ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.cylinder_mem_measurableCylinders
theorem cylinder_mem_measurableCylinders (s : Finset ι) (S : Set (∀ i : s, α i))
    (hS : MeasurableSet S) :
    cylinder s S ∈ measurableCylinders α 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] (s : Finset ι) (S : Set ((i : { x // x ∈ s }) → α ↑i)) (hS : MeasurableSet S)
Goal: MeasureTheory.cylinder s S ∈ MeasureTheory.measurableCylinders α
-- MeasureTheory.inter_mem_measurableCylinders
theorem inter_mem_measurableCylinders (hs : s ∈ measurableCylinders α)
    (ht : t ∈ measurableCylinders α) :
    s ∩ t ∈ measurableCylinders α 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {s t : Set ((i : ι) → α i)} (hs : s ∈ MeasureTheory.measurableCylinders α) (ht : t ∈ MeasureTheory.measurableCylinders α)
Goal: s ∩ t ∈ MeasureTheory.measurableCylinders α
-- MeasureTheory.isPiSystem_measurableCylinders
theorem isPiSystem_measurableCylinders : IsPiSystem (measurableCylinders α) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)]
Goal: IsPiSystem (MeasureTheory.measurableCylinders α)
-- MeasureTheory.compl_mem_measurableCylinders
theorem compl_mem_measurableCylinders (hs : s ∈ measurableCylinders α) :
    sᶜ ∈ measurableCylinders α 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {s : Set ((i : ι) → α i)} (hs : s ∈ MeasureTheory.measurableCylinders α)
Goal: sᶜ ∈ MeasureTheory.measurableCylinders α
-- MeasureTheory.univ_mem_measurableCylinders
theorem univ_mem_measurableCylinders (α : ι → Type*) [∀ i, MeasurableSpace (α i)] :
    Set.univ ∈ measurableCylinders α 
Arguments: {ι : Type u_2} (α : ι → Type u_1) [(i : ι) → MeasurableSpace (α i)]
Goal: Set.univ ∈ MeasureTheory.measurableCylinders α
-- MeasureTheory.union_mem_measurableCylinders
theorem union_mem_measurableCylinders (hs : s ∈ measurableCylinders α)
    (ht : t ∈ measurableCylinders α) :
    s ∪ t ∈ measurableCylinders α 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {s t : Set ((i : ι) → α i)} (hs : s ∈ MeasureTheory.measurableCylinders α) (ht : t ∈ MeasureTheory.measurableCylinders α)
Goal: s ∪ t ∈ MeasureTheory.measurableCylinders α
-- MeasureTheory.diff_mem_measurableCylinders
theorem diff_mem_measurableCylinders (hs : s ∈ measurableCylinders α)
    (ht : t ∈ measurableCylinders α) :
    s \ t ∈ measurableCylinders α 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {s t : Set ((i : ι) → α i)} (hs : s ∈ MeasureTheory.measurableCylinders α) (ht : t ∈ MeasureTheory.measurableCylinders α)
Goal: s \ t ∈ MeasureTheory.measurableCylinders α
-- MeasureTheory.generateFrom_measurableCylinders
theorem generateFrom_measurableCylinders :
    MeasurableSpace.generateFrom (measurableCylinders α) = MeasurableSpace.pi 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)]
Goal: MeasurableSpace.generateFrom (MeasureTheory.measurableCylinders α) = MeasurableSpace.pi
-- groupCohomology.resolution.actionDiagonalSucc_hom_apply
theorem actionDiagonalSucc_hom_apply {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G) :
    (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) 
Arguments: {G : Type u} [Group G] {n : ℕ} (f : Fin (n + 1) → G)
Goal: (groupCohomology.resolution.actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f i.castSucc)⁻¹ * f i.succ)
-- groupCohomology.resolution.actionDiagonalSucc_inv_apply
theorem actionDiagonalSucc_inv_apply {G : Type u} [Group G] {n : ℕ} (g : G) (f : Fin n → G) :
    (actionDiagonalSucc G n).inv.hom (g, f) = (g • Fin.partialProd f : Fin (n + 1) → G) 
Arguments: {G : Type u} [Group G] {n : ℕ} (g : G) (f : Fin n → G)
