ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AntilipschitzWith.le_dimH_image
theorem le_dimH_image (hf : AntilipschitzWith K f) (s : Set X) : dimH s ≤ dimH (f '' s) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {K : ℝ≥0} {f : X → Y} (hf : AntilipschitzWith K f) (s : Set X)
Goal: dimH s ≤ dimH (f '' s)
-- Isometry.dimH_image
theorem Isometry.dimH_image (hf : Isometry f) (s : Set X) : dimH (f '' s) = dimH s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {f : X → Y} (hf : Isometry f) (s : Set X)
Goal: dimH (f '' s) = dimH s
-- IsometryEquiv.dimH_image
@[simp]
theorem dimH_image (e : X ≃ᵢ Y) (s : Set X) : dimH (e '' s) = dimH s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (e : X ≃ᵢ Y) (s : Set X)
Goal: dimH (⇑e '' s) = dimH s
-- IsometryEquiv.dimH_preimage
@[simp]
theorem dimH_preimage (e : X ≃ᵢ Y) (s : Set Y) : dimH (e ⁻¹' s) = dimH s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (e : X ≃ᵢ Y) (s : Set Y)
Goal: dimH (⇑e ⁻¹' s) = dimH s
-- IsometryEquiv.dimH_univ
theorem dimH_univ (e : X ≃ᵢ Y) : dimH (univ : Set X) = dimH (univ : Set Y) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (e : X ≃ᵢ Y)
Goal: dimH Set.univ = dimH Set.univ
-- ContinuousLinearEquiv.dimH_image
@[simp]
theorem dimH_image (e : E ≃L[𝕜] F) (s : Set E) : dimH (e '' s) = dimH s 
Arguments: {𝕜 : Type u_4} {E : Type u_5} {F : Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (e : E ≃L[𝕜] F) (s : Set E)
Goal: dimH (⇑e '' s) = dimH s
-- ContinuousLinearEquiv.dimH_preimage
@[simp]
theorem dimH_preimage (e : E ≃L[𝕜] F) (s : Set F) : dimH (e ⁻¹' s) = dimH s 
Arguments: {𝕜 : Type u_4} {E : Type u_5} {F : Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (e : E ≃L[𝕜] F) (s : Set F)
Goal: dimH (⇑e ⁻¹' s) = dimH s
-- ContinuousLinearEquiv.dimH_univ
theorem dimH_univ (e : E ≃L[𝕜] F) : dimH (univ : Set E) = dimH (univ : Set F) 
Arguments: {𝕜 : Type u_4} {E : Type u_5} {F : Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (e : E ≃L[𝕜] F)
Goal: dimH Set.univ = dimH Set.univ
-- Real.dimH_ball_pi
theorem dimH_ball_pi (x : ι → ℝ) {r : ℝ} (hr : 0 < r) :
    dimH (Metric.ball x r) = Fintype.card ι 
Arguments: {ι : Type u_1} [Fintype ι] (x : ι → ℝ) {r : ℝ} (hr : 0 < r)
Goal: dimH (Metric.ball x r) = ↑(Fintype.card ι)
-- Real.dimH_ball_pi_fin
theorem dimH_ball_pi_fin {n : ℕ} (x : Fin n → ℝ) {r : ℝ} (hr : 0 < r) :
    dimH (Metric.ball x r) = n 
Arguments: {n : ℕ} (x : Fin n → ℝ) {r : ℝ} (hr : 0 < r)
Goal: dimH (Metric.ball x r) = ↑n
-- Real.dimH_univ_pi
theorem dimH_univ_pi (ι : Type*) [Fintype ι] : dimH (univ : Set (ι → ℝ)) = Fintype.card ι 
Arguments: (ι : Type u_5) [Fintype ι]
Goal: dimH Set.univ = ↑(Fintype.card ι)
-- Real.dimH_univ_pi_fin
theorem dimH_univ_pi_fin (n : ℕ) : dimH (univ : Set (Fin n → ℝ)) = n 
Arguments: (n : ℕ)
Goal: dimH Set.univ = ↑n
-- Real.dimH_of_mem_nhds
theorem dimH_of_mem_nhds {x : E} {s : Set E} (h : s ∈ 𝓝 x) : dimH s = finrank ℝ E 
