ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Continuous.tendsto_nhdsSet_nhds
lemma Continuous.tendsto_nhdsSet_nhds
    {y : Y} {f : X → Y} (h : Continuous f) (h' : EqOn f (fun _ ↦ y) s) :
    Tendsto f (𝓝ˢ s) (𝓝 y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {y : Y} {f : X → Y} (h : Continuous f) (h' : Set.EqOn f (fun x => y) s)
Goal: Filter.Tendsto f (𝓝ˢ s) (𝓝 y)
-- nhdsSet_inter_le
theorem nhdsSet_inter_le (s t : Set X) : 𝓝ˢ (s ∩ t) ≤ 𝓝ˢ s ⊓ 𝓝ˢ t 
Arguments: {X : Type u_1} [TopologicalSpace X] (s t : Set X)
Goal: 𝓝ˢ (s ∩ t) ≤ 𝓝ˢ s ⊓ 𝓝ˢ t
-- IsClosed.nhdsSet_le_sup
theorem IsClosed.nhdsSet_le_sup (h : IsClosed t) : 𝓝ˢ s ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓟 (tᶜ) 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : Set X) {t : Set X} (h : IsClosed t)
Goal: 𝓝ˢ s ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓟 tᶜ
-- IsClosed.nhdsSet_le_sup'
theorem IsClosed.nhdsSet_le_sup' (h : IsClosed t) :
    𝓝ˢ s ≤ 𝓝ˢ (t ∩ s) ⊔ 𝓟 (tᶜ) 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : Set X) {t : Set X} (h : IsClosed t)
Goal: 𝓝ˢ s ≤ 𝓝ˢ (t ∩ s) ⊔ 𝓟 tᶜ
-- Filter.Eventually.eventually_nhdsSet
theorem Filter.Eventually.eventually_nhdsSet {p : X → Prop} (h : ∀ᶠ y in 𝓝ˢ s, p y) :
    ∀ᶠ y in 𝓝ˢ s, ∀ᶠ x in 𝓝 y, p x 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} {p : X → Prop} (h : ∀ᶠ (y : X) in 𝓝ˢ s, p y)
Goal: ∀ᶠ (y : X) in 𝓝ˢ s, ∀ᶠ (x : X) in 𝓝 y, p x
-- Filter.Eventually.union_nhdsSet
theorem Filter.Eventually.union_nhdsSet {p : X → Prop} :
    (∀ᶠ x in 𝓝ˢ (s ∪ t), p x) ↔ (∀ᶠ x in 𝓝ˢ s, p x) ∧ ∀ᶠ x in 𝓝ˢ t, p x 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} {p : X → Prop}
Goal: (∀ᶠ (x : X) in 𝓝ˢ (s ∪ t), p x) ↔ (∀ᶠ (x : X) in 𝓝ˢ s, p x) ∧ ∀ᶠ (x : X) in 𝓝ˢ t, p x
-- Filter.Eventually.union
theorem Filter.Eventually.union {p : X → Prop} (hs : ∀ᶠ x in 𝓝ˢ s, p x) (ht : ∀ᶠ x in 𝓝ˢ t, p x) :
    ∀ᶠ x in 𝓝ˢ (s ∪ t), p x 
Arguments: {X : Type u_1} [TopologicalSpace X] {s t : Set X} {p : X → Prop} (hs : ∀ᶠ (x : X) in 𝓝ˢ s, p x) (ht : ∀ᶠ (x : X) in 𝓝ˢ t, p x)
Goal: ∀ᶠ (x : X) in 𝓝ˢ (s ∪ t), p x
-- nhdsSet_iUnion
theorem nhdsSet_iUnion {ι : Sort*} (s : ι → Set X) : 𝓝ˢ (⋃ i, s i) = ⨆ i, 𝓝ˢ (s i) 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Sort u_3} (s : ι → Set X)
Goal: 𝓝ˢ (⋃ i, s i) = ⨆ i, 𝓝ˢ (s i)
-- eventually_nhdsSet_iUnion₂
theorem eventually_nhdsSet_iUnion₂ {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {P : X → Prop} :
    (∀ᶠ x in 𝓝ˢ (⋃ (i) (_ : p i), s i), P x) ↔ ∀ i, p i → ∀ᶠ x in 𝓝ˢ (s i), P x 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Sort u_3} {p : ι → Prop} {s : ι → Set X} {P : X → Prop}
Goal: (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, ⋃ (_ : p i), s i), P x) ↔ ∀ (i : ι), p i → ∀ᶠ (x : X) in 𝓝ˢ (s i), P x
-- eventually_nhdsSet_iUnion
theorem eventually_nhdsSet_iUnion {ι : Sort*} {s : ι → Set X} {P : X → Prop} :
    (∀ᶠ x in 𝓝ˢ (⋃ i, s i), P x) ↔ ∀ i, ∀ᶠ x in 𝓝ˢ (s i), P x 
Arguments: {X : Type u_1} [TopologicalSpace X] {ι : Sort u_3} {s : ι → Set X} {P : X → Prop}
Goal: (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, s i), P x) ↔ ∀ (i : ι), ∀ᶠ (x : X) in 𝓝ˢ (s i), P x
-- Set.restrictPreimage_inducing
theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
    Inducing (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : Inducing f)
Goal: Inducing (s.restrictPreimage f)
-- Set.restrictPreimage_embedding
theorem Set.restrictPreimage_embedding (s : Set β) (h : Embedding f) :
    Embedding (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : Embedding f)
Goal: Embedding (s.restrictPreimage f)
-- Set.restrictPreimage_openEmbedding
theorem Set.restrictPreimage_openEmbedding (s : Set β) (h : OpenEmbedding f) :
    OpenEmbedding (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : OpenEmbedding f)
Goal: OpenEmbedding (s.restrictPreimage f)
-- Set.restrictPreimage_closedEmbedding
theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f) :
    ClosedEmbedding (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (h : ClosedEmbedding f)
Goal: ClosedEmbedding (s.restrictPreimage f)
-- IsClosedMap.restrictPreimage
theorem IsClosedMap.restrictPreimage (H : IsClosedMap f) (s : Set β) :
    IsClosedMap (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (H : IsClosedMap f) (s : Set β)
Goal: IsClosedMap (s.restrictPreimage f)
-- Set.restrictPreimage_isClosedMap
@[deprecated (since := "2024-04-02")]
theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
    IsClosedMap (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (H : IsClosedMap f)
Goal: IsClosedMap (s.restrictPreimage f)
-- IsOpenMap.restrictPreimage
theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) :
    IsOpenMap (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (H : IsOpenMap f) (s : Set β)
Goal: IsOpenMap (s.restrictPreimage f)
-- Set.restrictPreimage_isOpenMap
@[deprecated (since := "2024-04-02")]
theorem Set.restrictPreimage_isOpenMap (s : Set β) (H : IsOpenMap f) :
    IsOpenMap (s.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (s : Set β) (H : IsOpenMap f)
Goal: IsOpenMap (s.restrictPreimage f)
-- isOpen_iff_inter_of_iSup_eq_top
theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) 
Arguments: {β : Type u_2} [TopologicalSpace β] {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (s : Set β)
Goal: IsOpen s ↔ ∀ (i : ι), IsOpen (s ∩ ↑(U i))
-- isOpen_iff_coe_preimage_of_iSup_eq_top
theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
    IsOpen s ↔ ∀ i, IsOpen ((↑) ⁻¹' s : Set (U i)) 
Arguments: {β : Type u_2} [TopologicalSpace β] {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (s : Set β)
Goal: IsOpen s ↔ ∀ (i : ι), IsOpen (Subtype.val ⁻¹' s)
-- isClosed_iff_coe_preimage_of_iSup_eq_top
theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
    IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) 
Arguments: {β : Type u_2} [TopologicalSpace β] {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (s : Set β)
Goal: IsClosed s ↔ ∀ (i : ι), IsClosed (Subtype.val ⁻¹' s)
-- isLocallyClosed_iff_coe_preimage_of_iSup_eq_top
theorem isLocallyClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
    IsLocallyClosed s ↔ ∀ i, IsLocallyClosed ((↑) ⁻¹' s : Set (U i)) 
Arguments: {β : Type u_2} [TopologicalSpace β] {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (s : Set β)
Goal: IsLocallyClosed s ↔ ∀ (i : ι), IsLocallyClosed (Subtype.val ⁻¹' s)
