ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- holderWith_one
@[simp]
theorem holderWith_one {C : ℝ≥0} {f : X → Y} : HolderWith C 1 f ↔ LipschitzWith C f 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C : ℝ≥0} {f : X → Y}
Goal: HolderWith C 1 f ↔ LipschitzWith C f
-- holderWith_id
theorem holderWith_id : HolderWith 1 1 (id : X → X) 
Arguments: {X : Type u_1} [PseudoEMetricSpace X]
Goal: HolderWith 1 1 id
-- HolderWith.holderOnWith
protected theorem HolderWith.holderOnWith {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f)
    (s : Set X) : HolderOnWith C r f s 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f) (s : Set X)
Goal: HolderOnWith C r f s
-- HolderOnWith.edist_le
theorem edist_le (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) :
    edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s)
Goal: edist (f x) (f y) ≤ ↑C * edist x y ^ ↑r
-- HolderOnWith.edist_le_of_le
theorem edist_le_of_le (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) {d : ℝ≥0∞}
    (hd : edist x y ≤ d) : edist (f x) (f y) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (h : HolderOnWith C r f s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) {d : ℝ≥0∞} (hd : edist x y ≤ d)
Goal: edist (f x) (f y) ≤ ↑C * d ^ ↑r
-- HolderOnWith.comp
theorem comp {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) {Cf rf : ℝ≥0}
    {f : X → Y} (hf : HolderOnWith Cf rf f s) (hst : MapsTo f s t) :
    HolderOnWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] {s : Set X} {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) {Cf rf : ℝ≥0} {f : X → Y} (hf : HolderOnWith Cf rf f s) (hst : Set.MapsTo f s t)
Goal: HolderOnWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f) s
-- HolderOnWith.comp_holderWith
theorem comp_holderWith {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t)
    {Cf rf : ℝ≥0} {f : X → Y} (hf : HolderWith Cf rf f) (ht : ∀ x, f x ∈ t) :
    HolderWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) {Cf rf : ℝ≥0} {f : X → Y} (hf : HolderWith Cf rf f) (ht : ∀ (x : X), f x ∈ t)
Goal: HolderWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f)
-- HolderOnWith.uniformContinuousOn
protected theorem uniformContinuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) :
    UniformContinuousOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (hf : HolderOnWith C r f s) (h0 : 0 < r)
Goal: UniformContinuousOn f s
-- HolderOnWith.continuousOn
protected theorem continuousOn (hf : HolderOnWith C r f s) (h0 : 0 < r) : ContinuousOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (hf : HolderOnWith C r f s) (h0 : 0 < r)
Goal: ContinuousOn f s
-- HolderOnWith.mono
protected theorem mono (hf : HolderOnWith C r f s) (ht : t ⊆ s) : HolderOnWith C r f t 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s t : Set X} (hf : HolderOnWith C r f s) (ht : t ⊆ s)
Goal: HolderOnWith C r f t
-- HolderOnWith.ediam_image_le_of_le
theorem ediam_image_le_of_le (hf : HolderOnWith C r f s) {d : ℝ≥0∞} (hd : EMetric.diam s ≤ d) :
    EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (hf : HolderOnWith C r f s) {d : ℝ≥0∞} (hd : EMetric.diam s ≤ d)
Goal: EMetric.diam (f '' s) ≤ ↑C * d ^ ↑r
-- HolderOnWith.ediam_image_le
theorem ediam_image_le (hf : HolderOnWith C r f s) :
    EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (hf : HolderOnWith C r f s)
Goal: EMetric.diam (f '' s) ≤ ↑C * EMetric.diam s ^ ↑r
-- HolderOnWith.ediam_image_le_of_subset
theorem ediam_image_le_of_subset (hf : HolderOnWith C r f s) (ht : t ⊆ s) :
