ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoxIntegral.IntegrationParams.henstock_le_mcShane
theorem henstock_le_mcShane : Henstock ≤ McShane 
Goal: BoxIntegral.IntegrationParams.Henstock ≤ BoxIntegral.IntegrationParams.McShane
-- BoxIntegral.IntegrationParams.gp_le
theorem gp_le : GP ≤ l 
Arguments: {l : BoxIntegral.IntegrationParams}
Goal: BoxIntegral.IntegrationParams.GP ≤ l
-- BoxIntegral.IntegrationParams.MemBaseSet.isHenstock
structure MemBaseSet (l : IntegrationParams) (I : Box ι) (c : ℝ≥0) (r : (ι → ℝ) → Ioi (0 : ℝ))
    (π : TaggedPrepartition I) : Prop where
  protected isSubordinate : π.IsSubordinate r
  protected isHenstock : l.bHenstock → π.IsHenstock
  protected distortion_le : l.bDistortion → π.distortion ≤ c
  protected exists_compl : l.bDistortion → ∃ π' : Prepartition I,
    π'.iUnion = ↑I \ π.iUnion ∧ π'.distortion ≤ c
Arguments: {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : ℝ≥0} {r : (ι → ℝ) → ↑(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π)
Goal: l.bHenstock = true → π.IsHenstock
-- BoxIntegral.IntegrationParams.MemBaseSet.isSubordinate
structure MemBaseSet (l : IntegrationParams) (I : Box ι) (c : ℝ≥0) (r : (ι → ℝ) → Ioi (0 : ℝ))
    (π : TaggedPrepartition I) : Prop where
  protected isSubordinate : π.IsSubordinate r
  protected isHenstock : l.bHenstock → π.IsHenstock
  protected distortion_le : l.bDistortion → π.distortion ≤ c
  protected exists_compl : l.bDistortion → ∃ π' : Prepartition I,
    π'.iUnion = ↑I \ π.iUnion ∧ π'.distortion ≤ c
Arguments: {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : ℝ≥0} {r : (ι → ℝ) → ↑(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π)
Goal: π.IsSubordinate r
-- BoxIntegral.IntegrationParams.MemBaseSet.exists_compl
structure MemBaseSet (l : IntegrationParams) (I : Box ι) (c : ℝ≥0) (r : (ι → ℝ) → Ioi (0 : ℝ))
    (π : TaggedPrepartition I) : Prop where
  protected isSubordinate : π.IsSubordinate r
  protected isHenstock : l.bHenstock → π.IsHenstock
  protected distortion_le : l.bDistortion → π.distortion ≤ c
  protected exists_compl : l.bDistortion → ∃ π' : Prepartition I,
    π'.iUnion = ↑I \ π.iUnion ∧ π'.distortion ≤ c
Arguments: {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : ℝ≥0} {r : (ι → ℝ) → ↑(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π)
Goal: l.bDistortion = true → ∃ π', π'.iUnion = ↑I \ π.iUnion ∧ π'.distortion ≤ c
-- BoxIntegral.IntegrationParams.MemBaseSet.distortion_le
structure MemBaseSet (l : IntegrationParams) (I : Box ι) (c : ℝ≥0) (r : (ι → ℝ) → Ioi (0 : ℝ))
    (π : TaggedPrepartition I) : Prop where
  protected isSubordinate : π.IsSubordinate r
  protected isHenstock : l.bHenstock → π.IsHenstock
  protected distortion_le : l.bDistortion → π.distortion ≤ c
  protected exists_compl : l.bDistortion → ∃ π' : Prepartition I,
    π'.iUnion = ↑I \ π.iUnion ∧ π'.distortion ≤ c
Arguments: {ι : Type u_1} [Fintype ι] {l : BoxIntegral.IntegrationParams} {I : BoxIntegral.Box ι} {c : ℝ≥0} {r : (ι → ℝ) → ↑(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} (self : l.MemBaseSet I c r π)
