ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- lipschitzWith_iff_dist_le_mul
theorem lipschitzWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
    {f : α → β} : LipschitzWith K f ↔ ∀ x y, dist (f x) (f y) ≤ K * dist x y 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β}
Goal: LipschitzWith K f ↔ ∀ (x y : α), dist (f x) (f y) ≤ ↑K * dist x y
-- lipschitzOnWith_iff_dist_le_mul
theorem lipschitzOnWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
    {s : Set α} {f : α → β} :
    LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ K * dist x y 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β}
Goal: LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑K * dist x y
-- LipschitzWith.of_dist_le'
protected theorem of_dist_le' {K : ℝ} (h : ∀ x y, dist (f x) (f y) ≤ K * dist x y) :
    LipschitzWith (Real.toNNReal K) f 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {K : ℝ} (h : ∀ (x y : α), dist (f x) (f y) ≤ K * dist x y)
Goal: LipschitzWith K.toNNReal f
-- LipschitzWith.mk_one
protected theorem mk_one (h : ∀ x y, dist (f x) (f y) ≤ dist x y) : LipschitzWith 1 f 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (h : ∀ (x y : α), dist (f x) (f y) ≤ dist x y)
Goal: LipschitzWith 1 f
-- LipschitzWith.of_le_add_mul'
protected theorem of_le_add_mul' {f : α → ℝ} (K : ℝ) (h : ∀ x y, f x ≤ f y + K * dist x y) :
    LipschitzWith (Real.toNNReal K) f 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : α → ℝ} (K : ℝ) (h : ∀ (x y : α), f x ≤ f y + K * dist x y)
Goal: LipschitzWith K.toNNReal f
-- LipschitzWith.of_le_add_mul
protected theorem of_le_add_mul {f : α → ℝ} (K : ℝ≥0) (h : ∀ x y, f x ≤ f y + K * dist x y) :
    LipschitzWith K f 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : α → ℝ} (K : ℝ≥0) (h : ∀ (x y : α), f x ≤ f y + ↑K * dist x y)
Goal: LipschitzWith K f
-- LipschitzWith.of_le_add
protected theorem of_le_add {f : α → ℝ} (h : ∀ x y, f x ≤ f y + dist x y) : LipschitzWith 1 f 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : α → ℝ} (h : ∀ (x y : α), f x ≤ f y + dist x y)
Goal: LipschitzWith 1 f
-- LipschitzWith.le_add_mul
protected theorem le_add_mul {f : α → ℝ} {K : ℝ≥0} (h : LipschitzWith K f) (x y) :
    f x ≤ f y + K * dist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : α → ℝ} {K : ℝ≥0} (h : LipschitzWith K f) (x y : α)
Goal: f x ≤ f y + ↑K * dist x y
-- LipschitzWith.iff_le_add_mul
protected theorem iff_le_add_mul {f : α → ℝ} {K : ℝ≥0} :
    LipschitzWith K f ↔ ∀ x y, f x ≤ f y + K * dist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : α → ℝ} {K : ℝ≥0}
Goal: LipschitzWith K f ↔ ∀ (x y : α), f x ≤ f y + ↑K * dist x y
-- LipschitzWith.nndist_le
theorem nndist_le (hf : LipschitzWith K f) (x y : α) : nndist (f x) (f y) ≤ K * nndist x y 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) (x y : α)
Goal: nndist (f x) (f y) ≤ K * nndist x y
-- LipschitzWith.dist_le_mul_of_le
theorem dist_le_mul_of_le (hf : LipschitzWith K f) (hr : dist x y ≤ r) : dist (f x) (f y) ≤ K * r 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} {x y : α} {r : ℝ} (hf : LipschitzWith K f) (hr : dist x y ≤ r)
Goal: dist (f x) (f y) ≤ ↑K * r
-- LipschitzWith.mapsTo_closedBall
theorem mapsTo_closedBall (hf : LipschitzWith K f) (x : α) (r : ℝ) :
    MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) (K * r)) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) (x : α) (r : ℝ)
Goal: Set.MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) (↑K * r))
-- LipschitzWith.dist_lt_mul_of_lt
theorem dist_lt_mul_of_lt (hf : LipschitzWith K f) (hK : K ≠ 0) (hr : dist x y < r) :
    dist (f x) (f y) < K * r 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} {x y : α} {r : ℝ} (hf : LipschitzWith K f) (hK : K ≠ 0) (hr : dist x y < r)
Goal: dist (f x) (f y) < ↑K * r
-- LipschitzWith.mapsTo_ball
theorem mapsTo_ball (hf : LipschitzWith K f) (hK : K ≠ 0) (x : α) (r : ℝ) :
    MapsTo f (Metric.ball x r) (Metric.ball (f x) (K * r)) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) (hK : K ≠ 0) (x : α) (r : ℝ)
Goal: Set.MapsTo f (Metric.ball x r) (Metric.ball (f x) (↑K * r))
-- LipschitzWith.coe_toLocallyBoundedMap
