mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case neg
ι✝ : Type u_1
inst✝¹ : Fintype ι✝
I✝ : Box ι✝
c₁✝ c₂✝ : ℝ≥0
l✝ : IntegrationParams
r₁✝ r₂✝ : (ι✝ → ℝ) → ↑(Set.Ioi 0)
π₁✝ π₂✝ : TaggedPrepartition I✝
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c₁ c₂ : ℝ≥0
l : IntegrationParams
r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ r₁ π₁
h₂ : l.MemBaseSet I c₂ r₂ π₂
hU : π₁.iUnion = π₂.iUnion
hc : c₁ ≤ c₂
hD : ¬l.bDistortion = true
⊢ ∃ π,
    π.iUnion = ↑I \ π₁.iUnion ∧ (l.bDistortion = true → π.distortion ≤ c₁) ∧ (l.bDistortion = true → π.distortion ≤ c₂)

Tactic:
·

Goal state after:
case neg
ι✝ : Type u_1
inst✝¹ : Fintype ι✝
I✝ : Box ι✝
c₁✝ c₂✝ : ℝ≥0
l✝ : IntegrationParams
r₁✝ r₂✝ : (ι✝ → ℝ) → ↑(Set.Ioi 0)
π₁✝ π₂✝ : TaggedPrepartition I✝
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c₁ c₂ : ℝ≥0
l : IntegrationParams
r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ r₁ π₁
h₂ : l.MemBaseSet I c₂ r₂ π₂
hU : π₁.iUnion = π₂.iUnion
hc : c₁ ≤ c₂
hD : ¬l.bDistortion = true
⊢ ∃ π,
    π.iUnion = ↑I \ π₁.iUnion ∧ (l.bDistortion = true → π.distortion ≤ c₁) ∧ (l.bDistortion = true → π.distortion ≤ c₂)
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_1
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := π.filter fun J ↦ ¬p J
this : Disjoint π₁.iUnion π₂.iUnion
⊢ (π₁.disjUnion π₂.toPrepartition this).iUnion = ↑I \ (π.filter p).iUnion

pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := π.filter fun J ↦ ¬p J
this : Disjoint π₁.iUnion π₂.iUnion
⊢ (π₁.disjUnion π₂.toPrepartition this).distortion ≤ c

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := π.filter fun J ↦ ¬p J
this : Disjoint π₁.iUnion π₂.iUnion
⊢ (π₁.disjUnion π₂.toPrepartition this).iUnion = ↑I \ (π.filter p).iUnion
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_1.mp
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := ⋯
this : Disjoint π₁.iUnion π₂.iUnion
h : (π.filter p).iUnion ⊆ π.iUnion
x : ι → ℝ
⊢ x ∈ ↑I \ π.iUnion ∪ π.iUnion \ (π.filter p).iUnion → x ∈ ↑I \ (π.filter p).iUnion

pp: case refine_1.mpr
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := ⋯
this : Disjoint π₁.iUnion π₂.iUnion
h : (π.filter p).iUnion ⊆ π.iUnion
x : ι → ℝ
⊢ x ∈ ↑I \ (π.filter p).iUnion → x ∈ ↑I \ π.iUnion ∪ π.iUnion \ (π.filter p).iUnion

Tactic:
·

Goal state after:
case refine_1.mp
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := ⋯
this : Disjoint π₁.iUnion π₂.iUnion
h : (π.filter p).iUnion ⊆ π.iUnion
x : ι → ℝ
⊢ x ∈ ↑I \ π.iUnion ∪ π.iUnion \ (π.filter p).iUnion → x ∈ ↑I \ (π.filter p).iUnion
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_1.mpr
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := ⋯
this : Disjoint π₁.iUnion π₂.iUnion
h : (π.filter p).iUnion ⊆ π.iUnion
x : ι → ℝ
⊢ x ∈ ↑I \ (π.filter p).iUnion → x ∈ ↑I \ π.iUnion ∪ π.iUnion \ (π.filter p).iUnion

Tactic:
·

Goal state after:
case refine_1.mpr
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := ⋯
this : Disjoint π₁.iUnion π₂.iUnion
h : (π.filter p).iUnion ⊆ π.iUnion
x : ι → ℝ
⊢ x ∈ ↑I \ (π.filter p).iUnion → x ∈ ↑I \ π.iUnion ∪ π.iUnion \ (π.filter p).iUnion
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := π.filter fun J ↦ ¬p J
this : Disjoint π₁.iUnion π₂.iUnion
⊢ (π₁.disjUnion π₂.toPrepartition this).distortion ≤ c

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := π.filter fun J ↦ ¬p J
this : Disjoint π₁.iUnion π₂.iUnion
⊢ (π₁.disjUnion π₂.toPrepartition this).distortion ≤ c
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_1
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
r : (ι → ℝ) → ↑(Set.Ioi 0)
π : Prepartition I
πi : (J : Box ι) → 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
hD : l.bDistortion = true
⊢ (π.biUnionTagged πi).distortion ≤ c

pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
r : (ι → ℝ) → ↑(Set.Ioi 0)
π : Prepartition I
πi : (J : Box ι) → 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
hD : l.bDistortion = true
⊢ ∃ π', π'.iUnion = ↑I \ (π.biUnionTagged πi).iUnion ∧ π'.distortion ≤ c