Goal: l.bDistortion = true → π.distortion ≤ c
-- BoxIntegral.IntegrationParams.rCond_of_bRiemann_eq_false
theorem rCond_of_bRiemann_eq_false {ι} (l : IntegrationParams) (hl : l.bRiemann = false)
    {r : (ι → ℝ) → Ioi (0 : ℝ)} : l.RCond r 
Arguments: {ι : Type u_2} (l : BoxIntegral.IntegrationParams) (hl : l.bRiemann = false) {r : (ι → ℝ) → ↑(Set.Ioi 0)}
Goal: l.RCond r
-- BoxIntegral.IntegrationParams.toFilter_inf_iUnion_eq
theorem toFilter_inf_iUnion_eq (l : IntegrationParams) (I : Box ι) (π₀ : Prepartition I) :
    l.toFilter I ⊓ 𝓟 { π | π.iUnion = π₀.iUnion } = l.toFilteriUnion I π₀ 
Arguments: {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (π₀ : BoxIntegral.Prepartition I)
Goal: l.toFilter I ⊓ 𝓟 {π | π.iUnion = π₀.iUnion} = l.toFilteriUnion I π₀
-- BoxIntegral.IntegrationParams.MemBaseSet.mono'
theorem MemBaseSet.mono' (I : Box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) {π : TaggedPrepartition I}
    (hr : ∀ J ∈ π, r₁ (π.tag J) ≤ r₂ (π.tag J)) (hπ : l₁.MemBaseSet I c₁ r₁ π) :
    l₂.MemBaseSet I c₂ r₂ π 
Arguments: {ι : Type u_1} [Fintype ι] {c₁ c₂ : ℝ≥0} {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} {l₁ l₂ : BoxIntegral.IntegrationParams} (I : BoxIntegral.Box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) {π : BoxIntegral.TaggedPrepartition I} (hr : ∀ J ∈ π, r₁ (π.tag J) ≤ r₂ (π.tag J)) (hπ : l₁.MemBaseSet I c₁ r₁ π)
Goal: l₂.MemBaseSet I c₂ r₂ π
-- BoxIntegral.IntegrationParams.MemBaseSet.mono
@[mono]
theorem MemBaseSet.mono (I : Box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) {π : TaggedPrepartition I}
    (hr : ∀ x ∈ Box.Icc I, r₁ x ≤ r₂ x) (hπ : l₁.MemBaseSet I c₁ r₁ π) : l₂.MemBaseSet I c₂ r₂ π 
Arguments: {ι : Type u_1} [Fintype ι] {c₁ c₂ : ℝ≥0} {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} {l₁ l₂ : BoxIntegral.IntegrationParams} (I : BoxIntegral.Box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) {π : BoxIntegral.TaggedPrepartition I} (hr : ∀ x ∈ BoxIntegral.Box.Icc I, r₁ x ≤ r₂ x) (hπ : l₁.MemBaseSet I c₁ r₁ π)
Goal: l₂.MemBaseSet I c₂ r₂ π
-- BoxIntegral.IntegrationParams.MemBaseSet.exists_common_compl
theorem MemBaseSet.exists_common_compl (h₁ : l.MemBaseSet I c₁ r₁ π₁) (h₂ : l.MemBaseSet I c₂ r₂ π₂)
    (hU : π₁.iUnion = π₂.iUnion) :
    ∃ π : Prepartition I, π.iUnion = ↑I \ π₁.iUnion ∧
      (l.bDistortion → π.distortion ≤ c₁) ∧ (l.bDistortion → π.distortion ≤ c₂) 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c₁ c₂ : ℝ≥0} {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} {l : BoxIntegral.IntegrationParams} (h₁ : l.MemBaseSet I c₁ r₁ π₁) (h₂ : l.MemBaseSet I c₂ r₂ π₂) (hU : π₁.iUnion = π₂.iUnion)
Goal: ∃ π, π.iUnion = ↑I \ π₁.iUnion ∧ (l.bDistortion = true → π.distortion ≤ c₁) ∧ (l.bDistortion = true → π.distortion ≤ c₂)
-- BoxIntegral.IntegrationParams.MemBaseSet.unionComplToSubordinate
protected theorem MemBaseSet.unionComplToSubordinate (hπ₁ : l.MemBaseSet I c r₁ π₁)
