mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Finite ι
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
⊢ μ (fundamentalDomain b) ≠ 0

Tactic:
by
  convert! (ZSpan.isAddFundamentalDomain b μ).measure_ne_zero (NeZero.ne μ)
  exact inferInstanceAs <| VAddInvariantMeasure (span ℤ (Set.range b)).toAddSubgroup E μ

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
this : parallelepiped ⇑b \ fundamentalDomain b ⊆ ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))
⊢ μ (parallelepiped ⇑b \ fundamentalDomain b) = 0

Tactic:
refine measure_mono_null this
      (measure_iUnion_null_iff.mpr fun i ↦ Measure.addHaar_affineSubspace μ _ ?_)

Goal state after:
E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
this : parallelepiped ⇑b \ fundamentalDomain b ⊆ ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))
i : ι
⊢ AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))) ≠ ⊤
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case hb
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
r : ℝ
hr : r ≠ 0
x y : E
s t : Set E
⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) ≠ ∞ ∨ μ t ≠ 0

Tactic:
simp only [ENNReal.ofReal_ne_top, true_or, Ne, not_false_iff]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
⊢ ↑⊥ = univ.pi fun i ↦ Ioc (l i) (u i)

Tactic:
rcases not_forall.mp h with ⟨i, hi⟩

Goal state after:
case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
i : ι
hi : ¬l i < u i
⊢ ↑⊥ = univ.pi fun i ↦ Ioc (l i) (u i)
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: ι : Type u_1
I✝ J I : Box ι
s : Set ι
i : ι
⊢ s.piecewise (fun i ↦ (I.lower i + I.upper i) / 2) I.lower i <
    s.piecewise I.upper (fun i ↦ (I.lower i + I.upper i) / 2) i

Tactic:
dsimp only [Set.piecewise]

Goal state after:
ι : Type u_1
I✝ J I : Box ι
s : Set ι
i : ι
⊢ (if i ∈ s then (I.lower i + I.upper i) / 2 else I.lower i) < if i ∈ s then I.upper i else (I.lower i + I.upper i) / 2
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: ι : Type u_1
I : Box ι
s : Set ι
y : ι → ℝ
i : ι
⊢ y i ∈
      Ioc (s.piecewise (fun i ↦ (I.lower i + I.upper i) / 2) I.lower i)
        (s.piecewise I.upper (fun i ↦ (I.lower i + I.upper i) / 2) i) ↔
    y i ∈ Ioc (I.lower i) (I.upper i) ∧ ((I.lower i + I.upper i) / 2 < y i ↔ i ∈ s)

Tactic:
dsimp only [Set.piecewise]

Goal state after:
ι : Type u_1
I : Box ι
s : Set ι
y : ι → ℝ
i : ι
⊢ y i ∈
      Ioc (if i ∈ s then (I.lower i + I.upper i) / 2 else I.lower i)
        (if i ∈ s then I.upper i else (I.lower i + I.upper i) / 2) ↔
    y i ∈ Ioc (I.lower i) (I.upper i) ∧ ((I.lower i + I.upper i) / 2 < y i ↔ i ∈ s)
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: case mp
ι : Type u_1
I : Box ι
π₁ π₂ : Prepartition I
H : π₁ ≤ π₂
J : Box ι
hJ : J ∈ π₁
J' : Box ι
hJ' : J' ∈ π₂
J'' : Box ι
hJ'' : J'' ∈ π₂
Hle : J ≤ J''
x : ι → ℝ
hx : x ∈ ↑J
hx' : x ∈ ↑J'
⊢ J ≤ J'

Tactic:
rwa [π₂.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx)]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
boxes : Finset (WithBot (Box ι))
le_of_mem : ∀ J ∈ boxes, J ≤ ↑I
pairwise_disjoint : (↑boxes).Pairwise Disjoint
H : ∀ J ∈ boxes, J ≠ ⊥ → ∃ J' ∈ π, J ≤ ↑J'
J : Box ι
hJ : ↑J ∈ boxes
⊢ ∃ J' ∈ π, J ≤ J'