@[simp]
theorem coe_toLocallyBoundedMap (hf : LipschitzWith K f) : ⇑(hf.toLocallyBoundedMap f) = f 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f)
Goal: ⇑(LipschitzWith.toLocallyBoundedMap f hf) = f
-- LipschitzWith.comap_cobounded_le
theorem comap_cobounded_le (hf : LipschitzWith K f) :
    comap f (Bornology.cobounded β) ≤ Bornology.cobounded α 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f)
Goal: Filter.comap f (Bornology.cobounded β) ≤ Bornology.cobounded α
-- LipschitzWith.isBounded_image
theorem isBounded_image (hf : LipschitzWith K f) {s : Set α} (hs : IsBounded s) :
    IsBounded (f '' s) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) {s : Set α} (hs : Bornology.IsBounded s)
Goal: Bornology.IsBounded (f '' s)
-- LipschitzWith.diam_image_le
theorem diam_image_le (hf : LipschitzWith K f) (s : Set α) (hs : IsBounded s) :
    Metric.diam (f '' s) ≤ K * Metric.diam s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) (s : Set α) (hs : Bornology.IsBounded s)
Goal: Metric.diam (f '' s) ≤ ↑K * Metric.diam s
-- LipschitzWith.dist_left
protected theorem dist_left (y : α) : LipschitzWith 1 (dist · y) 
Arguments: {α : Type u} [PseudoMetricSpace α] (y : α)
Goal: LipschitzWith 1 fun x => dist x y
-- LipschitzWith.dist_right
protected theorem dist_right (x : α) : LipschitzWith 1 (dist x) 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: LipschitzWith 1 (dist x)
-- LipschitzWith.dist
protected theorem dist : LipschitzWith 2 (Function.uncurry <| @dist α _) 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: LipschitzWith 2 (Function.uncurry dist)
-- LipschitzWith.dist_iterate_succ_le_geometric
theorem dist_iterate_succ_le_geometric {f : α → α} (hf : LipschitzWith K f) (x n) :
    dist (f^[n] x) (f^[n + 1] x) ≤ dist x (f x) * (K : ℝ) ^ n 
Arguments: {α : Type u} [PseudoMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : LipschitzWith K f) (x : α) (n : ℕ)
Goal: dist (f^[n] x) (f^[n + 1] x) ≤ dist x (f x) * ↑K ^ n
-- lipschitzWith_max
theorem _root_.lipschitzWith_max : LipschitzWith 1 fun p : ℝ × ℝ => max p.1 p.2 
Goal: LipschitzWith 1 fun p => max p.1 p.2
-- lipschitzWith_min
theorem _root_.lipschitzWith_min : LipschitzWith 1 fun p : ℝ × ℝ => min p.1 p.2 
Goal: LipschitzWith 1 fun p => min p.1 p.2
-- Real.lipschitzWith_toNNReal
lemma _root_.Real.lipschitzWith_toNNReal : LipschitzWith 1 Real.toNNReal 
Goal: LipschitzWith 1 Real.toNNReal
-- LipschitzWith.cauchySeq_comp
lemma cauchySeq_comp (hf : LipschitzWith K f) {u : ℕ → α} (hu : CauchySeq u) :
    CauchySeq (f ∘ u) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {f : α → β} (hf : LipschitzWith K f) {u : ℕ → α} (hu : CauchySeq u)
Goal: CauchySeq (f ∘ u)
-- LipschitzWith.max
protected theorem max (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
    LipschitzWith (max Kf Kg) fun x => max (f x) (g x) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f g : α → ℝ} {Kf Kg : ℝ≥0} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (max Kf Kg) fun x => max (f x) (g x)
-- LipschitzWith.min
protected theorem min (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
    LipschitzWith (max Kf Kg) fun x => min (f x) (g x) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f g : α → ℝ} {Kf Kg : ℝ≥0} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (max Kf Kg) fun x => min (f x) (g x)
-- LipschitzWith.max_const
theorem max_const (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => max (f x) a 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} {Kf : ℝ≥0} (hf : LipschitzWith Kf f) (a : ℝ)
Goal: LipschitzWith Kf fun x => max (f x) a
-- LipschitzWith.const_max
theorem const_max (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => max a (f x) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} {Kf : ℝ≥0} (hf : LipschitzWith Kf f) (a : ℝ)
Goal: LipschitzWith Kf fun x => max a (f x)
-- LipschitzWith.min_const
theorem min_const (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => min (f x) a 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} {Kf : ℝ≥0} (hf : LipschitzWith Kf f) (a : ℝ)
Goal: LipschitzWith Kf fun x => min (f x) a
-- LipschitzWith.const_min
