ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- VectorBundleCore.trivializationAt_coordChange_eq
@[simp, mfld_simps]
theorem trivializationAt_coordChange_eq {b₀ b₁ b : B}
    (hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet ∩ (trivializationAt F Z.Fiber b₁).baseSet)
    (v : F) :
    (trivializationAt F Z.Fiber b₀).coordChangeL R (trivializationAt F Z.Fiber b₁) b v =
      Z.coordChange (Z.indexAt b₀) (Z.indexAt b₁) b v 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} [NontriviallyNormedField R] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] {ι : Type u_5} (Z : VectorBundleCore R B F ι) {b₀ b₁ b : B} (hb : b ∈ (trivializationAt F Z.Fiber b₀).baseSet ∩ (trivializationAt F Z.Fiber b₁).baseSet) (v : F)
Goal: (Trivialization.coordChangeL R (trivializationAt F Z.Fiber b₀) (trivializationAt F Z.Fiber b₁) b) v =
  (Z.coordChange (Z.indexAt b₀) (Z.indexAt b₁) b) v
-- VectorPrebundle.mem_base_pretrivializationAt
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure VectorPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivialization_linear' : ∀ e, e ∈ pretrivializationAtlas → e.IsLinear R
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  exists_coordChange : ∀ᵉ (e ∈ pretrivializationAtlas) (e' ∈ pretrivializationAtlas),
    ∃ f : B → F →L[R] F, ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
      ∀ᵉ (b ∈ e.baseSet ∩ e'.baseSet) (v : F), f b v = (e' ⟨b, e.symm b v⟩).2
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ .mk b)
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (self : VectorPrebundle R F E) (x : B)
Goal: x ∈ (self.pretrivializationAt x).baseSet
-- VectorPrebundle.totalSpaceMk_inducing
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure VectorPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivialization_linear' : ∀ e, e ∈ pretrivializationAtlas → e.IsLinear R
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  exists_coordChange : ∀ᵉ (e ∈ pretrivializationAtlas) (e' ∈ pretrivializationAtlas),
    ∃ f : B → F →L[R] F, ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
      ∀ᵉ (b ∈ e.baseSet ∩ e'.baseSet) (v : F), f b v = (e' ⟨b, e.symm b v⟩).2
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ .mk b)
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (self : VectorPrebundle R F E) (b : B)
Goal: Inducing (↑(self.pretrivializationAt b) ∘ Bundle.TotalSpace.mk b)
-- VectorPrebundle.pretrivialization_linear'
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure VectorPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivialization_linear' : ∀ e, e ∈ pretrivializationAtlas → e.IsLinear R
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  exists_coordChange : ∀ᵉ (e ∈ pretrivializationAtlas) (e' ∈ pretrivializationAtlas),
    ∃ f : B → F →L[R] F, ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
      ∀ᵉ (b ∈ e.baseSet ∩ e'.baseSet) (v : F), f b v = (e' ⟨b, e.symm b v⟩).2
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ .mk b)
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (self : VectorPrebundle R F E) (e : Pretrivialization F Bundle.TotalSpace.proj)
Goal: e ∈ self.pretrivializationAtlas → Pretrivialization.IsLinear R e
-- VectorPrebundle.exists_coordChange
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure VectorPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivialization_linear' : ∀ e, e ∈ pretrivializationAtlas → e.IsLinear R