Arguments: {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {x : E} {s : Set E} (h : s ∈ 𝓝 x)
Goal: dimH s = ↑(FiniteDimensional.finrank ℝ E)
-- Real.dimH_of_nonempty_interior
theorem dimH_of_nonempty_interior {s : Set E} (h : (interior s).Nonempty) : dimH s = finrank ℝ E 
Arguments: {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} (h : (interior s).Nonempty)
Goal: dimH s = ↑(FiniteDimensional.finrank ℝ E)
-- Real.dimH_univ_eq_finrank
theorem dimH_univ_eq_finrank : dimH (univ : Set E) = finrank ℝ E 
Arguments: (E : Type u_4) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: dimH Set.univ = ↑(FiniteDimensional.finrank ℝ E)
-- Real.dimH_univ
theorem dimH_univ : dimH (univ : Set ℝ) = 1 
Goal: dimH Set.univ = 1
-- Real.hausdorffMeasure_of_finrank_lt
lemma hausdorffMeasure_of_finrank_lt [MeasurableSpace E] [BorelSpace E] {d : ℝ}
    (hd : finrank ℝ E < d) : (μH[d] : Measure E) = 0 
Arguments: {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {d : ℝ} (hd : ↑(FiniteDimensional.finrank ℝ E) < d)
Goal: μH[d] = 0
-- dense_compl_of_dimH_lt_finrank
theorem dense_compl_of_dimH_lt_finrank {s : Set E} (hs : dimH s < finrank ℝ E) : Dense sᶜ 
Arguments: {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} (hs : dimH s < ↑(FiniteDimensional.finrank ℝ E))
Goal: Dense sᶜ
-- ContDiffOn.dimH_image_le
theorem ContDiffOn.dimH_image_le {f : E → F} {s t : Set E} (hf : ContDiffOn ℝ 1 f s)
    (hc : Convex ℝ s) (ht : t ⊆ s) : dimH (f '' t) ≤ dimH t 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {s t : Set E} (hf : ContDiffOn ℝ 1 f s) (hc : Convex ℝ s) (ht : t ⊆ s)
Goal: dimH (f '' t) ≤ dimH t
-- ContDiff.dimH_range_le
theorem ContDiff.dimH_range_le {f : E → F} (h : ContDiff ℝ 1 f) : dimH (range f) ≤ finrank ℝ E 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} (h : ContDiff ℝ 1 f)
Goal: dimH (Set.range f) ≤ ↑(FiniteDimensional.finrank ℝ E)
-- ContDiffOn.dense_compl_image_of_dimH_lt_finrank
theorem ContDiffOn.dense_compl_image_of_dimH_lt_finrank [FiniteDimensional ℝ F] {f : E → F}
    {s t : Set E} (h : ContDiffOn ℝ 1 f s) (hc : Convex ℝ s) (ht : t ⊆ s)
    (htF : dimH t < finrank ℝ F) : Dense (f '' t)ᶜ 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {f : E → F} {s t : Set E} (h : ContDiffOn ℝ 1 f s) (hc : Convex ℝ s) (ht : t ⊆ s) (htF : dimH t < ↑(FiniteDimensional.finrank ℝ F))
Goal: Dense (f '' t)ᶜ
-- ContDiff.dense_compl_range_of_finrank_lt_finrank
theorem ContDiff.dense_compl_range_of_finrank_lt_finrank [FiniteDimensional ℝ F] {f : E → F}
    (h : ContDiff ℝ 1 f) (hEF : finrank ℝ E < finrank ℝ F) : Dense (range f)ᶜ 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {f : E → F} (h : ContDiff ℝ 1 f) (hEF : FiniteDimensional.finrank ℝ E < FiniteDimensional.finrank ℝ F)
Goal: Dense (Set.range f)ᶜ
-- TopCat.Presheaf.isSheaf_unit
theorem isSheaf_unit (F : Presheaf (CategoryTheory.Discrete Unit) X) : F.IsSheaf 
