕜]
    [TopologicalSpace 𝕜] [OrderTopology 𝕜] {r : 𝕜} (h₁ : 0 ≤ r) (h₂ : r < 1) :
    Tendsto (fun n : ℕ ↦ r ^ n) atTop (𝓝 0) 
Arguments: {𝕜 : Type u_4} [LinearOrderedField 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {r : 𝕜} (h₁ : 0 ≤ r) (h₂ : r < 1)
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝 0)
-- tendsto_pow_atTop_nhds_zero_iff
@[simp] theorem tendsto_pow_atTop_nhds_zero_iff {𝕜 : Type*} [LinearOrderedField 𝕜] [Archimedean 𝕜]
    [TopologicalSpace 𝕜] [OrderTopology 𝕜] {r : 𝕜} :
    Tendsto (fun n : ℕ ↦ r ^ n) atTop (𝓝 0) ↔ |r| < 1 
Arguments: {𝕜 : Type u_4} [LinearOrderedField 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {r : 𝕜}
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝 0) ↔ |r| < 1
-- tendsto_pow_atTop_nhdsWithin_zero_of_lt_one
theorem tendsto_pow_atTop_nhdsWithin_zero_of_lt_one {𝕜 : Type*} [LinearOrderedField 𝕜]
    [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {r : 𝕜} (h₁ : 0 < r) (h₂ : r < 1) :
    Tendsto (fun n : ℕ ↦ r ^ n) atTop (𝓝[>] 0) 
Arguments: {𝕜 : Type u_4} [LinearOrderedField 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {r : 𝕜} (h₁ : 0 < r) (h₂ : r < 1)
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝[>] 0)
-- uniformity_basis_dist_pow_of_lt_one
theorem uniformity_basis_dist_pow_of_lt_one {α : Type*} [PseudoMetricSpace α] {r : ℝ} (h₀ : 0 < r)
    (h₁ : r < 1) :
    (uniformity α).HasBasis (fun _ : ℕ ↦ True) fun k ↦ { p : α × α | dist p.1 p.2 < r ^ k } 
Arguments: {α : Type u_4} [PseudoMetricSpace α] {r : ℝ} (h₀ : 0 < r) (h₁ : r < 1)
Goal: (uniformity α).HasBasis (fun x => True) fun k => {p | dist p.1 p.2 < r ^ k}
-- geom_lt
theorem geom_lt {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) {n : ℕ} (hn : 0 < n)
    (h : ∀ k < n, c * u k < u (k + 1)) : c ^ n * u 0 < u n 
Arguments: {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) {n : ℕ} (hn : 0 < n) (h : ∀ k < n, c * u k < u (k + 1))
Goal: c ^ n * u 0 < u n
-- geom_le
theorem geom_le {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, c * u k ≤ u (k + 1)) :
    c ^ n * u 0 ≤ u n 
Arguments: {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, c * u k ≤ u (k + 1))
Goal: c ^ n * u 0 ≤ u n
-- lt_geom
theorem lt_geom {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) {n : ℕ} (hn : 0 < n)
    (h : ∀ k < n, u (k + 1) < c * u k) : u n < c ^ n * u 0 
Arguments: {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) {n : ℕ} (hn : 0 < n) (h : ∀ k < n, u (k + 1) < c * u k)
Goal: u n < c ^ n * u 0
-- le_geom
theorem le_geom {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, u (k + 1) ≤ c * u k) :
    u n ≤ c ^ n * u 0 
Arguments: {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, u (k + 1) ≤ c * u k)
Goal: u n ≤ c ^ n * u 0
-- tendsto_atTop_of_geom_le
theorem tendsto_atTop_of_geom_le {v : ℕ → ℝ} {c : ℝ} (h₀ : 0 < v 0) (hc : 1 < c)