    EMetric.diam (f '' t) ≤ (C : ℝ≥0∞) * EMetric.diam t ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s t : Set X} (hf : HolderOnWith C r f s) (ht : t ⊆ s)
Goal: EMetric.diam (f '' t) ≤ ↑C * EMetric.diam t ^ ↑r
-- HolderOnWith.ediam_image_le_of_subset_of_le
theorem ediam_image_le_of_subset_of_le (hf : HolderOnWith C r f s) (ht : t ⊆ s) {d : ℝ≥0∞}
    (hd : EMetric.diam t ≤ d) : EMetric.diam (f '' t) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s t : Set X} (hf : HolderOnWith C r f s) (ht : t ⊆ s) {d : ℝ≥0∞} (hd : EMetric.diam t ≤ d)
Goal: EMetric.diam (f '' t) ≤ ↑C * d ^ ↑r
-- HolderOnWith.ediam_image_inter_le_of_le
theorem ediam_image_inter_le_of_le (hf : HolderOnWith C r f s) {d : ℝ≥0∞}
    (hd : EMetric.diam t ≤ d) : EMetric.diam (f '' (t ∩ s)) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s t : Set X} (hf : HolderOnWith C r f s) {d : ℝ≥0∞} (hd : EMetric.diam t ≤ d)
Goal: EMetric.diam (f '' (t ∩ s)) ≤ ↑C * d ^ ↑r
-- HolderOnWith.ediam_image_inter_le
theorem ediam_image_inter_le (hf : HolderOnWith C r f s) (t : Set X) :
    EMetric.diam (f '' (t ∩ s)) ≤ (C : ℝ≥0∞) * EMetric.diam t ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (hf : HolderOnWith C r f s) (t : Set X)
Goal: EMetric.diam (f '' (t ∩ s)) ≤ ↑C * EMetric.diam t ^ ↑r
-- HolderWith.edist_le
theorem edist_le (h : HolderWith C r f) (x y : X) :
    edist (f x) (f y) ≤ (C : ℝ≥0∞) * edist x y ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f) (x y : X)
Goal: edist (f x) (f y) ≤ ↑C * edist x y ^ ↑r
-- HolderWith.edist_le_of_le
theorem edist_le_of_le (h : HolderWith C r f) {x y : X} {d : ℝ≥0∞} (hd : edist x y ≤ d) :
    edist (f x) (f y) ≤ (C : ℝ≥0∞) * d ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f) {x y : X} {d : ℝ≥0∞} (hd : edist x y ≤ d)
Goal: edist (f x) (f y) ≤ ↑C * d ^ ↑r
-- HolderWith.comp
theorem comp {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0} {f : X → Y}
    (hf : HolderWith Cf rf f) : HolderWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0} {f : X → Y} (hf : HolderWith Cf rf f)
Goal: HolderWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f)
-- HolderWith.comp_holderOnWith
theorem comp_holderOnWith {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0}
    {f : X → Y} {s : Set X} (hf : HolderOnWith Cf rf f s) :
    HolderOnWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [PseudoEMetricSpace Z] {Cg rg : ℝ≥0} {g : Y → Z} (hg : HolderWith Cg rg g) {Cf rf : ℝ≥0} {f : X → Y} {s : Set X} (hf : HolderOnWith Cf rf f s)
Goal: HolderOnWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f) s
-- HolderWith.uniformContinuous
protected theorem uniformContinuous (hf : HolderWith C r f) (h0 : 0 < r) : UniformContinuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) (h0 : 0 < r)
Goal: UniformContinuous f
-- HolderWith.continuous
protected theorem continuous (hf : HolderWith C r f) (h0 : 0 < r) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) (h0 : 0 < r)
Goal: Continuous f
-- HolderWith.ediam_image_le
theorem ediam_image_le (hf : HolderWith C r f) (s : Set X) :
    EMetric.diam (f '' s) ≤ (C : ℝ≥0∞) * EMetric.diam s ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) (s : Set X)
Goal: EMetric.diam (f '' s) ≤ ↑C * EMetric.diam s ^ ↑r
-- HolderWith.nndist_le_of_le
theorem nndist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ≥0} (hd : nndist x y ≤ d) :
    nndist (f x) (f y) ≤ C * d ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) {x y : X} {d : ℝ≥0} (hd : nndist x y ≤ d)
Goal: nndist (f x) (f y) ≤ C * d ^ ↑r
