ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- tsum_mul_right
theorem tsum_mul_right [T2Space α] : ∑' x, f x * a = (∑' x, f x) * a 
Arguments: {ι : Type u_1} {α : Type u_4} [DivisionSemiring α] [TopologicalSpace α] [TopologicalSemiring α] {f : ι → α} {a : α} [T2Space α]
Goal: ∑' (x : ι), f x * a = (∑' (x : ι), f x) * a
-- tsum_div_const
theorem tsum_div_const [T2Space α] : ∑' x, f x / a = (∑' x, f x) / a 
Arguments: {ι : Type u_1} {α : Type u_4} [DivisionSemiring α] [TopologicalSpace α] [TopologicalSemiring α] {f : ι → α} {a : α} [T2Space α]
Goal: ∑' (x : ι), f x / a = (∑' (x : ι), f x) / a
-- HasSum.mul_eq
theorem HasSum.mul_eq (hf : HasSum f s) (hg : HasSum g t)
    (hfg : HasSum (fun x : ι × κ ↦ f x.1 * g x.2) u) : s * t = u 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_4} [TopologicalSpace α] [T3Space α] [NonUnitalNonAssocSemiring α] [TopologicalSemiring α] {f : ι → α} {g : κ → α} {s t u : α} (hf : HasSum f s) (hg : HasSum g t) (hfg : HasSum (fun x => f x.1 * g x.2) u)
Goal: s * t = u
-- HasSum.mul
theorem HasSum.mul (hf : HasSum f s) (hg : HasSum g t)
    (hfg : Summable fun x : ι × κ ↦ f x.1 * g x.2) :
    HasSum (fun x : ι × κ ↦ f x.1 * g x.2) (s * t) 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_4} [TopologicalSpace α] [T3Space α] [NonUnitalNonAssocSemiring α] [TopologicalSemiring α] {f : ι → α} {g : κ → α} {s t : α} (hf : HasSum f s) (hg : HasSum g t) (hfg : Summable fun x => f x.1 * g x.2)
Goal: HasSum (fun x => f x.1 * g x.2) (s * t)
-- tsum_mul_tsum
theorem tsum_mul_tsum (hf : Summable f) (hg : Summable g)
    (hfg : Summable fun x : ι × κ ↦ f x.1 * g x.2) :
    ((∑' x, f x) * ∑' y, g y) = ∑' z : ι × κ, f z.1 * g z.2 
Arguments: {ι : Type u_1} {κ : Type u_2} {α : Type u_4} [TopologicalSpace α] [T3Space α] [NonUnitalNonAssocSemiring α] [TopologicalSemiring α] {f : ι → α} {g : κ → α} (hf : Summable f) (hg : Summable g) (hfg : Summable fun x => f x.1 * g x.2)
Goal: (∑' (x : ι), f x) * ∑' (y : κ), g y = ∑' (z : ι × κ), f z.1 * g z.2
-- summable_mul_prod_iff_summable_mul_sigma_antidiagonal
theorem summable_mul_prod_iff_summable_mul_sigma_antidiagonal :
    (Summable fun x : A × A ↦ f x.1 * g x.2) ↔
      Summable fun x : Σn : A, antidiagonal n ↦ f (x.2 : A × A).1 * g (x.2 : A × A).2 
Arguments: {α : Type u_4} {A : Type u_5} [AddCommMonoid A] [Finset.HasAntidiagonal A] [TopologicalSpace α] [NonUnitalNonAssocSemiring α] {f g : A → α}
Goal: (Summable fun x => f x.1 * g x.2) ↔ Summable fun x => f (↑x.snd).1 * g (↑x.snd).2
-- summable_sum_mul_antidiagonal_of_summable_mul
theorem summable_sum_mul_antidiagonal_of_summable_mul
    (h : Summable fun x : A × A ↦ f x.1 * g x.2) :
    Summable fun n ↦ ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 
Arguments: {α : Type u_4} {A : Type u_5} [AddCommMonoid A] [Finset.HasAntidiagonal A] [TopologicalSpace α] [NonUnitalNonAssocSemiring α] {f g : A → α} [T3Space α] [TopologicalSemiring α] (h : Summable fun x => f x.1 * g x.2)
Goal: Summable fun n => ∑ kl ∈ Finset.antidiagonal n, f kl.1 * g kl.2
-- tsum_mul_tsum_eq_tsum_sum_antidiagonal
theorem tsum_mul_tsum_eq_tsum_sum_antidiagonal (hf : Summable f) (hg : Summable g)
    (hfg : Summable fun x : A × A ↦ f x.1 * g x.2) :
    ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 
Arguments: {α : Type u_4} {A : Type u_5} [AddCommMonoid A] [Finset.HasAntidiagonal A] [TopologicalSpace α] [NonUnitalNonAssocSemiring α] {f g : A → α} [T3Space α] [TopologicalSemiring α] (hf : Summable f) (hg : Summable g) (hfg : Summable fun x => f x.1 * g x.2)