    (hu : ∀ n, c * v n ≤ v (n + 1)) : Tendsto v atTop atTop 
Arguments: {v : ℕ → ℝ} {c : ℝ} (h₀ : 0 < v 0) (hc : 1 < c) (hu : ∀ (n : ℕ), c * v n ≤ v (n + 1))
Goal: Filter.Tendsto v Filter.atTop Filter.atTop
-- NNReal.tendsto_pow_atTop_nhds_zero_of_lt_one
theorem NNReal.tendsto_pow_atTop_nhds_zero_of_lt_one {r : ℝ≥0} (hr : r < 1) :
    Tendsto (fun n : ℕ ↦ r ^ n) atTop (𝓝 0) 
Arguments: {r : ℝ≥0} (hr : r < 1)
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝 0)
-- NNReal.tendsto_pow_atTop_nhds_zero_iff
@[simp]
protected theorem NNReal.tendsto_pow_atTop_nhds_zero_iff {r : ℝ≥0} :
    Tendsto (fun n : ℕ => r ^ n) atTop (𝓝 0) ↔ r < 1 
Arguments: {r : ℝ≥0}
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝 0) ↔ r < 1
-- ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one
theorem ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one {r : ℝ≥0∞} (hr : r < 1) :
    Tendsto (fun n : ℕ ↦ r ^ n) atTop (𝓝 0) 
Arguments: {r : ℝ≥0∞} (hr : r < 1)
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝 0)
-- ENNReal.tendsto_pow_atTop_nhds_zero_iff
@[simp]
protected theorem ENNReal.tendsto_pow_atTop_nhds_zero_iff {r : ℝ≥0∞} :
    Tendsto (fun n : ℕ => r ^ n) atTop (𝓝 0) ↔ r < 1 
Arguments: {r : ℝ≥0∞}
Goal: Filter.Tendsto (fun n => r ^ n) Filter.atTop (𝓝 0) ↔ r < 1
-- hasSum_geometric_of_lt_one
theorem hasSum_geometric_of_lt_one {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1) :
    HasSum (fun n : ℕ ↦ r ^ n) (1 - r)⁻¹ 
Arguments: {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1)
Goal: HasSum (fun n => r ^ n) (1 - r)⁻¹
-- summable_geometric_of_lt_one
theorem summable_geometric_of_lt_one {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1) :
    Summable fun n : ℕ ↦ r ^ n 
Arguments: {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1)
Goal: Summable fun n => r ^ n
-- tsum_geometric_of_lt_one
theorem tsum_geometric_of_lt_one {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1) : ∑' n : ℕ, r ^ n = (1 - r)⁻¹ 
Arguments: {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1)
Goal: ∑' (n : ℕ), r ^ n = (1 - r)⁻¹
-- hasSum_geometric_two
theorem hasSum_geometric_two : HasSum (fun n : ℕ ↦ ((1 : ℝ) / 2) ^ n) 2 
Goal: HasSum (fun n => (1 / 2) ^ n) 2
-- summable_geometric_two
theorem summable_geometric_two : Summable fun n : ℕ ↦ ((1 : ℝ) / 2) ^ n 
Goal: Summable fun n => (1 / 2) ^ n
-- summable_geometric_two_encode
theorem summable_geometric_two_encode {ι : Type*} [Encodable ι] :
    Summable fun i : ι ↦ (1 / 2 : ℝ) ^ Encodable.encode i 
Arguments: {ι : Type u_4} [Encodable ι]
Goal: Summable fun i => (1 / 2) ^ Encodable.encode i
-- tsum_geometric_two
theorem tsum_geometric_two : (∑' n : ℕ, ((1 : ℝ) / 2) ^ n) = 2 
Goal: ∑' (n : ℕ), (1 / 2) ^ n = 2
-- sum_geometric_two_le
