mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
K : Set B
hK : IsCompact K
hf : ∀ x ∉ K, f x = 0
x : C
hx : x ∉ ⇑ψ '' K
⊢ ∫ (a : A), (H.pullback f (Function.invFun (⇑ψ) x)) a ∂μA = 0

Tactic:
suffices ∀ a : A, f (Function.invFun ψ x * φ a) = 0 by simp [this, pullback_def]

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
K : Set B
hK : IsCompact K
hf : ∀ x ∉ K, f x = 0
x : C
hx : x ∉ ⇑ψ '' K
⊢ ∀ (a : A), f (Function.invFun (⇑ψ) x * φ a) = 0
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_2
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
eq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
eq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ^ p) ^ (1 / p) = 0 ↔ x = 0
nm_zero : (∑ i, ‖0 i‖ ^ p) ^ (1 / p) = 0
nm_neg : ∀ (x : ι → ℂ), (∑ i, ‖(-x) i‖ ^ p) ^ (1 / p) = (∑ i, ‖x i‖ ^ p) ^ (1 / p)
nm_add : ∀ (x y : ι → ℂ), (∑ i, ‖(x + y) i‖ ^ p) ^ (1 / p) ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p) + (∑ i, ‖y i‖ ^ p) ^ (1 / p)
nm_smul : ∀ (r : ℝ) (x : ι → ℂ), (∑ i, ‖(r • x) i‖ ^ p) ^ (1 / p) ≤ ‖r‖ * (∑ i, ‖x i‖ ^ p) ^ (1 / p)
x✝ : ι → ℂ
⊢ ∑ i, ‖x✝ i‖ ^ p < 1 ↔ (∑ i, ‖x✝ i‖ ^ p) ^ (1 / p) < 1

Tactic:
rw [rpow_lt_one_iff' _ (one_div_pos.mpr h₁)]
    exact Finset.sum_nonneg' (fun _ => rpow_nonneg (norm_nonneg _) _)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: case inr
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)

Tactic:
have h_le : (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖})
      ≤ fun r ↦ ∑ i, ⨆ μ ∈ S, μ {x | r / √(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖} := by
    intro r
    calc ⨆ μ ∈ S, μ {x | r < ‖x‖}
    _ ≤ ⨆ μ ∈ S, μ (⋃ i, {x : E | r / √(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}) := by
      gcongr with μ hμS
      intro x hx
      simp only [Set.mem_ofPred_eq, Set.mem_iUnion] at hx ⊢
      have hx' : r < √(Fintype.card ι) * ⨆ i, ‖⟪b i, x⟫_𝕜‖ :=
        hx.trans_le (b.norm_le_card_mul_iSup_norm_inner x)
      rw [← div_lt_iff₀' (by positivity)] at hx'
      by_contra! h_le
      exact lt_irrefl (r / √(Fintype.card ι)) (hx'.trans_le (ciSup_le h_le))
    _ ≤ ⨆ μ ∈ S, ∑ i, μ {x : E | r / √(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖} := by
      gcongr with μ hμS
      exact measure_iUnion_fintype_le μ _
    _ ≤ ∑ i, ⨆ μ ∈ S, μ {x | r / √(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖} := by
      refine iSup_le fun μ ↦ (iSup_le fun hμS ↦ ?_)
      gcongr with i
      exact le_biSup (fun μ ↦ μ {x | r / √(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}) hμS

Goal state after:
case inr
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
h_le : (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) ≤ fun r ↦ ∑ i, ⨆ μ ∈ S, μ {x | r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
⊢ IsTightMeasureSet (Set.range μ)

Tactic:
refine isTightMeasureSet_of_inner_tendsto 𝕜 fun z ↦ ?_
  simp_rw [iSup_range]
  refine Nat.tendsto_iSup_of_tendsto_limsup (fun n ↦ ?_) (h z) (fun n u v huv ↦ by gcongr)
  have h_tight : IsTightMeasureSet {(μ n).map (fun x ↦ ⟪z, x⟫_𝕜)} := isTightMeasureSet_singleton
  rw [isTightMeasureSet_iff_tendsto_measure_norm_gt] at h_tight
  have h_map r : (μ n).map (fun x ↦ ⟪z, x⟫_𝕜) {x | r < ‖x‖} = μ n {x | r < ‖⟪z, x⟫_𝕜‖} := by
    rw [Measure.map_apply (by fun_prop)]
    · simp
    · exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop)
  simpa [h_map] using h_tight

