ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ENNReal.tsum_add_one_eq_top
theorem tsum_add_one_eq_top {f : ℕ → ℝ≥0∞} (hf : ∑' n, f n = ∞) (hf0 : f 0 ≠ ∞) :
    ∑' n, f (n + 1) = ∞ 
Arguments: {f : ℕ → ℝ≥0∞} (hf : ∑' (n : ℕ), f n = ⊤) (hf0 : f 0 ≠ ⊤)
Goal: ∑' (n : ℕ), f (n + 1) = ⊤
-- ENNReal.finite_const_le_of_tsum_ne_top
theorem finite_const_le_of_tsum_ne_top {ι : Type*} {a : ι → ℝ≥0∞} (tsum_ne_top : ∑' i, a i ≠ ∞)
    {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0) : { i : ι | ε ≤ a i }.Finite 
Arguments: {ι : Type u_4} {a : ι → ℝ≥0∞} (tsum_ne_top : ∑' (i : ι), a i ≠ ⊤) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0)
Goal: {i | ε ≤ a i}.Finite
-- ENNReal.finset_card_const_le_le_of_tsum_le
theorem finset_card_const_le_le_of_tsum_le {ι : Type*} {a : ι → ℝ≥0∞} {c : ℝ≥0∞} (c_ne_top : c ≠ ∞)
    (tsum_le_c : ∑' i, a i ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0) :
    ∃ hf : { i : ι | ε ≤ a i }.Finite, ↑hf.toFinset.card ≤ c / ε 
Arguments: {ι : Type u_4} {a : ι → ℝ≥0∞} {c : ℝ≥0∞} (c_ne_top : c ≠ ⊤) (tsum_le_c : ∑' (i : ι), a i ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0)
Goal: ∃ (hf : {i | ε ≤ a i}.Finite), ↑hf.toFinset.card ≤ c / ε
-- ENNReal.tsum_fiberwise
theorem tsum_fiberwise (f : β → ℝ≥0∞) (g : β → γ) :
    ∑' x, ∑' b : g ⁻¹' {x}, f b = ∑' i, f i 
Arguments: {β : Type u_2} {γ : Type u_3} (f : β → ℝ≥0∞) (g : β → γ)
Goal: ∑' (x : γ) (b : ↑(g ⁻¹' {x})), f ↑b = ∑' (i : β), f i
-- ENNReal.tendsto_toReal_iff
theorem tendsto_toReal_iff {ι} {fi : Filter ι} {f : ι → ℝ≥0∞} (hf : ∀ i, f i ≠ ∞) {x : ℝ≥0∞}
    (hx : x ≠ ∞) : Tendsto (fun n => (f n).toReal) fi (𝓝 x.toReal) ↔ Tendsto f fi (𝓝 x) 
Arguments: {ι : Type u_4} {fi : Filter ι} {f : ι → ℝ≥0∞} (hf : ∀ (i : ι), f i ≠ ⊤) {x : ℝ≥0∞} (hx : x ≠ ⊤)
Goal: Filter.Tendsto (fun n => (f n).toReal) fi (𝓝 x.toReal) ↔ Filter.Tendsto f fi (𝓝 x)
-- ENNReal.tsum_coe_ne_top_iff_summable_coe
theorem tsum_coe_ne_top_iff_summable_coe {f : α → ℝ≥0} :
    (∑' a, (f a : ℝ≥0∞)) ≠ ∞ ↔ Summable fun a => (f a : ℝ) 
Arguments: {α : Type u_1} {f : α → ℝ≥0}
Goal: ∑' (a : α), ↑(f a) ≠ ⊤ ↔ Summable fun a => ↑(f a)
-- ENNReal.tsum_coe_eq_top_iff_not_summable_coe
theorem tsum_coe_eq_top_iff_not_summable_coe {f : α → ℝ≥0} :
    (∑' a, (f a : ℝ≥0∞)) = ∞ ↔ ¬Summable fun a => (f a : ℝ) 
Arguments: {α : Type u_1} {f : α → ℝ≥0}
Goal: ∑' (a : α), ↑(f a) = ⊤ ↔ ¬Summable fun a => ↑(f a)
-- ENNReal.hasSum_toReal
theorem hasSum_toReal {f : α → ℝ≥0∞} (hsum : ∑' x, f x ≠ ∞) :
    HasSum (fun x => (f x).toReal) (∑' x, (f x).toReal) 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (hsum : ∑' (x : α), f x ≠ ⊤)
Goal: HasSum (fun x => (f x).toReal) (∑' (x : α), (f x).toReal)
-- ENNReal.summable_toReal
theorem summable_toReal {f : α → ℝ≥0∞} (hsum : ∑' x, f x ≠ ∞) : Summable fun x => (f x).toReal 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (hsum : ∑' (x : α), f x ≠ ⊤)
Goal: Summable fun x => (f x).toReal
-- NNReal.tsum_eq_toNNReal_tsum
theorem tsum_eq_toNNReal_tsum {f : β → ℝ≥0} : ∑' b, f b = (∑' b, (f b : ℝ≥0∞)).toNNReal 