Arguments: {X : TopCat} (F : TopCat.Presheaf (CategoryTheory.Discrete Unit) X)
Goal: F.IsSheaf
-- TopCat.Presheaf.isSheaf_iso_iff
theorem isSheaf_iso_iff {F G : Presheaf C X} (α : F ≅ G) : F.IsSheaf ↔ G.IsSheaf 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} {F G : TopCat.Presheaf C X} (α : F ≅ G)
Goal: F.IsSheaf ↔ G.IsSheaf
-- TopCat.Presheaf.isSheaf_of_iso
theorem isSheaf_of_iso {F G : Presheaf C X} (α : F ≅ G) (h : F.IsSheaf) : G.IsSheaf 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} {F G : TopCat.Presheaf C X} (α : F ≅ G) (h : F.IsSheaf)
Goal: G.IsSheaf
-- TopCat.Sheaf.id_app
theorem id_app (F : Sheaf C X) (t) : (𝟙 F : F ⟶ F).1.app t = 𝟙 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : TopCat) (F : TopCat.Sheaf C X) (t : (TopologicalSpace.Opens ↑X)ᵒᵖ)
Goal: (𝟙 F).val.app t = 𝟙 (F.val.obj t)
-- TopCat.Sheaf.comp_app
theorem comp_app {F G H : Sheaf C X} (f : F ⟶ G) (g : G ⟶ H) (t) :
    (f ≫ g).1.app t = f.1.app t ≫ g.1.app t 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : TopCat) {F G H : TopCat.Sheaf C X} (f : F ⟶ G) (g : G ⟶ H) (t : (TopologicalSpace.Opens ↑X)ᵒᵖ)
Goal: (f ≫ g).val.app t = f.val.app t ≫ g.val.app t
-- Metric.mem_thickening_iff_infEdist_lt
theorem mem_thickening_iff_infEdist_lt : x ∈ thickening δ s ↔ infEdist x s < ENNReal.ofReal δ 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} {s : Set α} {x : α}
Goal: x ∈ Metric.thickening δ s ↔ EMetric.infEdist x s < ENNReal.ofReal δ
-- Metric.eventually_not_mem_thickening_of_infEdist_pos
lemma eventually_not_mem_thickening_of_infEdist_pos {E : Set α} {x : α} (h : x ∉ closure E) :
    ∀ᶠ δ in 𝓝 (0 : ℝ), x ∉ Metric.thickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {E : Set α} {x : α} (h : x ∉ closure E)
Goal: ∀ᶠ (δ : ℝ) in 𝓝 0, x ∉ Metric.thickening δ E
-- Metric.thickening_eq_preimage_infEdist
theorem thickening_eq_preimage_infEdist (δ : ℝ) (E : Set α) :
    thickening δ E = (infEdist · E) ⁻¹' Iio (ENNReal.ofReal δ) 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: Metric.thickening δ E = (fun x => EMetric.infEdist x E) ⁻¹' Set.Iio (ENNReal.ofReal δ)
-- Metric.isOpen_thickening
theorem isOpen_thickening {δ : ℝ} {E : Set α} : IsOpen (thickening δ E) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} {E : Set α}
Goal: IsOpen (Metric.thickening δ E)
-- Metric.thickening_empty
@[simp]
theorem thickening_empty (δ : ℝ) : thickening δ (∅ : Set α) = ∅ 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ)
Goal: Metric.thickening δ ∅ = ∅
-- Metric.thickening_of_nonpos
theorem thickening_of_nonpos (hδ : δ ≤ 0) (s : Set α) : thickening δ s = ∅ 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} (hδ : δ ≤ 0) (s : Set α)
Goal: Metric.thickening δ s = ∅
-- Metric.thickening_mono
theorem thickening_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
    thickening δ₁ E ⊆ thickening δ₂ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α)