Goal state after:
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
⊢ IsTightMeasureSet (Set.range μ)

Tactic:
refine isTightMeasureSet_range_of_tendsto_limsup_inner 𝕜 fun y ↦ ?_
  by_cases hy : y = 0
  · simp only [hy, inner_zero_left]
    refine (tendsto_congr' ?_).mpr tendsto_const_nhds
    filter_upwards [eventually_ge_atTop 0] with r hr
    simp [not_lt.mpr hr]
  have h' : Tendsto (fun r : ℝ ↦ limsup (fun n ↦ μ n {x | ‖y‖⁻¹ * r < ‖⟪(‖y‖⁻¹ : 𝕜) • y, x⟫_𝕜‖})
      atTop) atTop (𝓝 0) := by
    specialize h ((‖y‖⁻¹ : 𝕜) • y) ?_
    · simp only [norm_smul, norm_inv, norm_algebraMap', Real.norm_eq_abs, abs_norm]
      rw [inv_mul_cancel₀ (by positivity)]
    exact h.comp <| (tendsto_const_mul_atTop_of_pos (by positivity)).mpr tendsto_id
  convert! h' using 7 with r n x
  rw [inner_smul_left]
  simp only [map_inv₀, RCLike.conj_ofReal, norm_mul, norm_inv, norm_algebraMap', norm_norm]
  rw [mul_lt_mul_iff_right₀]
  positivity

Goal state after:
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
ε : ℝ≥0
hε : 0 < ε
h : preVariationFun f s ≠ ∞
ε' : ℝ≥0 := min ε (preVariationFun f s).toNNReal
⊢ ∃ P, preVariationFun f s ≤ ∑ p ∈ P.parts, f ↑p + ↑ε

Tactic:
have hε' : ε' ≤ preVariationFun f s := by simp_all [ε']

Goal state after:
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
ε : ℝ≥0
hε : 0 < ε
h : preVariationFun f s ≠ ∞
ε' : ℝ≥0 := min ε (preVariationFun f s).toNNReal
hε' : ↑ε' ≤ preVariationFun f s
⊢ ∃ P, preVariationFun f s ≤ ∑ p ∈ P.parts, f ↑p + ↑ε
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case pos
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : NormalSpace E
this✝ : (↑μ).InnerRegular
g : E → ℝ
g_cont : Continuous[inst✝², _] g
gK : EqOn g 0 (K n)
g_nonneg : 0 ≤ g
g_le : g ≤ 1
this : Tendsto (fun ρ ↦ ∫ (x : E), g x ∂↑ρ) (↑f) (𝓝 (∫ (x : E), g x ∂↑μ))
ρ : FiniteMeasure E
hρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
x : E
hx : x ∈ (K n)ᶜ
⊢ g x ≤ (K n)ᶜ.indicator 1 x

Tactic:
simpa [hx] using g_le x

Goal state after:
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
n : ℕ
hn : ¬n = 0
ε' : ℝ≥0
hε' : 0 < ε'
hsnetop : preVariationFun f (⋃ i, s i) < ∞
ε : ℝ≥0 := ε' / ↑n
hε : 0 < ε
hs'' : ∀ (i : ℕ), preVariationFun f (s i) ≠ ∞
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
hP : ∀ (i : ℕ), preVariationFun f (s i) ≤ ∑ p ∈ (P i).parts, f ↑p + ↑ε
⊢ ∑ x ∈ Finset.range n, ∑ p ∈ (P x).parts, f ↑p + ↑(∑ i ∈ Finset.range n, ε) =
    ∑ x ∈ Finset.range n, ∑ p ∈ (P x).parts, f ↑p + ↑ε'

Tactic:
simp [show n * ε = ε' by field]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case hbc.refine_1.refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
h_le_Ioc : ∀ x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - f (x • z)‖ ≤ ε / 4
hf_meas : Measurable f
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)), ‖‖1 - f (a • z)‖‖ ≤ ε / 4

Tactic:
simpa using ae_restrict_of_forall_mem measurableSet_Ioc h_le_Ioc

Goal state after:
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case iUnion
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
⊢ ∃ t ∈ 𝒜, μ.real ((⋃ i, f i) ∆ t) < ε