Arguments: {β : Type u_2} {f : β → ℝ≥0}
Goal: ∑' (b : β), f b = (∑' (b : β), ↑(f b)).toNNReal
-- NNReal.exists_le_hasSum_of_le
theorem exists_le_hasSum_of_le {f g : β → ℝ≥0} {r : ℝ≥0} (hgf : ∀ b, g b ≤ f b) (hfr : HasSum f r) :
    ∃ p ≤ r, HasSum g p 
Arguments: {β : Type u_2} {f g : β → ℝ≥0} {r : ℝ≥0} (hgf : ∀ (b : β), g b ≤ f b) (hfr : HasSum f r)
Goal: ∃ p ≤ r, HasSum g p
-- NNReal.summable_of_le
theorem summable_of_le {f g : β → ℝ≥0} (hgf : ∀ b, g b ≤ f b) : Summable f → Summable g
  
Arguments: {β : Type u_2} {f g : β → ℝ≥0} (hgf : ∀ (b : β), g b ≤ f b)
Goal: Summable f → Summable g
-- Summable.countable_support_nnreal
theorem _root_.Summable.countable_support_nnreal (f : α → ℝ≥0) (h : Summable f) :
    f.support.Countable 
Arguments: {α : Type u_1} (f : α → ℝ≥0) (h : Summable f)
Goal: (Function.support f).Countable
-- NNReal.hasSum_iff_tendsto_nat
theorem hasSum_iff_tendsto_nat {f : ℕ → ℝ≥0} {r : ℝ≥0} :
    HasSum f r ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 r) 
Arguments: {f : ℕ → ℝ≥0} {r : ℝ≥0}
Goal: HasSum f r ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 r)
-- NNReal.not_summable_iff_tendsto_nat_atTop
theorem not_summable_iff_tendsto_nat_atTop {f : ℕ → ℝ≥0} :
    ¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop 
Arguments: {f : ℕ → ℝ≥0}
Goal: ¬Summable f ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop Filter.atTop
-- NNReal.summable_iff_not_tendsto_nat_atTop
theorem summable_iff_not_tendsto_nat_atTop {f : ℕ → ℝ≥0} :
    Summable f ↔ ¬Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop 
Arguments: {f : ℕ → ℝ≥0}
Goal: Summable f ↔ ¬Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop Filter.atTop
-- NNReal.summable_of_sum_range_le
theorem summable_of_sum_range_le {f : ℕ → ℝ≥0} {c : ℝ≥0}
    (h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : Summable f 
Arguments: {f : ℕ → ℝ≥0} {c : ℝ≥0} (h : ∀ (n : ℕ), ∑ i ∈ Finset.range n, f i ≤ c)
Goal: Summable f
-- NNReal.tsum_le_of_sum_range_le
theorem tsum_le_of_sum_range_le {f : ℕ → ℝ≥0} {c : ℝ≥0}
    (h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : ∑' n, f n ≤ c 
Arguments: {f : ℕ → ℝ≥0} {c : ℝ≥0} (h : ∀ (n : ℕ), ∑ i ∈ Finset.range n, f i ≤ c)
Goal: ∑' (n : ℕ), f n ≤ c
-- NNReal.tsum_comp_le_tsum_of_inj
theorem tsum_comp_le_tsum_of_inj {β : Type*} {f : α → ℝ≥0} (hf : Summable f) {i : β → α}
    (hi : Function.Injective i) : (∑' x, f (i x)) ≤ ∑' x, f x 
Arguments: {α : Type u_1} {β : Type u_4} {f : α → ℝ≥0} (hf : Summable f) {i : β → α} (hi : Function.Injective i)
Goal: ∑' (x : β), f (i x) ≤ ∑' (x : α), f x
-- NNReal.summable_sigma
theorem summable_sigma {β : α → Type*} {f : (Σ x, β x) → ℝ≥0} :
    Summable f ↔ (∀ x, Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' y, f ⟨x, y⟩ 
Arguments: {α : Type u_1} {β : α → Type u_4} {f : (x : α) × β x → ℝ≥0}
Goal: Summable f ↔ (∀ (x : α), Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' (y : β x), f ⟨x, y⟩
-- NNReal.indicator_summable
