ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContractingWith.fixedPoint_isFixedPt
theorem fixedPoint_isFixedPt : IsFixedPt f (fixedPoint f hf) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) [Nonempty α] [CompleteSpace α]
Goal: Function.IsFixedPt f (ContractingWith.fixedPoint f hf)
-- ContractingWith.fixedPoint_unique
theorem fixedPoint_unique {x} (hx : IsFixedPt f x) : x = fixedPoint f hf 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) [Nonempty α] [CompleteSpace α] {x : α} (hx : Function.IsFixedPt f x)
Goal: x = ContractingWith.fixedPoint f hf
-- ContractingWith.dist_fixedPoint_le
theorem dist_fixedPoint_le (x) : dist x (fixedPoint f hf) ≤ dist x (f x) / (1 - K) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) [Nonempty α] [CompleteSpace α] (x : α)
Goal: dist x (ContractingWith.fixedPoint f hf) ≤ dist x (f x) / (1 - ↑K)
-- ContractingWith.aposteriori_dist_iterate_fixedPoint_le
theorem aposteriori_dist_iterate_fixedPoint_le (x n) :
    dist (f^[n] x) (fixedPoint f hf) ≤ dist (f^[n] x) (f^[n + 1] x) / (1 - K) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) [Nonempty α] [CompleteSpace α] (x : α) (n : ℕ)
Goal: dist (f^[n] x) (ContractingWith.fixedPoint f hf) ≤ dist (f^[n] x) (f^[n + 1] x) / (1 - ↑K)
-- ContractingWith.apriori_dist_iterate_fixedPoint_le
theorem apriori_dist_iterate_fixedPoint_le (x n) :
    dist (f^[n] x) (fixedPoint f hf) ≤ dist x (f x) * (K : ℝ) ^ n / (1 - K) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) [Nonempty α] [CompleteSpace α] (x : α) (n : ℕ)
Goal: dist (f^[n] x) (ContractingWith.fixedPoint f hf) ≤ dist x (f x) * ↑K ^ n / (1 - ↑K)
-- ContractingWith.tendsto_iterate_fixedPoint
theorem tendsto_iterate_fixedPoint (x) :
    Tendsto (fun n ↦ f^[n] x) atTop (𝓝 <| fixedPoint f hf) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) [Nonempty α] [CompleteSpace α] (x : α)
Goal: Filter.Tendsto (fun n => f^[n] x) Filter.atTop (𝓝 (ContractingWith.fixedPoint f hf))
-- ContractingWith.fixedPoint_lipschitz_in_map
theorem fixedPoint_lipschitz_in_map {g : α → α} (hg : ContractingWith K g) {C}
    (hfg : ∀ z, dist (f z) (g z) ≤ C) : dist (fixedPoint f hf) (fixedPoint g hg) ≤ C / (1 - K) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) [Nonempty α] [CompleteSpace α] {g : α → α} (hg : ContractingWith K g) {C : ℝ} (hfg : ∀ (z : α), dist (f z) (g z) ≤ C)
Goal: dist (ContractingWith.fixedPoint f hf) (ContractingWith.fixedPoint g hg) ≤ C / (1 - ↑K)
-- ContractingWith.isFixedPt_fixedPoint_iterate
theorem isFixedPt_fixedPoint_iterate {n : ℕ} (hf : ContractingWith K f^[n]) :
    IsFixedPt f (hf.fixedPoint f^[n]) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} [Nonempty α] [CompleteSpace α] {n : ℕ} (hf : ContractingWith K f^[n])
Goal: Function.IsFixedPt f (ContractingWith.fixedPoint f^[n] hf)
-- Equicontinuous.comap_uniformFun_eq
theorem Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicontinuous F) :
    (UniformFun.uniformSpace X α).comap F =
    (Pi.uniformSpace _).comap F 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [CompactSpace X] (F_eqcont : Equicontinuous F)
Goal: UniformSpace.comap F (UniformFun.uniformSpace X α) = UniformSpace.comap F (Pi.uniformSpace fun i => α)
-- Equicontinuous.uniformInducing_uniformFun_iff_pi
lemma Equicontinuous.uniformInducing_uniformFun_iff_pi [UniformSpace ι] [CompactSpace X]
    (F_eqcont : Equicontinuous F) :
    UniformInducing (UniformFun.ofFun ∘ F) ↔ UniformInducing F 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [UniformSpace ι] [CompactSpace X] (F_eqcont : Equicontinuous F)
Goal: UniformInducing (⇑UniformFun.ofFun ∘ F) ↔ UniformInducing F
-- Equicontinuous.inducing_uniformFun_iff_pi