Goal: (∑' (n : A), f n) * ∑' (n : A), g n = ∑' (n : A), ∑ kl ∈ Finset.antidiagonal n, f kl.1 * g kl.2
-- summable_sum_mul_range_of_summable_mul
theorem summable_sum_mul_range_of_summable_mul (h : Summable fun x : ℕ × ℕ ↦ f x.1 * g x.2) :
    Summable fun n ↦ ∑ k ∈ range (n + 1), f k * g (n - k) 
Arguments: {α : Type u_4} [TopologicalSpace α] [NonUnitalNonAssocSemiring α] {f g : ℕ → α} [T3Space α] [TopologicalSemiring α] (h : Summable fun x => f x.1 * g x.2)
Goal: Summable fun n => ∑ k ∈ Finset.range (n + 1), f k * g (n - k)
-- tsum_mul_tsum_eq_tsum_sum_range
theorem tsum_mul_tsum_eq_tsum_sum_range (hf : Summable f) (hg : Summable g)
    (hfg : Summable fun x : ℕ × ℕ ↦ f x.1 * g x.2) :
    ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ k ∈ range (n + 1), f k * g (n - k) 
Arguments: {α : Type u_4} [TopologicalSpace α] [NonUnitalNonAssocSemiring α] {f g : ℕ → α} [T3Space α] [TopologicalSemiring α] (hf : Summable f) (hg : Summable g) (hfg : Summable fun x => f x.1 * g x.2)
Goal: (∑' (n : ℕ), f n) * ∑' (n : ℕ), g n = ∑' (n : ℕ), ∑ k ∈ Finset.range (n + 1), f k * g (n - k)
-- PseudoMetricSpace.dist_ofPreNNDist
theorem dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
    (dist_comm : ∀ x y, d x y = d y x) (x y : X) :
    @dist X (@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm)) x
        y =
      ↑(⨅ l : List X, ((x::l).zipWith d (l ++ [y])).sum : ℝ≥0) 
Arguments: {X : Type u_1} (d : X → X → ℝ≥0) (dist_self : ∀ (x : X), d x x = 0) (dist_comm : ∀ (x y : X), d x y = d y x) (x y : X)
Goal: dist x y = ↑(⨅ l, (List.zipWith d (x :: l) (l ++ [y])).sum)
-- PseudoMetricSpace.dist_ofPreNNDist_le
theorem dist_ofPreNNDist_le (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
    (dist_comm : ∀ x y, d x y = d y x) (x y : X) :
    @dist X (@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm)) x
        y ≤
      d x y 
Arguments: {X : Type u_1} (d : X → X → ℝ≥0) (dist_self : ∀ (x : X), d x x = 0) (dist_comm : ∀ (x y : X), d x y = d y x) (x y : X)
Goal: dist x y ≤ ↑(d x y)
-- PseudoMetricSpace.le_two_mul_dist_ofPreNNDist
theorem le_two_mul_dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
    (dist_comm : ∀ x y, d x y = d y x)
    (hd : ∀ x₁ x₂ x₃ x₄, d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))) (x y : X) :
    ↑(d x y) ≤ 2 * @dist X
      (@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm)) x y 
Arguments: {X : Type u_1} (d : X → X → ℝ≥0) (dist_self : ∀ (x : X), d x x = 0) (dist_comm : ∀ (x y : X), d x y = d y x) (hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))) (x y : X)
Goal: ↑(d x y) ≤ 2 * dist x y
-- UniformSpace.metrizable_uniformity
protected theorem UniformSpace.metrizable_uniformity (X : Type*) [UniformSpace X]
    [IsCountablyGenerated (𝓤 X)] : ∃ I : PseudoMetricSpace X, I.toUniformSpace = ‹_› 
Arguments: (X : Type u_2) [UniformSpace X] [(𝓤 X).IsCountablyGenerated]
Goal: ∃ I, PseudoMetricSpace.toUniformSpace = inst✝
-- UniformSpace.metrizableSpace
theorem UniformSpace.metrizableSpace [UniformSpace X] [IsCountablyGenerated (𝓤 X)] [T0Space X] :
    TopologicalSpace.MetrizableSpace X 
Arguments: {X : Type u_1} [UniformSpace X] [(𝓤 X).IsCountablyGenerated] [T0Space X]
Goal: TopologicalSpace.MetrizableSpace X
-- TotallyBounded.isSeparable
lemma TotallyBounded.isSeparable [UniformSpace X] [i : IsCountablyGenerated (𝓤 X)]
    {s : Set X} (h : TotallyBounded s) : TopologicalSpace.IsSeparable s 
Arguments: {X : Type u_1} [UniformSpace X] [i : (𝓤 X).IsCountablyGenerated] {s : Set X} (h : TotallyBounded s)
Goal: TopologicalSpace.IsSeparable s
-- hasSum_iff_hasSum_compl