theorem indicator_summable {f : α → ℝ≥0} (hf : Summable f) (s : Set α) :
    Summable (s.indicator f) 
Arguments: {α : Type u_1} {f : α → ℝ≥0} (hf : Summable f) (s : Set α)
Goal: Summable (s.indicator f)
-- NNReal.tsum_indicator_ne_zero
theorem tsum_indicator_ne_zero {f : α → ℝ≥0} (hf : Summable f) {s : Set α} (h : ∃ a ∈ s, f a ≠ 0) :
    (∑' x, (s.indicator f) x) ≠ 0 
Arguments: {α : Type u_1} {f : α → ℝ≥0} (hf : Summable f) {s : Set α} (h : ∃ a ∈ s, f a ≠ 0)
Goal: ∑' (x : α), s.indicator f x ≠ 0
-- NNReal.tendsto_sum_nat_add
theorem tendsto_sum_nat_add (f : ℕ → ℝ≥0) : Tendsto (fun i => ∑' k, f (k + i)) atTop (𝓝 0) 
Arguments: (f : ℕ → ℝ≥0)
Goal: Filter.Tendsto (fun i => ∑' (k : ℕ), f (k + i)) Filter.atTop (𝓝 0)
-- NNReal.hasSum_lt
nonrec theorem hasSum_lt {f g : α → ℝ≥0} {sf sg : ℝ≥0} {i : α} (h : ∀ a : α, f a ≤ g a)
    (hi : f i < g i) (hf : HasSum f sf) (hg : HasSum g sg) : sf < sg 
Arguments: {α : Type u_1} {f g : α → ℝ≥0} {sf sg : ℝ≥0} {i : α} (h : ∀ (a : α), f a ≤ g a) (hi : f i < g i) (hf : HasSum f sf) (hg : HasSum g sg)
Goal: sf < sg
-- NNReal.hasSum_strict_mono
@[mono]
theorem hasSum_strict_mono {f g : α → ℝ≥0} {sf sg : ℝ≥0} (hf : HasSum f sf) (hg : HasSum g sg)
    (h : f < g) : sf < sg 
Arguments: {α : Type u_1} {f g : α → ℝ≥0} {sf sg : ℝ≥0} (hf : HasSum f sf) (hg : HasSum g sg) (h : f < g)
Goal: sf < sg
-- NNReal.tsum_lt_tsum
theorem tsum_lt_tsum {f g : α → ℝ≥0} {i : α} (h : ∀ a : α, f a ≤ g a) (hi : f i < g i)
    (hg : Summable g) : ∑' n, f n < ∑' n, g n 
Arguments: {α : Type u_1} {f g : α → ℝ≥0} {i : α} (h : ∀ (a : α), f a ≤ g a) (hi : f i < g i) (hg : Summable g)
Goal: ∑' (n : α), f n < ∑' (n : α), g n
-- NNReal.tsum_strict_mono
@[mono]
theorem tsum_strict_mono {f g : α → ℝ≥0} (hg : Summable g) (h : f < g) : ∑' n, f n < ∑' n, g n 
Arguments: {α : Type u_1} {f g : α → ℝ≥0} (hg : Summable g) (h : f < g)
Goal: ∑' (n : α), f n < ∑' (n : α), g n
-- NNReal.tsum_pos
theorem tsum_pos {g : α → ℝ≥0} (hg : Summable g) (i : α) (hi : 0 < g i) : 0 < ∑' b, g b 
Arguments: {α : Type u_1} {g : α → ℝ≥0} (hg : Summable g) (i : α) (hi : 0 < g i)
Goal: 0 < ∑' (b : α), g b
-- NNReal.tsum_eq_add_tsum_ite
theorem tsum_eq_add_tsum_ite {f : α → ℝ≥0} (hf : Summable f) (i : α) :
    ∑' x, f x = f i + ∑' x, ite (x = i) 0 (f x) 
Arguments: {α : Type u_1} {f : α → ℝ≥0} (hf : Summable f) (i : α)
Goal: ∑' (x : α), f x = f i + ∑' (x : α), if x = i then 0 else f x
-- ENNReal.tsum_toNNReal_eq
theorem tsum_toNNReal_eq {f : α → ℝ≥0∞} (hf : ∀ a, f a ≠ ∞) :
    (∑' a, f a).toNNReal = ∑' a, (f a).toNNReal 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (hf : ∀ (a : α), f a ≠ ⊤)
Goal: (∑' (a : α), f a).toNNReal = ∑' (a : α), (f a).toNNReal
-- ENNReal.tsum_toReal_eq
theorem tsum_toReal_eq {f : α → ℝ≥0∞} (hf : ∀ a, f a ≠ ∞) :
    (∑' a, f a).toReal = ∑' a, (f a).toReal 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (hf : ∀ (a : α), f a ≠ ⊤)
Goal: (∑' (a : α), f a).toReal = ∑' (a : α), (f a).toReal
-- ENNReal.tendsto_sum_nat_add
theorem tendsto_sum_nat_add (f : ℕ → ℝ≥0∞) (hf : ∑' i, f i ≠ ∞) :