lemma Equicontinuous.inducing_uniformFun_iff_pi [TopologicalSpace ι] [CompactSpace X]
    (F_eqcont : Equicontinuous F) :
    Inducing (UniformFun.ofFun ∘ F) ↔ Inducing F 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [TopologicalSpace ι] [CompactSpace X] (F_eqcont : Equicontinuous F)
Goal: Inducing (⇑UniformFun.ofFun ∘ F) ↔ Inducing F
-- Equicontinuous.tendsto_uniformFun_iff_pi
theorem Equicontinuous.tendsto_uniformFun_iff_pi [CompactSpace X]
    (F_eqcont : Equicontinuous F) (ℱ : Filter ι) (f : X → α) :
    Tendsto (UniformFun.ofFun ∘ F) ℱ (𝓝 <| UniformFun.ofFun f) ↔
    Tendsto F ℱ (𝓝 f) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [CompactSpace X] (F_eqcont : Equicontinuous F) (ℱ : Filter ι) (f : X → α)
Goal: Filter.Tendsto (⇑UniformFun.ofFun ∘ F) ℱ (𝓝 (UniformFun.ofFun f)) ↔ Filter.Tendsto F ℱ (𝓝 f)
-- EquicontinuousOn.comap_uniformOnFun_eq
theorem EquicontinuousOn.comap_uniformOnFun_eq {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
    (UniformOnFun.uniformSpace X α 𝔖).comap F =
    (Pi.uniformSpace _).comap ((⋃₀ 𝔖).restrict ∘ F) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
Goal: UniformSpace.comap F (UniformOnFun.uniformSpace X α 𝔖) =
  UniformSpace.comap ((⋃₀ 𝔖).restrict ∘ F) (Pi.uniformSpace fun i => α)
-- EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi'
lemma EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi' [UniformSpace ι]
    {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
    UniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
    UniformInducing ((⋃₀ 𝔖).restrict ∘ F) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [UniformSpace ι] {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
Goal: UniformInducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ↔ UniformInducing ((⋃₀ 𝔖).restrict ∘ F)
-- EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi
lemma EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi [UniformSpace ι]
    {𝔖 : Set (Set X)} (𝔖_covers : ⋃₀ 𝔖 = univ) (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
    UniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
    UniformInducing F 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [UniformSpace ι] {𝔖 : Set (Set X)} (𝔖_covers : ⋃₀ 𝔖 = Set.univ) (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
Goal: UniformInducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ↔ UniformInducing F
-- EquicontinuousOn.inducing_uniformOnFun_iff_pi'
lemma EquicontinuousOn.inducing_uniformOnFun_iff_pi' [TopologicalSpace ι]
    {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
    Inducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
    Inducing ((⋃₀ 𝔖).restrict ∘ F) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [TopologicalSpace ι] {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
Goal: Inducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ↔ Inducing ((⋃₀ 𝔖).restrict ∘ F)
-- EquicontinuousOn.inducing_uniformOnFun_iff_pi
lemma EquicontinuousOn.inducing_uniformOnFun_iff_pi [TopologicalSpace ι]
    {𝔖 : Set (Set X)} (𝔖_covers : ⋃₀ 𝔖 = univ) (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
    Inducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
    Inducing F 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [TopologicalSpace ι] {𝔖 : Set (Set X)} (𝔖_covers : ⋃₀ 𝔖 = Set.univ) (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
Goal: Inducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ↔ Inducing F
-- EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
theorem EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
    {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α) :
