ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoxIntegral.HasIntegral.smul
theorem HasIntegral.smul (hf : HasIntegral I l f vol y) (c : ℝ) :
    HasIntegral I l (c • f) vol (c • y) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} {y : F} (hf : BoxIntegral.HasIntegral I l f vol y) (c : ℝ)
Goal: BoxIntegral.HasIntegral I l (c • f) vol (c • y)
-- BoxIntegral.Integrable.smul
theorem Integrable.smul (hf : Integrable I l f vol) (c : ℝ) : Integrable I l (c • f) vol 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (hf : BoxIntegral.Integrable I l f vol) (c : ℝ)
Goal: BoxIntegral.Integrable I l (c • f) vol
-- BoxIntegral.Integrable.of_smul
theorem Integrable.of_smul {c : ℝ} (hf : Integrable I l (c • f) vol) (hc : c ≠ 0) :
    Integrable I l f vol 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} {c : ℝ} (hf : BoxIntegral.Integrable I l (c • f) vol) (hc : c ≠ 0)
Goal: BoxIntegral.Integrable I l f vol
-- BoxIntegral.integral_smul
@[simp]
theorem integral_smul (c : ℝ) : integral I l (fun x => c • f x) vol = c • integral I l f vol 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (c : ℝ)
Goal: BoxIntegral.integral I l (fun x => c • f x) vol = c • BoxIntegral.integral I l f vol
-- BoxIntegral.integral_nonneg
theorem integral_nonneg {g : ℝⁿ → ℝ} (hg : ∀ x ∈ Box.Icc I, 0 ≤ g x) (μ : Measure ℝⁿ)
    [IsLocallyFiniteMeasure μ] : 0 ≤ integral I l g μ.toBoxAdditive.toSMul 
Arguments: {ι : Type u} {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {g : (ι → ℝ) → ℝ} (hg : ∀ x ∈ BoxIntegral.Box.Icc I, 0 ≤ g x) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: 0 ≤ BoxIntegral.integral I l g μ.toBoxAdditive.toSMul
-- BoxIntegral.norm_integral_le_of_norm_le
theorem norm_integral_le_of_norm_le {g : ℝⁿ → ℝ} (hle : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ g x)
    (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ] (hg : Integrable I l g μ.toBoxAdditive.toSMul) :
    ‖(integral I l f μ.toBoxAdditive.toSMul : E)‖ ≤ integral I l g μ.toBoxAdditive.toSMul 
Arguments: {ι : Type u} {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {g : (ι → ℝ) → ℝ} (hle : ∀ x ∈ BoxIntegral.Box.Icc I, ‖f x‖ ≤ g x) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] (hg : BoxIntegral.Integrable I l g μ.toBoxAdditive.toSMul)
Goal: ‖BoxIntegral.integral I l f μ.toBoxAdditive.toSMul‖ ≤ BoxIntegral.integral I l g μ.toBoxAdditive.toSMul
-- BoxIntegral.norm_integral_le_of_le_const
theorem norm_integral_le_of_le_const {c : ℝ}
    (hc : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ c) (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ] :
    ‖(integral I l f μ.toBoxAdditive.toSMul : E)‖ ≤ (μ I).toReal * c 
Arguments: {ι : Type u} {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {c : ℝ} (hc : ∀ x ∈ BoxIntegral.Box.Icc I, ‖f x‖ ≤ c) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: ‖BoxIntegral.integral I l f μ.toBoxAdditive.toSMul‖ ≤ (μ ↑I).toReal * c
-- BoxIntegral.Integrable.convergenceR_cond
theorem convergenceR_cond (h : Integrable I l f vol) (ε : ℝ) (c : ℝ≥0) :
    l.RCond (h.convergenceR ε c) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (h : BoxIntegral.Integrable I l f vol) (ε : ℝ) (c : ℝ≥0)
Goal: l.RCond (h.convergenceR ε c)
-- BoxIntegral.Integrable.dist_integralSum_integral_le_of_memBaseSet
theorem dist_integralSum_integral_le_of_memBaseSet (h : Integrable I l f vol) (h₀ : 0 < ε)
    (hπ : l.MemBaseSet I c (h.convergenceR ε c) π) (hπp : π.IsPartition) :
    dist (integralSum f vol π) (integral I l f vol) ≤ ε 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} {c : ℝ≥0} {ε : ℝ} (h : BoxIntegral.Integrable I l f vol) (h₀ : 0 < ε) (hπ : l.MemBaseSet I c (h.convergenceR ε c) π) (hπp : π.IsPartition)
Goal: dist (BoxIntegral.integralSum f vol π) (BoxIntegral.integral I l f vol) ≤ ε
-- BoxIntegral.Integrable.dist_integralSum_le_of_memBaseSet