theorem sum_geometric_two_le (n : ℕ) : (∑ i ∈ range n, (1 / (2 : ℝ)) ^ i) ≤ 2 
Arguments: (n : ℕ)
Goal: ∑ i ∈ Finset.range n, (1 / 2) ^ i ≤ 2
-- tsum_geometric_inv_two
theorem tsum_geometric_inv_two : (∑' n : ℕ, (2 : ℝ)⁻¹ ^ n) = 2 
Goal: ∑' (n : ℕ), 2⁻¹ ^ n = 2
-- tsum_geometric_inv_two_ge
theorem tsum_geometric_inv_two_ge (n : ℕ) :
    (∑' i, ite (n ≤ i) ((2 : ℝ)⁻¹ ^ i) 0) = 2 * 2⁻¹ ^ n 
Arguments: (n : ℕ)
Goal: (∑' (i : ℕ), if n ≤ i then 2⁻¹ ^ i else 0) = 2 * 2⁻¹ ^ n
-- hasSum_geometric_two'
theorem hasSum_geometric_two' (a : ℝ) : HasSum (fun n : ℕ ↦ a / 2 / 2 ^ n) a 
Arguments: (a : ℝ)
Goal: HasSum (fun n => a / 2 / 2 ^ n) a
-- summable_geometric_two'
theorem summable_geometric_two' (a : ℝ) : Summable fun n : ℕ ↦ a / 2 / 2 ^ n 
Arguments: (a : ℝ)
Goal: Summable fun n => a / 2 / 2 ^ n
-- tsum_geometric_two'
theorem tsum_geometric_two' (a : ℝ) : ∑' n : ℕ, a / 2 / 2 ^ n = a 
Arguments: (a : ℝ)
Goal: ∑' (n : ℕ), a / 2 / 2 ^ n = a
-- NNReal.hasSum_geometric
theorem NNReal.hasSum_geometric {r : ℝ≥0} (hr : r < 1) : HasSum (fun n : ℕ ↦ r ^ n) (1 - r)⁻¹ 
Arguments: {r : ℝ≥0} (hr : r < 1)
Goal: HasSum (fun n => r ^ n) (1 - r)⁻¹
-- NNReal.summable_geometric
theorem NNReal.summable_geometric {r : ℝ≥0} (hr : r < 1) : Summable fun n : ℕ ↦ r ^ n 
Arguments: {r : ℝ≥0} (hr : r < 1)
Goal: Summable fun n => r ^ n
-- tsum_geometric_nnreal
theorem tsum_geometric_nnreal {r : ℝ≥0} (hr : r < 1) : ∑' n : ℕ, r ^ n = (1 - r)⁻¹ 
Arguments: {r : ℝ≥0} (hr : r < 1)
Goal: ∑' (n : ℕ), r ^ n = (1 - r)⁻¹
-- ENNReal.tsum_geometric
@[simp]
theorem ENNReal.tsum_geometric (r : ℝ≥0∞) : ∑' n : ℕ, r ^ n = (1 - r)⁻¹ 
Arguments: (r : ℝ≥0∞)
Goal: ∑' (n : ℕ), r ^ n = (1 - r)⁻¹
-- ENNReal.tsum_geometric_add_one
theorem ENNReal.tsum_geometric_add_one (r : ℝ≥0∞) : ∑' n : ℕ, r ^ (n + 1) = r * (1 - r)⁻¹ 
Arguments: (r : ℝ≥0∞)
Goal: ∑' (n : ℕ), r ^ (n + 1) = r * (1 - r)⁻¹
-- cauchySeq_of_edist_le_geometric
theorem cauchySeq_of_edist_le_geometric : CauchySeq f 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (r C : ℝ≥0∞) (hr : r < 1) (hC : C ≠ ⊤) {f : ℕ → α} (hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n)
Goal: CauchySeq f
-- edist_le_of_edist_le_geometric_of_tendsto
theorem edist_le_of_edist_le_geometric_of_tendsto {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
    edist (f n) a ≤ C * r ^ n / (1 - r) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (r C : ℝ≥0∞) {f : ℕ → α} (hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (n : ℕ)
Goal: edist (f n) a ≤ C * r ^ n / (1 - r)
-- edist_le_of_edist_le_geometric_of_tendsto₀
theorem edist_le_of_edist_le_geometric_of_tendsto₀ {a : α} (ha : Tendsto f atTop (𝓝 a)) :
    edist (f 0) a ≤ C / (1 - r) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (r C : ℝ≥0∞) {f : ℕ → α} (hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: edist (f 0) a ≤ C / (1 - r)