    Tendsto (UniformOnFun.ofFun 𝔖 ∘ F) ℱ (𝓝 <| UniformOnFun.ofFun 𝔖 f) ↔
    Tendsto ((⋃₀ 𝔖).restrict ∘ F) ℱ (𝓝 <| (⋃₀ 𝔖).restrict f) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α)
Goal: Filter.Tendsto (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((UniformOnFun.ofFun 𝔖) f)) ↔
  Filter.Tendsto ((⋃₀ 𝔖).restrict ∘ F) ℱ (𝓝 ((⋃₀ 𝔖).restrict f))
-- EquicontinuousOn.tendsto_uniformOnFun_iff_pi
theorem EquicontinuousOn.tendsto_uniformOnFun_iff_pi
    {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = univ)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α) :
    Tendsto (UniformOnFun.ofFun 𝔖 ∘ F) ℱ (𝓝 <| UniformOnFun.ofFun 𝔖 f) ↔
    Tendsto F ℱ (𝓝 f) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = Set.univ) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α)
Goal: Filter.Tendsto (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ (𝓝 ((UniformOnFun.ofFun 𝔖) f)) ↔ Filter.Tendsto F ℱ (𝓝 f)
-- EquicontinuousOn.isClosed_range_pi_of_uniformOnFun'
theorem EquicontinuousOn.isClosed_range_pi_of_uniformOnFun'
    {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
    (H : IsClosed (range <| UniformOnFun.ofFun 𝔖 ∘ F)) :
    IsClosed (range <| (⋃₀ 𝔖).restrict ∘ F) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (H : IsClosed (Set.range (⇑(UniformOnFun.ofFun 𝔖) ∘ F)))
Goal: IsClosed (Set.range ((⋃₀ 𝔖).restrict ∘ F))
-- EquicontinuousOn.isClosed_range_uniformOnFun_iff_pi
theorem EquicontinuousOn.isClosed_range_uniformOnFun_iff_pi
    {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = univ)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
    IsClosed (range <| UniformOnFun.ofFun 𝔖 ∘ F) ↔
    IsClosed (range F) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = Set.univ) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
Goal: IsClosed (Set.range (⇑(UniformOnFun.ofFun 𝔖) ∘ F)) ↔ IsClosed (Set.range F)
-- ArzelaAscoli.compactSpace_of_closed_inducing'
theorem ArzelaAscoli.compactSpace_of_closed_inducing' [TopologicalSpace ι] {𝔖 : Set (Set X)}
    (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_ind : Inducing (UniformOnFun.ofFun 𝔖 ∘ F))
    (F_cl : IsClosed <| range <| UniformOnFun.ofFun 𝔖 ∘ F)
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
    (F_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ i, F i x ∈ Q) :
    CompactSpace ι 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [TopologicalSpace ι] {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_ind : Inducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F)) (F_cl : IsClosed (Set.range (⇑(UniformOnFun.ofFun 𝔖) ∘ F))) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (F_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ (i : ι), F i x ∈ Q)
Goal: CompactSpace ι
-- ArzelaAscoli.compactSpace_of_closedEmbedding
theorem ArzelaAscoli.compactSpace_of_closedEmbedding [TopologicalSpace ι] {𝔖 : Set (Set X)}
    (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_clemb : ClosedEmbedding (UniformOnFun.ofFun 𝔖 ∘ F))
    (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
    (F_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ i, F i x ∈ Q) :
    CompactSpace ι 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [TopologicalSpace ι] {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_clemb : ClosedEmbedding (⇑(UniformOnFun.ofFun 𝔖) ∘ F)) (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (F_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ (i : ι), F i x ∈ Q)
Goal: CompactSpace ι
-- ArzelaAscoli.isCompact_closure_of_closedEmbedding
theorem ArzelaAscoli.isCompact_closure_of_closedEmbedding [TopologicalSpace ι] [T2Space α]
    {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