-- HolderWith.nndist_le
theorem nndist_le (hf : HolderWith C r f) (x y : X) :
    nndist (f x) (f y) ≤ C * nndist x y ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) (x y : X)
Goal: nndist (f x) (f y) ≤ C * nndist x y ^ ↑r
-- HolderWith.dist_le_of_le
theorem dist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ} (hd : dist x y ≤ d) :
    dist (f x) (f y) ≤ C * d ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) {x y : X} {d : ℝ} (hd : dist x y ≤ d)
Goal: dist (f x) (f y) ≤ ↑C * d ^ ↑r
-- HolderWith.dist_le
theorem dist_le (hf : HolderWith C r f) (x y : X) : dist (f x) (f y) ≤ C * dist x y ^ (r : ℝ) 
Arguments: {X : Type u_1} {Y : Type u_2} [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (hf : HolderWith C r f) (x y : X)
Goal: dist (f x) (f y) ≤ ↑C * dist x y ^ ↑r
-- continuous_thickenedIndicatorAux
theorem continuous_thickenedIndicatorAux {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
    Continuous (thickenedIndicatorAux δ E) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α)
Goal: Continuous (thickenedIndicatorAux δ E)
-- thickenedIndicatorAux_le_one
theorem thickenedIndicatorAux_le_one (δ : ℝ) (E : Set α) (x : α) :
    thickenedIndicatorAux δ E x ≤ 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α) (x : α)
Goal: thickenedIndicatorAux δ E x ≤ 1
-- thickenedIndicatorAux_lt_top
theorem thickenedIndicatorAux_lt_top {δ : ℝ} {E : Set α} {x : α} :
    thickenedIndicatorAux δ E x < ∞ 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} {E : Set α} {x : α}
Goal: thickenedIndicatorAux δ E x < ⊤
-- thickenedIndicatorAux_closure_eq
theorem thickenedIndicatorAux_closure_eq (δ : ℝ) (E : Set α) :
    thickenedIndicatorAux δ (closure E) = thickenedIndicatorAux δ E 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: thickenedIndicatorAux δ (closure E) = thickenedIndicatorAux δ E
-- thickenedIndicatorAux_one
theorem thickenedIndicatorAux_one (δ : ℝ) (E : Set α) {x : α} (x_in_E : x ∈ E) :
    thickenedIndicatorAux δ E x = 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α) {x : α} (x_in_E : x ∈ E)
Goal: thickenedIndicatorAux δ E x = 1
-- thickenedIndicatorAux_one_of_mem_closure
theorem thickenedIndicatorAux_one_of_mem_closure (δ : ℝ) (E : Set α) {x : α}
    (x_mem : x ∈ closure E) : thickenedIndicatorAux δ E x = 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α) {x : α} (x_mem : x ∈ closure E)
Goal: thickenedIndicatorAux δ E x = 1
-- thickenedIndicatorAux_zero
theorem thickenedIndicatorAux_zero {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
    (x_out : x ∉ thickening δ E) : thickenedIndicatorAux δ E x = 0 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α} (x_out : x ∉ Metric.thickening δ E)
Goal: thickenedIndicatorAux δ E x = 0
-- thickenedIndicatorAux_mono
theorem thickenedIndicatorAux_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
    thickenedIndicatorAux δ₁ E ≤ thickenedIndicatorAux δ₂ E 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α)
Goal: thickenedIndicatorAux δ₁ E ≤ thickenedIndicatorAux δ₂ E
-- indicator_le_thickenedIndicatorAux
theorem indicator_le_thickenedIndicatorAux (δ : ℝ) (E : Set α) :
    (E.indicator fun _ => (1 : ℝ≥0∞)) ≤ thickenedIndicatorAux δ E 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: (E.indicator fun x => 1) ≤ thickenedIndicatorAux δ E
-- thickenedIndicatorAux_subset
theorem thickenedIndicatorAux_subset (δ : ℝ) {E₁ E₂ : Set α} (subset : E₁ ⊆ E₂) :
    thickenedIndicatorAux δ E₁ ≤ thickenedIndicatorAux δ E₂ 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] (δ : ℝ) {E₁ E₂ : Set α} (subset : E₁ ⊆ E₂)