    (hle : ∀ x ∈ Box.Icc I, r₂ x ≤ r₁ x) {π₂ : Prepartition I} (hU : π₂.iUnion = ↑I \ π₁.iUnion)
    (hc : l.bDistortion → π₂.distortion ≤ c) :
    l.MemBaseSet I c r₁ (π₁.unionComplToSubordinate π₂ hU r₂) 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : ℝ≥0} {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} {π₁ : BoxIntegral.TaggedPrepartition I} {l : BoxIntegral.IntegrationParams} (hπ₁ : l.MemBaseSet I c r₁ π₁) (hle : ∀ x ∈ BoxIntegral.Box.Icc I, r₂ x ≤ r₁ x) {π₂ : BoxIntegral.Prepartition I} (hU : π₂.iUnion = ↑I \ π₁.iUnion) (hc : l.bDistortion = true → π₂.distortion ≤ c)
Goal: l.MemBaseSet I c r₁ (π₁.unionComplToSubordinate π₂ hU r₂)
-- BoxIntegral.IntegrationParams.MemBaseSet.filter
protected theorem MemBaseSet.filter (hπ : l.MemBaseSet I c r π) (p : Box ι → Prop) :
    l.MemBaseSet I c r (π.filter p) 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : ℝ≥0} {r : (ι → ℝ) → ↑(Set.Ioi 0)} {π : BoxIntegral.TaggedPrepartition I} {l : BoxIntegral.IntegrationParams} (hπ : l.MemBaseSet I c r π) (p : BoxIntegral.Box ι → Prop)
Goal: l.MemBaseSet I c r (π.filter p)
-- BoxIntegral.IntegrationParams.biUnionTagged_memBaseSet
theorem biUnionTagged_memBaseSet {π : Prepartition I} {πi : ∀ J, TaggedPrepartition J}
    (h : ∀ J ∈ π, l.MemBaseSet J c r (πi J)) (hp : ∀ J ∈ π, (πi J).IsPartition)
    (hc : l.bDistortion → π.compl.distortion ≤ c) : l.MemBaseSet I c r (π.biUnionTagged πi) 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : ℝ≥0} {r : (ι → ℝ) → ↑(Set.Ioi 0)} {l : BoxIntegral.IntegrationParams} {π : BoxIntegral.Prepartition I} {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J} (h : ∀ J ∈ π, l.MemBaseSet J c r (πi J)) (hp : ∀ J ∈ π, (πi J).IsPartition) (hc : l.bDistortion = true → π.compl.distortion ≤ c)
Goal: l.MemBaseSet I c r (π.biUnionTagged πi)
-- BoxIntegral.IntegrationParams.RCond.mono
@[mono]
theorem RCond.mono {ι : Type*} {r : (ι → ℝ) → Ioi (0 : ℝ)} (h : l₁ ≤ l₂) (hr : l₂.RCond r) :
    l₁.RCond r 
Arguments: {l₁ l₂ : BoxIntegral.IntegrationParams} {ι : Type u_2} {r : (ι → ℝ) → ↑(Set.Ioi 0)} (h : l₁ ≤ l₂) (hr : l₂.RCond r)
Goal: l₁.RCond r
-- BoxIntegral.IntegrationParams.RCond.min
nonrec theorem RCond.min {ι : Type*} {r₁ r₂ : (ι → ℝ) → Ioi (0 : ℝ)} (h₁ : l.RCond r₁)
    (h₂ : l.RCond r₂) : l.RCond fun x => min (r₁ x) (r₂ x) 
Arguments: {l : BoxIntegral.IntegrationParams} {ι : Type u_2} {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} (h₁ : l.RCond r₁) (h₂ : l.RCond r₂)
Goal: l.RCond fun x => min (r₁ x) (r₂ x)
-- BoxIntegral.IntegrationParams.toFilterDistortion_mono
@[mono]
theorem toFilterDistortion_mono (I : Box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂) :
    l₁.toFilterDistortion I c₁ ≤ l₂.toFilterDistortion I c₂ 
Arguments: {ι : Type u_1} [Fintype ι] {c₁ c₂ : ℝ≥0} {l₁ l₂ : BoxIntegral.IntegrationParams} (I : BoxIntegral.Box ι) (h : l₁ ≤ l₂) (hc : c₁ ≤ c₂)
Goal: l₁.toFilterDistortion I c₁ ≤ l₂.toFilterDistortion I c₂
-- BoxIntegral.IntegrationParams.toFilter_mono