Tactic:
by
    simpa only [WithBot.coe_le_coe] using H J hJ WithBot.coe_ne_bot

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
⊢ ∀ (i j : ι), True

Tactic:
intros

Goal state after:
ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i✝ j✝ : ι
⊢ True
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: case neg
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
S_open : ∀ r > 0, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (S r)
x : E
hx : x ∈ K
n : Set F
hn₁ : f ⁻¹' n ∈ 𝓝[D] x
hn₂ : ediam n < ε
r : ℝ≥0∞
r0 : r > 0
hr : eball x r ∩ D ⊆ f ⁻¹' n
this : ∀ (r' : ℝ), ENNReal.ofReal r' ≤ r → ediam (f '' (eball x (ENNReal.ofReal r') ∩ D)) ≤ ε
r_top : ¬r = ∞
r' : ℝ
hr' : 0 < r' ∧ r' < r.toReal
⊢ ∃ i, 0 < i ∧ x ∈ S i

Tactic:
use r', hr'.1, r.toReal, hr'.2, this r.toReal ofReal_toReal_le

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B x✝¹ : Box ι
hI₁ : x✝¹ ∈ ↑(Finset.image (fun ν ↦ box n ν) (admissibleIndex n B))
x✝ : Box ι
hI₂ : x✝ ∈ ↑(Finset.image (fun ν ↦ box n ν) (admissibleIndex n B))
h : x✝¹ ≠ x✝
⊢ Function.onFun Disjoint Box.toSet x✝¹ x✝

Tactic:
by
    obtain ⟨_, _, rfl⟩ := Finset.mem_image.mp hI₁
    obtain ⟨_, _, rfl⟩ := Finset.mem_image.mp hI₂
    exact disjoint.mp fun x ↦ h (congrArg (box n) x)

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case hbc.refine_2
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ε
A : μ (s ∩ Box.Icc I) ≠ ∞
B : μ (s ∩ ↑I) ≠ ∞
F : Set (ι → ℝ)
hFs : F ⊆ s ∩ Box.Icc I
hFc : IsClosed F
hμF : μ ((s ∩ Box.Icc I) \ F) < ↑ε
U : Set (ι → ℝ)
hsU : s ∩ Box.Icc I ⊆ U
hUo : IsOpen U
hUt : μ U < ∞
hμU : μ (U \ (s ∩ Box.Icc I)) < ↑ε
rs : (ι → ℝ) → ↑(Set.Ioi 0)
hrsU : ∀ x ∈ s ∩ Box.Icc I, closedBall x ↑(rs x) ⊆ U
rs' : (ι → ℝ) → ↑(Set.Ioi 0)
hrs'F : ∀ x ∈ Box.Icc I \ s, closedBall x ↑(rs' x) ⊆ Fᶜ
r : (ι → ℝ) → ↑(Set.Ioi 0) := s.piecewise rs rs'
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c ((fun x ↦ r) c) π
hπp : π.IsPartition
t : Set (ι → ℝ) := (π.filter fun x ↦ π.tag x ∈ s).iUnion
htU : t ⊆ U ∩ ↑I
hμt : μ t ≠ ∞
x : ι → ℝ
hxt : x ∉ t
hxs : x ∈ s
hxI : x ∈ ↑I
J : Box ι
hJπ : J ∈ π.toPrepartition
hxJ : x ∈ J
hxF : π.tag J ∉ s
⊢ x ∉ F