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  exists_coordChange : ∀ᵉ (e ∈ pretrivializationAtlas) (e' ∈ pretrivializationAtlas),
    ∃ f : B → F →L[R] F, ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
      ∀ᵉ (b ∈ e.baseSet ∩ e'.baseSet) (v : F), f b v = (e' ⟨b, e.symm b v⟩).2
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ .mk b)
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (self : VectorPrebundle R F E) (e : Pretrivialization F Bundle.TotalSpace.proj)
Goal: e ∈ self.pretrivializationAtlas →
  ∀ e' ∈ self.pretrivializationAtlas,
    ∃ f,
      ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
        ∀ b ∈ e.baseSet ∩ e'.baseSet, ∀ (v : F), (f b) v = (↑e' { proj := b, snd := e.symm b v }).2
-- VectorPrebundle.pretrivialization_mem_atlas
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure VectorPrebundle where
  pretrivializationAtlas : Set (Pretrivialization F (π F E))
  pretrivialization_linear' : ∀ e, e ∈ pretrivializationAtlas → e.IsLinear R
  pretrivializationAt : B → Pretrivialization F (π F E)
  mem_base_pretrivializationAt : ∀ x : B, x ∈ (pretrivializationAt x).baseSet
  pretrivialization_mem_atlas : ∀ x : B, pretrivializationAt x ∈ pretrivializationAtlas
  exists_coordChange : ∀ᵉ (e ∈ pretrivializationAtlas) (e' ∈ pretrivializationAtlas),
    ∃ f : B → F →L[R] F, ContinuousOn f (e.baseSet ∩ e'.baseSet) ∧
      ∀ᵉ (b ∈ e.baseSet ∩ e'.baseSet) (v : F), f b v = (e' ⟨b, e.symm b v⟩).2
  totalSpaceMk_inducing : ∀ b : B, Inducing (pretrivializationAt b ∘ .mk b)
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (self : VectorPrebundle R F E) (x : B)
Goal: self.pretrivializationAt x ∈ self.pretrivializationAtlas
-- VectorPrebundle.continuousOn_coordChange
theorem continuousOn_coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π F E)}
    (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) :
    ContinuousOn (a.coordChange he he') (e.baseSet ∩ e'.baseSet) 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E) {e e' : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas)
Goal: ContinuousOn (a.coordChange he he') (e.baseSet ∩ e'.baseSet)
-- VectorPrebundle.coordChange_apply
theorem coordChange_apply (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π F E)}
    (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B}
    (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
    a.coordChange he he' b v = (e' ⟨b, e.symm b v⟩).2 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E) {e e' : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F)
Goal: (a.coordChange he he' b) v = (↑e' { proj := b, snd := e.symm b v }).2
-- VectorPrebundle.mk_coordChange
theorem mk_coordChange (a : VectorPrebundle R F E) {e e' : Pretrivialization F (π F E)}
    (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B}
    (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
    (b, a.coordChange he he' b v) = e' ⟨b, e.symm b v⟩ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E) {e e' : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F)
Goal: (b, (a.coordChange he he' b) v) = ↑e' { proj := b, snd := e.symm b v }
-- VectorPrebundle.linear_trivializationOfMemPretrivializationAtlas
theorem linear_trivializationOfMemPretrivializationAtlas (a : VectorPrebundle R F E)