@[mono]
theorem toFilter_mono (I : Box ι) {l₁ l₂ : IntegrationParams} (h : l₁ ≤ l₂) :
    l₁.toFilter I ≤ l₂.toFilter I 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) {l₁ l₂ : BoxIntegral.IntegrationParams} (h : l₁ ≤ l₂)
Goal: l₁.toFilter I ≤ l₂.toFilter I
-- BoxIntegral.IntegrationParams.toFilteriUnion_mono
@[mono]
theorem toFilteriUnion_mono (I : Box ι) {l₁ l₂ : IntegrationParams} (h : l₁ ≤ l₂)
    (π₀ : Prepartition I) : l₁.toFilteriUnion I π₀ ≤ l₂.toFilteriUnion I π₀ 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) {l₁ l₂ : BoxIntegral.IntegrationParams} (h : l₁ ≤ l₂) (π₀ : BoxIntegral.Prepartition I)
Goal: l₁.toFilteriUnion I π₀ ≤ l₂.toFilteriUnion I π₀
-- BoxIntegral.IntegrationParams.toFilteriUnion_congr
theorem toFilteriUnion_congr (I : Box ι) (l : IntegrationParams) {π₁ π₂ : Prepartition I}
    (h : π₁.iUnion = π₂.iUnion) : l.toFilteriUnion I π₁ = l.toFilteriUnion I π₂ 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) (l : BoxIntegral.IntegrationParams) {π₁ π₂ : BoxIntegral.Prepartition I} (h : π₁.iUnion = π₂.iUnion)
Goal: l.toFilteriUnion I π₁ = l.toFilteriUnion I π₂
-- BoxIntegral.IntegrationParams.hasBasis_toFilterDistortion
theorem hasBasis_toFilterDistortion (l : IntegrationParams) (I : Box ι) (c : ℝ≥0) :
    (l.toFilterDistortion I c).HasBasis l.RCond fun r => { π | l.MemBaseSet I c r π } 
Arguments: {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (c : ℝ≥0)
Goal: (l.toFilterDistortion I c).HasBasis l.RCond fun r => {π | l.MemBaseSet I c r π}
-- BoxIntegral.IntegrationParams.hasBasis_toFilterDistortioniUnion
theorem hasBasis_toFilterDistortioniUnion (l : IntegrationParams) (I : Box ι) (c : ℝ≥0)
    (π₀ : Prepartition I) :
    (l.toFilterDistortioniUnion I c π₀).HasBasis l.RCond fun r =>
      { π | l.MemBaseSet I c r π ∧ π.iUnion = π₀.iUnion } 
Arguments: {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (c : ℝ≥0) (π₀ : BoxIntegral.Prepartition I)
Goal: (l.toFilterDistortioniUnion I c π₀).HasBasis l.RCond fun r => {π | l.MemBaseSet I c r π ∧ π.iUnion = π₀.iUnion}
-- BoxIntegral.IntegrationParams.hasBasis_toFilteriUnion
theorem hasBasis_toFilteriUnion (l : IntegrationParams) (I : Box ι) (π₀ : Prepartition I) :
    (l.toFilteriUnion I π₀).HasBasis (fun r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ) => ∀ c, l.RCond (r c))
      fun r => { π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.iUnion = π₀.iUnion } 
Arguments: {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (π₀ : BoxIntegral.Prepartition I)
Goal: (l.toFilteriUnion I π₀).HasBasis (fun r => ∀ (c : ℝ≥0), l.RCond (r c)) fun r =>
  {π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.iUnion = π₀.iUnion}
-- BoxIntegral.IntegrationParams.hasBasis_toFilteriUnion_top
theorem hasBasis_toFilteriUnion_top (l : IntegrationParams) (I : Box ι) :
    (l.toFilteriUnion I ⊤).HasBasis (fun r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ) => ∀ c, l.RCond (r c))
      fun r => { π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.IsPartition } 