    Tendsto (fun i => ∑' k, f (k + i)) atTop (𝓝 0) 
Arguments: (f : ℕ → ℝ≥0∞) (hf : ∑' (i : ℕ), f i ≠ ⊤)
Goal: Filter.Tendsto (fun i => ∑' (k : ℕ), f (k + i)) Filter.atTop (𝓝 0)
-- ENNReal.tsum_le_of_sum_range_le
theorem tsum_le_of_sum_range_le {f : ℕ → ℝ≥0∞} {c : ℝ≥0∞}
    (h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : ∑' n, f n ≤ c 
Arguments: {f : ℕ → ℝ≥0∞} {c : ℝ≥0∞} (h : ∀ (n : ℕ), ∑ i ∈ Finset.range n, f i ≤ c)
Goal: ∑' (n : ℕ), f n ≤ c
-- ENNReal.hasSum_lt
theorem hasSum_lt {f g : α → ℝ≥0∞} {sf sg : ℝ≥0∞} {i : α} (h : ∀ a : α, f a ≤ g a) (hi : f i < g i)
    (hsf : sf ≠ ∞) (hf : HasSum f sf) (hg : HasSum g sg) : sf < sg 
Arguments: {α : Type u_1} {f g : α → ℝ≥0∞} {sf sg : ℝ≥0∞} {i : α} (h : ∀ (a : α), f a ≤ g a) (hi : f i < g i) (hsf : sf ≠ ⊤) (hf : HasSum f sf) (hg : HasSum g sg)
Goal: sf < sg
-- ENNReal.tsum_lt_tsum
theorem tsum_lt_tsum {f g : α → ℝ≥0∞} {i : α} (hfi : tsum f ≠ ∞) (h : ∀ a : α, f a ≤ g a)
    (hi : f i < g i) : ∑' x, f x < ∑' x, g x 
Arguments: {α : Type u_1} {f g : α → ℝ≥0∞} {i : α} (hfi : tsum f ≠ ⊤) (h : ∀ (a : α), f a ≤ g a) (hi : f i < g i)
Goal: ∑' (x : α), f x < ∑' (x : α), g x
-- tsum_comp_le_tsum_of_inj
theorem tsum_comp_le_tsum_of_inj {β : Type*} {f : α → ℝ} (hf : Summable f) (hn : ∀ a, 0 ≤ f a)
    {i : β → α} (hi : Function.Injective i) : tsum (f ∘ i) ≤ tsum f 
Arguments: {α : Type u_1} {β : Type u_4} {f : α → ℝ} (hf : Summable f) (hn : ∀ (a : α), 0 ≤ f a) {i : β → α} (hi : Function.Injective i)
Goal: tsum (f ∘ i) ≤ tsum f
-- Summable.of_nonneg_of_le
theorem Summable.of_nonneg_of_le {f g : β → ℝ} (hg : ∀ b, 0 ≤ g b) (hgf : ∀ b, g b ≤ f b)
    (hf : Summable f) : Summable g 
Arguments: {β : Type u_2} {f g : β → ℝ} (hg : ∀ (b : β), 0 ≤ g b) (hgf : ∀ (b : β), g b ≤ f b) (hf : Summable f)
Goal: Summable g
-- Summable.toNNReal
theorem Summable.toNNReal {f : α → ℝ} (hf : Summable f) : Summable fun n => (f n).toNNReal 
Arguments: {α : Type u_1} {f : α → ℝ} (hf : Summable f)
Goal: Summable fun n => (f n).toNNReal
-- Summable.countable_support_ennreal
theorem _root_.Summable.countable_support_ennreal {f : α → ℝ≥0∞} (h : ∑' (i : α), f i ≠ ∞) :
    f.support.Countable 
Arguments: {α : Type u_1} {f : α → ℝ≥0∞} (h : ∑' (i : α), f i ≠ ⊤)
Goal: (Function.support f).Countable
-- hasSum_iff_tendsto_nat_of_nonneg
theorem hasSum_iff_tendsto_nat_of_nonneg {f : ℕ → ℝ} (hf : ∀ i, 0 ≤ f i) (r : ℝ) :
    HasSum f r ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 r) 
Arguments: {f : ℕ → ℝ} (hf : ∀ (i : ℕ), 0 ≤ f i) (r : ℝ)
Goal: HasSum f r ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 r)
-- ENNReal.ofReal_tsum_of_nonneg
theorem ENNReal.ofReal_tsum_of_nonneg {f : α → ℝ} (hf_nonneg : ∀ n, 0 ≤ f n) (hf : Summable f) :
    ENNReal.ofReal (∑' n, f n) = ∑' n, ENNReal.ofReal (f n) 
Arguments: {α : Type u_1} {f : α → ℝ} (hf_nonneg : ∀ (n : α), 0 ≤ f n) (hf : Summable f)
Goal: ENNReal.ofReal (∑' (n : α), f n) = ∑' (n : α), ENNReal.ofReal (f n)
-- edist_ne_top_of_mem_ball
theorem edist_ne_top_of_mem_ball {a : β} {r : ℝ≥0∞} (x y : ball a r) : edist x.1 y.1 ≠ ∞ 