theorem const_min (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => min a (f x) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} {Kf : ℝ≥0} (hf : LipschitzWith Kf f) (a : ℝ)
Goal: LipschitzWith Kf fun x => min a (f x)
-- LipschitzWith.projIcc
protected theorem projIcc {a b : ℝ} (h : a ≤ b) : LipschitzWith 1 (projIcc a b h) 
Arguments: {a b : ℝ} (h : a ≤ b)
Goal: LipschitzWith 1 (Set.projIcc a b h)
-- LipschitzOnWith.of_dist_le'
protected theorem of_dist_le' {K : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ K * dist x y) :
    LipschitzOnWith (Real.toNNReal K) f s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {s : Set α} {f : α → β} {K : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ K * dist x y)
Goal: LipschitzOnWith K.toNNReal f s
-- LipschitzOnWith.mk_one
protected theorem mk_one (h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y) :
    LipschitzOnWith 1 f s 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {s : Set α} {f : α → β} (h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y)
Goal: LipschitzOnWith 1 f s
-- LipschitzOnWith.of_le_add_mul'
protected theorem of_le_add_mul' {f : α → ℝ} (K : ℝ)
    (h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + K * dist x y) : LipschitzOnWith (Real.toNNReal K) f s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {f : α → ℝ} (K : ℝ) (h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + K * dist x y)
Goal: LipschitzOnWith K.toNNReal f s
-- LipschitzOnWith.of_le_add_mul
protected theorem of_le_add_mul {f : α → ℝ} (K : ℝ≥0)
    (h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + K * dist x y) : LipschitzOnWith K f s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {f : α → ℝ} (K : ℝ≥0) (h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + ↑K * dist x y)
Goal: LipschitzOnWith K f s
-- LipschitzOnWith.of_le_add
protected theorem of_le_add {f : α → ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + dist x y) :
    LipschitzOnWith 1 f s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {f : α → ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + dist x y)
Goal: LipschitzOnWith 1 f s
-- LipschitzOnWith.le_add_mul
protected theorem le_add_mul {f : α → ℝ} {K : ℝ≥0} (h : LipschitzOnWith K f s) {x : α} (hx : x ∈ s)
    {y : α} (hy : y ∈ s) : f x ≤ f y + K * dist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {f : α → ℝ} {K : ℝ≥0} (h : LipschitzOnWith K f s) {x : α} (hx : x ∈ s) {y : α} (hy : y ∈ s)
Goal: f x ≤ f y + ↑K * dist x y
-- LipschitzOnWith.iff_le_add_mul
protected theorem iff_le_add_mul {f : α → ℝ} {K : ℝ≥0} :
    LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + K * dist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {f : α → ℝ} {K : ℝ≥0}
Goal: LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, f x ≤ f y + ↑K * dist x y
-- LipschitzOnWith.isBounded_image2
theorem isBounded_image2 (f : α → β → γ) {K₁ K₂ : ℝ≥0} {s : Set α} {t : Set β}
    (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
    (hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (fun a => f a b) s)
    (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t) : Bornology.IsBounded (Set.image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [PseudoMetricSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] (f : α → β → γ) {K₁ K₂ : ℝ≥0} {s : Set α} {t : Set β} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t) (hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (fun a => f a b) s) (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t)
Goal: Bornology.IsBounded (Set.image2 f s t)
-- LipschitzOnWith.cauchySeq_comp
lemma cauchySeq_comp (hf : LipschitzOnWith K f s)
    {u : ℕ → α} (hu : CauchySeq u) (h'u : range u ⊆ s) :
    CauchySeq (f ∘ u) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} (hf : LipschitzOnWith K f s) {u : ℕ → α} (hu : CauchySeq u) (h'u : Set.range u ⊆ s)
Goal: CauchySeq (f ∘ u)
-- LocallyLipschitz.min
protected lemma min (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) :
    LocallyLipschitz (fun x => min (f x) (g x)) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f g : α → ℝ} (hf : LocallyLipschitz f) (hg : LocallyLipschitz g)