-- cauchySeq_of_edist_le_geometric_two
theorem cauchySeq_of_edist_le_geometric_two : CauchySeq f 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (C : ℝ≥0∞) (hC : C ≠ ⊤) {f : ℕ → α} (hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n)
Goal: CauchySeq f
-- edist_le_of_edist_le_geometric_two_of_tendsto
theorem edist_le_of_edist_le_geometric_two_of_tendsto (n : ℕ) : edist (f n) a ≤ 2 * C / 2 ^ n 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (C : ℝ≥0∞) {f : ℕ → α} (hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (n : ℕ)
Goal: edist (f n) a ≤ 2 * C / 2 ^ n
-- edist_le_of_edist_le_geometric_two_of_tendsto₀
theorem edist_le_of_edist_le_geometric_two_of_tendsto₀ : edist (f 0) a ≤ 2 * C 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (C : ℝ≥0∞) {f : ℕ → α} (hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: edist (f 0) a ≤ 2 * C
-- aux_hasSum_of_le_geometric
theorem aux_hasSum_of_le_geometric : HasSum (fun n : ℕ ↦ C * r ^ n) (C / (1 - r)) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {r C : ℝ} (hr : r < 1) {f : ℕ → α} (hu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n)
Goal: HasSum (fun n => C * r ^ n) (C / (1 - r))
-- cauchySeq_of_le_geometric
theorem cauchySeq_of_le_geometric : CauchySeq f 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (r C : ℝ) (hr : r < 1) {f : ℕ → α} (hu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n)
Goal: CauchySeq f
-- dist_le_of_le_geometric_of_tendsto₀
theorem dist_le_of_le_geometric_of_tendsto₀ {a : α} (ha : Tendsto f atTop (𝓝 a)) :
    dist (f 0) a ≤ C / (1 - r) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (r C : ℝ) (hr : r < 1) {f : ℕ → α} (hu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: dist (f 0) a ≤ C / (1 - r)
-- dist_le_of_le_geometric_of_tendsto
theorem dist_le_of_le_geometric_of_tendsto {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
    dist (f n) a ≤ C * r ^ n / (1 - r) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (r C : ℝ) (hr : r < 1) {f : ℕ → α} (hu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (n : ℕ)
Goal: dist (f n) a ≤ C * r ^ n / (1 - r)
-- cauchySeq_of_le_geometric_two
theorem cauchySeq_of_le_geometric_two : CauchySeq f 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (C : ℝ) {f : ℕ → α} (hu₂ : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C / 2 / 2 ^ n)
Goal: CauchySeq f
-- dist_le_of_le_geometric_two_of_tendsto₀
theorem dist_le_of_le_geometric_two_of_tendsto₀ {a : α} (ha : Tendsto f atTop (𝓝 a)) :
    dist (f 0) a ≤ C 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (C : ℝ) {f : ℕ → α} (hu₂ : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C / 2 / 2 ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: dist (f 0) a ≤ C
-- dist_le_of_le_geometric_two_of_tendsto