Arguments: {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι)
Goal: (l.toFilteriUnion I ⊤).HasBasis (fun r => ∀ (c : ℝ≥0), l.RCond (r c)) fun r =>
  {π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.IsPartition}
-- BoxIntegral.IntegrationParams.hasBasis_toFilter
theorem hasBasis_toFilter (l : IntegrationParams) (I : Box ι) :
    (l.toFilter I).HasBasis (fun r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ) => ∀ c, l.RCond (r c))
      fun r => { π | ∃ c, l.MemBaseSet I c (r c) π } 
Arguments: {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι)
Goal: (l.toFilter I).HasBasis (fun r => ∀ (c : ℝ≥0), l.RCond (r c)) fun r => {π | ∃ c, l.MemBaseSet I c (r c) π}
-- BoxIntegral.IntegrationParams.tendsto_embedBox_toFilteriUnion_top
theorem tendsto_embedBox_toFilteriUnion_top (l : IntegrationParams) (h : I ≤ J) :
    Tendsto (TaggedPrepartition.embedBox I J h) (l.toFilteriUnion I ⊤)
      (l.toFilteriUnion J (Prepartition.single J I h)) 
Arguments: {ι : Type u_1} [Fintype ι] {I J : BoxIntegral.Box ι} (l : BoxIntegral.IntegrationParams) (h : I ≤ J)
Goal: Filter.Tendsto (⇑(BoxIntegral.TaggedPrepartition.embedBox I J h)) (l.toFilteriUnion I ⊤)
  (l.toFilteriUnion J (BoxIntegral.Prepartition.single J I h))
-- BoxIntegral.IntegrationParams.exists_memBaseSet_le_iUnion_eq
theorem exists_memBaseSet_le_iUnion_eq (l : IntegrationParams) (π₀ : Prepartition I)
    (hc₁ : π₀.distortion ≤ c) (hc₂ : π₀.compl.distortion ≤ c) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    ∃ π, l.MemBaseSet I c r π ∧ π.toPrepartition ≤ π₀ ∧ π.iUnion = π₀.iUnion 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {c : ℝ≥0} (l : BoxIntegral.IntegrationParams) (π₀ : BoxIntegral.Prepartition I) (hc₁ : π₀.distortion ≤ c) (hc₂ : π₀.compl.distortion ≤ c) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: ∃ π, l.MemBaseSet I c r π ∧ π.toPrepartition ≤ π₀ ∧ π.iUnion = π₀.iUnion
-- BoxIntegral.IntegrationParams.exists_memBaseSet_isPartition
theorem exists_memBaseSet_isPartition (l : IntegrationParams) (I : Box ι) (hc : I.distortion ≤ c)
    (r : (ι → ℝ) → Ioi (0 : ℝ)) : ∃ π, l.MemBaseSet I c r π ∧ π.IsPartition 
Arguments: {ι : Type u_1} [Fintype ι] {c : ℝ≥0} (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (hc : I.distortion ≤ c) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: ∃ π, l.MemBaseSet I c r π ∧ π.IsPartition
-- BoxIntegral.IntegrationParams.toFilterDistortioniUnion_neBot
theorem toFilterDistortioniUnion_neBot (l : IntegrationParams) (I : Box ι) (π₀ : Prepartition I)
    (hc₁ : π₀.distortion ≤ c) (hc₂ : π₀.compl.distortion ≤ c) :
    (l.toFilterDistortioniUnion I c π₀).NeBot 
Arguments: {ι : Type u_1} [Fintype ι] {c : ℝ≥0} (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι) (π₀ : BoxIntegral.Prepartition I) (hc₁ : π₀.distortion ≤ c) (hc₂ : π₀.compl.distortion ≤ c)
Goal: (l.toFilterDistortioniUnion I c π₀).NeBot
-- BoxIntegral.IntegrationParams.eventually_isPartition