Goal: Metric.thickening δ₁ E ⊆ Metric.thickening δ₂ E
-- Metric.thickening_subset_of_subset
theorem thickening_subset_of_subset (δ : ℝ) {E₁ E₂ : Set α} (h : E₁ ⊆ E₂) :
    thickening δ E₁ ⊆ thickening δ E₂ 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) {E₁ E₂ : Set α} (h : E₁ ⊆ E₂)
Goal: Metric.thickening δ E₁ ⊆ Metric.thickening δ E₂
-- Metric.mem_thickening_iff_exists_edist_lt
theorem mem_thickening_iff_exists_edist_lt {δ : ℝ} (E : Set α) (x : α) :
    x ∈ thickening δ E ↔ ∃ z ∈ E, edist x z < ENNReal.ofReal δ 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} (E : Set α) (x : α)
Goal: x ∈ Metric.thickening δ E ↔ ∃ z ∈ E, edist x z < ENNReal.ofReal δ
-- Metric.frontier_thickening_subset
theorem frontier_thickening_subset (E : Set α) {δ : ℝ} :
    frontier (thickening δ E) ⊆ { x : α | infEdist x E = ENNReal.ofReal δ } 
Arguments: {α : Type u} [PseudoEMetricSpace α] (E : Set α) {δ : ℝ}
Goal: frontier (Metric.thickening δ E) ⊆ {x | EMetric.infEdist x E = ENNReal.ofReal δ}
-- Metric.frontier_thickening_disjoint
theorem frontier_thickening_disjoint (A : Set α) :
    Pairwise (Disjoint on fun r : ℝ => frontier (thickening r A)) 
Arguments: {α : Type u} [PseudoEMetricSpace α] (A : Set α)
Goal: Pairwise (Disjoint on fun r => frontier (Metric.thickening r A))
-- Metric.subset_compl_thickening_compl_thickening_self
lemma subset_compl_thickening_compl_thickening_self (δ : ℝ) (E : Set α) :
    E ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: E ⊆ (Metric.thickening δ (Metric.thickening δ E)ᶜ)ᶜ
-- Metric.thickening_compl_thickening_self_subset_compl
lemma thickening_compl_thickening_self_subset_compl (δ : ℝ) (E : Set α) :
    thickening δ (thickening δ E)ᶜ ⊆ Eᶜ 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: Metric.thickening δ (Metric.thickening δ E)ᶜ ⊆ Eᶜ
-- Metric.mem_thickening_iff_infDist_lt
theorem mem_thickening_iff_infDist_lt {E : Set X} {x : X} (h : E.Nonempty) :
    x ∈ thickening δ E ↔ infDist x E < δ 
Arguments: {δ : ℝ} {X : Type u} [PseudoMetricSpace X] {E : Set X} {x : X} (h : E.Nonempty)
Goal: x ∈ Metric.thickening δ E ↔ Metric.infDist x E < δ
-- Metric.mem_thickening_iff
theorem mem_thickening_iff {E : Set X} {x : X} : x ∈ thickening δ E ↔ ∃ z ∈ E, dist x z < δ 
Arguments: {δ : ℝ} {X : Type u} [PseudoMetricSpace X] {E : Set X} {x : X}
Goal: x ∈ Metric.thickening δ E ↔ ∃ z ∈ E, dist x z < δ
-- Metric.thickening_singleton
@[simp]
theorem thickening_singleton (δ : ℝ) (x : X) : thickening δ ({x} : Set X) = ball x δ 
Arguments: {X : Type u} [PseudoMetricSpace X] (δ : ℝ) (x : X)
Goal: Metric.thickening δ {x} = Metric.ball x δ
-- Metric.ball_subset_thickening
theorem ball_subset_thickening {x : X} {E : Set X} (hx : x ∈ E) (δ : ℝ) :
    ball x δ ⊆ thickening δ E 
Arguments: {X : Type u} [PseudoMetricSpace X] {x : X} {E : Set X} (hx : x ∈ E) (δ : ℝ)
Goal: Metric.ball x δ ⊆ Metric.thickening δ E