Arguments: {β : Type u_2} [EMetricSpace β] {a : β} {r : ℝ≥0∞} (x y : ↑(EMetric.ball a r))
Goal: edist ↑x ↑y ≠ ⊤
-- nhds_eq_nhds_emetric_ball
theorem nhds_eq_nhds_emetric_ball (a x : β) (r : ℝ≥0∞) (h : x ∈ ball a r) :
    𝓝 x = map ((↑) : ball a r → β) (𝓝 ⟨x, h⟩) 
Arguments: {β : Type u_2} [EMetricSpace β] (a x : β) (r : ℝ≥0∞) (h : x ∈ EMetric.ball a r)
Goal: 𝓝 x = Filter.map Subtype.val (𝓝 ⟨x, h⟩)
-- tendsto_iff_edist_tendsto_0
theorem tendsto_iff_edist_tendsto_0 {l : Filter β} {f : β → α} {y : α} :
    Tendsto f l (𝓝 y) ↔ Tendsto (fun x => edist (f x) y) l (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] {l : Filter β} {f : β → α} {y : α}
Goal: Filter.Tendsto f l (𝓝 y) ↔ Filter.Tendsto (fun x => edist (f x) y) l (𝓝 0)
-- EMetric.cauchySeq_iff_le_tendsto_0
theorem EMetric.cauchySeq_iff_le_tendsto_0 [Nonempty β] [SemilatticeSup β] {s : β → α} :
    CauchySeq s ↔ ∃ b : β → ℝ≥0∞, (∀ n m N : β, N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧
      Tendsto b atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [Nonempty β] [SemilatticeSup β] {s : β → α}
Goal: CauchySeq s ↔ ∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Filter.Tendsto b Filter.atTop (𝓝 0)
-- continuous_of_le_add_edist
theorem continuous_of_le_add_edist {f : α → ℝ≥0∞} (C : ℝ≥0∞) (hC : C ≠ ∞)
    (h : ∀ x y, f x ≤ f y + C * edist x y) : Continuous f 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {f : α → ℝ≥0∞} (C : ℝ≥0∞) (hC : C ≠ ⊤) (h : ∀ (x y : α), f x ≤ f y + C * edist x y)
Goal: Continuous f
-- continuous_edist
theorem continuous_edist : Continuous fun p : α × α => edist p.1 p.2 
Arguments: {α : Type u_1} [PseudoEMetricSpace α]
Goal: Continuous fun p => edist p.1 p.2
-- Continuous.edist
@[continuity, fun_prop]
theorem Continuous.edist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun b => edist (f b) (g b) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun b => edist (f b) (g b)
-- Filter.Tendsto.edist
theorem Filter.Tendsto.edist {f g : β → α} {x : Filter β} {a b : α} (hf : Tendsto f x (𝓝 a))
    (hg : Tendsto g x (𝓝 b)) : Tendsto (fun x => edist (f x) (g x)) x (𝓝 (edist a b)) 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoEMetricSpace α] {f g : β → α} {x : Filter β} {a b : α} (hf : Filter.Tendsto f x (𝓝 a)) (hg : Filter.Tendsto g x (𝓝 b))
Goal: Filter.Tendsto (fun x => edist (f x) (g x)) x (𝓝 (edist a b))
-- cauchySeq_of_edist_le_of_summable
theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
    (hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : Summable d) : CauchySeq f 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0) (hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ ↑(d n)) (hd : Summable d)
Goal: CauchySeq f
-- cauchySeq_of_edist_le_of_tsum_ne_top
theorem cauchySeq_of_edist_le_of_tsum_ne_top {f : ℕ → α} (d : ℕ → ℝ≥0∞)
    (hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : tsum d ≠ ∞) : CauchySeq f 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0∞) (hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ d n) (hd : tsum d ≠ ⊤)