Goal: (groupCohomology.resolution.actionDiagonalSucc G n).inv.hom (g, f) = g • Fin.partialProd f
-- groupCohomology.resolution.diagonalSucc_hom_single
theorem diagonalSucc_hom_single (f : Gⁿ⁺¹) (a : k) :
    (diagonalSucc k G n).hom.hom (single f a) =
      single (f 0) 1 ⊗ₜ single (fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) a 
Arguments: {k G : Type u} [CommRing k] {n : ℕ} [Group G] (f : Fin (n + 1) → G) (a : k)
Goal: (groupCohomology.resolution.diagonalSucc k G n).hom.hom (Finsupp.single f a) =
  Finsupp.single (f 0) 1 ⊗ₜ[k] Finsupp.single (fun i => (f i.castSucc)⁻¹ * f i.succ) a
-- groupCohomology.resolution.diagonalSucc_inv_single_single
theorem diagonalSucc_inv_single_single (g : G) (f : Gⁿ) (a b : k) :
    (diagonalSucc k G n).inv.hom (Finsupp.single g a ⊗ₜ Finsupp.single f b) =
      single (g • partialProd f) (a * b) 
Arguments: {k G : Type u} [CommRing k] {n : ℕ} [Group G] (g : G) (f : Fin n → G) (a b : k)
Goal: (groupCohomology.resolution.diagonalSucc k G n).inv.hom (Finsupp.single g a ⊗ₜ[k] Finsupp.single f b) =
  Finsupp.single (g • Fin.partialProd f) (a * b)
-- groupCohomology.resolution.diagonalSucc_inv_single_left
theorem diagonalSucc_inv_single_left (g : G) (f : Gⁿ →₀ k) (r : k) :
    (diagonalSucc k G n).inv.hom (Finsupp.single g r ⊗ₜ f) =
      Finsupp.lift (Gⁿ⁺¹ →₀ k) k Gⁿ (fun f => single (g • partialProd f) r) f 
Arguments: {k G : Type u} [CommRing k] {n : ℕ} [Group G] (g : G) (f : (Fin n → G) →₀ k) (r : k)
Goal: (groupCohomology.resolution.diagonalSucc k G n).inv.hom (Finsupp.single g r ⊗ₜ[k] f) =
  ((Finsupp.lift ((Fin (n + 1) → G) →₀ k) k (Fin n → G)) fun f => Finsupp.single (g • Fin.partialProd f) r) f
-- groupCohomology.resolution.diagonalSucc_inv_single_right
theorem diagonalSucc_inv_single_right (g : G →₀ k) (f : Gⁿ) (r : k) :
    (diagonalSucc k G n).inv.hom (g ⊗ₜ Finsupp.single f r) =
      Finsupp.lift _ k G (fun a => single (a • partialProd f) r) g 
Arguments: {k G : Type u} [CommRing k] {n : ℕ} [Group G] (g : G →₀ k) (f : Fin n → G) (r : k)
Goal: (groupCohomology.resolution.diagonalSucc k G n).inv.hom (g ⊗ₜ[k] Finsupp.single f r) =
  ((Finsupp.lift ((Fin (n + 1) → G) →₀ k) k G) fun a => Finsupp.single (a • Fin.partialProd f) r) g
-- groupCohomology.resolution.ofMulAction_free
theorem ofMulAction_free :
    Module.Free (MonoidAlgebra k G) (ofMulAction k G (Fin (n + 1) → G)).asModule 
Arguments: (k G : Type u) [CommRing k] (n : ℕ) [Group G]
Goal: Module.Free (MonoidAlgebra k G) (Representation.ofMulAction k G (Fin (n + 1) → G)).asModule
-- Rep.diagonalHomEquiv_apply
theorem diagonalHomEquiv_apply (f : Rep.ofMulAction k G (Fin (n + 1) → G) ⟶ A) (x : Fin n → G) :
    diagonalHomEquiv n A f x = f.hom (Finsupp.single (Fin.partialProd x) 1) 
Arguments: {k G : Type u} [CommRing k] {n : ℕ} [Group G] {A : Rep k G} (f : Rep.ofMulAction k G (Fin (n + 1) → G) ⟶ A) (x : Fin n → G)
Goal: (Rep.diagonalHomEquiv n A) f x = f.hom (Finsupp.single (Fin.partialProd x) 1)
-- Rep.diagonalHomEquiv_symm_apply
theorem diagonalHomEquiv_symm_apply (f : (Fin n → G) → A) (x : Fin (n + 1) → G) :
    ((diagonalHomEquiv n A).symm f).hom (Finsupp.single x 1) =
      A.ρ (x 0) (f fun i : Fin n => (x (Fin.castSucc i))⁻¹ * x i.succ) 
Arguments: {k G : Type u} [CommRing k] {n : ℕ} [Group G] {A : Rep k G} (f : (Fin n → G) → CoeSort.coe A) (x : Fin (n + 1) → G)
Goal: ((Rep.diagonalHomEquiv n A).symm f).hom (Finsupp.single x 1) = (A.ρ (x 0)) (f fun i => (x i.castSucc)⁻¹ * x i.succ)
-- Rep.diagonalHomEquiv_symm_partialProd_succ
theorem diagonalHomEquiv_symm_partialProd_succ (f : (Fin n → G) → A) (g : Fin (n + 1) → G)
    (a : Fin (n + 1)) :
    ((diagonalHomEquiv n A).symm f).hom (Finsupp.single (Fin.partialProd g ∘ a.succ.succAbove) 1)