-- isOpenMap_iff_isOpenMap_of_iSup_eq_top
theorem isOpenMap_iff_isOpenMap_of_iSup_eq_top :
    IsOpenMap f ↔ ∀ i, IsOpenMap ((U i).1.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤)
Goal: IsOpenMap f ↔ ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)
-- isClosedMap_iff_isClosedMap_of_iSup_eq_top
theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
    IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤)
Goal: IsClosedMap f ↔ ∀ (i : ι), IsClosedMap ((U i).carrier.restrictPreimage f)
-- inducing_iff_inducing_of_iSup_eq_top
theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
    Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (h : Continuous f)
Goal: Inducing f ↔ ∀ (i : ι), Inducing ((U i).carrier.restrictPreimage f)
-- embedding_iff_embedding_of_iSup_eq_top
theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
    Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (h : Continuous f)
Goal: Embedding f ↔ ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f)
-- openEmbedding_iff_openEmbedding_of_iSup_eq_top
theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
    OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (h : Continuous f)
Goal: OpenEmbedding f ↔ ∀ (i : ι), OpenEmbedding ((U i).carrier.restrictPreimage f)
-- closedEmbedding_iff_closedEmbedding_of_iSup_eq_top
theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
    ClosedEmbedding f ↔ ∀ i, ClosedEmbedding ((U i).1.restrictPreimage f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {ι : Type u_3} {U : ι → TopologicalSpace.Opens β} (hU : iSup U = ⊤) (h : Continuous f)
Goal: ClosedEmbedding f ↔ ∀ (i : ι), ClosedEmbedding ((U i).carrier.restrictPreimage f)
-- EReal.denseRange_ratCast
lemma denseRange_ratCast : DenseRange (fun r : ℚ ↦ ((r : ℝ) : EReal)) 
Goal: DenseRange fun r => ↑↑r
-- EReal.embedding_coe
theorem embedding_coe : Embedding ((↑) : ℝ → EReal) 
Goal: Embedding Real.toEReal
-- EReal.openEmbedding_coe
theorem openEmbedding_coe : OpenEmbedding ((↑) : ℝ → EReal) 
Goal: OpenEmbedding Real.toEReal
-- EReal.tendsto_coe
@[norm_cast]
theorem tendsto_coe {α : Type*} {f : Filter α} {m : α → ℝ} {a : ℝ} :
    Tendsto (fun a => (m a : EReal)) f (𝓝 ↑a) ↔ Tendsto m f (𝓝 a) 
Arguments: {α : Type u_2} {f : Filter α} {m : α → ℝ} {a : ℝ}
Goal: Filter.Tendsto (fun a => ↑(m a)) f (𝓝 ↑a) ↔ Filter.Tendsto m f (𝓝 a)
-- continuous_coe_real_ereal
theorem _root_.continuous_coe_real_ereal : Continuous ((↑) : ℝ → EReal) 
Goal: Continuous Real.toEReal
-- EReal.continuous_coe_iff
theorem continuous_coe_iff {f : α → ℝ} : (Continuous fun a => (f a : EReal)) ↔ Continuous f 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ}
Goal: (Continuous fun a => ↑(f a)) ↔ Continuous f
-- EReal.nhds_coe
theorem nhds_coe {r : ℝ} : 𝓝 (r : EReal) = (𝓝 r).map (↑) 
Arguments: {r : ℝ}
Goal: 𝓝 ↑r = Filter.map Real.toEReal (𝓝 r)
-- EReal.nhds_coe_coe
theorem nhds_coe_coe {r p : ℝ} :
    𝓝 ((r : EReal), (p : EReal)) = (𝓝 (r, p)).map fun p : ℝ × ℝ => (↑p.1, ↑p.2) 
Arguments: {r p : ℝ}
Goal: 𝓝 (↑r, ↑p) = Filter.map (fun p => (↑p.1, ↑p.2)) (𝓝 (r, p))
-- EReal.tendsto_toReal
theorem tendsto_toReal {a : EReal} (ha : a ≠ ⊤) (h'a : a ≠ ⊥) :