Goal: CauchySeq f
-- EMetric.isClosed_ball
theorem EMetric.isClosed_ball {a : α} {r : ℝ≥0∞} : IsClosed (closedBall a r) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {a : α} {r : ℝ≥0∞}
Goal: IsClosed (EMetric.closedBall a r)
-- EMetric.diam_closure
@[simp]
theorem EMetric.diam_closure (s : Set α) : diam (closure s) = diam s 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (s : Set α)
Goal: EMetric.diam (closure s) = EMetric.diam s
-- Metric.diam_closure
@[simp]
theorem Metric.diam_closure {α : Type*} [PseudoMetricSpace α] (s : Set α) :
    Metric.diam (closure s) = diam s 
Arguments: {α : Type u_4} [PseudoMetricSpace α] (s : Set α)
Goal: Metric.diam (closure s) = Metric.diam s
-- isClosed_setOf_lipschitzOnWith
theorem isClosed_setOf_lipschitzOnWith {α β} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0)
    (s : Set α) : IsClosed { f : α → β | LipschitzOnWith K f s } 
Arguments: {α : Type u_4} {β : Type u_5} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0) (s : Set α)
Goal: IsClosed {f | LipschitzOnWith K f s}
-- isClosed_setOf_lipschitzWith
theorem isClosed_setOf_lipschitzWith {α β} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0) :
    IsClosed { f : α → β | LipschitzWith K f } 
Arguments: {α : Type u_4} {β : Type u_5} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0)
Goal: IsClosed {f | LipschitzWith K f}
-- Real.ediam_eq
theorem ediam_eq {s : Set ℝ} (h : Bornology.IsBounded s) :
    EMetric.diam s = ENNReal.ofReal (sSup s - sInf s) 
Arguments: {s : Set ℝ} (h : Bornology.IsBounded s)
Goal: EMetric.diam s = ENNReal.ofReal (sSup s - sInf s)
-- Real.diam_eq
theorem diam_eq {s : Set ℝ} (h : Bornology.IsBounded s) : Metric.diam s = sSup s - sInf s 
Arguments: {s : Set ℝ} (h : Bornology.IsBounded s)
Goal: Metric.diam s = sSup s - sInf s
-- Real.ediam_Ioo
@[simp]
theorem ediam_Ioo (a b : ℝ) : EMetric.diam (Ioo a b) = ENNReal.ofReal (b - a) 
Arguments: (a b : ℝ)
Goal: EMetric.diam (Set.Ioo a b) = ENNReal.ofReal (b - a)
-- Real.ediam_Icc
@[simp]
theorem ediam_Icc (a b : ℝ) : EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) 
Arguments: (a b : ℝ)
Goal: EMetric.diam (Set.Icc a b) = ENNReal.ofReal (b - a)
-- Real.ediam_Ico
@[simp]
theorem ediam_Ico (a b : ℝ) : EMetric.diam (Ico a b) = ENNReal.ofReal (b - a) 
Arguments: (a b : ℝ)
Goal: EMetric.diam (Set.Ico a b) = ENNReal.ofReal (b - a)
-- Real.ediam_Ioc
@[simp]
theorem ediam_Ioc (a b : ℝ) : EMetric.diam (Ioc a b) = ENNReal.ofReal (b - a) 
Arguments: (a b : ℝ)
Goal: EMetric.diam (Set.Ioc a b) = ENNReal.ofReal (b - a)
-- Real.diam_Icc
theorem diam_Icc {a b : ℝ} (h : a ≤ b) : Metric.diam (Icc a b) = b - a 
Arguments: {a b : ℝ} (h : a ≤ b)
Goal: Metric.diam (Set.Icc a b) = b - a
-- Real.diam_Ico
theorem diam_Ico {a b : ℝ} (h : a ≤ b) : Metric.diam (Ico a b) = b - a 
Arguments: {a b : ℝ} (h : a ≤ b)
Goal: Metric.diam (Set.Ico a b) = b - a
-- Real.diam_Ioc
theorem diam_Ioc {a b : ℝ} (h : a ≤ b) : Metric.diam (Ioc a b) = b - a 