theorem dist_integralSum_le_of_memBaseSet (h : Integrable I l f vol) (hpos₁ : 0 < ε₁)
    (hpos₂ : 0 < ε₂) (h₁ : l.MemBaseSet I c₁ (h.convergenceR ε₁ c₁) π₁)
    (h₂ : l.MemBaseSet I c₂ (h.convergenceR ε₂ c₂) π₂) (HU : π₁.iUnion = π₂.iUnion) :
    dist (integralSum f vol π₁) (integralSum f vol π₂) ≤ ε₁ + ε₂ 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} {c₁ c₂ : ℝ≥0} {ε₁ ε₂ : ℝ} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} (h : BoxIntegral.Integrable I l f vol) (hpos₁ : 0 < ε₁) (hpos₂ : 0 < ε₂) (h₁ : l.MemBaseSet I c₁ (h.convergenceR ε₁ c₁) π₁) (h₂ : l.MemBaseSet I c₂ (h.convergenceR ε₂ c₂) π₂) (HU : π₁.iUnion = π₂.iUnion)
Goal: dist (BoxIntegral.integralSum f vol π₁) (BoxIntegral.integralSum f vol π₂) ≤ ε₁ + ε₂
-- BoxIntegral.Integrable.tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity
theorem tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity (h : Integrable I l f vol) :
    Tendsto (fun π : TaggedPrepartition I × TaggedPrepartition I =>
      (integralSum f vol π.1, integralSum f vol π.2))
        ((l.toFilter I ×ˢ l.toFilter I) ⊓ 𝓟 {π | π.1.iUnion = π.2.iUnion}) (𝓤 F) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (h : BoxIntegral.Integrable I l f vol)
Goal: Filter.Tendsto (fun π => (BoxIntegral.integralSum f vol π.1, BoxIntegral.integralSum f vol π.2))
  (l.toFilter I ×ˢ l.toFilter I ⊓ 𝓟 {π | π.1.iUnion = π.2.iUnion}) (𝓤 F)
-- BoxIntegral.Integrable.cauchy_map_integralSum_toFilteriUnion
theorem cauchy_map_integralSum_toFilteriUnion (h : Integrable I l f vol) (π₀ : Prepartition I) :
    Cauchy ((l.toFilteriUnion I π₀).map (integralSum f vol)) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (h : BoxIntegral.Integrable I l f vol) (π₀ : BoxIntegral.Prepartition I)
Goal: Cauchy (Filter.map (BoxIntegral.integralSum f vol) (l.toFilteriUnion I π₀))
-- BoxIntegral.Integrable.to_subbox_aux
theorem to_subbox_aux (h : Integrable I l f vol) (hJ : J ≤ I) :
    ∃ y : F, HasIntegral J l f vol y ∧
      Tendsto (integralSum f vol) (l.toFilteriUnion I (Prepartition.single I J hJ)) (𝓝 y) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I J : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) (hJ : J ≤ I)
Goal: ∃ y,
  BoxIntegral.HasIntegral J l f vol y ∧
    Filter.Tendsto (BoxIntegral.integralSum f vol) (l.toFilteriUnion I (BoxIntegral.Prepartition.single I J hJ)) (𝓝 y)
-- BoxIntegral.Integrable.to_subbox
theorem to_subbox (h : Integrable I l f vol) (hJ : J ≤ I) : Integrable J l f vol 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I J : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) (hJ : J ≤ I)
Goal: BoxIntegral.Integrable J l f vol
-- BoxIntegral.Integrable.tendsto_integralSum_toFilteriUnion_single
theorem tendsto_integralSum_toFilteriUnion_single (h : Integrable I l f vol) (hJ : J ≤ I) :
    Tendsto (integralSum f vol) (l.toFilteriUnion I (Prepartition.single I J hJ))
      (𝓝 <| integral J l f vol) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I J : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) (hJ : J ≤ I)
Goal: Filter.Tendsto (BoxIntegral.integralSum f vol) (l.toFilteriUnion I (BoxIntegral.Prepartition.single I J hJ))
  (𝓝 (BoxIntegral.integral J l f vol))
-- BoxIntegral.Integrable.dist_integralSum_sum_integral_le_of_memBaseSet_of_iUnion_eq
theorem dist_integralSum_sum_integral_le_of_memBaseSet_of_iUnion_eq (h : Integrable I l f vol)
    (h0 : 0 < ε) (hπ : l.MemBaseSet I c (h.convergenceR ε c) π) {π₀ : Prepartition I}
    (hU : π.iUnion = π₀.iUnion) :
    dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integral J l f vol) ≤ ε 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} {c : ℝ≥0} {ε : ℝ} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) (h0 : 0 < ε) (hπ : l.MemBaseSet I c (h.convergenceR ε c) π) {π₀ : BoxIntegral.Prepartition I} (hU : π.iUnion = π₀.iUnion)
Goal: dist (BoxIntegral.integralSum f vol π) (∑ J ∈ π₀.boxes, BoxIntegral.integral J l f vol) ≤ ε