theorem dist_le_of_le_geometric_two_of_tendsto {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
    dist (f n) a ≤ C / 2 ^ n 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (C : ℝ) {f : ℕ → α} (hu₂ : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C / 2 / 2 ^ n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (n : ℕ)
Goal: dist (f n) a ≤ C / 2 ^ n
-- summable_one_div_pow_of_le
theorem summable_one_div_pow_of_le {m : ℝ} {f : ℕ → ℕ} (hm : 1 < m) (fi : ∀ i, i ≤ f i) :
    Summable fun i ↦ 1 / m ^ f i 
Arguments: {m : ℝ} {f : ℕ → ℕ} (hm : 1 < m) (fi : ∀ (i : ℕ), i ≤ f i)
Goal: Summable fun i => 1 / m ^ f i
-- Set.Countable.exists_pos_hasSum_le
theorem Set.Countable.exists_pos_hasSum_le {ι : Type*} {s : Set ι} (hs : s.Countable) {ε : ℝ}
    (hε : 0 < ε) : ∃ ε' : ι → ℝ, (∀ i, 0 < ε' i) ∧ ∃ c, HasSum (fun i : s ↦ ε' i) c ∧ c ≤ ε 
Arguments: {ι : Type u_4} {s : Set ι} (hs : s.Countable) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε
-- Set.Countable.exists_pos_forall_sum_le
theorem Set.Countable.exists_pos_forall_sum_le {ι : Type*} {s : Set ι} (hs : s.Countable) {ε : ℝ}
    (hε : 0 < ε) : ∃ ε' : ι → ℝ,
    (∀ i, 0 < ε' i) ∧ ∀ t : Finset ι, ↑t ⊆ s → ∑ i ∈ t, ε' i ≤ ε 
Arguments: {ι : Type u_4} {s : Set ι} (hs : s.Countable) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∀ (t : Finset ι), ↑t ⊆ s → ∑ i ∈ t, ε' i ≤ ε
-- NNReal.exists_pos_sum_of_countable
theorem exists_pos_sum_of_countable {ε : ℝ≥0} (hε : ε ≠ 0) (ι) [Countable ι] :
    ∃ ε' : ι → ℝ≥0, (∀ i, 0 < ε' i) ∧ ∃ c, HasSum ε' c ∧ c < ε 
Arguments: {ε : ℝ≥0} (hε : ε ≠ 0) (ι : Type u_4) [Countable ι]
Goal: ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum ε' c ∧ c < ε
-- ENNReal.exists_pos_sum_of_countable
theorem exists_pos_sum_of_countable {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι) [Countable ι] :
    ∃ ε' : ι → ℝ≥0, (∀ i, 0 < ε' i) ∧ (∑' i, (ε' i : ℝ≥0∞)) < ε 
Arguments: {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι : Type u_4) [Countable ι]
Goal: ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∑' (i : ι), ↑(ε' i) < ε
-- ENNReal.exists_pos_sum_of_countable'
theorem exists_pos_sum_of_countable' {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι) [Countable ι] :
    ∃ ε' : ι → ℝ≥0∞, (∀ i, 0 < ε' i) ∧ ∑' i, ε' i < ε 
Arguments: {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι : Type u_4) [Countable ι]
Goal: ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∑' (i : ι), ε' i < ε
-- ENNReal.exists_pos_tsum_mul_lt_of_countable
theorem exists_pos_tsum_mul_lt_of_countable {ε : ℝ≥0∞} (hε : ε ≠ 0) {ι} [Countable ι] (w : ι → ℝ≥0∞)
    (hw : ∀ i, w i ≠ ∞) : ∃ δ : ι → ℝ≥0, (∀ i, 0 < δ i) ∧ (∑' i, (w i * δ i : ℝ≥0∞)) < ε 
Arguments: {ε : ℝ≥0∞} (hε : ε ≠ 0) {ι : Type u_4} [Countable ι] (w : ι → ℝ≥0∞) (hw : ∀ (i : ι), w i ≠ ⊤)
Goal: ∃ δ, (∀ (i : ι), 0 < δ i) ∧ ∑' (i : ι), w i * ↑(δ i) < ε