      = f (Fin.contractNth a (· * ·) g) 
Arguments: {k G : Type u} [CommRing k] {n : ℕ} [Group G] {A : Rep k G} (f : (Fin n → G) → CoeSort.coe A) (g : Fin (n + 1) → G) (a : Fin (n + 1))
Goal: ((Rep.diagonalHomEquiv n A).symm f).hom (Finsupp.single (Fin.partialProd g ∘ a.succ.succAbove) 1) =
  f (a.contractNth (fun x x_1 => x * x_1) g)
-- groupCohomology.resolution.d_of
@[simp]
theorem d_of {G : Type u} {n : ℕ} (c : Fin (n + 1) → G) :
    d k G n (Finsupp.single c 1) =
      Finset.univ.sum fun p : Fin (n + 1) =>
        Finsupp.single (c ∘ p.succAbove) ((-1 : k) ^ (p : ℕ)) 
Arguments: {k : Type u} [CommRing k] {G : Type u} {n : ℕ} (c : Fin (n + 1) → G)
Goal: (groupCohomology.resolution.d k G n) (Finsupp.single c 1) =
  ∑ p : Fin (n + 1), Finsupp.single (c ∘ p.succAbove) ((-1) ^ ↑p)
-- groupCohomology.resolution.d_eq
theorem d_eq (n : ℕ) : ((groupCohomology.resolution k G).d (n + 1) n).hom = d k G (n + 1) 
Arguments: (k G : Type u) [CommRing k] [Monoid G] (n : ℕ)
Goal: ((groupCohomology.resolution k G).d (n + 1) n).hom = groupCohomology.resolution.d k G (n + 1)
-- groupCohomology.resolution.forget₂ToModuleCatHomotopyEquiv_f_0_eq
theorem forget₂ToModuleCatHomotopyEquiv_f_0_eq :
    (forget₂ToModuleCatHomotopyEquiv k G).1.f 0 = (forget₂ (Rep k G) _).map (ε k G) 
Arguments: (k G : Type u) [CommRing k] [Monoid G]
Goal: (groupCohomology.resolution.forget₂ToModuleCatHomotopyEquiv k G).hom.f 0 =
  (CategoryTheory.forget₂ (Rep k G) (ModuleCat k)).map (groupCohomology.resolution.ε k G)
-- groupCohomology.resolution.d_comp_ε
theorem d_comp_ε : (groupCohomology.resolution k G).d 1 0 ≫ ε k G = 0 
Arguments: (k G : Type u) [CommRing k] [Monoid G]
Goal: (groupCohomology.resolution k G).d 1 0 ≫ groupCohomology.resolution.ε k G = 0
-- groupCohomology.resolution.εToSingle₀_comp_eq
theorem εToSingle₀_comp_eq :
    ((forget₂ _ (ModuleCat.{u} k)).mapHomologicalComplex _).map (εToSingle₀ k G) ≫
        (HomologicalComplex.singleMapHomologicalComplex _ _ _).hom.app _ =
      (forget₂ToModuleCatHomotopyEquiv k G).hom 
Arguments: (k G : Type u) [CommRing k] [Monoid G]
Goal: ((CategoryTheory.forget₂ (Rep k G) (ModuleCat k)).mapHomologicalComplex (ComplexShape.down ℕ)).map
      (groupCohomology.resolution.εToSingle₀ k G) ≫
    (HomologicalComplex.singleMapHomologicalComplex (CategoryTheory.forget₂ (Rep k G) (ModuleCat k))
            (ComplexShape.down ℕ) 0).hom.app
      (Rep.trivial k G k) =
  (groupCohomology.resolution.forget₂ToModuleCatHomotopyEquiv k G).hom
-- groupCohomology.resolution.quasiIso_forget₂_εToSingle₀
theorem quasiIso_forget₂_εToSingle₀ :
    QuasiIso (((forget₂ _ (ModuleCat.{u} k)).mapHomologicalComplex _).map (εToSingle₀ k G)) 
Arguments: (k G : Type u) [CommRing k] [Monoid G]
Goal: QuasiIso
  (((CategoryTheory.forget₂ (Rep k G) (ModuleCat k)).mapHomologicalComplex (ComplexShape.down ℕ)).map
    (groupCohomology.resolution.εToSingle₀ k G))
-- Int.measurable_floor
theorem Int.measurable_floor [OpensMeasurableSpace R] : Measurable (Int.floor : R → ℤ) 
Arguments: {R : Type u_2} [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R]
Goal: Measurable Int.floor
-- Measurable.floor
@[measurability]
theorem Measurable.floor [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f) :
    Measurable fun x => ⌊f x⌋ 
Arguments: {α : Type u_1} {R : Type u_2} [MeasurableSpace α] [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f)
Goal: Measurable fun x => ⌊f x⌋
-- Int.measurable_ceil
theorem Int.measurable_ceil [OpensMeasurableSpace R] : Measurable (Int.ceil : R → ℤ) 
Arguments: {R : Type u_2} [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R]