-- BoxIntegral.Integrable.dist_integralSum_sum_integral_le_of_memBaseSet
theorem dist_integralSum_sum_integral_le_of_memBaseSet (h : Integrable I l f vol) (h0 : 0 < ε)
    (hπ : l.MemBaseSet I c (h.convergenceR ε c) π) :
    dist (integralSum f vol π) (∑ J ∈ π.boxes, integral J l f vol) ≤ ε 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} {c : ℝ≥0} {ε : ℝ} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) (h0 : 0 < ε) (hπ : l.MemBaseSet I c (h.convergenceR ε c) π)
Goal: dist (BoxIntegral.integralSum f vol π) (∑ J ∈ π.boxes, BoxIntegral.integral J l f vol) ≤ ε
-- BoxIntegral.Integrable.tendsto_integralSum_sum_integral
theorem tendsto_integralSum_sum_integral (h : Integrable I l f vol) (π₀ : Prepartition I) :
    Tendsto (integralSum f vol) (l.toFilteriUnion I π₀)
      (𝓝 <| ∑ J ∈ π₀.boxes, integral J l f vol) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) (π₀ : BoxIntegral.Prepartition I)
Goal: Filter.Tendsto (BoxIntegral.integralSum f vol) (l.toFilteriUnion I π₀)
  (𝓝 (∑ J ∈ π₀.boxes, BoxIntegral.integral J l f vol))
-- BoxIntegral.Integrable.sum_integral_congr
theorem sum_integral_congr (h : Integrable I l f vol) {π₁ π₂ : Prepartition I}
    (hU : π₁.iUnion = π₂.iUnion) :
    ∑ J ∈ π₁.boxes, integral J l f vol = ∑ J ∈ π₂.boxes, integral J l f vol 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) {π₁ π₂ : BoxIntegral.Prepartition I} (hU : π₁.iUnion = π₂.iUnion)
Goal: ∑ J ∈ π₁.boxes, BoxIntegral.integral J l f vol = ∑ J ∈ π₂.boxes, BoxIntegral.integral J l f vol
-- BoxIntegral.Integrable.toBoxAdditive_apply
@[simps]
def toBoxAdditive (h : Integrable I l f vol) : ι →ᵇᵃ[I] F 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} [CompleteSpace F] (h : BoxIntegral.Integrable I l f vol) (J : BoxIntegral.Box ι)
Goal: h.toBoxAdditive J = BoxIntegral.integral J l f vol
-- BoxIntegral.integrable_of_bounded_and_ae_continuousWithinAt
theorem integrable_of_bounded_and_ae_continuousWithinAt [CompleteSpace E] {I : Box ι} {f : ℝⁿ → E}
    (hb : ∃ C : ℝ, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C) (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ]
    (hc : ∀ᵐ x ∂(μ.restrict (Box.Icc I)), ContinuousWithinAt f (Box.Icc I) x) :
    Integrable I l f μ.toBoxAdditive.toSMul 
Arguments: {ι : Type u} {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] [Fintype ι] (l : BoxIntegral.IntegrationParams) [CompleteSpace E] {I : BoxIntegral.Box ι} {f : (ι → ℝ) → E} (hb : ∃ C, ∀ x ∈ BoxIntegral.Box.Icc I, ‖f x‖ ≤ C) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] (hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (BoxIntegral.Box.Icc I), ContinuousWithinAt f (BoxIntegral.Box.Icc I) x)
Goal: BoxIntegral.Integrable I l f μ.toBoxAdditive.toSMul
-- BoxIntegral.integrable_of_bounded_and_ae_continuous
theorem integrable_of_bounded_and_ae_continuous [CompleteSpace E] {I : Box ι} {f : ℝⁿ → E}
    (hb : ∃ C : ℝ, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C) (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ]
    (hc : ∀ᵐ x ∂μ, ContinuousAt f x) : Integrable I l f μ.toBoxAdditive.toSMul 
Arguments: {ι : Type u} {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] [Fintype ι] (l : BoxIntegral.IntegrationParams) [CompleteSpace E] {I : BoxIntegral.Box ι} {f : (ι → ℝ) → E} (hb : ∃ C, ∀ x ∈ BoxIntegral.Box.Icc I, ‖f x‖ ≤ C) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] (hc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x)
Goal: BoxIntegral.Integrable I l f μ.toBoxAdditive.toSMul
-- BoxIntegral.integrable_of_continuousOn
theorem integrable_of_continuousOn [CompleteSpace E] {I : Box ι} {f : ℝⁿ → E}
    (hc : ContinuousOn f (Box.Icc I)) (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ] :
    Integrable.{u, v, v} I l f μ.toBoxAdditive.toSMul 