    (F_clemb : ClosedEmbedding (UniformOnFun.ofFun 𝔖 ∘ F))
    {s : Set ι} (s_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn (F ∘ ((↑) : s → ι)) K)
    (s_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ i ∈ s, F i x ∈ Q) :
    IsCompact (closure s) 
Arguments: {ι : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α} [TopologicalSpace ι] [T2Space α] {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_clemb : ClosedEmbedding (⇑(UniformOnFun.ofFun 𝔖) ∘ F)) {s : Set ι} (s_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn (F ∘ Subtype.val) K) (s_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ i ∈ s, F i x ∈ Q)
Goal: IsCompact (closure s)
-- ArzelaAscoli.isCompact_of_equicontinuous
theorem ArzelaAscoli.isCompact_of_equicontinuous
    (S : Set C(X, α)) (hS1 : IsCompact (ContinuousMap.toFun '' S))
    (hS2 : Equicontinuous ((↑) : S → X → α)) : IsCompact S 
Arguments: {X : Type u_2} {α : Type u_4} [TopologicalSpace X] [UniformSpace α] (S : Set C(X, α)) (hS1 : IsCompact (ContinuousMap.toFun '' S)) (hS2 : Equicontinuous fun x => ⇑↑x)
Goal: IsCompact S
-- exists_pos_lt_subset_ball
theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
    ∃ r' ∈ Ioo 0 r, s ⊆ ball x r' 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [ProperSpace α] {x : α} {r : ℝ} {s : Set α} (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ Metric.ball x r)
Goal: ∃ r' ∈ Set.Ioo 0 r, s ⊆ Metric.ball x r'
-- exists_lt_subset_ball
theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' 
Arguments: {α : Type u_1} [PseudoMetricSpace α] [ProperSpace α] {x : α} {r : ℝ} {s : Set α} (hs : IsClosed s) (h : s ⊆ Metric.ball x r)
Goal: ∃ r' < r, s ⊆ Metric.ball x r'
-- Metric.exists_isLocalMin_mem_ball
theorem Metric.exists_isLocalMin_mem_ball [TopologicalSpace β]
    [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
    (hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
    (hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z 
Arguments: {α : Type u_1} {β : Type u_2} [PseudoMetricSpace α] [ProperSpace α] [TopologicalSpace β] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ} (hf : ContinuousOn f (Metric.closedBall a r)) (hz : z ∈ Metric.closedBall a r) (hf1 : ∀ z' ∈ Metric.sphere a r, f z < f z')
Goal: ∃ z ∈ Metric.ball a r, IsLocalMin f z
-- Metric.isBounded_closedBall
theorem isBounded_closedBall : IsBounded (closedBall x r) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ}
Goal: Bornology.IsBounded (Metric.closedBall x r)
-- Metric.isBounded_ball
theorem isBounded_ball : IsBounded (ball x r) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ}
Goal: Bornology.IsBounded (Metric.ball x r)
-- Metric.isBounded_sphere
theorem isBounded_sphere : IsBounded (sphere x r) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ}
Goal: Bornology.IsBounded (Metric.sphere x r)
-- Metric.isBounded_iff_subset_closedBall
theorem isBounded_iff_subset_closedBall (c : α) : IsBounded s ↔ ∃ r, s ⊆ closedBall c r 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (c : α)
Goal: Bornology.IsBounded s ↔ ∃ r, s ⊆ Metric.closedBall c r
-- Bornology.IsBounded.subset_closedBall
theorem _root_.Bornology.IsBounded.subset_closedBall (h : IsBounded s) (c : α) :
    ∃ r, s ⊆ closedBall c r 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : Bornology.IsBounded s) (c : α)
Goal: ∃ r, s ⊆ Metric.closedBall c r
-- Bornology.IsBounded.subset_ball_lt
theorem _root_.Bornology.IsBounded.subset_ball_lt (h : IsBounded s) (a : ℝ) (c : α) :
    ∃ r, a < r ∧ s ⊆ ball c r 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : Bornology.IsBounded s) (a : ℝ) (c : α)
Goal: ∃ r, a < r ∧ s ⊆ Metric.ball c r