Goal: Measurable Int.ceil
-- Measurable.ceil
@[measurability]
theorem Measurable.ceil [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f) :
    Measurable fun x => ⌈f x⌉ 
Arguments: {α : Type u_1} {R : Type u_2} [MeasurableSpace α] [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f)
Goal: Measurable fun x => ⌈f x⌉
-- measurable_fract
theorem measurable_fract [BorelSpace R] : Measurable (Int.fract : R → R) 
Arguments: {R : Type u_2} [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [BorelSpace R]
Goal: Measurable Int.fract
-- Measurable.fract
@[measurability]
theorem Measurable.fract [BorelSpace R] {f : α → R} (hf : Measurable f) :
    Measurable fun x => Int.fract (f x) 
Arguments: {α : Type u_1} {R : Type u_2} [MeasurableSpace α] [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [BorelSpace R] {f : α → R} (hf : Measurable f)
Goal: Measurable fun x => Int.fract (f x)
-- MeasurableSet.image_fract
theorem MeasurableSet.image_fract [BorelSpace R] {s : Set R} (hs : MeasurableSet s) :
    MeasurableSet (Int.fract '' s) 
Arguments: {R : Type u_2} [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [BorelSpace R] {s : Set R} (hs : MeasurableSet s)
Goal: MeasurableSet (Int.fract '' s)
-- Nat.measurable_floor
theorem Nat.measurable_floor : Measurable (Nat.floor : R → ℕ) 
Arguments: {R : Type u_2} [LinearOrderedSemiring R] [FloorSemiring R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R]
Goal: Measurable Nat.floor
-- Measurable.nat_floor
@[measurability]
theorem Measurable.nat_floor (hf : Measurable f) : Measurable fun x => ⌊f x⌋₊ 
Arguments: {α : Type u_1} {R : Type u_2} [MeasurableSpace α] [LinearOrderedSemiring R] [FloorSemiring R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f)
Goal: Measurable fun x => ⌊f x⌋₊
-- Nat.measurable_ceil
theorem Nat.measurable_ceil : Measurable (Nat.ceil : R → ℕ) 
Arguments: {R : Type u_2} [LinearOrderedSemiring R] [FloorSemiring R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R]
Goal: Measurable Nat.ceil
-- Measurable.nat_ceil
@[measurability]
theorem Measurable.nat_ceil (hf : Measurable f) : Measurable fun x => ⌈f x⌉₊ 
Arguments: {α : Type u_1} {R : Type u_2} [MeasurableSpace α] [LinearOrderedSemiring R] [FloorSemiring R] [TopologicalSpace R] [OrderTopology R] [MeasurableSpace R] [OpensMeasurableSpace R] {f : α → R} (hf : Measurable f)
Goal: Measurable fun x => ⌈f x⌉₊
-- MeasCat.coe_of
@[simp]
theorem coe_of (X : Type u) [MeasurableSpace X] : (of X : Type u) = X 
Arguments: (X : Type u) [MeasurableSpace X]
Goal: ↑(MeasCat.of X) = X
-- IsUnifLocDoublingMeasure.vitaliFamily_def
irreducible_def vitaliFamily (K : ℝ) : VitaliFamily μ 
Arguments: {α : Type u_2} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] (K : ℝ)
Goal: IsUnifLocDoublingMeasure.vitaliFamily μ K =
  let R := IsUnifLocDoublingMeasure.scalingScaleOf μ (max (4 * K + 3) 3);
  let_fun Rpos := ⋯;
  let_fun A := ⋯;
  (Vitali.vitaliFamily μ (IsUnifLocDoublingMeasure.scalingConstantOf μ (max (4 * K + 3) 3)) A).enlarge (R / 4) ⋯
-- IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mul
theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r)
    (rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r) (rpos : 0 < r)
Goal: Metric.closedBall y r ∈ (IsUnifLocDoublingMeasure.vitaliFamily μ K).setsAt x
-- IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt
theorem tendsto_closedBall_filterAt {K : ℝ} {x : α} {ι : Type*} {l : Filter ι} (w : ι → α)
    (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0)) (xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)) :