Tactic:
refine hrs'F _ ⟨π.tag_mem_Icc J, hxF⟩ ?_

Goal state after:
case hbc.refine_2
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ε
A : μ (s ∩ Box.Icc I) ≠ ∞
B : μ (s ∩ ↑I) ≠ ∞
F : Set (ι → ℝ)
hFs : F ⊆ s ∩ Box.Icc I
hFc : IsClosed F
hμF : μ ((s ∩ Box.Icc I) \ F) < ↑ε
U : Set (ι → ℝ)
hsU : s ∩ Box.Icc I ⊆ U
hUo : IsOpen U
hUt : μ U < ∞
hμU : μ (U \ (s ∩ Box.Icc I)) < ↑ε
rs : (ι → ℝ) → ↑(Set.Ioi 0)
hrsU : ∀ x ∈ s ∩ Box.Icc I, closedBall x ↑(rs x) ⊆ U
rs' : (ι → ℝ) → ↑(Set.Ioi 0)
hrs'F : ∀ x ∈ Box.Icc I \ s, closedBall x ↑(rs' x) ⊆ Fᶜ
r : (ι → ℝ) → ↑(Set.Ioi 0) := s.piecewise rs rs'
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c ((fun x ↦ r) c) π
hπp : π.IsPartition
t : Set (ι → ℝ) := (π.filter fun x ↦ π.tag x ∈ s).iUnion
htU : t ⊆ U ∩ ↑I
hμt : μ t ≠ ∞
x : ι → ℝ
hxt : x ∉ t
hxs : x ∈ s
hxI : x ∈ ↑I
J : Box ι
hJπ : J ∈ π.toPrepartition
hxJ : x ∈ J
hxF : π.tag J ∉ s
⊢ x ∈ closedBall (π.tag J) ↑(rs' (π.tag J))
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[μ.restrict ↑I] 0
hl : l.bRiemann = false
ε : ℝ
ε0 : ε > 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π → π.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π) 0 ≤ ε

Tactic:
lift ε to ℝ≥0 using ε0.lt.le

Goal state after:
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[μ.restrict ↑I] 0
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : ↑ε > 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π → π.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π) 0 ≤ ↑ε
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : SimpleFunc (ι → ℝ) E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
⊢ HasIntegral I l (⇑f) μ.toBoxAdditive.toSMul (integral (μ.restrict ↑I) f)

Tactic:
by
  induction f using MeasureTheory.SimpleFunc.induction with
  | @const y s hs =>
    simpa [hs] using! BoxIntegral.hasIntegralIndicatorConst l hl hs I y μ
  | @add f g _ hfi hgi =>
    borelize E; have := Fact.mk (I.measure_coe_lt_top μ)
    rw [integral_add]
    exacts [hfi.add hgi, integrable_iff.2 fun _ _ => measure_lt_top _ _,
      integrable_iff.2 fun _ _ => measure_lt_top _ _]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this✝ : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
this : Tendsto (fun n ↦ ∫⁻ (x : ι → ℝ) in ↑I, ↑‖(f n) x - g x‖₊ ∂μ) atTop (𝓝 0)
⊢ ∃ N, ∫ (x : ι → ℝ) in ↑I, ‖(f N) x - g x‖ ∂μ ≤ ↑ε

Tactic:
refine (this.eventually (ge_mem_nhds ε0')).exists.imp fun N hN => ?_

Goal state after:
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this✝ : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
this : Tendsto (fun n ↦ ∫⁻ (x : ι → ℝ) in ↑I, ↑‖(f n) x - g x‖₊ ∂μ) atTop (𝓝 0)
N : ℕ
hN : ∫⁻ (x : ι → ℝ) in ↑I, ↑‖(f N) x - g x‖₊ ∂μ ≤ ↑ε
⊢ ∫ (x : ι → ℝ) in ↑I, ‖(f N) x - g x‖ ∂μ ≤ ↑ε
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
⊢ ∃ N, ∫ (x : ι → ℝ) in ↑I, ‖(f N) x - g x‖ ∂μ ≤ ↑ε

Tactic:
have : Tendsto (fun n => ∫⁻ x in I, ‖f n x - g x‖₊ ∂μ) atTop (𝓝 0) :=
      SimpleFunc.tendsto_approxOn_range_L1_enorm hg.measurable hgi
    refine (this.eventually (ge_mem_nhds ε0')).exists.imp fun N hN => ?_
    exact integral_coe_le_of_lintegral_coe_le hN

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
h : Integrable I l f vol
π₀ : Prepartition I
⊢ Filter.map (fun p ↦ (integralSum f vol p.1, integralSum f vol p.2)) (toFilteriUnion I π₀ ×ˢ toFilteriUnion I π₀) ≤ 𝓤 F

Tactic:
← toFilter_inf_iUnion_eq,

Goal state after:
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
h : Integrable I l f vol
π₀ : Prepartition I
⊢ Filter.map (fun p ↦ (integralSum f vol p.1, integralSum f vol p.2))
      ((l.toFilter I ⊓ 𝓟 {π | π.iUnion = π₀.iUnion}) ×ˢ (l.toFilter I ⊓ 𝓟 {π | π.iUnion = π₀.iUnion})) ≤
    𝓤 F
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case refine_2
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
      (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) ≤
    ↑ε

