ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoxIntegral.TaggedPrepartition.mem_mk
@[simp]
theorem mem_mk (π : Prepartition I) (f h) : J ∈ mk π f h ↔ J ∈ π 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (f : BoxIntegral.Box ι → ι → ℝ) (h : ∀ (J : BoxIntegral.Box ι), f J ∈ BoxIntegral.Box.Icc I)
Goal: J ∈ { toPrepartition := π, tag := f, tag_mem_Icc := h } ↔ J ∈ π
-- BoxIntegral.TaggedPrepartition.iUnion_def
theorem iUnion_def : π.iUnion = ⋃ J ∈ π, ↑J 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I)
Goal: π.iUnion = ⋃ J ∈ π, ↑J
-- BoxIntegral.TaggedPrepartition.iUnion_mk
@[simp]
theorem iUnion_mk (π : Prepartition I) (f h) : (mk π f h).iUnion = π.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (f : BoxIntegral.Box ι → ι → ℝ) (h : ∀ (J : BoxIntegral.Box ι), f J ∈ BoxIntegral.Box.Icc I)
Goal: { toPrepartition := π, tag := f, tag_mem_Icc := h }.iUnion = π.iUnion
-- BoxIntegral.TaggedPrepartition.iUnion_toPrepartition
@[simp]
theorem iUnion_toPrepartition : π.toPrepartition.iUnion = π.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I)
Goal: π.iUnion = π.iUnion
-- BoxIntegral.TaggedPrepartition.mem_iUnion
@[simp]
theorem mem_iUnion : x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) {x : ι → ℝ}
Goal: x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J
-- BoxIntegral.TaggedPrepartition.subset_iUnion
theorem subset_iUnion (h : J ∈ π) : ↑J ⊆ π.iUnion 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) (h : J ∈ π)
Goal: ↑J ⊆ π.iUnion
-- BoxIntegral.TaggedPrepartition.iUnion_subset
theorem iUnion_subset : π.iUnion ⊆ I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I)
Goal: π.iUnion ⊆ ↑I
-- BoxIntegral.TaggedPrepartition.isPartition_iff_iUnion_eq
theorem isPartition_iff_iUnion_eq : IsPartition π ↔ π.iUnion = I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I)
Goal: π.IsPartition ↔ π.iUnion = ↑I
-- BoxIntegral.TaggedPrepartition.filter_boxes_val
@[simps! (config := .asFn)]
def filter (p : Box ι → Prop) : TaggedPrepartition I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) (p : BoxIntegral.Box ι → Prop)
Goal: (π.filter p).boxes.val = Multiset.filter p π.boxes.val
-- BoxIntegral.TaggedPrepartition.filter_tag
@[simps! (config := .asFn)]
def filter (p : Box ι → Prop) : TaggedPrepartition I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) (p : BoxIntegral.Box ι → Prop)
Goal: (π.filter p).tag = π.tag
-- BoxIntegral.TaggedPrepartition.mem_filter
@[simp]
theorem mem_filter {p : Box ι → Prop} : J ∈ π.filter p ↔ J ∈ π ∧ p J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) {p : BoxIntegral.Box ι → Prop}
Goal: J ∈ π.filter p ↔ J ∈ π ∧ p J
-- BoxIntegral.TaggedPrepartition.iUnion_filter_not
@[simp]
theorem iUnion_filter_not (π : TaggedPrepartition I) (p : Box ι → Prop) :
    (π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) (p : BoxIntegral.Box ι → Prop)
Goal: (π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion
-- BoxIntegral.Prepartition.mem_biUnionTagged
@[simp]
theorem mem_biUnionTagged (π : Prepartition I) {πi : ∀ J, TaggedPrepartition J} :
    J ∈ π.biUnionTagged πi ↔ ∃ J' ∈ π, J ∈ πi J' 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J}