theorem hasSum_iff_hasSum_compl (f : β → α) (a : α) :
    HasSum (toCompl ∘ f) a ↔ HasSum f a 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] (f : β → α) (a : α)
Goal: HasSum (⇑UniformSpace.Completion.toCompl ∘ f) (↑α a) ↔ HasSum f a
-- summable_iff_summable_compl_and_tsum_mem
theorem summable_iff_summable_compl_and_tsum_mem (f : β → α) :
    Summable f ↔ Summable (toCompl ∘ f) ∧ ∑' i, toCompl (f i) ∈ Set.range toCompl 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] (f : β → α)
Goal: Summable f ↔
  Summable (⇑UniformSpace.Completion.toCompl ∘ f) ∧
    ∑' (i : β), UniformSpace.Completion.toCompl (f i) ∈ Set.range ⇑UniformSpace.Completion.toCompl
-- summable_iff_cauchySeq_finset_and_tsum_mem
theorem summable_iff_cauchySeq_finset_and_tsum_mem (f : β → α) :
    Summable f ↔ CauchySeq (fun s : Finset β ↦ ∑ b in s, f b) ∧
      ∑' i, toCompl (f i) ∈ Set.range toCompl 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] (f : β → α)
Goal: Summable f ↔
  (CauchySeq fun s => ∑ b ∈ s, f b) ∧
    ∑' (i : β), UniformSpace.Completion.toCompl (f i) ∈ Set.range ⇑UniformSpace.Completion.toCompl
-- Summable.toCompl_tsum
theorem Summable.toCompl_tsum {f : β → α} (hf : Summable f) : ∑' i, toCompl (f i) = ∑' i, f i 
Arguments: {α : Type u_1} {β : Type u_2} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α] {f : β → α} (hf : Summable f)
Goal: ∑' (i : β), UniformSpace.Completion.toCompl (f i) = ↑α (∑' (i : β), f i)
-- cauchySeq_of_dist_le_of_summable
theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
    (hd : Summable d) : CauchySeq f 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ) (hf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n) (hd : Summable d)
Goal: CauchySeq f
-- cauchySeq_of_summable_dist
theorem cauchySeq_of_summable_dist (h : Summable fun n ↦ dist (f n) (f n.succ)) : CauchySeq f 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {f : ℕ → α} (h : Summable fun n => dist (f n) (f n.succ))
Goal: CauchySeq f
-- dist_le_tsum_of_dist_le_of_tendsto
theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
    (hd : Summable d) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
    dist (f n) a ≤ ∑' m, d (n + m) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ) (hf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n) (hd : Summable d) {a : α} (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (n : ℕ)
Goal: dist (f n) a ≤ ∑' (m : ℕ), d (n + m)
-- dist_le_tsum_of_dist_le_of_tendsto₀
theorem dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
    (hd : Summable d) (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ tsum d 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {f : ℕ → α} {a : α} (d : ℕ → ℝ) (hf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n) (hd : Summable d) (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: dist (f 0) a ≤ tsum d
-- dist_le_tsum_dist_of_tendsto
theorem dist_le_tsum_dist_of_tendsto (h : Summable fun n ↦ dist (f n) (f n.succ))
    (ha : Tendsto f atTop (𝓝 a)) (n) : dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {f : ℕ → α} {a : α} (h : Summable fun n => dist (f n) (f n.succ)) (ha : Filter.Tendsto f Filter.atTop (𝓝 a)) (n : ℕ)
Goal: dist (f n) a ≤ ∑' (m : ℕ), dist (f (n + m)) (f (n + m).succ)
-- dist_le_tsum_dist_of_tendsto₀
theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n ↦ dist (f n) (f n.succ))
    (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {f : ℕ → α} {a : α} (h : Summable fun n => dist (f n) (f n.succ)) (ha : Filter.Tendsto f Filter.atTop (𝓝 a))
Goal: dist (f 0) a ≤ ∑' (n : ℕ), dist (f n) (f n.succ)