    Tendsto EReal.toReal (𝓝 a) (𝓝 a.toReal) 
Arguments: {a : EReal} (ha : a ≠ ⊤) (h'a : a ≠ ⊥)
Goal: Filter.Tendsto EReal.toReal (𝓝 a) (𝓝 a.toReal)
-- EReal.continuousOn_toReal
theorem continuousOn_toReal : ContinuousOn EReal.toReal ({⊥, ⊤}ᶜ : Set EReal) 
Goal: ContinuousOn EReal.toReal {⊥, ⊤}ᶜ
-- EReal.embedding_coe_ennreal
theorem embedding_coe_ennreal : Embedding ((↑) : ℝ≥0∞ → EReal) 
Goal: Embedding ENNReal.toEReal
-- EReal.closedEmbedding_coe_ennreal
theorem closedEmbedding_coe_ennreal : ClosedEmbedding ((↑) : ℝ≥0∞ → EReal) 
Goal: ClosedEmbedding ENNReal.toEReal
-- EReal.tendsto_coe_ennreal
@[norm_cast]
theorem tendsto_coe_ennreal {α : Type*} {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} :
    Tendsto (fun a => (m a : EReal)) f (𝓝 ↑a) ↔ Tendsto m f (𝓝 a) 
Arguments: {α : Type u_2} {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞}
Goal: Filter.Tendsto (fun a => ↑(m a)) f (𝓝 ↑a) ↔ Filter.Tendsto m f (𝓝 a)
-- continuous_coe_ennreal_ereal
theorem _root_.continuous_coe_ennreal_ereal : Continuous ((↑) : ℝ≥0∞ → EReal) 
Goal: Continuous ENNReal.toEReal
-- EReal.continuous_coe_ennreal_iff
theorem continuous_coe_ennreal_iff {f : α → ℝ≥0∞} :
    (Continuous fun a => (f a : EReal)) ↔ Continuous f 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ≥0∞}
Goal: (Continuous fun a => ↑(f a)) ↔ Continuous f
-- EReal.nhds_top
theorem nhds_top : 𝓝 (⊤ : EReal) = ⨅ (a) (_ : a ≠ ⊤), 𝓟 (Ioi a) 
Goal: 𝓝 ⊤ = ⨅ a, ⨅ (_ : a ≠ ⊤), 𝓟 (Set.Ioi a)
-- EReal.nhds_top_basis
nonrec theorem nhds_top_basis : (𝓝 (⊤ : EReal)).HasBasis (fun _ : ℝ ↦ True) (Ioi ·) 
Goal: (𝓝 ⊤).HasBasis (fun x => True) fun x => Set.Ioi ↑x
-- EReal.nhds_top'
theorem nhds_top' : 𝓝 (⊤ : EReal) = ⨅ a : ℝ, 𝓟 (Ioi ↑a) 
Goal: 𝓝 ⊤ = ⨅ a, 𝓟 (Set.Ioi ↑a)
-- EReal.mem_nhds_top_iff
theorem mem_nhds_top_iff {s : Set EReal} : s ∈ 𝓝 (⊤ : EReal) ↔ ∃ y : ℝ, Ioi (y : EReal) ⊆ s 
Arguments: {s : Set EReal}
Goal: s ∈ 𝓝 ⊤ ↔ ∃ y, Set.Ioi ↑y ⊆ s
-- EReal.tendsto_nhds_top_iff_real
theorem tendsto_nhds_top_iff_real {α : Type*} {m : α → EReal} {f : Filter α} :
    Tendsto m f (𝓝 ⊤) ↔ ∀ x : ℝ, ∀ᶠ a in f, ↑x < m a 
Arguments: {α : Type u_2} {m : α → EReal} {f : Filter α}
Goal: Filter.Tendsto m f (𝓝 ⊤) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, ↑x < m a
-- EReal.nhds_bot
theorem nhds_bot : 𝓝 (⊥ : EReal) = ⨅ (a) (_ : a ≠ ⊥), 𝓟 (Iio a) 
Goal: 𝓝 ⊥ = ⨅ a, ⨅ (_ : a ≠ ⊥), 𝓟 (Set.Iio a)
-- EReal.nhds_bot_basis
theorem nhds_bot_basis : (𝓝 (⊥ : EReal)).HasBasis (fun _ : ℝ ↦ True) (Iio ·) 
Goal: (𝓝 ⊥).HasBasis (fun x => True) fun x => Set.Iio ↑x
-- EReal.nhds_bot'
theorem nhds_bot' : 𝓝 (⊥ : EReal) = ⨅ a : ℝ, 𝓟 (Iio ↑a) 
Goal: 𝓝 ⊥ = ⨅ a, 𝓟 (Set.Iio ↑a)
-- EReal.mem_nhds_bot_iff
theorem mem_nhds_bot_iff {s : Set EReal} : s ∈ 𝓝 (⊥ : EReal) ↔ ∃ y : ℝ, Iio (y : EReal) ⊆ s 
Arguments: {s : Set EReal}
Goal: s ∈ 𝓝 ⊥ ↔ ∃ y, Set.Iio ↑y ⊆ s
-- EReal.tendsto_nhds_bot_iff_real
theorem tendsto_nhds_bot_iff_real {α : Type*} {m : α → EReal} {f : Filter α} :
    Tendsto m f (𝓝 ⊥) ↔ ∀ x : ℝ, ∀ᶠ a in f, m a < x 
Arguments: {α : Type u_2} {m : α → EReal} {f : Filter α}
Goal: Filter.Tendsto m f (𝓝 ⊥) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, m a < ↑x
-- EReal.liminf_le_liminf
lemma liminf_le_liminf (h : u ≤ᶠ[f] v) :