-- Bornology.IsBounded.subset_ball
theorem _root_.Bornology.IsBounded.subset_ball (h : IsBounded s) (c : α) : ∃ r, s ⊆ ball c r 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : Bornology.IsBounded s) (c : α)
Goal: ∃ r, s ⊆ Metric.ball c r
-- Metric.isBounded_iff_subset_ball
theorem isBounded_iff_subset_ball (c : α) : IsBounded s ↔ ∃ r, s ⊆ ball c r 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (c : α)
Goal: Bornology.IsBounded s ↔ ∃ r, s ⊆ Metric.ball c r
-- Bornology.IsBounded.subset_closedBall_lt
theorem _root_.Bornology.IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
    ∃ r, a < r ∧ s ⊆ closedBall c r 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : Bornology.IsBounded s) (a : ℝ) (c : α)
Goal: ∃ r, a < r ∧ s ⊆ Metric.closedBall c r
-- Metric.isBounded_closure_of_isBounded
theorem isBounded_closure_of_isBounded (h : IsBounded s) : IsBounded (closure s) 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : Bornology.IsBounded s)
Goal: Bornology.IsBounded (closure s)
-- Bornology.IsBounded.closure
protected theorem _root_.Bornology.IsBounded.closure (h : IsBounded s) : IsBounded (closure s) 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : Bornology.IsBounded s)
Goal: Bornology.IsBounded (closure s)
-- Metric.isBounded_closure_iff
@[simp]
theorem isBounded_closure_iff : IsBounded (closure s) ↔ IsBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: Bornology.IsBounded (closure s) ↔ Bornology.IsBounded s
-- Metric.hasBasis_cobounded_compl_closedBall
theorem hasBasis_cobounded_compl_closedBall (c : α) :
    (cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (closedBall c r)ᶜ) 
Arguments: {α : Type u} [PseudoMetricSpace α] (c : α)
Goal: (Bornology.cobounded α).HasBasis (fun x => True) fun r => (Metric.closedBall c r)ᶜ
-- Metric.hasBasis_cobounded_compl_ball
theorem hasBasis_cobounded_compl_ball (c : α) :
    (cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (ball c r)ᶜ) 
Arguments: {α : Type u} [PseudoMetricSpace α] (c : α)
Goal: (Bornology.cobounded α).HasBasis (fun x => True) fun r => (Metric.ball c r)ᶜ
-- Metric.comap_dist_right_atTop
@[simp]
theorem comap_dist_right_atTop (c : α) : comap (dist · c) atTop = cobounded α 
Arguments: {α : Type u} [PseudoMetricSpace α] (c : α)
Goal: Filter.comap (fun x => dist x c) Filter.atTop = Bornology.cobounded α
-- Metric.comap_dist_left_atTop
@[simp]
theorem comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α 
Arguments: {α : Type u} [PseudoMetricSpace α] (c : α)
Goal: Filter.comap (dist c) Filter.atTop = Bornology.cobounded α
-- Metric.tendsto_dist_right_atTop_iff
@[simp]
theorem tendsto_dist_right_atTop_iff (c : α) {f : β → α} {l : Filter β} :
    Tendsto (fun x ↦ dist (f x) c) l atTop ↔ Tendsto f l (cobounded α) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] (c : α) {f : β → α} {l : Filter β}
Goal: Filter.Tendsto (fun x => dist (f x) c) l Filter.atTop ↔ Filter.Tendsto f l (Bornology.cobounded α)
-- Metric.tendsto_dist_left_atTop_iff
@[simp]
theorem tendsto_dist_left_atTop_iff (c : α) {f : β → α} {l : Filter β} :
    Tendsto (fun x ↦ dist c (f x)) l atTop ↔ Tendsto f l (cobounded α) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] (c : α) {f : β → α} {l : Filter β}
Goal: Filter.Tendsto (fun x => dist c (f x)) l Filter.atTop ↔ Filter.Tendsto f l (Bornology.cobounded α)
-- Metric.tendsto_dist_right_cobounded_atTop
theorem tendsto_dist_right_cobounded_atTop (c : α) : Tendsto (dist · c) (cobounded α) atTop 
Arguments: {α : Type u} [PseudoMetricSpace α] (c : α)
Goal: Filter.Tendsto (fun x => dist x c) (Bornology.cobounded α) Filter.atTop
-- Metric.tendsto_dist_left_cobounded_atTop