Arguments: {ι : Type u} {E : Type v} [NormedAddCommGroup E] [NormedSpace ℝ E] [Fintype ι] (l : BoxIntegral.IntegrationParams) [CompleteSpace E] {I : BoxIntegral.Box ι} {f : (ι → ℝ) → E} (hc : ContinuousOn f (BoxIntegral.Box.Icc I)) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: BoxIntegral.Integrable I l f μ.toBoxAdditive.toSMul
-- BoxIntegral.HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO
theorem HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO (hl : l.bRiemann = false)
    (B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : Set ℝⁿ) (hs : s.Countable)
    (hlH : s.Nonempty → l.bHenstock = true)
    (H₁ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > (0 : ℝ),
      ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → x ∈ Box.Icc J →
        (l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε)
    (H₂ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I \ s, ∀ ε > (0 : ℝ),
      ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → (l.bHenstock → x ∈ Box.Icc J) →
        (l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε * B J) :
    HasIntegral I l f vol (g I) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (hl : l.bRiemann = false) (B : ι →ᵇᵃ[↑I] ℝ) (hB0 : ∀ (J : BoxIntegral.Box ι), 0 ≤ B J) (g : ι →ᵇᵃ[↑I] F) (s : Set (ι → ℝ)) (hs : s.Countable) (hlH : s.Nonempty → l.bHenstock = true) (H₁ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ BoxIntegral.Box.Icc I ∩ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            BoxIntegral.Box.Icc J ⊆ Metric.closedBall x δ →
              x ∈ BoxIntegral.Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε) (H₂ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ BoxIntegral.Box.Icc I \ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            BoxIntegral.Box.Icc J ⊆ Metric.closedBall x δ →
              (l.bHenstock = true → x ∈ BoxIntegral.Box.Icc J) →
                (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J)
Goal: BoxIntegral.HasIntegral I l f vol (g I)
-- BoxIntegral.HasIntegral.of_le_Henstock_of_forall_isLittleO
theorem HasIntegral.of_le_Henstock_of_forall_isLittleO (hl : l ≤ Henstock) (B : ι →ᵇᵃ[I] ℝ)
    (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : Set ℝⁿ) (hs : s.Countable)
    (H₁ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > (0 : ℝ),
      ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → x ∈ Box.Icc J →
        (l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε)
    (H₂ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I \ s, ∀ ε > (0 : ℝ),
      ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → x ∈ Box.Icc J →
        (l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε * B J) :
    HasIntegral I l f vol (g I) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (hl : l ≤ BoxIntegral.IntegrationParams.Henstock) (B : ι →ᵇᵃ[↑I] ℝ) (hB0 : ∀ (J : BoxIntegral.Box ι), 0 ≤ B J) (g : ι →ᵇᵃ[↑I] F) (s : Set (ι → ℝ)) (hs : s.Countable) (H₁ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ BoxIntegral.Box.Icc I ∩ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            BoxIntegral.Box.Icc J ⊆ Metric.closedBall x δ →
              x ∈ BoxIntegral.Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε) (H₂ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ BoxIntegral.Box.Icc I \ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            BoxIntegral.Box.Icc J ⊆ Metric.closedBall x δ →
              x ∈ BoxIntegral.Box.Icc J →
                (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J)
Goal: BoxIntegral.HasIntegral I l f vol (g I)
-- BoxIntegral.HasIntegral.mcShane_of_forall_isLittleO
theorem HasIntegral.mcShane_of_forall_isLittleO (B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J)
    (g : ι →ᵇᵃ[I] F) (H : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I, ∀ ε > (0 : ℝ), ∃ δ > 0, ∀ J ≤ I,
      Box.Icc J ⊆ Metric.closedBall x δ → dist (vol J (f x)) (g J) ≤ ε * B J) :
    HasIntegral I McShane f vol (g I) 