    liminf u f ≤ liminf v f 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} (h : u ≤ᶠ[f] v)
Goal: Filter.liminf u f ≤ Filter.liminf v f
-- EReal.limsup_le_limsup
lemma limsup_le_limsup (h : u ≤ᶠ[f] v) :
    limsup u f ≤ limsup v f 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} (h : u ≤ᶠ[f] v)
Goal: Filter.limsup u f ≤ Filter.limsup v f
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.limsup_add_le_of_lt
private lemma limsup_add_le_of_lt (ha : limsup u f < a) (hb : limsup v f < b) :
    limsup (u + v) f ≤ a + b 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} {a b : EReal} (ha : Filter.limsup u f < a) (hb : Filter.limsup v f < b)
Goal: Filter.limsup (u + v) f ≤ a + b
-- EReal.limsup_add_lt_of_lt
lemma limsup_add_lt_of_lt (ha : limsup u f < a) (hb : limsup v f < b) :
    limsup (u + v) f < a + b 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} {a b : EReal} (ha : Filter.limsup u f < a) (hb : Filter.limsup v f < b)
Goal: Filter.limsup (u + v) f < a + b
-- EReal.limsup_add_bot_of_ne_top
lemma limsup_add_bot_of_ne_top (h : limsup u f = ⊥) (h' : limsup v f ≠ ⊤) :
    limsup (u + v) f = ⊥ 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} (h : Filter.limsup u f = ⊥) (h' : Filter.limsup v f ≠ ⊤)
Goal: Filter.limsup (u + v) f = ⊥
-- EReal.limsup_add_le_add_limsup
lemma limsup_add_le_add_limsup
    (h : limsup u f ≠ ⊥ ∨ limsup v f ≠ ⊤) (h' : limsup u f ≠ ⊤ ∨ limsup v f ≠ ⊥) :
    limsup (u + v) f ≤ (limsup u f) + (limsup v f) 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} (h : Filter.limsup u f ≠ ⊥ ∨ Filter.limsup v f ≠ ⊤) (h' : Filter.limsup u f ≠ ⊤ ∨ Filter.limsup v f ≠ ⊥)
Goal: Filter.limsup (u + v) f ≤ Filter.limsup u f + Filter.limsup v f
-- EReal.limsup_add_le_of_le
lemma limsup_add_le_of_le (ha : limsup u f < a) (hb : limsup v f ≤ b) :
    limsup (u + v) f ≤ a + b 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} {a b : EReal} (ha : Filter.limsup u f < a) (hb : Filter.limsup v f ≤ b)
Goal: Filter.limsup (u + v) f ≤ a + b
-- EReal.liminf_neg
lemma liminf_neg : liminf (- v) f = - limsup v f 
Arguments: {α : Type u_2} {f : Filter α} {v : α → EReal}
Goal: Filter.liminf (-v) f = -Filter.limsup v f
-- EReal.limsup_neg
lemma limsup_neg : limsup (- v) f = - liminf v f 
Arguments: {α : Type u_2} {f : Filter α} {v : α → EReal}
Goal: Filter.limsup (-v) f = -Filter.liminf v f
-- EReal.liminf_add_gt_of_gt
lemma liminf_add_gt_of_gt (ha : a < liminf u f) (hb : b < liminf v f) :
    a + b < liminf (u + v) f 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} {a b : EReal} (ha : a < Filter.liminf u f) (hb : b < Filter.liminf v f)
Goal: a + b < Filter.liminf (u + v) f
-- EReal.liminf_add_top_of_ne_bot
lemma liminf_add_top_of_ne_bot (h : liminf u f = ⊤) (h' : liminf v f ≠ ⊥) :
    liminf (u + v) f = ⊤ 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal} (h : Filter.liminf u f = ⊤) (h' : Filter.liminf v f ≠ ⊥)
Goal: Filter.liminf (u + v) f = ⊤
-- EReal.add_liminf_le_liminf_add
lemma add_liminf_le_liminf_add : (liminf u f) + (liminf v f) ≤ liminf (u + v) f 
Arguments: {α : Type u_2} {f : Filter α} {u v : α → EReal}
Goal: Filter.liminf u f + Filter.liminf v f ≤ Filter.liminf (u + v) f