theorem tendsto_dist_left_cobounded_atTop (c : α) : Tendsto (dist c) (cobounded α) atTop 
Arguments: {α : Type u} [PseudoMetricSpace α] (c : α)
Goal: Filter.Tendsto (dist c) (Bornology.cobounded α) Filter.atTop
-- TotallyBounded.isBounded
theorem _root_.TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : TotallyBounded s)
Goal: Bornology.IsBounded s
-- IsCompact.isBounded
theorem _root_.IsCompact.isBounded {s : Set α} (h : IsCompact s) : IsBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : IsCompact s)
Goal: Bornology.IsBounded s
-- Metric.cobounded_le_cocompact
theorem cobounded_le_cocompact : cobounded α ≤ cocompact α 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: Bornology.cobounded α ≤ Filter.cocompact α
-- Metric.isCobounded_iff_closedBall_compl_subset
theorem isCobounded_iff_closedBall_compl_subset {s : Set α} (c : α) :
    IsCobounded s ↔ ∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (c : α)
Goal: Bornology.IsCobounded s ↔ ∃ r, (Metric.closedBall c r)ᶜ ⊆ s
-- Bornology.IsCobounded.closedBall_compl_subset
theorem _root_.Bornology.IsCobounded.closedBall_compl_subset {s : Set α} (hs : IsCobounded s)
    (c : α) : ∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : Bornology.IsCobounded s) (c : α)
Goal: ∃ r, (Metric.closedBall c r)ᶜ ⊆ s
-- Metric.closedBall_compl_subset_of_mem_cocompact
theorem closedBall_compl_subset_of_mem_cocompact {s : Set α} (hs : s ∈ cocompact α) (c : α) :
    ∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : s ∈ Filter.cocompact α) (c : α)
Goal: ∃ r, (Metric.closedBall c r)ᶜ ⊆ s
-- Metric.mem_cocompact_of_closedBall_compl_subset
theorem mem_cocompact_of_closedBall_compl_subset [ProperSpace α] (c : α)
    (h : ∃ r, (closedBall c r)ᶜ ⊆ s) : s ∈ cocompact α 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [ProperSpace α] (c : α) (h : ∃ r, (Metric.closedBall c r)ᶜ ⊆ s)
Goal: s ∈ Filter.cocompact α
-- Metric.mem_cocompact_iff_closedBall_compl_subset
theorem mem_cocompact_iff_closedBall_compl_subset [ProperSpace α] (c : α) :
    s ∈ cocompact α ↔ ∃ r, (closedBall c r)ᶜ ⊆ s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [ProperSpace α] (c : α)
Goal: s ∈ Filter.cocompact α ↔ ∃ r, (Metric.closedBall c r)ᶜ ⊆ s
-- Metric.isBounded_range_iff
theorem isBounded_range_iff {f : β → α} : IsBounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α}
Goal: Bornology.IsBounded (Set.range f) ↔ ∃ C, ∀ (x y : β), dist (f x) (f y) ≤ C
-- Metric.isBounded_image_iff
theorem isBounded_image_iff {f : β → α} {s : Set β} :
    IsBounded (f '' s) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ C 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} {s : Set β}
Goal: Bornology.IsBounded (f '' s) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ C
-- Metric.isBounded_range_of_tendsto_cofinite_uniformity
theorem isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
    (hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : IsBounded (range f) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} (hf : Filter.Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α))
Goal: Bornology.IsBounded (Set.range f)
-- Metric.isBounded_range_of_cauchy_map_cofinite
theorem isBounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
    IsBounded (range f) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} (hf : Cauchy (Filter.map f Filter.cofinite))
Goal: Bornology.IsBounded (Set.range f)
-- CauchySeq.isBounded_range
theorem _root_.CauchySeq.isBounded_range {f : ℕ → α} (hf : CauchySeq f) : IsBounded (range f) 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : ℕ → α} (hf : CauchySeq f)
Goal: Bornology.IsBounded (Set.range f)