-- factorial_tendsto_atTop
theorem factorial_tendsto_atTop : Tendsto Nat.factorial atTop atTop 
Goal: Filter.Tendsto Nat.factorial Filter.atTop Filter.atTop
-- tendsto_factorial_div_pow_self_atTop
theorem tendsto_factorial_div_pow_self_atTop :
    Tendsto (fun n ↦ n ! / (n : ℝ) ^ n : ℕ → ℝ) atTop (𝓝 0) 
Goal: Filter.Tendsto (fun n => ↑n ! / ↑n ^ n) Filter.atTop (𝓝 0)
-- tendsto_nat_floor_atTop
theorem tendsto_nat_floor_atTop {α : Type*} [LinearOrderedSemiring α] [FloorSemiring α] :
    Tendsto (fun x : α ↦ ⌊x⌋₊) atTop atTop 
Arguments: {α : Type u_4} [LinearOrderedSemiring α] [FloorSemiring α]
Goal: Filter.Tendsto (fun x => ⌊x⌋₊) Filter.atTop Filter.atTop
-- tendsto_nat_ceil_atTop
lemma tendsto_nat_ceil_atTop {α : Type*} [LinearOrderedSemiring α] [FloorSemiring α] :
    Tendsto (fun x : α ↦ ⌈x⌉₊) atTop atTop 
Arguments: {α : Type u_4} [LinearOrderedSemiring α] [FloorSemiring α]
Goal: Filter.Tendsto (fun x => ⌈x⌉₊) Filter.atTop Filter.atTop
-- tendsto_nat_floor_mul_atTop
lemma tendsto_nat_floor_mul_atTop {α : Type _} [LinearOrderedSemifield α] [FloorSemiring α]
    [Archimedean α] (a : α) (ha : 0 < a) : Tendsto (fun (x : ℕ) => ⌊a * x⌋₊) atTop atTop 
Arguments: {α : Type u_4} [LinearOrderedSemifield α] [FloorSemiring α] [Archimedean α] (a : α) (ha : 0 < a)
Goal: Filter.Tendsto (fun x => ⌊a * ↑x⌋₊) Filter.atTop Filter.atTop
-- tendsto_nat_floor_mul_div_atTop
theorem tendsto_nat_floor_mul_div_atTop {a : R} (ha : 0 ≤ a) :
    Tendsto (fun x ↦ (⌊a * x⌋₊ : R) / x) atTop (𝓝 a) 
Arguments: {R : Type u_4} [TopologicalSpace R] [LinearOrderedField R] [OrderTopology R] [FloorRing R] {a : R} (ha : 0 ≤ a)
Goal: Filter.Tendsto (fun x => ↑⌊a * x⌋₊ / x) Filter.atTop (𝓝 a)
-- tendsto_nat_floor_div_atTop
theorem tendsto_nat_floor_div_atTop : Tendsto (fun x ↦ (⌊x⌋₊ : R) / x) atTop (𝓝 1) 
Arguments: {R : Type u_4} [TopologicalSpace R] [LinearOrderedField R] [OrderTopology R] [FloorRing R]
Goal: Filter.Tendsto (fun x => ↑⌊x⌋₊ / x) Filter.atTop (𝓝 1)
-- tendsto_nat_ceil_mul_div_atTop
theorem tendsto_nat_ceil_mul_div_atTop {a : R} (ha : 0 ≤ a) :
    Tendsto (fun x ↦ (⌈a * x⌉₊ : R) / x) atTop (𝓝 a) 
Arguments: {R : Type u_4} [TopologicalSpace R] [LinearOrderedField R] [OrderTopology R] [FloorRing R] {a : R} (ha : 0 ≤ a)
Goal: Filter.Tendsto (fun x => ↑⌈a * x⌉₊ / x) Filter.atTop (𝓝 a)
-- tendsto_nat_ceil_div_atTop
theorem tendsto_nat_ceil_div_atTop : Tendsto (fun x ↦ (⌈x⌉₊ : R) / x) atTop (𝓝 1) 
Arguments: {R : Type u_4} [TopologicalSpace R] [LinearOrderedField R] [OrderTopology R] [FloorRing R]
Goal: Filter.Tendsto (fun x => ↑⌈x⌉₊ / x) Filter.atTop (𝓝 1)
-- Nat.tendsto_div_const_atTop
lemma Nat.tendsto_div_const_atTop {n : ℕ} (hn : n ≠ 0) : Tendsto (· / n) atTop atTop 
Arguments: {n : ℕ} (hn : n ≠ 0)
Goal: Filter.Tendsto (fun x => x / n) Filter.atTop Filter.atTop