Arguments: {a b : ℝ} (h : a ≤ b)
Goal: Metric.diam (Set.Ioc a b) = b - a
-- Real.diam_Ioo
theorem diam_Ioo {a b : ℝ} (h : a ≤ b) : Metric.diam (Ioo a b) = b - a 
Arguments: {a b : ℝ} (h : a ≤ b)
Goal: Metric.diam (Set.Ioo a b) = b - a
-- edist_le_tsum_of_edist_le_of_tendsto
theorem edist_le_tsum_of_edist_le_of_tendsto {f : ℕ → α} (d : ℕ → ℝ≥0∞)
    (hf : ∀ n, edist (f n) (f n.succ) ≤ d n) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
    edist (f n) a ≤ ∑' m, d (n + m) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0∞) (hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ d n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (n : ℕ)
Goal: edist (f n) a ≤ ∑' (m : ℕ), d (n + m)
-- edist_le_tsum_of_edist_le_of_tendsto₀
theorem edist_le_tsum_of_edist_le_of_tendsto₀ {f : ℕ → α} (d : ℕ → ℝ≥0∞)
    (hf : ∀ n, edist (f n) (f n.succ) ≤ d n) {a : α} (ha : Tendsto f atTop (𝓝 a)) :
    edist (f 0) a ≤ ∑' m, d m 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0∞) (hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ d n) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: edist (f 0) a ≤ ∑' (m : ℕ), d m
-- ENNReal.truncateToReal_eq_toReal
lemma truncateToReal_eq_toReal {t x : ℝ≥0∞} (t_ne_top : t ≠ ∞) (x_le : x ≤ t) :
    truncateToReal t x = x.toReal 
Arguments: {t x : ℝ≥0∞} (t_ne_top : t ≠ ⊤) (x_le : x ≤ t)
Goal: t.truncateToReal x = x.toReal
-- ENNReal.truncateToReal_le
lemma truncateToReal_le {t : ℝ≥0∞} (t_ne_top : t ≠ ∞) {x : ℝ≥0∞} :
    truncateToReal t x ≤ t.toReal 
Arguments: {t : ℝ≥0∞} (t_ne_top : t ≠ ⊤) {x : ℝ≥0∞}
Goal: t.truncateToReal x ≤ t.toReal
-- ENNReal.truncateToReal_nonneg
lemma truncateToReal_nonneg {t x : ℝ≥0∞} : 0 ≤ truncateToReal t x 
Arguments: {t x : ℝ≥0∞}
Goal: 0 ≤ t.truncateToReal x
-- ENNReal.monotone_truncateToReal
lemma monotone_truncateToReal {t : ℝ≥0∞} (t_ne_top : t ≠ ∞) : Monotone (truncateToReal t) 
Arguments: {t : ℝ≥0∞} (t_ne_top : t ≠ ⊤)
Goal: Monotone t.truncateToReal
-- ENNReal.continuous_truncateToReal
lemma continuous_truncateToReal {t : ℝ≥0∞} (t_ne_top : t ≠ ∞) : Continuous (truncateToReal t) 
Arguments: {t : ℝ≥0∞} (t_ne_top : t ≠ ⊤)
Goal: Continuous t.truncateToReal
-- ENNReal.limsup_sub_const
lemma limsup_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :
    Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c 
Arguments: {ι : Type u_4} (F : Filter ι) [F.NeBot] (f : ι → ℝ≥0∞) (c : ℝ≥0∞)
Goal: Filter.limsup (fun i => f i - c) F = Filter.limsup f F - c
-- ENNReal.liminf_sub_const
lemma liminf_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :
    Filter.liminf (fun i ↦ f i - c) F = Filter.liminf f F - c 
Arguments: {ι : Type u_4} (F : Filter ι) [F.NeBot] (f : ι → ℝ≥0∞) (c : ℝ≥0∞)
Goal: Filter.liminf (fun i => f i - c) F = Filter.liminf f F - c
-- ENNReal.limsup_const_sub
lemma limsup_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞)
    {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
    Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F 