theorem eventually_isPartition (l : IntegrationParams) (I : Box ι) :
    ∀ᶠ π in l.toFilteriUnion I ⊤, TaggedPrepartition.IsPartition π 
Arguments: {ι : Type u_1} [Fintype ι] (l : BoxIntegral.IntegrationParams) (I : BoxIntegral.Box ι)
Goal: ∀ᶠ (π : BoxIntegral.TaggedPrepartition I) in l.toFilteriUnion I ⊤, π.IsPartition
-- BoxIntegral.Box.coe_splitLower
@[simp]
theorem coe_splitLower : (splitLower I i x : Set (ι → ℝ)) = ↑I ∩ { y | y i ≤ x } 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: ↑(I.splitLower i x) = ↑I ∩ {y | y i ≤ x}
-- BoxIntegral.Box.splitLower_le
theorem splitLower_le : I.splitLower i x ≤ I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: I.splitLower i x ≤ ↑I
-- BoxIntegral.Box.splitLower_eq_bot
@[simp]
theorem splitLower_eq_bot {i x} : I.splitLower i x = ⊥ ↔ x ≤ I.lower i 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: I.splitLower i x = ⊥ ↔ x ≤ I.lower i
-- BoxIntegral.Box.splitLower_eq_self
@[simp]
theorem splitLower_eq_self : I.splitLower i x = I ↔ I.upper i ≤ x 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: I.splitLower i x = ↑I ↔ I.upper i ≤ x
-- BoxIntegral.Box.splitLower_def
theorem splitLower_def [DecidableEq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
    (h' : ∀ j, I.lower j < update I.upper i x j :=
      (forall_update_iff I.upper fun j y => I.lower j < y).2
        ⟨h.1, fun j _ => I.lower_lt_upper _⟩) :
    I.splitLower i x = (⟨I.lower, update I.upper i x, h'⟩ : Box ι) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [DecidableEq ι] {i : ι} {x : ℝ} (h : x ∈ Set.Ioo (I.lower i) (I.upper i)) (h' : ∀ (j : ι), I.lower j < Function.update I.upper i x j := ⋯)
Goal: I.splitLower i x = ↑{ lower := I.lower, upper := Function.update I.upper i x, lower_lt_upper := h' }
-- BoxIntegral.Box.coe_splitUpper
@[simp]
theorem coe_splitUpper : (splitUpper I i x : Set (ι → ℝ)) = ↑I ∩ { y | x < y i } 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: ↑(I.splitUpper i x) = ↑I ∩ {y | x < y i}
-- BoxIntegral.Box.splitUpper_le
theorem splitUpper_le : I.splitUpper i x ≤ I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: I.splitUpper i x ≤ ↑I
-- BoxIntegral.Box.splitUpper_eq_bot
@[simp]
theorem splitUpper_eq_bot {i x} : I.splitUpper i x = ⊥ ↔ I.upper i ≤ x 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: I.splitUpper i x = ⊥ ↔ I.upper i ≤ x
-- BoxIntegral.Box.splitUpper_eq_self
@[simp]
theorem splitUpper_eq_self : I.splitUpper i x = I ↔ x ≤ I.lower i 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: I.splitUpper i x = ↑I ↔ x ≤ I.lower i
-- BoxIntegral.Box.splitUpper_def
theorem splitUpper_def [DecidableEq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
    (h' : ∀ j, update I.lower i x j < I.upper j :=
      (forall_update_iff I.lower fun j y => y < I.upper j).2
        ⟨h.2, fun j _ => I.lower_lt_upper _⟩) :
    I.splitUpper i x = (⟨update I.lower i x, I.upper, h'⟩ : Box ι) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [DecidableEq ι] {i : ι} {x : ℝ} (h : x ∈ Set.Ioo (I.lower i) (I.upper i)) (h' : ∀ (j : ι), Function.update I.lower i x j < I.upper j := ⋯)