    {e : Pretrivialization F (π F E)} (he : e ∈ a.pretrivializationAtlas) :
    letI := a.totalSpaceTopology
    Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he) 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E) {e : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas)
Goal: Trivialization.IsLinear R (a.trivializationOfMemPretrivializationAtlas he)
-- VectorPrebundle.mem_trivialization_at_source
theorem mem_trivialization_at_source (b : B) (x : E b) :
    ⟨b, x⟩ ∈ (a.pretrivializationAt b).source 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E) (b : B) (x : E b)
Goal: { proj := b, snd := x } ∈ (a.pretrivializationAt b).source
-- VectorPrebundle.totalSpaceMk_preimage_source
@[simp]
theorem totalSpaceMk_preimage_source (b : B) :
    .mk b ⁻¹' (a.pretrivializationAt b).source = univ 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E) (b : B)
Goal: Bundle.TotalSpace.mk b ⁻¹' (a.pretrivializationAt b).source = Set.univ
-- VectorPrebundle.continuous_totalSpaceMk
@[continuity]
theorem continuous_totalSpaceMk (b : B) :
    Continuous[_, a.totalSpaceTopology] (.mk b) 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E) (b : B)
Goal: Continuous (Bundle.TotalSpace.mk b)
-- VectorPrebundle.toVectorBundle
theorem toVectorBundle : @VectorBundle R _ F E _ _ _ _ _ _ a.totalSpaceTopology _ a.toFiberBundle 
Arguments: {R : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [NontriviallyNormedField R] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module R (E x)] [NormedAddCommGroup F] [NormedSpace R F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle R F E)
Goal: VectorBundle R F E
-- ContinuousLinearMap.inCoordinates_eq
theorem inCoordinates_eq (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y)
    (hx : x ∈ (trivializationAt F E x₀).baseSet) (hy : y ∈ (trivializationAt F' E' y₀).baseSet) :
    inCoordinates F E F' E' x₀ x y₀ y ϕ =
      ((trivializationAt F' E' y₀).continuousLinearEquivAt 𝕜₂ y hy : E' y →L[𝕜₂] F').comp
        (ϕ.comp <|
          (((trivializationAt F E x₀).continuousLinearEquivAt 𝕜₁ x hx).symm : F →L[𝕜₁] E x)) 
Arguments: {B : Type u_2} {F : Type u_3} (E : B → Type u_4) [(x : B) → AddCommMonoid (E x)] [NormedAddCommGroup F] [TopologicalSpace B] [(x : B) → TopologicalSpace (E x)] {𝕜₁ : Type u_5} {𝕜₂ : Type u_6} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ : 𝕜₁ →+* 𝕜₂} {B' : Type u_7} [TopologicalSpace B'] [NormedSpace 𝕜₁ F] [(x : B) → Module 𝕜₁ (E x)] [TopologicalSpace (Bundle.TotalSpace F E)] {F' : Type u_8} [NormedAddCommGroup F'] [NormedSpace 𝕜₂ F'] (E' : B' → Type u_9) [(x : B') → AddCommMonoid (E' x)] [(x : B') → Module 𝕜₂ (E' x)] [TopologicalSpace (Bundle.TotalSpace F' E')] [FiberBundle F E] [VectorBundle 𝕜₁ F E] [(x : B') → TopologicalSpace (E' x)] [FiberBundle F' E'] [VectorBundle 𝕜₂ F' E'] (x₀ x : B) (y₀ y : B') (ϕ : E x →SL[σ] E' y) (hx : x ∈ (trivializationAt F E x₀).baseSet) (hy : y ∈ (trivializationAt F' E' y₀).baseSet)
Goal: ContinuousLinearMap.inCoordinates F E F' E' x₀ x y₀ y ϕ =
  (↑(Trivialization.continuousLinearEquivAt 𝕜₂ (trivializationAt F' E' y₀) y hy)).comp
    (ϕ.comp ↑(Trivialization.continuousLinearEquivAt 𝕜₁ (trivializationAt F E x₀) x hx).symm)
-- VectorBundleCore.inCoordinates_eq
protected theorem _root_.VectorBundleCore.inCoordinates_eq {ι ι'} (Z : VectorBundleCore 𝕜₁ B F ι)