-- not_summable_iff_tendsto_nat_atTop_of_nonneg
theorem not_summable_iff_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
    ¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop 
Arguments: {f : ℕ → ℝ} (hf : ∀ (n : ℕ), 0 ≤ f n)
Goal: ¬Summable f ↔ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop Filter.atTop
-- summable_iff_not_tendsto_nat_atTop_of_nonneg
theorem summable_iff_not_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
    Summable f ↔ ¬Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop 
Arguments: {f : ℕ → ℝ} (hf : ∀ (n : ℕ), 0 ≤ f n)
Goal: Summable f ↔ ¬Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop Filter.atTop
-- summable_sigma_of_nonneg
theorem summable_sigma_of_nonneg {β : α → Type*} {f : (Σ x, β x) → ℝ} (hf : ∀ x, 0 ≤ f x) :
    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_3} {f : (x : α) × β x → ℝ} (hf : ∀ (x : (x : α) × β x), 0 ≤ f x)
Goal: Summable f ↔ (∀ (x : α), Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' (y : β x), f ⟨x, y⟩
-- summable_partition
lemma summable_partition {α β : Type*} {f : β → ℝ} (hf : 0 ≤ f) {s : α  → Set β}
    (hs : ∀ i, ∃! j, i ∈ s j) : Summable f ↔
      (∀ j, Summable fun i : s j ↦ f i) ∧ Summable fun j ↦ ∑' i : s j, f i 
Arguments: {α : Type u_3} {β : Type u_4} {f : β → ℝ} (hf : 0 ≤ f) {s : α → Set β} (hs : ∀ (i : β), ∃! j, i ∈ s j)
Goal: Summable f ↔ (∀ (j : α), Summable fun i => f ↑i) ∧ Summable fun j => ∑' (i : ↑(s j)), f ↑i
-- summable_prod_of_nonneg
theorem summable_prod_of_nonneg {f : (α × β) → ℝ} (hf : 0 ≤ f) :
    Summable f ↔ (∀ x, Summable fun y ↦ f (x, y)) ∧ Summable fun x ↦ ∑' y, f (x, y) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α × β → ℝ} (hf : 0 ≤ f)
Goal: Summable f ↔ (∀ (x : α), Summable fun y => f (x, y)) ∧ Summable fun x => ∑' (y : β), f (x, y)
-- summable_of_sum_le
theorem summable_of_sum_le {ι : Type*} {f : ι → ℝ} {c : ℝ} (hf : 0 ≤ f)
    (h : ∀ u : Finset ι, ∑ x ∈ u, f x ≤ c) : Summable f 
Arguments: {ι : Type u_3} {f : ι → ℝ} {c : ℝ} (hf : 0 ≤ f) (h : ∀ (u : Finset ι), ∑ x ∈ u, f x ≤ c)
Goal: Summable f
-- summable_of_sum_range_le
theorem summable_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
    (h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : Summable f 
Arguments: {f : ℕ → ℝ} {c : ℝ} (hf : ∀ (n : ℕ), 0 ≤ f n) (h : ∀ (n : ℕ), ∑ i ∈ Finset.range n, f i ≤ c)
Goal: Summable f
-- Real.tsum_le_of_sum_range_le
theorem Real.tsum_le_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
    (h : ∀ n, ∑ i ∈ Finset.range n, f i ≤ c) : ∑' n, f n ≤ c 
Arguments: {f : ℕ → ℝ} {c : ℝ} (hf : ∀ (n : ℕ), 0 ≤ f n) (h : ∀ (n : ℕ), ∑ i ∈ Finset.range n, f i ≤ c)
Goal: ∑' (n : ℕ), f n ≤ c
-- tsum_lt_tsum_of_nonneg
theorem tsum_lt_tsum_of_nonneg {i : ℕ} {f g : ℕ → ℝ} (h0 : ∀ b : ℕ, 0 ≤ f b)
    (h : ∀ b : ℕ, f b ≤ g b) (hi : f i < g i) (hg : Summable g) : ∑' n, f n < ∑' n, g n 
Arguments: {i : ℕ} {f g : ℕ → ℝ} (h0 : ∀ (b : ℕ), 0 ≤ f b) (h : ∀ (b : ℕ), f b ≤ g b) (hi : f i < g i) (hg : Summable g)
Goal: ∑' (n : ℕ), f n < ∑' (n : ℕ), g n
-- CompactIccSpace.isCompact_Icc
class CompactIccSpace (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
    isCompact_Icc : ∀ {a b : α}, IsCompact (Icc a b)
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : CompactIccSpace α] {a b : α}
Goal: IsCompact (Set.Icc a b)
-- CompactIccSpace.mk'
lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]
    (h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] (h : ∀ {a b : α}, a ≤ b → IsCompact (Set.Icc a b))
Goal: CompactIccSpace α
-- CompactIccSpace.mk''
lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]
    (h : ∀ {a b : α}, a < b → IsCompact (Icc a b)) : CompactIccSpace α 
Arguments: {α : Type u_1} [TopologicalSpace α] [PartialOrder α] (h : ∀ {a b : α}, a < b → IsCompact (Set.Icc a b))
Goal: CompactIccSpace α
-- isCompact_uIcc
theorem isCompact_uIcc {α : Type*} [LinearOrder α] [TopologicalSpace α] [CompactIccSpace α]
    {a b : α} : IsCompact (uIcc a b) 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [CompactIccSpace α] {a b : α}
Goal: IsCompact (Set.uIcc a b)
-- IsCompact.exists_isLeast
theorem IsCompact.exists_isLeast [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : ∃ x, IsLeast s x 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: ∃ x, IsLeast s x
-- IsCompact.exists_isGreatest
theorem IsCompact.exists_isGreatest [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : ∃ x, IsGreatest s x 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: ∃ x, IsGreatest s x
-- IsCompact.exists_isGLB
theorem IsCompact.exists_isGLB [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : ∃ x ∈ s, IsGLB s x 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: ∃ x ∈ s, IsGLB s x
-- IsCompact.exists_isLUB
theorem IsCompact.exists_isLUB [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : ∃ x ∈ s, IsLUB s x 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: ∃ x ∈ s, IsLUB s x
-- cocompact_le_atBot_atTop
theorem cocompact_le_atBot_atTop [CompactIccSpace α] :
    cocompact α ≤ atBot ⊔ atTop 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [CompactIccSpace α]
Goal: Filter.cocompact α ≤ Filter.atBot ⊔ Filter.atTop
-- cocompact_le_atBot
theorem cocompact_le_atBot [OrderTop α] [CompactIccSpace α] :
    cocompact α ≤ atBot 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderTop α] [CompactIccSpace α]
Goal: Filter.cocompact α ≤ Filter.atBot
-- cocompact_le_atTop
theorem cocompact_le_atTop [OrderBot α] [CompactIccSpace α] :
    cocompact α ≤ atTop 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [OrderBot α] [CompactIccSpace α]
Goal: Filter.cocompact α ≤ Filter.atTop
-- atBot_le_cocompact
theorem atBot_le_cocompact [NoMinOrder α] [ClosedIicTopology α] :
    atBot ≤ cocompact α 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [NoMinOrder α] [ClosedIicTopology α]
Goal: Filter.atBot ≤ Filter.cocompact α
-- atTop_le_cocompact
theorem atTop_le_cocompact [NoMaxOrder α] [ClosedIciTopology α] :
    atTop ≤ cocompact α 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [NoMaxOrder α] [ClosedIciTopology α]
Goal: Filter.atTop ≤ Filter.cocompact α
-- atBot_atTop_le_cocompact
theorem atBot_atTop_le_cocompact [NoMinOrder α] [NoMaxOrder α]
    [OrderClosedTopology α] : atBot ⊔ atTop ≤ cocompact α 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [NoMinOrder α] [NoMaxOrder α] [OrderClosedTopology α]
Goal: Filter.atBot ⊔ Filter.atTop ≤ Filter.cocompact α
-- cocompact_eq_atBot_atTop
@[simp 900]
theorem cocompact_eq_atBot_atTop [NoMaxOrder α] [NoMinOrder α]
    [OrderClosedTopology α] [CompactIccSpace α] : cocompact α = atBot ⊔ atTop 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [NoMaxOrder α] [NoMinOrder α] [OrderClosedTopology α] [CompactIccSpace α]
Goal: Filter.cocompact α = Filter.atBot ⊔ Filter.atTop
-- cocompact_eq_atBot
@[simp]
theorem cocompact_eq_atBot [NoMinOrder α] [OrderTop α]
    [ClosedIicTopology α] [CompactIccSpace α] : cocompact α = atBot 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [NoMinOrder α] [OrderTop α] [ClosedIicTopology α] [CompactIccSpace α]
Goal: Filter.cocompact α = Filter.atBot
-- cocompact_eq_atTop
@[simp]