Tactic:
← π.sum_fiberwise fun J => Nx (π.tag J),

Goal state after:
case refine_2
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist
      (∑ y ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes,
        ∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = y).boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
      (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) ≤
    ↑ε
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case refine_2
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist
      (∑ y ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes,
        ∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = y).boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
      (∑ y ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes,
        ∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = y).boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) ≤
    ↑c✝

Tactic:
refine
      (dist_sum_sum_le_of_le _ fun n hn => ?_).trans
        (sum_le_hasSum _ (fun n _ => (δ n).2) (NNReal.hasSum_coe.2 hδc))

Goal state after:
case refine_2
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
n : ℕ
hn : n ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes
⊢ dist (∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = n).boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
      (∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = n).boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) ≤
    ↑(δ n)
Module: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

Goal state before:
pp: case fst
U : Type u_4
V : Type u_5
inst✝⁹ : NormedAddCommGroup U
inst✝⁸ : InnerProductSpace ℝ U
inst✝⁷ : MeasurableSpace U
inst✝⁶ : BorelSpace U
inst✝⁵ : FiniteDimensional ℝ U
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
this : MeasurePreserving (⇑(volumePreservingSymmMeasurableEquivToLpProdAux U V)) volume volume
uv : WithLp 2 (U × V)
⊢ ((MeasurableEquiv.toLp 2 (U × V)).symm uv).1 = ((volumePreservingSymmMeasurableEquivToLpProdAux U V) uv).1

Tactic:
simp [volumePreservingSymmMeasurableEquivToLpProdAux, MeasurableEquiv.coe_sumPiEquivProdPi,
      MeasurableEquiv.prodCongr]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

Goal state before:
pp: case snd
U : Type u_4
V : Type u_5
inst✝⁹ : NormedAddCommGroup U
inst✝⁸ : InnerProductSpace ℝ U
inst✝⁷ : MeasurableSpace U
inst✝⁶ : BorelSpace U
inst✝⁵ : FiniteDimensional ℝ U
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
this : MeasurePreserving (⇑(volumePreservingSymmMeasurableEquivToLpProdAux U V)) volume volume
uv : WithLp 2 (U × V)
⊢ ((MeasurableEquiv.toLp 2 (U × V)).symm uv).2 = ((volumePreservingSymmMeasurableEquivToLpProdAux U V) uv).2

Tactic:
simp [volumePreservingSymmMeasurableEquivToLpProdAux, MeasurableEquiv.coe_sumPiEquivProdPi,
      MeasurableEquiv.prodCongr]

Goal state after:
Module: Mathlib.Analysis.SumOverResidueClass

Goal state before:
pp: f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
hs : Summable f
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε
⊢ False

Tactic:
obtain ⟨N, hN⟩ := this (|f n|) (abs_pos_of_neg hn)

Goal state after:
f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
hs : Summable f
this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε
N : ℕ
hN : ∀ n_1 ≥ N, |f n_1| < |f n|
⊢ False
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inl
p : ℝ
hp : 0 ≤ p
⊢ (Summable fun k ↦ (2 * (2 ^ p)⁻¹) ^ k) ↔ 1 < p

Tactic:
summable_geometric_iff_norm_lt_one

Goal state after:
case inl
p : ℝ
hp : 0 ≤ p
⊢ ‖2 * (2 ^ p)⁻¹‖ < 1 ↔ 1 < p
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case hf
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n i : ℕ
_hi : i ∈ Ioc k (max (k + 1) n)
⊢ i ∉ Ioo k n → 0 ≤ (↑i ^ 2)⁻¹

Tactic:
_hident

Goal state after:
case hf
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n i : ℕ
_hi : i ∈ Ioc k (max (k + 1) n)
_hident : i ∉ Ioo k n
⊢ 0 ≤ (↑i ^ 2)⁻¹
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case refine_4
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ x • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ⊆
    y • ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1}

Tactic:
← image_smul_set

Goal state after:
case refine_4
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' (x • {x | x ∈ X ∧ F x ≤ 1}) ⊆
    ⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' (y • {x | x ∈ X ∧ F x ≤ 1})