    (Z' : VectorBundleCore 𝕜₂ B' F' ι') {x₀ x : B} {y₀ y : B'} (ϕ : F →SL[σ] F')
    (hx : x ∈ Z.baseSet (Z.indexAt x₀)) (hy : y ∈ Z'.baseSet (Z'.indexAt y₀)) :
    inCoordinates F Z.Fiber F' Z'.Fiber x₀ x y₀ y ϕ =
      (Z'.coordChange (Z'.indexAt y) (Z'.indexAt y₀) y).comp
        (ϕ.comp <| Z.coordChange (Z.indexAt x₀) (Z.indexAt x) x) 
Arguments: {B : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [TopologicalSpace B] {𝕜₁ : Type u_5} {𝕜₂ : Type u_6} [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ : 𝕜₁ →+* 𝕜₂} {B' : Type u_7} [TopologicalSpace B'] [NormedSpace 𝕜₁ F] {F' : Type u_8} [NormedAddCommGroup F'] [NormedSpace 𝕜₂ F'] {ι : Type u_10} {ι' : Type u_11} (Z : VectorBundleCore 𝕜₁ B F ι) (Z' : VectorBundleCore 𝕜₂ B' F' ι') {x₀ x : B} {y₀ y : B'} (ϕ : F →SL[σ] F') (hx : x ∈ Z.baseSet (Z.indexAt x₀)) (hy : y ∈ Z'.baseSet (Z'.indexAt y₀))
Goal: ContinuousLinearMap.inCoordinates F Z.Fiber F' Z'.Fiber x₀ x y₀ y ϕ =
  (Z'.coordChange (Z'.indexAt y) (Z'.indexAt y₀) y).comp (ϕ.comp (Z.coordChange (Z.indexAt x₀) (Z.indexAt x) x))
-- dimH_def
theorem dimH_def (s : Set X) : dimH s = ⨆ (d : ℝ≥0) (_ : μH[d] s = ∞), (d : ℝ≥0∞) 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (s : Set X)
Goal: dimH s = ⨆ d, ⨆ (_ : μH[↑d] s = ⊤), ↑d
-- hausdorffMeasure_of_lt_dimH
theorem hausdorffMeasure_of_lt_dimH {s : Set X} {d : ℝ≥0} (h : ↑d < dimH s) : μH[d] s = ∞ 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} {d : ℝ≥0} (h : ↑d < dimH s)
Goal: μH[↑d] s = ⊤
-- dimH_le
theorem dimH_le {s : Set X} {d : ℝ≥0∞} (H : ∀ d' : ℝ≥0, μH[d'] s = ∞ → ↑d' ≤ d) : dimH s ≤ d 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} {d : ℝ≥0∞} (H : ∀ (d' : ℝ≥0), μH[↑d'] s = ⊤ → ↑d' ≤ d)
Goal: dimH s ≤ d
-- dimH_le_of_hausdorffMeasure_ne_top
theorem dimH_le_of_hausdorffMeasure_ne_top {s : Set X} {d : ℝ≥0} (h : μH[d] s ≠ ∞) : dimH s ≤ d 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} {d : ℝ≥0} (h : μH[↑d] s ≠ ⊤)
Goal: dimH s ≤ ↑d
-- le_dimH_of_hausdorffMeasure_eq_top
theorem le_dimH_of_hausdorffMeasure_eq_top {s : Set X} {d : ℝ≥0} (h : μH[d] s = ∞) :
    ↑d ≤ dimH s 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} {d : ℝ≥0} (h : μH[↑d] s = ⊤)
Goal: ↑d ≤ dimH s
-- hausdorffMeasure_of_dimH_lt
theorem hausdorffMeasure_of_dimH_lt {s : Set X} {d : ℝ≥0} (h : dimH s < d) : μH[d] s = 0 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} {d : ℝ≥0} (h : dimH s < ↑d)
Goal: μH[↑d] s = 0
-- measure_zero_of_dimH_lt
theorem measure_zero_of_dimH_lt {μ : Measure X} {d : ℝ≥0} (h : μ ≪ μH[d]) {s : Set X}
    (hd : dimH s < d) : μ s = 0 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {μ : MeasureTheory.Measure X} {d : ℝ≥0} (h : μ ≪ μH[↑d]) {s : Set X} (hd : dimH s < ↑d)
Goal: μ s = 0
-- le_dimH_of_hausdorffMeasure_ne_zero
theorem le_dimH_of_hausdorffMeasure_ne_zero {s : Set X} {d : ℝ≥0} (h : μH[d] s ≠ 0) : ↑d ≤ dimH s 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {s : Set X} {d : ℝ≥0} (h : μH[↑d] s ≠ 0)
Goal: ↑d ≤ dimH s
-- dimH_of_hausdorffMeasure_ne_zero_ne_top
theorem dimH_of_hausdorffMeasure_ne_zero_ne_top {d : ℝ≥0} {s : Set X} (h : μH[d] s ≠ 0)
    (h' : μH[d] s ≠ ∞) : dimH s = d 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {d : ℝ≥0} {s : Set X} (h : μH[↑d] s ≠ 0) (h' : μH[↑d] s ≠ ⊤)
Goal: dimH s = ↑d
-- dimH_mono
@[mono]