Goal: J ∈ π.biUnionTagged πi ↔ ∃ J' ∈ π, J ∈ πi J'
-- BoxIntegral.Prepartition.tag_biUnionTagged
theorem tag_biUnionTagged (π : Prepartition I) {πi : ∀ J, TaggedPrepartition J} (hJ : J ∈ π) {J'}
    (hJ' : J' ∈ πi J) : (π.biUnionTagged πi).tag J' = (πi J).tag J' 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J} (hJ : J ∈ π) {J' : BoxIntegral.Box ι} (hJ' : J' ∈ πi J)
Goal: (π.biUnionTagged πi).tag J' = (πi J).tag J'
-- BoxIntegral.Prepartition.iUnion_biUnionTagged
@[simp]
theorem iUnion_biUnionTagged (π : Prepartition I) (πi : ∀ J, TaggedPrepartition J) :
    (π.biUnionTagged πi).iUnion = ⋃ J ∈ π, (πi J).iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J)
Goal: (π.biUnionTagged πi).iUnion = ⋃ J ∈ π, (πi J).iUnion
-- BoxIntegral.Prepartition.forall_biUnionTagged
theorem forall_biUnionTagged (p : (ι → ℝ) → Box ι → Prop) (π : Prepartition I)
    (πi : ∀ J, TaggedPrepartition J) :
    (∀ J ∈ π.biUnionTagged πi, p ((π.biUnionTagged πi).tag J) J) ↔
      ∀ J ∈ π, ∀ J' ∈ πi J, p ((πi J).tag J') J' 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (p : (ι → ℝ) → BoxIntegral.Box ι → Prop) (π : BoxIntegral.Prepartition I) (πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J)
Goal: (∀ J ∈ π.biUnionTagged πi, p ((π.biUnionTagged πi).tag J) J) ↔ ∀ J ∈ π, ∀ J' ∈ πi J, p ((πi J).tag J') J'
-- BoxIntegral.Prepartition.IsPartition.biUnionTagged
theorem IsPartition.biUnionTagged {π : Prepartition I} (h : IsPartition π)
    {πi : ∀ J, TaggedPrepartition J} (hi : ∀ J ∈ π, (πi J).IsPartition) :
    (π.biUnionTagged πi).IsPartition 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.Prepartition I} (h : π.IsPartition) {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J} (hi : ∀ J ∈ π, (πi J).IsPartition)
Goal: (π.biUnionTagged πi).IsPartition
-- BoxIntegral.TaggedPrepartition.biUnionPrepartition_tag
@[simps (config := .asFn) tag]
def biUnionPrepartition (π : TaggedPrepartition I) (πi : ∀ J : Box ι, Prepartition J) :
    TaggedPrepartition I 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) (πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J)
Goal: (π.biUnionPrepartition πi).tag = fun J => π.tag (π.biUnionIndex πi J)
-- BoxIntegral.TaggedPrepartition.IsPartition.biUnionPrepartition
theorem IsPartition.biUnionPrepartition {π : TaggedPrepartition I} (h : IsPartition π)
    {πi : ∀ J, Prepartition J} (hi : ∀ J ∈ π, (πi J).IsPartition) :
    (π.biUnionPrepartition πi).IsPartition 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} (h : π.IsPartition) {πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J} (hi : ∀ J ∈ π, (πi J).IsPartition)
Goal: (π.biUnionPrepartition πi).IsPartition
-- BoxIntegral.TaggedPrepartition.infPrepartition_toPrepartition
@[simp]
theorem infPrepartition_toPrepartition (π : TaggedPrepartition I) (π' : Prepartition I) :
    (π.infPrepartition π').toPrepartition = π.toPrepartition ⊓ π' 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) (π' : BoxIntegral.Prepartition I)
Goal: (π.infPrepartition π').toPrepartition = π.toPrepartition ⊓ π'
-- BoxIntegral.TaggedPrepartition.mem_infPrepartition_comm
theorem mem_infPrepartition_comm :