-- Metric.thickening_eq_biUnion_ball
theorem thickening_eq_biUnion_ball {δ : ℝ} {E : Set X} : thickening δ E = ⋃ x ∈ E, ball x δ 
Arguments: {X : Type u} [PseudoMetricSpace X] {δ : ℝ} {E : Set X}
Goal: Metric.thickening δ E = ⋃ x ∈ E, Metric.ball x δ
-- Bornology.IsBounded.thickening
protected theorem _root_.Bornology.IsBounded.thickening {δ : ℝ} {E : Set X} (h : IsBounded E) :
    IsBounded (thickening δ E) 
Arguments: {X : Type u} [PseudoMetricSpace X] {δ : ℝ} {E : Set X} (h : Bornology.IsBounded E)
Goal: Bornology.IsBounded (Metric.thickening δ E)
-- Metric.mem_cthickening_iff
@[simp]
theorem mem_cthickening_iff : x ∈ cthickening δ s ↔ infEdist x s ≤ ENNReal.ofReal δ 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} {s : Set α} {x : α}
Goal: x ∈ Metric.cthickening δ s ↔ EMetric.infEdist x s ≤ ENNReal.ofReal δ
-- Metric.eventually_not_mem_cthickening_of_infEdist_pos
lemma eventually_not_mem_cthickening_of_infEdist_pos {E : Set α} {x : α} (h : x ∉ closure E) :
    ∀ᶠ δ in 𝓝 (0 : ℝ), x ∉ Metric.cthickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {E : Set α} {x : α} (h : x ∉ closure E)
Goal: ∀ᶠ (δ : ℝ) in 𝓝 0, x ∉ Metric.cthickening δ E
-- Metric.mem_cthickening_of_edist_le
theorem mem_cthickening_of_edist_le (x y : α) (δ : ℝ) (E : Set α) (h : y ∈ E)
    (h' : edist x y ≤ ENNReal.ofReal δ) : x ∈ cthickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (x y : α) (δ : ℝ) (E : Set α) (h : y ∈ E) (h' : edist x y ≤ ENNReal.ofReal δ)
Goal: x ∈ Metric.cthickening δ E
-- Metric.mem_cthickening_of_dist_le
theorem mem_cthickening_of_dist_le {α : Type*} [PseudoMetricSpace α] (x y : α) (δ : ℝ) (E : Set α)
    (h : y ∈ E) (h' : dist x y ≤ δ) : x ∈ cthickening δ E 
Arguments: {α : Type u_2} [PseudoMetricSpace α] (x y : α) (δ : ℝ) (E : Set α) (h : y ∈ E) (h' : dist x y ≤ δ)
Goal: x ∈ Metric.cthickening δ E
-- Metric.cthickening_eq_preimage_infEdist
theorem cthickening_eq_preimage_infEdist (δ : ℝ) (E : Set α) :
    cthickening δ E = (fun x => infEdist x E) ⁻¹' Iic (ENNReal.ofReal δ) 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: Metric.cthickening δ E = (fun x => EMetric.infEdist x E) ⁻¹' Set.Iic (ENNReal.ofReal δ)
-- Metric.isClosed_cthickening
theorem isClosed_cthickening {δ : ℝ} {E : Set α} : IsClosed (cthickening δ E) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} {E : Set α}
Goal: IsClosed (Metric.cthickening δ E)
-- Metric.cthickening_empty
@[simp]
theorem cthickening_empty (δ : ℝ) : cthickening δ (∅ : Set α) = ∅ 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ)
Goal: Metric.cthickening δ ∅ = ∅
-- Metric.cthickening_of_nonpos
theorem cthickening_of_nonpos {δ : ℝ} (hδ : δ ≤ 0) (E : Set α) : cthickening δ E = closure E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} (hδ : δ ≤ 0) (E : Set α)
Goal: Metric.cthickening δ E = closure E
-- Metric.cthickening_zero
@[simp]