theorem dimH_mono {s t : Set X} (h : s ⊆ t) : dimH s ≤ dimH t 
Arguments: {X : Type u_2} [EMetricSpace X] {s t : Set X} (h : s ⊆ t)
Goal: dimH s ≤ dimH t
-- dimH_subsingleton
theorem dimH_subsingleton {s : Set X} (h : s.Subsingleton) : dimH s = 0 
Arguments: {X : Type u_2} [EMetricSpace X] {s : Set X} (h : s.Subsingleton)
Goal: dimH s = 0
-- dimH_empty
@[simp]
theorem dimH_empty : dimH (∅ : Set X) = 0 
Arguments: {X : Type u_2} [EMetricSpace X]
Goal: dimH ∅ = 0
-- dimH_singleton
@[simp]
theorem dimH_singleton (x : X) : dimH ({x} : Set X) = 0 
Arguments: {X : Type u_2} [EMetricSpace X] (x : X)
Goal: dimH {x} = 0
-- dimH_iUnion
@[simp]
theorem dimH_iUnion {ι : Sort*} [Countable ι] (s : ι → Set X) :
    dimH (⋃ i, s i) = ⨆ i, dimH (s i) 
Arguments: {X : Type u_2} [EMetricSpace X] {ι : Sort u_4} [Countable ι] (s : ι → Set X)
Goal: dimH (⋃ i, s i) = ⨆ i, dimH (s i)
-- dimH_bUnion
@[simp]
theorem dimH_bUnion {s : Set ι} (hs : s.Countable) (t : ι → Set X) :
    dimH (⋃ i ∈ s, t i) = ⨆ i ∈ s, dimH (t i) 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {s : Set ι} (hs : s.Countable) (t : ι → Set X)
Goal: dimH (⋃ i ∈ s, t i) = ⨆ i ∈ s, dimH (t i)
-- dimH_sUnion
@[simp]
theorem dimH_sUnion {S : Set (Set X)} (hS : S.Countable) : dimH (⋃₀ S) = ⨆ s ∈ S, dimH s 
Arguments: {X : Type u_2} [EMetricSpace X] {S : Set (Set X)} (hS : S.Countable)
Goal: dimH (⋃₀ S) = ⨆ s ∈ S, dimH s
-- dimH_union
@[simp]
theorem dimH_union (s t : Set X) : dimH (s ∪ t) = max (dimH s) (dimH t) 
Arguments: {X : Type u_2} [EMetricSpace X] (s t : Set X)
Goal: dimH (s ∪ t) = max (dimH s) (dimH t)
-- dimH_countable
theorem dimH_countable {s : Set X} (hs : s.Countable) : dimH s = 0 
Arguments: {X : Type u_2} [EMetricSpace X] {s : Set X} (hs : s.Countable)
Goal: dimH s = 0
-- dimH_finite
theorem dimH_finite {s : Set X} (hs : s.Finite) : dimH s = 0 
Arguments: {X : Type u_2} [EMetricSpace X] {s : Set X} (hs : s.Finite)
Goal: dimH s = 0
-- dimH_coe_finset
@[simp]
theorem dimH_coe_finset (s : Finset X) : dimH (s : Set X) = 0 
Arguments: {X : Type u_2} [EMetricSpace X] (s : Finset X)
Goal: dimH ↑s = 0
-- exists_mem_nhdsWithin_lt_dimH_of_lt_dimH
theorem exists_mem_nhdsWithin_lt_dimH_of_lt_dimH {s : Set X} {r : ℝ≥0∞} (h : r < dimH s) :
    ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, r < dimH t 
Arguments: {X : Type u_2} [EMetricSpace X] [SecondCountableTopology X] {s : Set X} {r : ℝ≥0∞} (h : r < dimH s)
Goal: ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, r < dimH t
-- bsupr_limsup_dimH
theorem bsupr_limsup_dimH (s : Set X) : ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets = dimH s 
Arguments: {X : Type u_2} [EMetricSpace X] [SecondCountableTopology X] (s : Set X)
Goal: ⨆ x ∈ s, Filter.limsup dimH (𝓝[s] x).smallSets = dimH s
-- iSup_limsup_dimH
theorem iSup_limsup_dimH (s : Set X) : ⨆ x, limsup dimH (𝓝[s] x).smallSets = dimH s 
Arguments: {X : Type u_2} [EMetricSpace X] [SecondCountableTopology X] (s : Set X)
Goal: ⨆ x, Filter.limsup dimH (𝓝[s] x).smallSets = dimH s
-- HolderOnWith.dimH_image_le
theorem HolderOnWith.dimH_image_le (h : HolderOnWith C r f s) (hr : 0 < r) :
    dimH (f '' s) ≤ dimH s / r 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {C r : ℝ≥0} {f : X → Y} {s : Set X} (h : HolderOnWith C r f s) (hr : 0 < r)
Goal: dimH (f '' s) ≤ dimH s / ↑r
-- HolderWith.dimH_image_le