Arguments: {ι : Type u_4} (F : Filter ι) [F.NeBot] (f : ι → ℝ≥0∞) {c : ℝ≥0∞} (c_ne_top : c ≠ ⊤)
Goal: Filter.limsup (fun i => c - f i) F = c - Filter.liminf f F
-- ENNReal.liminf_const_sub
lemma liminf_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞)
    {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
    Filter.liminf (fun i ↦ c - f i) F = c - Filter.limsup f F 
Arguments: {ι : Type u_4} (F : Filter ι) [F.NeBot] (f : ι → ℝ≥0∞) {c : ℝ≥0∞} (c_ne_top : c ≠ ⊤)
Goal: Filter.liminf (fun i => c - f i) F = c - Filter.limsup f F
-- ENNReal.liminf_toReal_eq
lemma liminf_toReal_eq {ι : Type*} {F : Filter ι} [NeBot F] {b : ℝ≥0∞} (b_ne_top : b ≠ ∞)
    {xs : ι → ℝ≥0∞} (le_b : ∀ᶠ i in F, xs i ≤ b) :
    F.liminf (fun i ↦ (xs i).toReal) = (F.liminf xs).toReal 
Arguments: {ι : Type u_5} {F : Filter ι} [F.NeBot] {b : ℝ≥0∞} (b_ne_top : b ≠ ⊤) {xs : ι → ℝ≥0∞} (le_b : ∀ᶠ (i : ι) in F, xs i ≤ b)
Goal: Filter.liminf (fun i => (xs i).toReal) F = (Filter.liminf xs F).toReal
-- ENNReal.limsup_toReal_eq
lemma limsup_toReal_eq {ι : Type*} {F : Filter ι} [NeBot F] {b : ℝ≥0∞} (b_ne_top : b ≠ ∞)
    {xs : ι → ℝ≥0∞} (le_b : ∀ᶠ i in F, xs i ≤ b) :
    F.limsup (fun i ↦ (xs i).toReal) = (F.limsup xs).toReal 
Arguments: {ι : Type u_5} {F : Filter ι} [F.NeBot] {b : ℝ≥0∞} (b_ne_top : b ≠ ⊤) {xs : ι → ℝ≥0∞} (le_b : ∀ᶠ (i : ι) in F, xs i ≤ b)
Goal: Filter.limsup (fun i => (xs i).toReal) F = (Filter.limsup xs F).toReal
-- ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn
theorem tendsto_iff_forall_compact_tendstoUniformlyOn
    {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f} :
    Tendsto F p (𝓝 f) ↔ ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f : C(α, β)}
Goal: Filter.Tendsto F p (𝓝 f) ↔ ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a => (F i) a) (⇑f) p K
-- ContinuousMap.toUniformOnFun_toFun
@[simp]
theorem toUniformOnFun_toFun (f : C(α, β)) :
    UniformOnFun.toFun _ f.toUniformOnFunIsCompact = f 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] (f : C(α, β))
Goal: (UniformOnFun.toFun {K | IsCompact K}) f.toUniformOnFunIsCompact = ⇑f
-- ContinuousMap.uniformEmbedding_toUniformOnFunIsCompact
theorem uniformEmbedding_toUniformOnFunIsCompact :
    UniformEmbedding (toUniformOnFunIsCompact : C(α, β) → α →ᵤ[{K | IsCompact K}] β) 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β]
Goal: UniformEmbedding ContinuousMap.toUniformOnFunIsCompact
-- Filter.HasBasis.compactConvergenceUniformity
theorem _root_.Filter.HasBasis.compactConvergenceUniformity {ι : Type*} {pi : ι → Prop}
    {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :
    HasBasis (𝓤 C(α, β)) (fun p : Set α × ι => IsCompact p.1 ∧ pi p.2) fun p =>
      { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2 } 
Arguments: {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {ι : Type u_1} {pi : ι → Prop} {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s)
Goal: (𝓤 C(α, β)).HasBasis (fun p => IsCompact p.1 ∧ pi p.2) fun p => {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2}