theorem cthickening_zero (E : Set α) : cthickening 0 E = closure E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (E : Set α)
Goal: Metric.cthickening 0 E = closure E
-- Metric.cthickening_max_zero
theorem cthickening_max_zero (δ : ℝ) (E : Set α) : cthickening (max 0 δ) E = cthickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: Metric.cthickening (max 0 δ) E = Metric.cthickening δ E
-- Metric.cthickening_mono
theorem cthickening_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
    cthickening δ₁ E ⊆ cthickening δ₂ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α)
Goal: Metric.cthickening δ₁ E ⊆ Metric.cthickening δ₂ E
-- Metric.cthickening_singleton
@[simp]
theorem cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) {δ : ℝ} (hδ : 0 ≤ δ) :
    cthickening δ ({x} : Set α) = closedBall x δ 
Arguments: {α : Type u_2} [PseudoMetricSpace α] (x : α) {δ : ℝ} (hδ : 0 ≤ δ)
Goal: Metric.cthickening δ {x} = Metric.closedBall x δ
-- Metric.closedBall_subset_cthickening_singleton
theorem closedBall_subset_cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) (δ : ℝ) :
    closedBall x δ ⊆ cthickening δ ({x} : Set α) 
Arguments: {α : Type u_2} [PseudoMetricSpace α] (x : α) (δ : ℝ)
Goal: Metric.closedBall x δ ⊆ Metric.cthickening δ {x}
-- Metric.cthickening_subset_of_subset
theorem cthickening_subset_of_subset (δ : ℝ) {E₁ E₂ : Set α} (h : E₁ ⊆ E₂) :
    cthickening δ E₁ ⊆ cthickening δ E₂ 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) {E₁ E₂ : Set α} (h : E₁ ⊆ E₂)
Goal: Metric.cthickening δ E₁ ⊆ Metric.cthickening δ E₂
-- Metric.cthickening_subset_thickening
theorem cthickening_subset_thickening {δ₁ : ℝ≥0} {δ₂ : ℝ} (hlt : (δ₁ : ℝ) < δ₂) (E : Set α) :
    cthickening δ₁ E ⊆ thickening δ₂ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ₁ : ℝ≥0} {δ₂ : ℝ} (hlt : ↑δ₁ < δ₂) (E : Set α)
Goal: Metric.cthickening (↑δ₁) E ⊆ Metric.thickening δ₂ E
-- Metric.cthickening_subset_thickening'
theorem cthickening_subset_thickening' {δ₁ δ₂ : ℝ} (δ₂_pos : 0 < δ₂) (hlt : δ₁ < δ₂) (E : Set α) :
    cthickening δ₁ E ⊆ thickening δ₂ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ₁ δ₂ : ℝ} (δ₂_pos : 0 < δ₂) (hlt : δ₁ < δ₂) (E : Set α)
Goal: Metric.cthickening δ₁ E ⊆ Metric.thickening δ₂ E
-- Metric.thickening_subset_cthickening
theorem thickening_subset_cthickening (δ : ℝ) (E : Set α) : thickening δ E ⊆ cthickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: Metric.thickening δ E ⊆ Metric.cthickening δ E
-- Metric.thickening_subset_cthickening_of_le
theorem thickening_subset_cthickening_of_le {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
    thickening δ₁ E ⊆ cthickening δ₂ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α)
Goal: Metric.thickening δ₁ E ⊆ Metric.cthickening δ₂ E
-- Bornology.IsBounded.cthickening
theorem _root_.Bornology.IsBounded.cthickening {α : Type*} [PseudoMetricSpace α] {δ : ℝ} {E : Set α}