    Tendsto (fun j => closedBall (w j) (δ j)) l ((vitaliFamily μ K).filterAt x) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {K : ℝ} {x : α} {ι : Type u_2} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Filter.Tendsto δ l (𝓝[>] 0)) (xmem : ∀ᶠ (j : ι) in l, x ∈ Metric.closedBall (w j) (K * δ j))
Goal: Filter.Tendsto (fun j => Metric.closedBall (w j) (δ j)) l ((IsUnifLocDoublingMeasure.vitaliFamily μ K).filterAt x)
-- IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div
theorem ae_tendsto_measure_inter_div (S : Set α) (K : ℝ) : ∀ᵐ x ∂μ.restrict S,
    ∀ {ι : Type*} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0))
      (xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)),
      Tendsto (fun j => μ (S ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] (S : Set α) (K : ℝ)
Goal: ∀ᵐ (x : α) ∂μ.restrict S,
  ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Filter.Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, x ∈ Metric.closedBall (w j) (K * δ j)) →
        Filter.Tendsto (fun j => μ (S ∩ Metric.closedBall (w j) (δ j)) / μ (Metric.closedBall (w j) (δ j))) l (𝓝 1)
-- IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub
theorem ae_tendsto_average_norm_sub {f : α → E} (hf : LocallyIntegrable f μ) (K : ℝ) : ∀ᵐ x ∂μ,
    ∀ {ι : Type*} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0))
      (xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)),
      Tendsto (fun j => ⨍ y in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {E : Type u_2} [NormedAddCommGroup E] {f : α → E} (hf : MeasureTheory.LocallyIntegrable f μ) (K : ℝ)
Goal: ∀ᵐ (x : α) ∂μ,
  ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Filter.Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, x ∈ Metric.closedBall (w j) (K * δ j)) →
        Filter.Tendsto (fun j => ⨍ (y : α) in Metric.closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)
-- IsUnifLocDoublingMeasure.ae_tendsto_average
theorem ae_tendsto_average [NormedSpace ℝ E] [CompleteSpace E]
    {f : α → E} (hf : LocallyIntegrable f μ) (K : ℝ) : ∀ᵐ x ∂μ,
      ∀ {ι : Type*} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0))
        (xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)),
        Tendsto (fun j => ⨍ y in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x)) 
Arguments: {α : Type u_1} [MetricSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) [IsUnifLocDoublingMeasure μ] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} (hf : MeasureTheory.LocallyIntegrable f μ) (K : ℝ)
Goal: ∀ᵐ (x : α) ∂μ,
  ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Filter.Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, x ∈ Metric.closedBall (w j) (K * δ j)) →
        Filter.Tendsto (fun j => ⨍ (y : α) in Metric.closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x))
-- essSup_congr_ae
theorem essSup_congr_ae {f g : α → β} (hfg : f =ᵐ[μ] g) : essSup f μ = essSup g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLattice β] {f g : α → β} (hfg : f =ᵐ[μ] g)
Goal: essSup f μ = essSup g μ
-- essInf_congr_ae
theorem essInf_congr_ae {f g : α → β} (hfg : f =ᵐ[μ] g) : essInf f μ = essInf g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLattice β] {f g : α → β} (hfg : f =ᵐ[μ] g)
Goal: essInf f μ = essInf g μ
-- essSup_const'
@[simp]
theorem essSup_const' [NeZero μ] (c : β) : essSup (fun _ : α => c) μ = c 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLattice β] [NeZero μ] (c : β)
Goal: essSup (fun x => c) μ = c
-- essInf_const'
@[simp]
theorem essInf_const' [NeZero μ] (c : β) : essInf (fun _ : α => c) μ = c 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLattice β] [NeZero μ] (c : β)