theorem cocompact_eq_atTop [NoMaxOrder α] [OrderBot α]
    [ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [NoMaxOrder α] [OrderBot α] [ClosedIciTopology α] [CompactIccSpace α]
Goal: Filter.cocompact α = Filter.atTop
-- IsCompact.exists_isMinOn
theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s)
Goal: ∃ x ∈ s, IsMinOn f s x
-- IsCompact.exists_forall_le'
theorem IsCompact.exists_forall_le' [ClosedIicTopology α] [NoMaxOrder α] {f : β → α}
    {s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b) :
    ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] [NoMaxOrder α] {f : β → α} {s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b)
Goal: ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b
-- IsCompact.exists_forall_le
@[deprecated IsCompact.exists_isMinOn (since := "2023-02-06")]
theorem IsCompact.exists_forall_le [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s)
Goal: ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y
-- IsCompact.exists_isMaxOn
theorem IsCompact.exists_isMaxOn [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMaxOn f s x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s)
Goal: ∃ x ∈ s, IsMaxOn f s x
-- IsCompact.exists_forall_ge
@[deprecated IsCompact.exists_isMaxOn (since := "2023-02-06")]
theorem IsCompact.exists_forall_ge [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s)
Goal: ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x
-- ContinuousOn.exists_isMinOn'
theorem ContinuousOn.exists_isMinOn' [ClosedIicTopology α] {s : Set β} {f : β → α}
    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) : ∃ x ∈ s, IsMinOn f s x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {s : Set β} {f : β → α} (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ (x : β) in Filter.cocompact β ⊓ 𝓟 s, f x₀ ≤ f x)
Goal: ∃ x ∈ s, IsMinOn f s x
-- ContinuousOn.exists_forall_le'
@[deprecated ContinuousOn.exists_isMinOn' (since := "2023-02-06")]
theorem ContinuousOn.exists_forall_le' [ClosedIicTopology α] {s : Set β} {f : β → α}
    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {s : Set β} {f : β → α} (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ (x : β) in Filter.cocompact β ⊓ 𝓟 s, f x₀ ≤ f x)
Goal: ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y
-- ContinuousOn.exists_isMaxOn'
theorem ContinuousOn.exists_isMaxOn' [ClosedIciTopology α] {s : Set β} {f : β → α}
    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) : ∃ x ∈ s, IsMaxOn f s x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {s : Set β} {f : β → α} (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ (x : β) in Filter.cocompact β ⊓ 𝓟 s, f x ≤ f x₀)
Goal: ∃ x ∈ s, IsMaxOn f s x
-- ContinuousOn.exists_forall_ge'
@[deprecated ContinuousOn.exists_isMaxOn' (since := "2023-02-06")]
theorem ContinuousOn.exists_forall_ge' [ClosedIciTopology α] {s : Set β} {f : β → α}
    (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s)
    (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {s : Set β} {f : β → α} (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ (x : β) in Filter.cocompact β ⊓ 𝓟 s, f x ≤ f x₀)
Goal: ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x
-- Continuous.exists_forall_le'
theorem Continuous.exists_forall_le' [ClosedIicTopology α] {f : β → α} (hf : Continuous f)
    (x₀ : β) (h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ x : β, ∀ y : β, f x ≤ f y 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {f : β → α} (hf : Continuous f) (x₀ : β) (h : ∀ᶠ (x : β) in Filter.cocompact β, f x₀ ≤ f x)
Goal: ∃ x, ∀ (y : β), f x ≤ f y