    (h : IsBounded E) : IsBounded (cthickening δ E) 
Arguments: {α : Type u_2} [PseudoMetricSpace α] {δ : ℝ} {E : Set α} (h : Bornology.IsBounded E)
Goal: Bornology.IsBounded (Metric.cthickening δ E)
-- IsCompact.cthickening
protected theorem _root_.IsCompact.cthickening
    {α : Type*} [PseudoMetricSpace α] [ProperSpace α] {s : Set α}
    (hs : IsCompact s) {r : ℝ} : IsCompact (cthickening r s) 
Arguments: {α : Type u_2} [PseudoMetricSpace α] [ProperSpace α] {s : Set α} (hs : IsCompact s) {r : ℝ}
Goal: IsCompact (Metric.cthickening r s)
-- Metric.thickening_subset_interior_cthickening
theorem thickening_subset_interior_cthickening (δ : ℝ) (E : Set α) :
    thickening δ E ⊆ interior (cthickening δ E) 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: Metric.thickening δ E ⊆ interior (Metric.cthickening δ E)
-- Metric.closure_thickening_subset_cthickening
theorem closure_thickening_subset_cthickening (δ : ℝ) (E : Set α) :
    closure (thickening δ E) ⊆ cthickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: closure (Metric.thickening δ E) ⊆ Metric.cthickening δ E
-- Metric.closure_subset_cthickening
theorem closure_subset_cthickening (δ : ℝ) (E : Set α) : closure E ⊆ cthickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (E : Set α)
Goal: closure E ⊆ Metric.cthickening δ E
-- Metric.closure_subset_thickening
theorem closure_subset_thickening {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
    closure E ⊆ thickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α)
Goal: closure E ⊆ Metric.thickening δ E
-- Metric.self_subset_thickening
theorem self_subset_thickening {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) : E ⊆ thickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} (δ_pos : 0 < δ) (E : Set α)
Goal: E ⊆ Metric.thickening δ E
-- Metric.self_subset_cthickening
theorem self_subset_cthickening {δ : ℝ} (E : Set α) : E ⊆ cthickening δ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} (E : Set α)
Goal: E ⊆ Metric.cthickening δ E
-- Metric.thickening_mem_nhdsSet
theorem thickening_mem_nhdsSet (E : Set α) {δ : ℝ} (hδ : 0 < δ) : thickening δ E ∈ 𝓝ˢ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (E : Set α) {δ : ℝ} (hδ : 0 < δ)
Goal: Metric.thickening δ E ∈ 𝓝ˢ E
-- Metric.cthickening_mem_nhdsSet
theorem cthickening_mem_nhdsSet (E : Set α) {δ : ℝ} (hδ : 0 < δ) : cthickening δ E ∈ 𝓝ˢ E 
Arguments: {α : Type u} [PseudoEMetricSpace α] (E : Set α) {δ : ℝ} (hδ : 0 < δ)
Goal: Metric.cthickening δ E ∈ 𝓝ˢ E
-- Metric.thickening_union
@[simp]
theorem thickening_union (δ : ℝ) (s t : Set α) :
    thickening δ (s ∪ t) = thickening δ s ∪ thickening δ t 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (s t : Set α)
Goal: Metric.thickening δ (s ∪ t) = Metric.thickening δ s ∪ Metric.thickening δ t
-- Metric.cthickening_union
@[simp]
theorem cthickening_union (δ : ℝ) (s t : Set α) :