theorem dimH_image_le (h : HolderWith C r f) (hr : 0 < r) (s : Set X) :
    dimH (f '' s) ≤ dimH s / r 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f) (hr : 0 < r) (s : Set X)
Goal: dimH (f '' s) ≤ dimH s / ↑r
-- HolderWith.dimH_range_le
theorem dimH_range_le (h : HolderWith C r f) (hr : 0 < r) :
    dimH (range f) ≤ dimH (univ : Set X) / r 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {C r : ℝ≥0} {f : X → Y} (h : HolderWith C r f) (hr : 0 < r)
Goal: dimH (Set.range f) ≤ dimH Set.univ / ↑r
-- dimH_image_le_of_locally_holder_on
theorem dimH_image_le_of_locally_holder_on [SecondCountableTopology X] {r : ℝ≥0} {f : X → Y}
    (hr : 0 < r) {s : Set X} (hf : ∀ x ∈ s, ∃ C : ℝ≥0, ∃ t ∈ 𝓝[s] x, HolderOnWith C r f t) :
    dimH (f '' s) ≤ dimH s / r 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [SecondCountableTopology X] {r : ℝ≥0} {f : X → Y} (hr : 0 < r) {s : Set X} (hf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C r f t)
Goal: dimH (f '' s) ≤ dimH s / ↑r
-- dimH_range_le_of_locally_holder_on
theorem dimH_range_le_of_locally_holder_on [SecondCountableTopology X] {r : ℝ≥0} {f : X → Y}
    (hr : 0 < r) (hf : ∀ x : X, ∃ C : ℝ≥0, ∃ s ∈ 𝓝 x, HolderOnWith C r f s) :
    dimH (range f) ≤ dimH (univ : Set X) / r 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [SecondCountableTopology X] {r : ℝ≥0} {f : X → Y} (hr : 0 < r) (hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s)
Goal: dimH (Set.range f) ≤ dimH Set.univ / ↑r
-- LipschitzOnWith.dimH_image_le
theorem LipschitzOnWith.dimH_image_le (h : LipschitzOnWith K f s) : dimH (f '' s) ≤ dimH s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {K : ℝ≥0} {f : X → Y} {s : Set X} (h : LipschitzOnWith K f s)
Goal: dimH (f '' s) ≤ dimH s
-- LipschitzWith.dimH_image_le
theorem dimH_image_le (h : LipschitzWith K f) (s : Set X) : dimH (f '' s) ≤ dimH s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {K : ℝ≥0} {f : X → Y} (h : LipschitzWith K f) (s : Set X)
Goal: dimH (f '' s) ≤ dimH s
-- LipschitzWith.dimH_range_le
theorem dimH_range_le (h : LipschitzWith K f) : dimH (range f) ≤ dimH (univ : Set X) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] {K : ℝ≥0} {f : X → Y} (h : LipschitzWith K f)
Goal: dimH (Set.range f) ≤ dimH Set.univ
-- dimH_image_le_of_locally_lipschitzOn
theorem dimH_image_le_of_locally_lipschitzOn [SecondCountableTopology X] {f : X → Y} {s : Set X}
    (hf : ∀ x ∈ s, ∃ C : ℝ≥0, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t) : dimH (f '' s) ≤ dimH s 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [SecondCountableTopology X] {f : X → Y} {s : Set X} (hf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t)
Goal: dimH (f '' s) ≤ dimH s
-- dimH_range_le_of_locally_lipschitzOn
theorem dimH_range_le_of_locally_lipschitzOn [SecondCountableTopology X] {f : X → Y}
    (hf : ∀ x : X, ∃ C : ℝ≥0, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s) :
    dimH (range f) ≤ dimH (univ : Set X) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] [SecondCountableTopology X] {f : X → Y} (hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s)
Goal: dimH (Set.range f) ≤ dimH Set.univ
-- AntilipschitzWith.dimH_preimage_le
theorem dimH_preimage_le (hf : AntilipschitzWith K f) (s : Set Y) : dimH (f ⁻¹' s) ≤ dimH 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 Y)
Goal: dimH (f ⁻¹' s) ≤ dimH s