Goal: essInf (fun x => c) μ = c
-- essSup_const
theorem essSup_const (c : β) (hμ : μ ≠ 0) : essSup (fun _ : α => c) μ = c 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLattice β] (c : β) (hμ : μ ≠ 0)
Goal: essSup (fun x => c) μ = c
-- essInf_const
theorem essInf_const (c : β) (hμ : μ ≠ 0) : essInf (fun _ : α => c) μ = c 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLattice β] (c : β) (hμ : μ ≠ 0)
Goal: essInf (fun x => c) μ = c
-- essSup_eq_sInf
theorem essSup_eq_sInf {m : MeasurableSpace α} (μ : Measure α) (f : α → β) :
    essSup f μ = sInf { a | μ { x | a < f x } = 0 } 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder β] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → β)
Goal: essSup f μ = sInf {a | μ {x | a < f x} = 0}
-- essInf_eq_sSup
theorem essInf_eq_sSup {m : MeasurableSpace α} (μ : Measure α) (f : α → β) :
    essInf f μ = sSup { a | μ { x | f x < a } = 0 } 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLinearOrder β] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → β)
Goal: essInf f μ = sSup {a | μ {x | f x < a} = 0}
-- ae_lt_of_essSup_lt
theorem ae_lt_of_essSup_lt (hx : essSup f μ < x)
    (hf : IsBoundedUnder (· ≤ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, f y < x 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLinearOrder β] {x : β} {f : α → β} (hx : essSup f μ < x) (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) (MeasureTheory.ae μ) f := by isBoundedDefault)
Goal: ∀ᵐ (y : α) ∂μ, f y < x
-- ae_lt_of_lt_essInf
theorem ae_lt_of_lt_essInf (hx : x < essInf f μ)
    (hf : IsBoundedUnder (· ≥ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, x < f y 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLinearOrder β] {x : β} {f : α → β} (hx : x < essInf f μ) (hf : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) (MeasureTheory.ae μ) f := by isBoundedDefault)
Goal: ∀ᵐ (y : α) ∂μ, x < f y
-- ae_le_essSup
theorem ae_le_essSup
    (hf : IsBoundedUnder (· ≤ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, f y ≤ essSup f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLinearOrder β] {f : α → β} [TopologicalSpace β] [FirstCountableTopology β] [OrderTopology β] (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) (MeasureTheory.ae μ) f := by isBoundedDefault)
Goal: ∀ᵐ (y : α) ∂μ, f y ≤ essSup f μ
-- ae_essInf_le
theorem ae_essInf_le
    (hf : IsBoundedUnder (· ≥ ·) (ae μ) f := by isBoundedDefault) :
    ∀ᵐ y ∂μ, essInf f μ ≤ f y 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLinearOrder β] {f : α → β} [TopologicalSpace β] [FirstCountableTopology β] [OrderTopology β] (hf : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) (MeasureTheory.ae μ) f := by isBoundedDefault)
Goal: ∀ᵐ (y : α) ∂μ, essInf f μ ≤ f y
-- meas_essSup_lt
theorem meas_essSup_lt
    (hf : IsBoundedUnder (· ≤ ·) (ae μ) f := by isBoundedDefault) :
    μ { y | essSup f μ < f y } = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLinearOrder β] {f : α → β} [TopologicalSpace β] [FirstCountableTopology β] [OrderTopology β] (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) (MeasureTheory.ae μ) f := by isBoundedDefault)
Goal: μ {y | essSup f μ < f y} = 0
-- meas_lt_essInf
theorem meas_lt_essInf
    (hf : IsBoundedUnder (· ≥ ·) (ae μ) f := by isBoundedDefault) :
    μ { y | f y < essInf f μ } = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [ConditionallyCompleteLinearOrder β] {f : α → β} [TopologicalSpace β] [FirstCountableTopology β] [OrderTopology β] (hf : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) (MeasureTheory.ae μ) f := by isBoundedDefault)
Goal: μ {y | f y < essInf f μ} = 0