Goal: I.splitUpper i x = ↑{ lower := Function.update I.lower i x, upper := I.upper, lower_lt_upper := h' }
-- BoxIntegral.Box.disjoint_splitLower_splitUpper
theorem disjoint_splitLower_splitUpper (I : Box ι) (i : ι) (x : ℝ) :
    Disjoint (I.splitLower i x) (I.splitUpper i x) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (i : ι) (x : ℝ)
Goal: Disjoint (I.splitLower i x) (I.splitUpper i x)
-- BoxIntegral.Box.splitLower_ne_splitUpper
theorem splitLower_ne_splitUpper (I : Box ι) (i : ι) (x : ℝ) :
    I.splitLower i x ≠ I.splitUpper i x 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (i : ι) (x : ℝ)
Goal: I.splitLower i x ≠ I.splitUpper i x
-- BoxIntegral.Prepartition.mem_split_iff
@[simp]
theorem mem_split_iff : J ∈ split I i x ↔ ↑J = I.splitLower i x ∨ ↑J = I.splitUpper i x 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: J ∈ BoxIntegral.Prepartition.split I i x ↔ ↑J = I.splitLower i x ∨ ↑J = I.splitUpper i x
-- BoxIntegral.Prepartition.mem_split_iff'
theorem mem_split_iff' : J ∈ split I i x ↔
    (J : Set (ι → ℝ)) = ↑I ∩ { y | y i ≤ x } ∨ (J : Set (ι → ℝ)) = ↑I ∩ { y | x < y i } 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {i : ι} {x : ℝ}
Goal: J ∈ BoxIntegral.Prepartition.split I i x ↔ ↑J = ↑I ∩ {y | y i ≤ x} ∨ ↑J = ↑I ∩ {y | x < y i}
-- BoxIntegral.Prepartition.iUnion_split
@[simp]
theorem iUnion_split (I : Box ι) (i : ι) (x : ℝ) : (split I i x).iUnion = I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (i : ι) (x : ℝ)
Goal: (BoxIntegral.Prepartition.split I i x).iUnion = ↑I
-- BoxIntegral.Prepartition.isPartitionSplit
theorem isPartitionSplit (I : Box ι) (i : ι) (x : ℝ) : IsPartition (split I i x) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (i : ι) (x : ℝ)
Goal: (BoxIntegral.Prepartition.split I i x).IsPartition
-- BoxIntegral.Prepartition.sum_split_boxes
theorem sum_split_boxes {M : Type*} [AddCommMonoid M] (I : Box ι) (i : ι) (x : ℝ) (f : Box ι → M) :
    (∑ J ∈ (split I i x).boxes, f J) =
      (I.splitLower i x).elim' 0 f + (I.splitUpper i x).elim' 0 f 
Arguments: {ι : Type u_1} {M : Type u_3} [AddCommMonoid M] (I : BoxIntegral.Box ι) (i : ι) (x : ℝ) (f : BoxIntegral.Box ι → M)
Goal: ∑ J ∈ (BoxIntegral.Prepartition.split I i x).boxes, f J =
  Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x)
-- BoxIntegral.Prepartition.split_of_not_mem_Ioo
theorem split_of_not_mem_Ioo (h : x ∉ Ioo (I.lower i) (I.upper i)) : split I i x = ⊤ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {i : ι} {x : ℝ} (h : x ∉ Set.Ioo (I.lower i) (I.upper i))
Goal: BoxIntegral.Prepartition.split I i x = ⊤
-- BoxIntegral.Prepartition.coe_eq_of_mem_split_of_mem_le
theorem coe_eq_of_mem_split_of_mem_le {y : ι → ℝ} (h₁ : J ∈ split I i x) (h₂ : y ∈ J)
    (h₃ : y i ≤ x) : (J : Set (ι → ℝ)) = ↑I ∩ { y | y i ≤ x } 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {i : ι} {x : ℝ} {y : ι → ℝ} (h₁ : J ∈ BoxIntegral.Prepartition.split I i x) (h₂ : y ∈ J) (h₃ : y i ≤ x)