-- EReal.limsup_le_iff
lemma limsup_le_iff {b : EReal} : limsup u f ≤ b ↔ ∀ c : ℝ, b < c → ∀ᶠ a : α in f, u a ≤ c 
Arguments: {α : Type u_2} {f : Filter α} {u : α → EReal} {b : EReal}
Goal: Filter.limsup u f ≤ b ↔ ∀ (c : ℝ), b < ↑c → ∀ᶠ (a : α) in f, u a ≤ ↑c
-- EReal.continuousAt_add_coe_coe
theorem continuousAt_add_coe_coe (a b : ℝ) :
    ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, b) 
Arguments: (a b : ℝ)
Goal: ContinuousAt (fun p => p.1 + p.2) (↑a, ↑b)
-- EReal.continuousAt_add_top_coe
theorem continuousAt_add_top_coe (a : ℝ) :
    ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊤, a) 
Arguments: (a : ℝ)
Goal: ContinuousAt (fun p => p.1 + p.2) (⊤, ↑a)
-- EReal.continuousAt_add_coe_top
theorem continuousAt_add_coe_top (a : ℝ) :
    ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, ⊤) 
Arguments: (a : ℝ)
Goal: ContinuousAt (fun p => p.1 + p.2) (↑a, ⊤)
-- EReal.continuousAt_add_top_top
theorem continuousAt_add_top_top : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊤, ⊤) 
Goal: ContinuousAt (fun p => p.1 + p.2) (⊤, ⊤)
-- EReal.continuousAt_add_bot_coe
theorem continuousAt_add_bot_coe (a : ℝ) :
    ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, a) 
Arguments: (a : ℝ)
Goal: ContinuousAt (fun p => p.1 + p.2) (⊥, ↑a)
-- EReal.continuousAt_add_coe_bot
theorem continuousAt_add_coe_bot (a : ℝ) :
    ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, ⊥) 
Arguments: (a : ℝ)
Goal: ContinuousAt (fun p => p.1 + p.2) (↑a, ⊥)
-- EReal.continuousAt_add_bot_bot
theorem continuousAt_add_bot_bot : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, ⊥) 
Goal: ContinuousAt (fun p => p.1 + p.2) (⊥, ⊥)
-- EReal.continuousAt_add
theorem continuousAt_add {p : EReal × EReal} (h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥) (h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤) :
    ContinuousAt (fun p : EReal × EReal => p.1 + p.2) p 
Arguments: {p : EReal × EReal} (h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥) (h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤)
Goal: ContinuousAt (fun p => p.1 + p.2) p
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_swap
private lemma continuousAt_mul_swap {a b : EReal}
    (h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (b, a) 
Arguments: {a b : EReal} (h : ContinuousAt (fun p => p.1 * p.2) (a, b))
Goal: ContinuousAt (fun p => p.1 * p.2) (b, a)
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_symm1
private lemma continuousAt_mul_symm1 {a b : EReal}
    (h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (-a, b) 
Arguments: {a b : EReal} (h : ContinuousAt (fun p => p.1 * p.2) (a, b))
Goal: ContinuousAt (fun p => p.1 * p.2) (-a, b)
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_symm2
private lemma continuousAt_mul_symm2 {a b : EReal}
    (h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, -b) 
Arguments: {a b : EReal} (h : ContinuousAt (fun p => p.1 * p.2) (a, b))
Goal: ContinuousAt (fun p => p.1 * p.2) (a, -b)
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_symm3
private lemma continuousAt_mul_symm3 {a b : EReal}
    (h : ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b)) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (-a, -b) 