Goal: thickenedIndicatorAux δ E₁ ≤ thickenedIndicatorAux δ E₂
-- thickenedIndicatorAux_tendsto_indicator_closure
theorem thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
    (δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
    Tendsto (fun n => thickenedIndicatorAux (δseq n) E) atTop
      (𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0∞))) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δseq : ℕ → ℝ} (δseq_lim : Filter.Tendsto δseq Filter.atTop (𝓝 0)) (E : Set α)
Goal: Filter.Tendsto (fun n => thickenedIndicatorAux (δseq n) E) Filter.atTop (𝓝 ((closure E).indicator fun x => 1))
-- thickenedIndicator_apply
@[simps]
def thickenedIndicator {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) : α →ᵇ ℝ≥0 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) (x : α)
Goal: (thickenedIndicator δ_pos E) x = (thickenedIndicatorAux δ E x).toNNReal
-- thickenedIndicator_toFun
@[simps]
def thickenedIndicator {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) : α →ᵇ ℝ≥0 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) (x : α)
Goal: (thickenedIndicator δ_pos E) x = (thickenedIndicatorAux δ E x).toNNReal
-- thickenedIndicator.coeFn_eq_comp
theorem thickenedIndicator.coeFn_eq_comp {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
    ⇑(thickenedIndicator δ_pos E) = ENNReal.toNNReal ∘ thickenedIndicatorAux δ E 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α)
Goal: ⇑(thickenedIndicator δ_pos E) = ENNReal.toNNReal ∘ thickenedIndicatorAux δ E
-- thickenedIndicator_le_one
theorem thickenedIndicator_le_one {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) (x : α) :
    thickenedIndicator δ_pos E x ≤ 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) (x : α)
Goal: (thickenedIndicator δ_pos E) x ≤ 1
-- thickenedIndicator_one_of_mem_closure
theorem thickenedIndicator_one_of_mem_closure {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
    (x_mem : x ∈ closure E) : thickenedIndicator δ_pos E x = 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α} (x_mem : x ∈ closure E)
Goal: (thickenedIndicator δ_pos E) x = 1
-- one_le_thickenedIndicator_apply'
lemma one_le_thickenedIndicator_apply' {X : Type _} [PseudoEMetricSpace X]
    {δ : ℝ} (δ_pos : 0 < δ) {F : Set X} {x : X} (hxF : x ∈ closure F) :
    1 ≤ thickenedIndicator δ_pos F x 
Arguments: {X : Type u_2} [PseudoEMetricSpace X] {δ : ℝ} (δ_pos : 0 < δ) {F : Set X} {x : X} (hxF : x ∈ closure F)
Goal: 1 ≤ (thickenedIndicator δ_pos F) x
-- one_le_thickenedIndicator_apply
lemma one_le_thickenedIndicator_apply (X : Type _) [PseudoEMetricSpace X]
    {δ : ℝ} (δ_pos : 0 < δ) {F : Set X} {x : X} (hxF : x ∈ F) :
    1 ≤ thickenedIndicator δ_pos F x 
Arguments: (X : Type u_2) [PseudoEMetricSpace X] {δ : ℝ} (δ_pos : 0 < δ) {F : Set X} {x : X} (hxF : x ∈ F)
Goal: 1 ≤ (thickenedIndicator δ_pos F) x
-- thickenedIndicator_one
theorem thickenedIndicator_one {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α} (x_in_E : x ∈ E) :
    thickenedIndicator δ_pos E x = 1 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α} (x_in_E : x ∈ E)
Goal: (thickenedIndicator δ_pos E) x = 1
-- thickenedIndicator_zero
theorem thickenedIndicator_zero {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
    (x_out : x ∉ thickening δ E) : thickenedIndicator δ_pos E x = 0 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α} (x_out : x ∉ Metric.thickening δ E)
Goal: (thickenedIndicator δ_pos E) x = 0
-- indicator_le_thickenedIndicator