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
r : (ι → ℝ) → ↑(Set.Ioi 0)
π : Prepartition I
πi : (J : Box ι) → 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
hD : l.bDistortion = true
⊢ (π.biUnionTagged πi).distortion ≤ c
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
L'' : Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
this : Tendsto (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) (𝓝[>] 0) (𝓝 1)
r : ℝ
rpos : 0 < r
⊢ μ ({x} + r • t) ≠ 0

pp: case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
L'' : Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
this : Tendsto (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) (𝓝[>] 0) (𝓝 1)
r : ℝ
rpos : 0 < r
⊢ μ ({x} + r • t) ≠ ∞

Tactic:
·

Goal state after:
case refine_1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
L'' : Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
this : Tendsto (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) (𝓝[>] 0) (𝓝 1)
r : ℝ
rpos : 0 < r
⊢ μ ({x} + r • t) ≠ 0
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
r : (ι → ℝ) → ↑(Set.Ioi 0)
π : Prepartition I
πi : (J : Box ι) → 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
hD : l.bDistortion = true
⊢ ∃ π', π'.iUnion = ↑I \ (π.biUnionTagged πi).iUnion ∧ π'.distortion ≤ c

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
r : (ι → ℝ) → ↑(Set.Ioi 0)
π : Prepartition I
πi : (J : Box ι) → 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
hD : l.bDistortion = true
⊢ ∃ π', π'.iUnion = ↑I \ (π.biUnionTagged πi).iUnion ∧ π'.distortion ≤ c
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case e'_2
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι'
a : OrthonormalBasis ι' 𝕜 E
b : OrthonormalBasis ι 𝕜 E
i j : ι
⊢ ((a.toBasis.toMatrix ⇑b)ᴴ * a.toBasis.toMatrix ⇑b) i j = ⟪a.repr (b i), a.repr (b j)⟫

pp: case e'_3
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι'
a : OrthonormalBasis ι' 𝕜 E
b : OrthonormalBasis ι 𝕜 E
i j : ι
⊢ 1 i j = ⟪b i, b j⟫

Tactic:
·

Goal state after:
case e'_2
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι'
a : OrthonormalBasis ι' 𝕜 E
b : OrthonormalBasis ι 𝕜 E
i j : ι
⊢ ((a.toBasis.toMatrix ⇑b)ᴴ * a.toBasis.toMatrix ⇑b) i j = ⟪a.repr (b i), a.repr (b j)⟫
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case e'_3
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι'
a : OrthonormalBasis ι' 𝕜 E
b : OrthonormalBasis ι 𝕜 E
i j : ι
⊢ 1 i j = ⟪b i, b j⟫

Tactic:
·

Goal state after:
case e'_3
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι'
a : OrthonormalBasis ι' 𝕜 E
b : OrthonormalBasis ι 𝕜 E
i j : ι
⊢ 1 i j = ⟪b i, b j⟫
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
L'' : Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
this : Tendsto (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) (𝓝[>] 0) (𝓝 1)
r : ℝ
rpos : 0 < r
⊢ μ ({x} + r • t) ≠ ∞

Tactic:
·

Goal state after:
case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
L'' : Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
this : Tendsto (fun a ↦ μ ({x} + a • t) / μ ({x} + a • t) - μ (sᶜ ∩ ({x} + a • t)) / μ ({x} + a • t)) (𝓝[>] 0) (𝓝 1)
r : ℝ
rpos : 0 < r
⊢ μ ({x} + r • t) ≠ ∞
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a b : OrthonormalBasis ι ℝ F
⊢ a.toBasis.det ⇑b = 1 ∨ a.toBasis.det ⇑b = -1

Tactic:
rw [← sq_eq_one_iff]

Goal state after:
ι : Type u_1
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a b : OrthonormalBasis ι ℝ F
⊢ a.toBasis.det ⇑b ^ 2 = 1
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case hgf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0,
    μ (s ∩ closedBall x b) / μ (closedBall x b) ≤ μ (toMeasurable μ s ∩ closedBall x b) / μ (closedBall x b)

pp: case hfh
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (toMeasurable μ s ∩ closedBall x b) / μ (closedBall x b) ≤ 1

Tactic:
·

Goal state after:
case hgf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0,
    μ (s ∩ closedBall x b) / μ (closedBall x b) ≤ μ (toMeasurable μ s ∩ closedBall x b) / μ (closedBall x b)
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_1
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
r : (ι → ℝ) → ↑(Set.Ioi 0)
hr : l.RCond r
π : TaggedPrepartition I
hπ : l.MemBaseSet I c r π ∧ π.iUnion = ↑I
x✝ : l.bDistortion = true
⊢ ∃ π', π'.iUnion = ↑J \ ((embedBox I J h) π).iUnion ∧ π'.distortion ≤ max c π₀.compl.distortion

pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
r : (ι → ℝ) → ↑(Set.Ioi 0)
hr : l.RCond r
π : TaggedPrepartition I
hπ : l.MemBaseSet I c r π ∧ π.iUnion = ↑I
⊢ ((embedBox I J h) π).iUnion = π₀.iUnion