    J ∈ π₁.infPrepartition π₂.toPrepartition ↔ J ∈ π₂.infPrepartition π₁.toPrepartition 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I}
Goal: J ∈ π₁.infPrepartition π₂.toPrepartition ↔ J ∈ π₂.infPrepartition π₁.toPrepartition
-- BoxIntegral.TaggedPrepartition.IsPartition.infPrepartition
theorem IsPartition.infPrepartition (h₁ : π₁.IsPartition) {π₂ : Prepartition I}
    (h₂ : π₂.IsPartition) : (π₁.infPrepartition π₂).IsPartition 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ : BoxIntegral.TaggedPrepartition I} (h₁ : π₁.IsPartition) {π₂ : BoxIntegral.Prepartition I} (h₂ : π₂.IsPartition)
Goal: (π₁.infPrepartition π₂).IsPartition
-- BoxIntegral.TaggedPrepartition.isHenstock_biUnionTagged
@[simp]
theorem isHenstock_biUnionTagged {π : Prepartition I} {πi : ∀ J, TaggedPrepartition J} :
    IsHenstock (π.biUnionTagged πi) ↔ ∀ J ∈ π, (πi J).IsHenstock 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.Prepartition I} {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J}
Goal: (π.biUnionTagged πi).IsHenstock ↔ ∀ J ∈ π, (πi J).IsHenstock
-- BoxIntegral.TaggedPrepartition.IsHenstock.card_filter_tag_eq_le
theorem IsHenstock.card_filter_tag_eq_le [Fintype ι] (h : π.IsHenstock) (x : ι → ℝ) :
    (π.boxes.filter fun J => π.tag J = x).card ≤ 2 ^ Fintype.card ι 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} [Fintype ι] (h : π.IsHenstock) (x : ι → ℝ)
Goal: (Finset.filter (fun J => π.tag J = x) π.boxes).card ≤ 2 ^ Fintype.card ι
-- BoxIntegral.TaggedPrepartition.isSubordinate_biUnionTagged
@[simp]
theorem isSubordinate_biUnionTagged [Fintype ι] {π : Prepartition I}
    {πi : ∀ J, TaggedPrepartition J} :
    IsSubordinate (π.biUnionTagged πi) r ↔ ∀ J ∈ π, (πi J).IsSubordinate r 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {r : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] {π : BoxIntegral.Prepartition I} {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J}
Goal: (π.biUnionTagged πi).IsSubordinate r ↔ ∀ J ∈ π, (πi J).IsSubordinate r
-- BoxIntegral.TaggedPrepartition.IsSubordinate.biUnionPrepartition
theorem IsSubordinate.biUnionPrepartition [Fintype ι] (h : IsSubordinate π r)
    (πi : ∀ J, Prepartition J) : IsSubordinate (π.biUnionPrepartition πi) r 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} {r : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] (h : π.IsSubordinate r) (πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J)
Goal: (π.biUnionPrepartition πi).IsSubordinate r
-- BoxIntegral.TaggedPrepartition.IsSubordinate.infPrepartition
theorem IsSubordinate.infPrepartition [Fintype ι] (h : IsSubordinate π r) (π' : Prepartition I) :
    IsSubordinate (π.infPrepartition π') r 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} {r : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] (h : π.IsSubordinate r) (π' : BoxIntegral.Prepartition I)
Goal: (π.infPrepartition π').IsSubordinate r
-- BoxIntegral.TaggedPrepartition.IsSubordinate.mono'
theorem IsSubordinate.mono' [Fintype ι] {π : TaggedPrepartition I} (hr₁ : π.IsSubordinate r₁)
    (h : ∀ J ∈ π, r₁ (π.tag J) ≤ r₂ (π.tag J)) : π.IsSubordinate r₂ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] {π : BoxIntegral.TaggedPrepartition I} (hr₁ : π.IsSubordinate r₁) (h : ∀ J ∈ π, r₁ (π.tag J) ≤ r₂ (π.tag J))