theorem indicator_le_thickenedIndicator {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
    (E.indicator fun _ => (1 : ℝ≥0)) ≤ thickenedIndicator δ_pos E 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α)
Goal: (E.indicator fun x => 1) ≤ ⇑(thickenedIndicator δ_pos E)
-- thickenedIndicator_mono
theorem thickenedIndicator_mono {δ₁ δ₂ : ℝ} (δ₁_pos : 0 < δ₁) (δ₂_pos : 0 < δ₂) (hle : δ₁ ≤ δ₂)
    (E : Set α) : ⇑(thickenedIndicator δ₁_pos E) ≤ thickenedIndicator δ₂_pos E 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ₁ δ₂ : ℝ} (δ₁_pos : 0 < δ₁) (δ₂_pos : 0 < δ₂) (hle : δ₁ ≤ δ₂) (E : Set α)
Goal: ⇑(thickenedIndicator δ₁_pos E) ≤ ⇑(thickenedIndicator δ₂_pos E)
-- thickenedIndicator_subset
theorem thickenedIndicator_subset {δ : ℝ} (δ_pos : 0 < δ) {E₁ E₂ : Set α} (subset : E₁ ⊆ E₂) :
    ⇑(thickenedIndicator δ_pos E₁) ≤ thickenedIndicator δ_pos E₂ 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) {E₁ E₂ : Set α} (subset : E₁ ⊆ E₂)
Goal: ⇑(thickenedIndicator δ_pos E₁) ≤ ⇑(thickenedIndicator δ_pos E₂)
-- thickenedIndicator_tendsto_indicator_closure
theorem thickenedIndicator_tendsto_indicator_closure {δseq : ℕ → ℝ} (δseq_pos : ∀ n, 0 < δseq n)
    (δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
    Tendsto (fun n : ℕ => ((↑) : (α →ᵇ ℝ≥0) → α → ℝ≥0) (thickenedIndicator (δseq_pos n) E)) atTop
      (𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0))) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {δseq : ℕ → ℝ} (δseq_pos : ∀ (n : ℕ), 0 < δseq n) (δseq_lim : Filter.Tendsto δseq Filter.atTop (𝓝 0)) (E : Set α)
Goal: Filter.Tendsto (fun n => ⇑(thickenedIndicator ⋯ E)) Filter.atTop (𝓝 ((closure E).indicator fun x => 1))
-- mulIndicator_thickening_eventually_eq_mulIndicator_closure
@[to_additive "Pointwise, the indicators of δ-thickenings of a set eventually coincide
with the indicator of the set as δ>0 tends to zero."]
lemma mulIndicator_thickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ),
      (Metric.thickening δ E).mulIndicator f x = (closure E).mulIndicator f x 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [One β] (f : α → β) (E : Set α) (x : α)
Goal: ∀ᶠ (δ : ℝ) in 𝓝[>] 0, (Metric.thickening δ E).mulIndicator f x = (closure E).mulIndicator f x
-- indicator_thickening_eventually_eq_indicator_closure
@[to_additive "Pointwise, the indicators of δ-thickenings of a set eventually coincide
with the indicator of the set as δ>0 tends to zero."]
lemma mulIndicator_thickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :
    ∀ᶠ δ in 𝓝[>] (0 : ℝ),
      (Metric.thickening δ E).mulIndicator f x = (closure E).mulIndicator f x 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [Zero β] (f : α → β) (E : Set α) (x : α)
Goal: ∀ᶠ (δ : ℝ) in 𝓝[>] 0, (Metric.thickening δ E).indicator f x = (closure E).indicator f x
-- indicator_cthickening_eventually_eq_indicator_closure
@[to_additive "Pointwise, the indicators of closed δ-thickenings of a set eventually coincide
with the indicator of the set as δ tends to zero."]
lemma mulIndicator_cthickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :
    ∀ᶠ δ in 𝓝 (0 : ℝ),
      (Metric.cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [Zero β] (f : α → β) (E : Set α) (x : α)
Goal: ∀ᶠ (δ : ℝ) in 𝓝 0, (Metric.cthickening δ E).indicator f x = (closure E).indicator f x
-- mulIndicator_cthickening_eventually_eq_mulIndicator_closure
@[to_additive "Pointwise, the indicators of closed δ-thickenings of a set eventually coincide
with the indicator of the set as δ tends to zero."]