Goal: LocallyLipschitz fun x => min (f x) (g x)
-- LocallyLipschitz.max
protected lemma max (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) :
    LocallyLipschitz (fun x => max (f x) (g x)) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f g : α → ℝ} (hf : LocallyLipschitz f) (hg : LocallyLipschitz g)
Goal: LocallyLipschitz fun x => max (f x) (g x)
-- LocallyLipschitz.max_const
theorem max_const (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => max (f x) a 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} (hf : LocallyLipschitz f) (a : ℝ)
Goal: LocallyLipschitz fun x => max (f x) a
-- LocallyLipschitz.const_max
theorem const_max (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => max a (f x) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} (hf : LocallyLipschitz f) (a : ℝ)
Goal: LocallyLipschitz fun x => max a (f x)
-- LocallyLipschitz.min_const
theorem min_const (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => min (f x) a 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} (hf : LocallyLipschitz f) (a : ℝ)
Goal: LocallyLipschitz fun x => min (f x) a
-- LocallyLipschitz.const_min
theorem const_min (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => min a (f x) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {f : α → ℝ} (hf : LocallyLipschitz f) (a : ℝ)
Goal: LocallyLipschitz fun x => min a (f x)
-- continuousAt_of_locally_lipschitz
theorem continuousAt_of_locally_lipschitz {x : α} {r : ℝ} (hr : 0 < r) (K : ℝ)
    (h : ∀ y, dist y x < r → dist (f y) (f x) ≤ K * dist y x) : ContinuousAt f x 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {x : α} {r : ℝ} (hr : 0 < r) (K : ℝ) (h : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x)
Goal: ContinuousAt f x
-- LipschitzOnWith.extend_real
theorem LipschitzOnWith.extend_real {f : α → ℝ} {s : Set α} {K : ℝ≥0} (hf : LipschitzOnWith K f s) :
    ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn f g s 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : α → ℝ} {s : Set α} {K : ℝ≥0} (hf : LipschitzOnWith K f s)
Goal: ∃ g, LipschitzWith K g ∧ Set.EqOn f g s
-- LipschitzOnWith.extend_pi
theorem LipschitzOnWith.extend_pi [Fintype ι] {f : α → ι → ℝ} {s : Set α}
    {K : ℝ≥0} (hf : LipschitzOnWith K f s) : ∃ g : α → ι → ℝ, LipschitzWith K g ∧ EqOn f g s 
Arguments: {α : Type u} {ι : Type x} [PseudoMetricSpace α] [Fintype ι] {f : α → ι → ℝ} {s : Set α} {K : ℝ≥0} (hf : LipschitzOnWith K f s)
Goal: ∃ g, LipschitzWith K g ∧ Set.EqOn f g s
-- EMetric.eventually_nhds_zero_forall_closedBall_subset
theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
    ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) (x : X)
Goal: ∀ᶠ (p : ℝ≥0∞ × X) in 𝓝 0 ×ˢ 𝓝 x, ∀ (i : ι), p.2 ∈ K i → EMetric.closedBall p.2 p.1 ⊆ U i
-- EMetric.exists_forall_closedBall_subset_aux₁
theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
    (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
    ∃ r : ℝ, ∀ᶠ y in 𝓝 x,
      r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i } 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) (x : X)
Goal: ∃ r, ∀ᶠ (y : X) in 𝓝 x, r ∈ Set.Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ i, ⋂ (_ : y ∈ K i), {r | EMetric.closedBall y r ⊆ U i}
-- EMetric.exists_forall_closedBall_subset_aux₂
theorem exists_forall_closedBall_subset_aux₂ (y : X) :
    Convex ℝ
      (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i }) 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (y : X)
Goal: Convex ℝ (Set.Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ i, ⋂ (_ : y ∈ K i), {r | EMetric.closedBall y r ⊆ U i})
-- EMetric.exists_continuous_real_forall_closedBall_subset
theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧
      ∀ (i), ∀ x ∈ K i, closedBall x (ENNReal.ofReal <| δ x) ⊆ U i 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K)
Goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i
-- EMetric.exists_continuous_nnreal_forall_closedBall_subset
theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K)
Goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, EMetric.closedBall x ↑(δ x) ⊆ U i