Goal: π.IsSubordinate r₂
-- BoxIntegral.TaggedPrepartition.IsSubordinate.mono
theorem IsSubordinate.mono [Fintype ι] {π : TaggedPrepartition I} (hr₁ : π.IsSubordinate r₁)
    (h : ∀ x ∈ Box.Icc I, r₁ x ≤ r₂ x) : π.IsSubordinate r₂ 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] {π : BoxIntegral.TaggedPrepartition I} (hr₁ : π.IsSubordinate r₁) (h : ∀ x ∈ BoxIntegral.Box.Icc I, r₁ x ≤ r₂ x)
Goal: π.IsSubordinate r₂
-- BoxIntegral.TaggedPrepartition.IsSubordinate.diam_le
theorem IsSubordinate.diam_le [Fintype ι] {π : TaggedPrepartition I} (h : π.IsSubordinate r)
    (hJ : J ∈ π.boxes) : diam (Box.Icc J) ≤ 2 * r (π.tag J) 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {r : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] {π : BoxIntegral.TaggedPrepartition I} (h : π.IsSubordinate r) (hJ : J ∈ π.boxes)
Goal: Metric.diam (BoxIntegral.Box.Icc J) ≤ 2 * ↑(r (π.tag J))
-- BoxIntegral.TaggedPrepartition.single_tag
@[simps! (config := .asFn)]
def single (I J : Box ι) (hJ : J ≤ I) (x : ι → ℝ) (h : x ∈ Box.Icc I) : TaggedPrepartition I 
Arguments: {ι : Type u_1} (I J : BoxIntegral.Box ι) (hJ : J ≤ I) (x : ι → ℝ) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I J hJ x h).tag = fun x_1 => x
-- BoxIntegral.TaggedPrepartition.single_boxes_val
@[simps! (config := .asFn)]
def single (I J : Box ι) (hJ : J ≤ I) (x : ι → ℝ) (h : x ∈ Box.Icc I) : TaggedPrepartition I 
Arguments: {ι : Type u_1} (I J : BoxIntegral.Box ι) (hJ : J ≤ I) (x : ι → ℝ) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I J hJ x h).boxes.val = {J}
-- BoxIntegral.TaggedPrepartition.mem_single
@[simp]
theorem mem_single {J'} (hJ : J ≤ I) (h : x ∈ Box.Icc I) : J' ∈ single I J hJ x h ↔ J' = J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {x : ι → ℝ} {J' : BoxIntegral.Box ι} (hJ : J ≤ I) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: J' ∈ BoxIntegral.TaggedPrepartition.single I J hJ x h ↔ J' = J
-- BoxIntegral.TaggedPrepartition.isPartition_single_iff
theorem isPartition_single_iff (hJ : J ≤ I) (h : x ∈ Box.Icc I) :
    (single I J hJ x h).IsPartition ↔ J = I 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {x : ι → ℝ} (hJ : J ≤ I) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I J hJ x h).IsPartition ↔ J = I
-- BoxIntegral.TaggedPrepartition.isPartition_single
theorem isPartition_single (h : x ∈ Box.Icc I) : (single I I le_rfl x h).IsPartition 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {x : ι → ℝ} (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I I ⋯ x h).IsPartition
-- BoxIntegral.TaggedPrepartition.forall_mem_single
theorem forall_mem_single (p : (ι → ℝ) → Box ι → Prop) (hJ : J ≤ I) (h : x ∈ Box.Icc I) :
    (∀ J' ∈ single I J hJ x h, p ((single I J hJ x h).tag J') J') ↔ p x J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {x : ι → ℝ} (p : (ι → ℝ) → BoxIntegral.Box ι → Prop) (hJ : J ≤ I) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (∀ J' ∈ BoxIntegral.TaggedPrepartition.single I J hJ x h,
    p ((BoxIntegral.TaggedPrepartition.single I J hJ x h).tag J') J') ↔