lemma mulIndicator_cthickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :
    ∀ᶠ δ in 𝓝 (0 : ℝ),
      (Metric.cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [One β] (f : α → β) (E : Set α) (x : α)
Goal: ∀ᶠ (δ : ℝ) in 𝓝 0, (Metric.cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x
-- tendsto_mulIndicator_thickening_mulIndicator_closure
@[to_additive "The indicators of δ-thickenings of a set tend pointwise to the indicator of the
set, as δ>0 tends to zero."]
lemma tendsto_mulIndicator_thickening_mulIndicator_closure (f : α → β) (E : Set α) :
    Tendsto (fun δ ↦ (Metric.thickening δ E).mulIndicator f) (𝓝[>] 0)
      (𝓝 ((closure E).mulIndicator f)) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [One β] [TopologicalSpace β] (f : α → β) (E : Set α)
Goal: Filter.Tendsto (fun δ => (Metric.thickening δ E).mulIndicator f) (𝓝[>] 0) (𝓝 ((closure E).mulIndicator f))
-- tendsto_indicator_thickening_indicator_closure
@[to_additive "The indicators of δ-thickenings of a set tend pointwise to the indicator of the
set, as δ>0 tends to zero."]
lemma tendsto_mulIndicator_thickening_mulIndicator_closure (f : α → β) (E : Set α) :
    Tendsto (fun δ ↦ (Metric.thickening δ E).mulIndicator f) (𝓝[>] 0)
      (𝓝 ((closure E).mulIndicator f)) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [Zero β] [TopologicalSpace β] (f : α → β) (E : Set α)
Goal: Filter.Tendsto (fun δ => (Metric.thickening δ E).indicator f) (𝓝[>] 0) (𝓝 ((closure E).indicator f))
-- tendsto_indicator_cthickening_indicator_closure
@[to_additive "The indicators of closed δ-thickenings of a set tend pointwise to the indicator
of the set, as δ tends to zero."]
lemma tendsto_mulIndicator_cthickening_mulIndicator_closure (f : α → β) (E : Set α) :
    Tendsto (fun δ ↦ (Metric.cthickening δ E).mulIndicator f) (𝓝 0)
      (𝓝 ((closure E).mulIndicator f)) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [Zero β] [TopologicalSpace β] (f : α → β) (E : Set α)
Goal: Filter.Tendsto (fun δ => (Metric.cthickening δ E).indicator f) (𝓝 0) (𝓝 ((closure E).indicator f))
-- tendsto_mulIndicator_cthickening_mulIndicator_closure
@[to_additive "The indicators of closed δ-thickenings of a set tend pointwise to the indicator
of the set, as δ tends to zero."]
lemma tendsto_mulIndicator_cthickening_mulIndicator_closure (f : α → β) (E : Set α) :
    Tendsto (fun δ ↦ (Metric.cthickening δ E).mulIndicator f) (𝓝 0)
      (𝓝 ((closure E).mulIndicator f)) 
Arguments: {α : Type u_1} [PseudoEMetricSpace α] {β : Type u_2} [One β] [TopologicalSpace β] (f : α → β) (E : Set α)
Goal: Filter.Tendsto (fun δ => (Metric.cthickening δ E).mulIndicator f) (𝓝 0) (𝓝 ((closure E).mulIndicator f))
-- _private.Mathlib.Topology.MetricSpace.Perfect.0.Perfect.small_diam_aux
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε 
Arguments: {α : Type u_1} [MetricSpace α] {C : Set α} (hC : Perfect C) {ε : ENNReal} (ε_pos : 0 < ε) {x : α} (xC : x ∈ C)
Goal: let D := closure (EMetric.ball x (ε / 2) ∩ C);
Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε
-- Perfect.small_diam_splitting
theorem Perfect.small_diam_splitting (ε_pos : 0 < ε) :
    ∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
    (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁ 
Arguments: {α : Type u_1} [MetricSpace α] {C : Set α} (hC : Perfect C) {ε : ENNReal} (hnonempty : C.Nonempty) (ε_pos : 0 < ε)
Goal: ∃ C₀ C₁,
  (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
    (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