    cthickening δ (s ∪ t) = cthickening δ s ∪ cthickening δ t 
Arguments: {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (s t : Set α)
Goal: Metric.cthickening δ (s ∪ t) = Metric.cthickening δ s ∪ Metric.cthickening δ t
-- Metric.thickening_iUnion
@[simp]
theorem thickening_iUnion (δ : ℝ) (f : ι → Set α) :
    thickening δ (⋃ i, f i) = ⋃ i, thickening δ (f i) 
Arguments: {ι : Sort u_1} {α : Type u} [PseudoEMetricSpace α] (δ : ℝ) (f : ι → Set α)
Goal: Metric.thickening δ (⋃ i, f i) = ⋃ i, Metric.thickening δ (f i)
-- Metric.thickening_biUnion
lemma thickening_biUnion {ι : Type*} (δ : ℝ) (f : ι → Set α) (I : Set ι) :
    thickening δ (⋃ i ∈ I, f i) = ⋃ i ∈ I, thickening δ (f i) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {ι : Type u_2} (δ : ℝ) (f : ι → Set α) (I : Set ι)
Goal: Metric.thickening δ (⋃ i ∈ I, f i) = ⋃ i ∈ I, Metric.thickening δ (f i)
-- Metric.ediam_cthickening_le
theorem ediam_cthickening_le (ε : ℝ≥0) :
    EMetric.diam (cthickening ε s) ≤ EMetric.diam s + 2 * ε 
Arguments: {α : Type u} [PseudoEMetricSpace α] {s : Set α} (ε : ℝ≥0)
Goal: EMetric.diam (Metric.cthickening (↑ε) s) ≤ EMetric.diam s + 2 * ↑ε
-- Metric.ediam_thickening_le
theorem ediam_thickening_le (ε : ℝ≥0) : EMetric.diam (thickening ε s) ≤ EMetric.diam s + 2 * ε 
Arguments: {α : Type u} [PseudoEMetricSpace α] {s : Set α} (ε : ℝ≥0)
Goal: EMetric.diam (Metric.thickening (↑ε) s) ≤ EMetric.diam s + 2 * ↑ε
-- Metric.diam_cthickening_le
theorem diam_cthickening_le {α : Type*} [PseudoMetricSpace α] (s : Set α) (hε : 0 ≤ ε) :
    diam (cthickening ε s) ≤ diam s + 2 * ε 
Arguments: {ε : ℝ} {α : Type u_2} [PseudoMetricSpace α] (s : Set α) (hε : 0 ≤ ε)
Goal: Metric.diam (Metric.cthickening ε s) ≤ Metric.diam s + 2 * ε
-- Metric.diam_thickening_le
theorem diam_thickening_le {α : Type*} [PseudoMetricSpace α] (s : Set α) (hε : 0 ≤ ε) :
    diam (thickening ε s) ≤ diam s + 2 * ε 
Arguments: {ε : ℝ} {α : Type u_2} [PseudoMetricSpace α] (s : Set α) (hε : 0 ≤ ε)
Goal: Metric.diam (Metric.thickening ε s) ≤ Metric.diam s + 2 * ε
-- Metric.thickening_closure
@[simp]
theorem thickening_closure : thickening δ (closure s) = thickening δ s 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} {s : Set α}
Goal: Metric.thickening δ (closure s) = Metric.thickening δ s
-- Metric.cthickening_closure
@[simp]
theorem cthickening_closure : cthickening δ (closure s) = cthickening δ s 
Arguments: {α : Type u} [PseudoEMetricSpace α] {δ : ℝ} {s : Set α}
Goal: Metric.cthickening δ (closure s) = Metric.cthickening δ s
-- Disjoint.exists_thickenings
theorem _root_.Disjoint.exists_thickenings (hst : Disjoint s t) (hs : IsCompact s)
    (ht : IsClosed t) :
    ∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) 
Arguments: {α : Type u} [PseudoEMetricSpace α] {s t : Set α} (hst : Disjoint s t) (hs : IsCompact s) (ht : IsClosed t)
Goal: ∃ δ, 0 < δ ∧ Disjoint (Metric.thickening δ s) (Metric.thickening δ t)