  p x J
-- BoxIntegral.TaggedPrepartition.isHenstock_single_iff
@[simp]
theorem isHenstock_single_iff (hJ : J ≤ I) (h : x ∈ Box.Icc I) :
    IsHenstock (single I J hJ x h) ↔ x ∈ Box.Icc J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {x : ι → ℝ} (hJ : J ≤ I) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I J hJ x h).IsHenstock ↔ x ∈ BoxIntegral.Box.Icc J
-- BoxIntegral.TaggedPrepartition.isHenstock_single
theorem isHenstock_single (h : x ∈ Box.Icc I) : IsHenstock (single I I le_rfl x h) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {x : ι → ℝ} (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I I ⋯ x h).IsHenstock
-- BoxIntegral.TaggedPrepartition.isSubordinate_single
@[simp]
theorem isSubordinate_single [Fintype ι] (hJ : J ≤ I) (h : x ∈ Box.Icc I) :
    IsSubordinate (single I J hJ x h) r ↔ Box.Icc J ⊆ closedBall x (r x) 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {x : ι → ℝ} {r : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] (hJ : J ≤ I) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I J hJ x h).IsSubordinate r ↔ BoxIntegral.Box.Icc J ⊆ Metric.closedBall x ↑(r x)
-- BoxIntegral.TaggedPrepartition.iUnion_single
@[simp]
theorem iUnion_single (hJ : J ≤ I) (h : x ∈ Box.Icc I) : (single I J hJ x h).iUnion = J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {x : ι → ℝ} (hJ : J ≤ I) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I J hJ x h).iUnion = ↑J
-- BoxIntegral.TaggedPrepartition.disjUnion_boxes
@[simp]
theorem disjUnion_boxes (h : Disjoint π₁.iUnion π₂.iUnion) :
    (π₁.disjUnion π₂ h).boxes = π₁.boxes ∪ π₂.boxes 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} (h : Disjoint π₁.iUnion π₂.iUnion)
Goal: (π₁.disjUnion π₂ h).boxes = π₁.boxes ∪ π₂.boxes
-- BoxIntegral.TaggedPrepartition.mem_disjUnion
@[simp]
theorem mem_disjUnion (h : Disjoint π₁.iUnion π₂.iUnion) :
    J ∈ π₁.disjUnion π₂ h ↔ J ∈ π₁ ∨ J ∈ π₂ 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} (h : Disjoint π₁.iUnion π₂.iUnion)
Goal: J ∈ π₁.disjUnion π₂ h ↔ J ∈ π₁ ∨ J ∈ π₂
-- BoxIntegral.TaggedPrepartition.iUnion_disjUnion
@[simp]
theorem iUnion_disjUnion (h : Disjoint π₁.iUnion π₂.iUnion) :
    (π₁.disjUnion π₂ h).iUnion = π₁.iUnion ∪ π₂.iUnion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} (h : Disjoint π₁.iUnion π₂.iUnion)
Goal: (π₁.disjUnion π₂ h).iUnion = π₁.iUnion ∪ π₂.iUnion
-- BoxIntegral.TaggedPrepartition.disjUnion_tag_of_mem_left
theorem disjUnion_tag_of_mem_left (h : Disjoint π₁.iUnion π₂.iUnion) (hJ : J ∈ π₁) :
    (π₁.disjUnion π₂ h).tag J = π₁.tag J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} (h : Disjoint π₁.iUnion π₂.iUnion) (hJ : J ∈ π₁)
Goal: (π₁.disjUnion π₂ h).tag J = π₁.tag J
-- BoxIntegral.TaggedPrepartition.disjUnion_tag_of_mem_right
theorem disjUnion_tag_of_mem_right (h : Disjoint π₁.iUnion π₂.iUnion) (hJ : J ∈ π₂) :
    (π₁.disjUnion π₂ h).tag J = π₂.tag J 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} (h : Disjoint π₁.iUnion π₂.iUnion) (hJ : J ∈ π₂)
Goal: (π₁.disjUnion π₂ h).tag J = π₂.tag J
-- BoxIntegral.TaggedPrepartition.IsSubordinate.disjUnion
theorem IsSubordinate.disjUnion [Fintype ι] (h₁ : IsSubordinate π₁ r) (h₂ : IsSubordinate π₂ r)