Arguments: {a b : EReal} (h : ContinuousAt (fun p => p.1 * p.2) (a, b))
Goal: ContinuousAt (fun p => p.1 * p.2) (-a, -b)
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_coe_coe
private lemma continuousAt_mul_coe_coe (a b : ℝ) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b) 
Arguments: (a b : ℝ)
Goal: ContinuousAt (fun p => p.1 * p.2) (↑a, ↑b)
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_top_top
private lemma continuousAt_mul_top_top :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, ⊤) 
Goal: ContinuousAt (fun p => p.1 * p.2) (⊤, ⊤)
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_top_pos
private lemma continuousAt_mul_top_pos {a : ℝ} (h : 0 < a) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, a) 
Arguments: {a : ℝ} (h : 0 < a)
Goal: ContinuousAt (fun p => p.1 * p.2) (⊤, ↑a)
-- _private.Mathlib.Topology.Instances.EReal.0.EReal.continuousAt_mul_top_ne_zero
private lemma continuousAt_mul_top_ne_zero {a : ℝ} (h : a ≠ 0) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, a) 
Arguments: {a : ℝ} (h : a ≠ 0)
Goal: ContinuousAt (fun p => p.1 * p.2) (⊤, ↑a)
-- EReal.continuousAt_mul
theorem continuousAt_mul {p : EReal × EReal} (h₁ : p.1 ≠ 0 ∨ p.2 ≠ ⊥)
    (h₂ : p.1 ≠ 0 ∨ p.2 ≠ ⊤) (h₃ : p.1 ≠ ⊥ ∨ p.2 ≠ 0) (h₄ : p.1 ≠ ⊤ ∨ p.2 ≠ 0) :
    ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) p 
Arguments: {p : EReal × EReal} (h₁ : p.1 ≠ 0 ∨ p.2 ≠ ⊥) (h₂ : p.1 ≠ 0 ∨ p.2 ≠ ⊤) (h₃ : p.1 ≠ ⊥ ∨ p.2 ≠ 0) (h₄ : p.1 ≠ ⊤ ∨ p.2 ≠ 0)
Goal: ContinuousAt (fun p => p.1 * p.2) p
-- isLocalHomeomorphOn_iff_openEmbedding_restrict
theorem isLocalHomeomorphOn_iff_openEmbedding_restrict {f : X → Y} :
    IsLocalHomeomorphOn f s ↔ ∀ x ∈ s, ∃ U ∈ 𝓝 x, OpenEmbedding (U.restrict f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (s : Set X) {f : X → Y}
Goal: IsLocalHomeomorphOn f s ↔ ∀ x ∈ s, ∃ U ∈ 𝓝 x, OpenEmbedding (U.restrict f)
-- IsLocalHomeomorphOn.mk
theorem mk (h : ∀ x ∈ s, ∃ e : PartialHomeomorph X Y, x ∈ e.source ∧ Set.EqOn f e e.source) :
    IsLocalHomeomorphOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (s : Set X) (h : ∀ x ∈ s, ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source)
Goal: IsLocalHomeomorphOn f s
-- IsLocalHomeomorphOn.PartialHomeomorph.isLocalHomeomorphOn
lemma PartialHomeomorph.isLocalHomeomorphOn (e : PartialHomeomorph X Y) :
    IsLocalHomeomorphOn e e.source 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: IsLocalHomeomorphOn (↑e) e.source
-- IsLocalHomeomorphOn.mono
theorem mono {t : Set X} (hf : IsLocalHomeomorphOn f t) (hst : s ⊆ t) : IsLocalHomeomorphOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s t : Set X} (hf : IsLocalHomeomorphOn f t) (hst : s ⊆ t)
Goal: IsLocalHomeomorphOn f s
-- IsLocalHomeomorphOn.of_comp_left
theorem of_comp_left (hgf : IsLocalHomeomorphOn (g ∘ f) s) (hg : IsLocalHomeomorphOn g (f '' s))
    (cont : ∀ x ∈ s, ContinuousAt f x) : IsLocalHomeomorphOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {g : Y → Z} {f : X → Y} {s : Set X} (hgf : IsLocalHomeomorphOn (g ∘ f) s) (hg : IsLocalHomeomorphOn g (f '' s)) (cont : ∀ x ∈ s, ContinuousAt f x)
Goal: IsLocalHomeomorphOn f s