-- EMetric.exists_continuous_eNNReal_forall_closedBall_subset
theorem exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ≥0∞), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K)
Goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, EMetric.closedBall x (δ x) ⊆ U i
-- Metric.exists_continuous_nnreal_forall_closedBall_subset
theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
Arguments: {ι : Type u_1} {X : Type u_2} [MetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K)
Goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x ↑(δ x) ⊆ U i
-- Metric.exists_continuous_real_forall_closedBall_subset
theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
Arguments: {ι : Type u_1} {X : Type u_2} [MetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K)
Goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x (δ x) ⊆ U i
-- tendstoUniformlyOnFilter_iff_tendsto
theorem tendstoUniformlyOnFilter_iff_tendsto :
    TendstoUniformlyOnFilter F f p p' ↔
      Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ p') (𝓤 β) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {p' : Filter α}
Goal: TendstoUniformlyOnFilter F f p p' ↔ Filter.Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ p') (𝓤 β)
-- tendstoUniformlyOn_iff_tendstoUniformlyOnFilter
theorem tendstoUniformlyOn_iff_tendstoUniformlyOnFilter :
    TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι}
Goal: TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s)
-- tendstoUniformlyOn_iff_tendsto
theorem tendstoUniformlyOn_iff_tendsto {F : ι → α → β} {f : α → β} {p : Filter ι} {s : Set α} :
    TendstoUniformlyOn F f p s ↔
    Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} {s : Set α}
Goal: TendstoUniformlyOn F f p s ↔ Filter.Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β)
-- tendstoUniformlyOn_univ
theorem tendstoUniformlyOn_univ : TendstoUniformlyOn F f p univ ↔ TendstoUniformly F f p 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι}
Goal: TendstoUniformlyOn F f p Set.univ ↔ TendstoUniformly F f p
-- tendstoUniformly_iff_tendstoUniformlyOnFilter
theorem tendstoUniformly_iff_tendstoUniformlyOnFilter :
    TendstoUniformly F f p ↔ TendstoUniformlyOnFilter F f p ⊤ 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι}
Goal: TendstoUniformly F f p ↔ TendstoUniformlyOnFilter F f p ⊤
-- TendstoUniformly.tendstoUniformlyOnFilter
theorem TendstoUniformly.tendstoUniformlyOnFilter (h : TendstoUniformly F f p) :
    TendstoUniformlyOnFilter F f p ⊤ 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι} (h : TendstoUniformly F f p)
Goal: TendstoUniformlyOnFilter F f p ⊤
-- tendstoUniformlyOn_iff_tendstoUniformly_comp_coe
theorem tendstoUniformlyOn_iff_tendstoUniformly_comp_coe :
    TendstoUniformlyOn F f p s ↔ TendstoUniformly (fun i (x : s) => F i x) (f ∘ (↑)) p 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι}
Goal: TendstoUniformlyOn F f p s ↔ TendstoUniformly (fun i x => F i ↑x) (f ∘ Subtype.val) p
-- tendstoUniformly_iff_tendsto
theorem tendstoUniformly_iff_tendsto {F : ι → α → β} {f : α → β} {p : Filter ι} :
    TendstoUniformly F f p ↔ Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ ⊤) (𝓤 β) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {p : Filter ι}
Goal: TendstoUniformly F f p ↔ Filter.Tendsto (fun q => (f q.2, F q.1 q.2)) (p ×ˢ ⊤) (𝓤 β)
-- TendstoUniformlyOnFilter.tendsto_at
theorem TendstoUniformlyOnFilter.tendsto_at (h : TendstoUniformlyOnFilter F f p p')
    (hx : 𝓟 {x} ≤ p') : Tendsto (fun n => F n x) p <| 𝓝 (f x) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {x : α} {p : Filter ι} {p' : Filter α} (h : TendstoUniformlyOnFilter F f p p') (hx : 𝓟 {x} ≤ p')
Goal: Filter.Tendsto (fun n => F n x) p (𝓝 (f x))
-- TendstoUniformlyOn.tendsto_at
theorem TendstoUniformlyOn.tendsto_at (h : TendstoUniformlyOn F f p s) {x : α} (hx : x ∈ s) :
    Tendsto (fun n => F n x) p <| 𝓝 (f x) 
Arguments: {α : Type u} {β : Type v} {ι : Type x} [UniformSpace β] {F : ι → α → β} {f : α → β} {s : Set α} {p : Filter ι} (h : TendstoUniformlyOn F f p s) {x : α} (hx : x ∈ s)
Goal: Filter.Tendsto (fun n => F n x) p (𝓝 (f x))