    (h : Disjoint π₁.iUnion π₂.iUnion) : IsSubordinate (π₁.disjUnion π₂ h) r 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} {r : (ι → ℝ) → ↑(Set.Ioi 0)} [Fintype ι] (h₁ : π₁.IsSubordinate r) (h₂ : π₂.IsSubordinate r) (h : Disjoint π₁.iUnion π₂.iUnion)
Goal: (π₁.disjUnion π₂ h).IsSubordinate r
-- BoxIntegral.TaggedPrepartition.IsHenstock.disjUnion
theorem IsHenstock.disjUnion (h₁ : IsHenstock π₁) (h₂ : IsHenstock π₂)
    (h : Disjoint π₁.iUnion π₂.iUnion) : IsHenstock (π₁.disjUnion π₂ h) 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} (h₁ : π₁.IsHenstock) (h₂ : π₂.IsHenstock) (h : Disjoint π₁.iUnion π₂.iUnion)
Goal: (π₁.disjUnion π₂ h).IsHenstock
-- BoxIntegral.TaggedPrepartition.distortion_le_of_mem
theorem distortion_le_of_mem (h : J ∈ π) : J.distortion ≤ π.distortion 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) [Fintype ι] (h : J ∈ π)
Goal: J.distortion ≤ π.distortion
-- BoxIntegral.TaggedPrepartition.distortion_le_iff
theorem distortion_le_iff {c : ℝ≥0} : π.distortion ≤ c ↔ ∀ J ∈ π, Box.distortion J ≤ c 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) [Fintype ι] {c : ℝ≥0}
Goal: π.distortion ≤ c ↔ ∀ J ∈ π, J.distortion ≤ c
-- BoxIntegral.Prepartition.distortion_biUnionTagged
@[simp]
theorem _root_.BoxIntegral.Prepartition.distortion_biUnionTagged (π : Prepartition I)
    (πi : ∀ J, TaggedPrepartition J) :
    (π.biUnionTagged πi).distortion = π.boxes.sup fun J => (πi J).distortion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Fintype ι] (π : BoxIntegral.Prepartition I) (πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J)
Goal: (π.biUnionTagged πi).distortion = π.boxes.sup fun J => (πi J).distortion
-- BoxIntegral.TaggedPrepartition.distortion_biUnionPrepartition
@[simp]
theorem distortion_biUnionPrepartition (π : TaggedPrepartition I) (πi : ∀ J, Prepartition J) :
    (π.biUnionPrepartition πi).distortion = π.boxes.sup fun J => (πi J).distortion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} [Fintype ι] (π : BoxIntegral.TaggedPrepartition I) (πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J)
Goal: (π.biUnionPrepartition πi).distortion = π.boxes.sup fun J => (πi J).distortion
-- BoxIntegral.TaggedPrepartition.distortion_disjUnion
@[simp]
theorem distortion_disjUnion (h : Disjoint π₁.iUnion π₂.iUnion) :
    (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} {π₁ π₂ : BoxIntegral.TaggedPrepartition I} [Fintype ι] (h : Disjoint π₁.iUnion π₂.iUnion)
Goal: (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion
-- BoxIntegral.TaggedPrepartition.distortion_of_const
theorem distortion_of_const {c} (h₁ : π.boxes.Nonempty) (h₂ : ∀ J ∈ π, Box.distortion J = c) :
    π.distortion = c 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) [Fintype ι] {c : ℝ≥0} (h₁ : π.boxes.Nonempty) (h₂ : ∀ J ∈ π, J.distortion = c)
Goal: π.distortion = c
-- BoxIntegral.TaggedPrepartition.distortion_single
@[simp]
theorem distortion_single (hJ : J ≤ I) (h : x ∈ Box.Icc I) :
    distortion (single I J hJ x h) = J.distortion 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {x : ι → ℝ} [Fintype ι] (hJ : J ≤ I) (h : x ∈ BoxIntegral.Box.Icc I)