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
r : (ι → ℝ) → ↑(Set.Ioi 0)
hr : l.RCond r
π : TaggedPrepartition I
hπ : l.MemBaseSet I c r π ∧ π.iUnion = ↑I
x✝ : l.bDistortion = true
⊢ ∃ π', π'.iUnion = ↑J \ ((embedBox I J h) π).iUnion ∧ π'.distortion ≤ max c π₀.compl.distortion
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case hfh
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (toMeasurable μ s ∩ closedBall x b) / μ (closedBall x b) ≤ 1

Tactic:
·

Goal state after:
case hfh
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (toMeasurable μ s ∩ closedBall x b) / μ (closedBall x b) ≤ 1
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: case refine_2
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
r : (ι → ℝ) → ↑(Set.Ioi 0)
hr : l.RCond r
π : TaggedPrepartition I
hπ : l.MemBaseSet I c r π ∧ π.iUnion = ↑I
⊢ ((embedBox I J h) π).iUnion = π₀.iUnion

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
r : (ι → ℝ) → ↑(Set.Ioi 0)
hr : l.RCond r
π : TaggedPrepartition I
hπ : l.MemBaseSet I c r π ∧ π.iUnion = ↑I
⊢ ((embedBox I J h) π).iUnion = π₀.iUnion
Module: Mathlib.Analysis.BoxIntegral.Partition.Additive

Goal state before:
pp: case pos
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I✝ : Box ι
i : ι
inst✝ : Finite ι
f : Box ι → M
I₀ : WithTop (Box ι)
hf :
  ∀ (I : Box ι),
    ↑I ≤ I₀ →
      ∀ {i : ι} {x : ℝ},
        x ∈ Set.Ioo (I.lower i) (I.upper i) →
          Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) = f I
I : Box ι
hI : ↑I ≤ I₀
a : ι × ℝ
s : Finset (ι × ℝ)
a✝ : a ∉ s
ihs : ∑ J ∈ (splitMany I s).boxes, f J = f I
J' : Box ι
hJ' : J' ∈ (splitMany I s).boxes
h : a.2 ∈ Set.Ioo (J'.lower a.1) (J'.upper a.1)
⊢ ∑ J' ∈ (split J' a.1 a.2).boxes, f J' = f J'

pp: case neg
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I✝ : Box ι
i : ι
inst✝ : Finite ι
f : Box ι → M
I₀ : WithTop (Box ι)
hf :
  ∀ (I : Box ι),
    ↑I ≤ I₀ →
      ∀ {i : ι} {x : ℝ},
        x ∈ Set.Ioo (I.lower i) (I.upper i) →
          Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) = f I
I : Box ι
hI : ↑I ≤ I₀
a : ι × ℝ
s : Finset (ι × ℝ)
a✝ : a ∉ s
ihs : ∑ J ∈ (splitMany I s).boxes, f J = f I
J' : Box ι
hJ' : J' ∈ (splitMany I s).boxes
h : a.2 ∉ Set.Ioo (J'.lower a.1) (J'.upper a.1)
⊢ ∑ J' ∈ (split J' a.1 a.2).boxes, f J' = f J'

Tactic:
·

Goal state after:
case pos
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I✝ : Box ι
i : ι
inst✝ : Finite ι
f : Box ι → M
I₀ : WithTop (Box ι)
hf :
  ∀ (I : Box ι),
    ↑I ≤ I₀ →
      ∀ {i : ι} {x : ℝ},
        x ∈ Set.Ioo (I.lower i) (I.upper i) →
          Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) = f I
I : Box ι
hI : ↑I ≤ I₀
a : ι × ℝ
s : Finset (ι × ℝ)
a✝ : a ∉ s
ihs : ∑ J ∈ (splitMany I s).boxes, f J = f I
J' : Box ι
hJ' : J' ∈ (splitMany I s).boxes
h : a.2 ∈ Set.Ioo (J'.lower a.1) (J'.upper a.1)
⊢ ∑ J' ∈ (split J' a.1 a.2).boxes, f J' = f J'
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case refine_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ (span 𝕜 (range Subtype.val))ᗮ = ⊥

pp: case refine_2
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ v ⊆ ↑u

pp: case refine_3
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ ⇑(OrthonormalBasis.mkOfOrthogonalEqBot hu ?refine_1) = Subtype.val

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
hu₀_finite : u₀.Finite
u : Finset E := hu₀_finite.toFinset
fu : ↥u ≃ ↑u₀ := hu₀_finite.subtypeEquivToFinset.symm
hu : Orthonormal 𝕜 Subtype.val
⊢ (span 𝕜 (range Subtype.val))ᗮ = ⊥