Goal: ↑J = ↑I ∩ {y | y i ≤ x}
-- BoxIntegral.Prepartition.coe_eq_of_mem_split_of_lt_mem
theorem coe_eq_of_mem_split_of_lt_mem {y : ι → ℝ} (h₁ : J ∈ split I i x) (h₂ : y ∈ J)
    (h₃ : x < y i) : (J : Set (ι → ℝ)) = ↑I ∩ { y | x < y i } 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {i : ι} {x : ℝ} {y : ι → ℝ} (h₁ : J ∈ BoxIntegral.Prepartition.split I i x) (h₂ : y ∈ J) (h₃ : x < y i)
Goal: ↑J = ↑I ∩ {y | x < y i}
-- BoxIntegral.Prepartition.restrict_split
@[simp]
theorem restrict_split (h : I ≤ J) (i : ι) (x : ℝ) : (split J i x).restrict I = split I i x 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (h : I ≤ J) (i : ι) (x : ℝ)
Goal: (BoxIntegral.Prepartition.split J i x).restrict I = BoxIntegral.Prepartition.split I i x
-- BoxIntegral.Prepartition.inf_split
theorem inf_split (π : Prepartition I) (i : ι) (x : ℝ) :
    π ⊓ split I i x = π.biUnion fun J => split J i x 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (i : ι) (x : ℝ)
Goal: π ⊓ BoxIntegral.Prepartition.split I i x = π.biUnion fun J => BoxIntegral.Prepartition.split J i x
-- BoxIntegral.Prepartition.splitMany_empty
@[simp]
theorem splitMany_empty (I : Box ι) : splitMany I ∅ = ⊤ 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι)
Goal: BoxIntegral.Prepartition.splitMany I ∅ = ⊤
-- BoxIntegral.Prepartition.splitMany_insert
@[simp]
theorem splitMany_insert (I : Box ι) (s : Finset (ι × ℝ)) (p : ι × ℝ) :
    splitMany I (insert p s) = splitMany I s ⊓ split I p.1 p.2 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (s : Finset (ι × ℝ)) (p : ι × ℝ)
Goal: BoxIntegral.Prepartition.splitMany I (insert p s) =
  BoxIntegral.Prepartition.splitMany I s ⊓ BoxIntegral.Prepartition.split I p.1 p.2
-- BoxIntegral.Prepartition.splitMany_le_split
theorem splitMany_le_split (I : Box ι) {s : Finset (ι × ℝ)} {p : ι × ℝ} (hp : p ∈ s) :
    splitMany I s ≤ split I p.1 p.2 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) {s : Finset (ι × ℝ)} {p : ι × ℝ} (hp : p ∈ s)
Goal: BoxIntegral.Prepartition.splitMany I s ≤ BoxIntegral.Prepartition.split I p.1 p.2
-- BoxIntegral.Prepartition.isPartition_splitMany
theorem isPartition_splitMany (I : Box ι) (s : Finset (ι × ℝ)) : IsPartition (splitMany I s) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (s : Finset (ι × ℝ))
Goal: (BoxIntegral.Prepartition.splitMany I s).IsPartition
-- BoxIntegral.Prepartition.iUnion_splitMany
@[simp]
theorem iUnion_splitMany (I : Box ι) (s : Finset (ι × ℝ)) : (splitMany I s).iUnion = I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (s : Finset (ι × ℝ))
Goal: (BoxIntegral.Prepartition.splitMany I s).iUnion = ↑I
-- BoxIntegral.Prepartition.inf_splitMany
theorem inf_splitMany {I : Box ι} (π : Prepartition I) (s : Finset (ι × ℝ)) :
    π ⊓ splitMany I s = π.biUnion fun J => splitMany J s 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (s : Finset (ι × ℝ))
Goal: π ⊓ BoxIntegral.Prepartition.splitMany I s = π.biUnion fun J => BoxIntegral.Prepartition.splitMany J s