Goal: (BoxIntegral.TaggedPrepartition.single I J hJ x h).distortion = J.distortion
-- BoxIntegral.TaggedPrepartition.distortion_filter_le
theorem distortion_filter_le (p : Box ι → Prop) : (π.filter p).distortion ≤ π.distortion 
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (π : BoxIntegral.TaggedPrepartition I) [Fintype ι] (p : BoxIntegral.Box ι → Prop)
Goal: (π.filter p).distortion ≤ π.distortion
-- ContinuousLinearMap.rayleigh_smul
theorem rayleigh_smul (x : E) {c : 𝕜} (hc : c ≠ 0) :
    rayleighQuotient T (c • x) = rayleighQuotient T x 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →L[𝕜] E) (x : E) {c : 𝕜} (hc : c ≠ 0)
Goal: T.rayleighQuotient (c • x) = T.rayleighQuotient x
-- ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere
theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
    rayleighQuotient T '' {0}ᶜ = rayleighQuotient T '' sphere 0 r 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →L[𝕜] E) {r : ℝ} (hr : 0 < r)
Goal: T.rayleighQuotient '' {0}ᶜ = T.rayleighQuotient '' Metric.sphere 0 r
-- ContinuousLinearMap.iSup_rayleigh_eq_iSup_rayleigh_sphere
theorem iSup_rayleigh_eq_iSup_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
    ⨆ x : { x : E // x ≠ 0 }, rayleighQuotient T x =
      ⨆ x : sphere (0 : E) r, rayleighQuotient T x 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →L[𝕜] E) {r : ℝ} (hr : 0 < r)
Goal: ⨆ x, T.rayleighQuotient ↑x = ⨆ x, T.rayleighQuotient ↑x
-- ContinuousLinearMap.iInf_rayleigh_eq_iInf_rayleigh_sphere
theorem iInf_rayleigh_eq_iInf_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
    ⨅ x : { x : E // x ≠ 0 }, rayleighQuotient T x =
      ⨅ x : sphere (0 : E) r, rayleighQuotient T x 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (T : E →L[𝕜] E) {r : ℝ} (hr : 0 < r)
Goal: ⨅ x, T.rayleighQuotient ↑x = ⨅ x, T.rayleighQuotient ↑x
-- LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf
theorem _root_.LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf {T : F →L[ℝ] F}
    (hT : (T : F →ₗ[ℝ] F).IsSymmetric) (x₀ : F) :
    HasStrictFDerivAt T.reApplyInnerSelf (2 • (innerSL ℝ (T x₀))) x₀ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {T : F →L[ℝ] F} (hT : (↑T).IsSymmetric) (x₀ : F)
Goal: HasStrictFDerivAt T.reApplyInnerSelf (2 • (innerSL ℝ) (T x₀)) x₀
-- IsSelfAdjoint.linearly_dependent_of_isLocalExtrOn
theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
    (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
    ∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] {T : F →L[ℝ] F} (hT : IsSelfAdjoint T) {x₀ : F} (hextr : IsLocalExtrOn T.reApplyInnerSelf (Metric.sphere 0 ‖x₀‖) x₀)
Goal: ∃ a b, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0
-- IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn_real
theorem eq_smul_self_of_isLocalExtrOn_real (hT : IsSelfAdjoint T) {x₀ : F}
    (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
    T x₀ = T.rayleighQuotient x₀ • x₀ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] {T : F →L[ℝ] F} (hT : IsSelfAdjoint T) {x₀ : F} (hextr : IsLocalExtrOn T.reApplyInnerSelf (Metric.sphere 0 ‖x₀‖) x₀)
Goal: T x₀ = T.rayleighQuotient x₀ • x₀