Arguments: {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {I : BoxIntegral.Box ι} [Fintype ι] {f : (ι → ℝ) → E} {vol : ι →ᵇᵃ[⊤] E →L[ℝ] F} (B : ι →ᵇᵃ[↑I] ℝ) (hB0 : ∀ (J : BoxIntegral.Box ι), 0 ≤ B J) (g : ι →ᵇᵃ[↑I] F) (H :
  ℝ≥0 →
    ∀ x ∈ BoxIntegral.Box.Icc I,
      ∀ ε > 0, ∃ δ > 0, ∀ J ≤ I, BoxIntegral.Box.Icc J ⊆ Metric.closedBall x δ → dist ((vol J) (f x)) (g J) ≤ ε * B J)
Goal: BoxIntegral.HasIntegral I BoxIntegral.IntegrationParams.McShane f vol (g I)
-- StrictConvexSpace.strictConvex_closedBall
class StrictConvexSpace (𝕜 E : Type*) [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
  [NormedSpace 𝕜 E] : Prop where
  strictConvex_closedBall : ∀ r : ℝ, 0 < r → StrictConvex 𝕜 (closedBall (0 : E) r)
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [self : StrictConvexSpace 𝕜 E] (r : ℝ)
Goal: 0 < r → StrictConvex 𝕜 (Metric.closedBall 0 r)
-- strictConvex_closedBall
theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
    StrictConvex 𝕜 (closedBall x r) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ)
Goal: StrictConvex 𝕜 (Metric.closedBall x r)
-- StrictConvexSpace.of_strictConvex_closed_unit_ball
theorem StrictConvexSpace.of_strictConvex_closed_unit_ball [LinearMap.CompatibleSMul E E 𝕜 ℝ]
    (h : StrictConvex 𝕜 (closedBall (0 : E) 1)) : StrictConvexSpace 𝕜 E 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [LinearMap.CompatibleSMul E E 𝕜 ℝ] (h : StrictConvex 𝕜 (Metric.closedBall 0 1))
Goal: StrictConvexSpace 𝕜 E
-- StrictConvexSpace.of_norm_combo_lt_one
theorem StrictConvexSpace.of_norm_combo_lt_one
    (h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
    StrictConvexSpace ℝ E 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1)
Goal: StrictConvexSpace ℝ E
-- StrictConvexSpace.of_norm_combo_ne_one
theorem StrictConvexSpace.of_norm_combo_ne_one
    (h :
      ∀ x y : E,
        ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
    StrictConvexSpace ℝ E 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1)
Goal: StrictConvexSpace ℝ E
-- StrictConvexSpace.of_norm_add_ne_two
theorem StrictConvexSpace.of_norm_add_ne_two
    (h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2)
Goal: StrictConvexSpace ℝ E
-- StrictConvexSpace.of_pairwise_sphere_norm_ne_two
theorem StrictConvexSpace.of_pairwise_sphere_norm_ne_two
    (h : (sphere (0 : E) 1).Pairwise fun x y => ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : (Metric.sphere 0 1).Pairwise fun x y => ‖x + y‖ ≠ 2)
Goal: StrictConvexSpace ℝ E
-- StrictConvexSpace.of_norm_add
theorem StrictConvexSpace.of_norm_add
    (h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y) : StrictConvexSpace ℝ E 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y)
Goal: StrictConvexSpace ℝ E
-- combo_mem_ball_of_ne
theorem combo_mem_ball_of_ne (hx : x ∈ closedBall z r) (hy : y ∈ closedBall z r) (hne : x ≠ y)
    (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) : a • x + b • y ∈ ball z r 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y z : E} {a b r : ℝ} (hx : x ∈ Metric.closedBall z r) (hy : y ∈ Metric.closedBall z r) (hne : x ≠ y) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1)
Goal: a • x + b • y ∈ Metric.ball z r
-- openSegment_subset_ball_of_ne
theorem openSegment_subset_ball_of_ne (hx : x ∈ closedBall z r) (hy : y ∈ closedBall z r)
    (hne : x ≠ y) : openSegment ℝ x y ⊆ ball z r 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y z : E} {r : ℝ} (hx : x ∈ Metric.closedBall z r) (hy : y ∈ Metric.closedBall z r) (hne : x ≠ y)
Goal: openSegment ℝ x y ⊆ Metric.ball z r
-- norm_combo_lt_of_ne
theorem norm_combo_lt_of_ne (hx : ‖x‖ ≤ r) (hy : ‖y‖ ≤ r) (hne : x ≠ y) (ha : 0 < a) (hb : 0 < b)
    (hab : a + b = 1) : ‖a • x + b • y‖ < r 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E} {a b r : ℝ} (hx : ‖x‖ ≤ r) (hy : ‖y‖ ≤ r) (hne : x ≠ y) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1)
Goal: ‖a • x + b • y‖ < r