Tactic:
choose g g_mem hg using fun n ↦ (hf n).2 (ε / (2 * (N + 1))) (div_pos ε_pos (by linarith))

Goal state after:
case iUnion
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (f n ∆ t) < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n ↦ (μ (accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), μ.real (f n ∆ g n) < ε / (2 * (↑N + 1))
⊢ ∃ t ∈ 𝒜, μ.real ((⋃ i, f i) ∆ t) < ε
Module: Mathlib.MeasureTheory.Measure.SubFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν ξ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ - ν ≤ ξ
s : Set α
hs : IsHahnDecomposition μ ν s
h_le_s : μ.restrict s ≤ ξ.restrict s + ν.restrict s
⊢ μ.restrict sᶜ - ν.restrict sᶜ ≤ ξ.restrict sᶜ

Tactic:
← restrict_sub_eq_restrict_sub_restrict hs.measurableSet.compl

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν ξ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ - ν ≤ ξ
s : Set α
hs : IsHahnDecomposition μ ν s
h_le_s : μ.restrict s ≤ ξ.restrict s + ν.restrict s
⊢ (μ - ν).restrict sᶜ ≤ ξ.restrict sᶜ
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : NeZero μ
⊢ IsProbabilityMeasure μ.toFinite

Tactic:
apply ProbabilityTheory.cond_isProbabilityMeasure

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : NeZero μ
⊢ μ.toFiniteAux univ ≠ 0
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: case refine_1
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ t ∩ s ⊆ ⋃ i, (fun i ↦ f i ∩ s) i

Tactic:
rw [← iUnion_inter]

Goal state after:
case refine_1
α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
t : Set α
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_subset : t ⊆ ⋃ i, f i
A : ℕ → Finset (Set α) := fun i ↦ hC.disjointOfDiff ⋯ ⋯
h_diff_eq_sUnion : ∀ (i : ℕ), f i \ s = ⋃₀ ↑(A i)
h_m_eq : ∀ (i : ℕ), m (f i) = m (f i ∩ s) + ∑ u ∈ A i, m u
⊢ t ∩ s ⊆ (⋃ i, f i) ∩ s
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.inl
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this✝¹ : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = 0
this✝ : (μ.restrict μ.sigmaFiniteSet) s ≠ ∞
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε
this : μ (s ∆ (t ∩ μ.sigmaFiniteSet) \ μ.sigmaFiniteSet) = 0
⊢ μ (s ∆ (t ∩ μ.sigmaFiniteSet) ∩ μ.sigmaFiniteSet) + 0 < ENNReal.ofReal ε

Tactic:
add_zero,

Goal state after:
case refine_2.inl
X : Type u_1
E : Type u_2
m : MeasurableSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
p : ℝ≥0∞
one_le_p : Fact (1 ≤ p)
p_ne_top : Fact (p ≠ ∞)
𝒜✝ : Set (Set X)
inst✝¹ : CountablyGenerated X
inst✝ : SFinite μ
this✝¹ : IsSeparable (μ.restrict μ.sigmaFiniteSet)
𝒜 : Set (Set X)
count_𝒜 : 𝒜.Countable
h𝒜 : (μ.restrict μ.sigmaFiniteSet).MeasureDense 𝒜
ℬ : Set (Set X) := {x | ∃ s ∈ 𝒜, s ∩ μ.sigmaFiniteSet = x}
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
hs : (μ.restrict μ.sigmaFiniteSetᶜ) s = 0
this✝ : (μ.restrict μ.sigmaFiniteSet) s ≠ ∞
t : Set X
t_mem : t ∈ 𝒜
ht : (μ.restrict μ.sigmaFiniteSet) (s ∆ t) < ENNReal.ofReal ε
this : μ (s ∆ (t ∩ μ.sigmaFiniteSet) \ μ.sigmaFiniteSet) = 0
⊢ μ (s ∆ (t ∩ μ.sigmaFiniteSet) ∩ μ.sigmaFiniteSet) < ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.SpecificCodomains.Pi

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
E : ι → Type u_3
inst✝ : (i : ι) → NormedAddCommGroup (E i)
f : X → (i : ι) → E i
hf : ∀ (i : ι), MemLp (fun x ↦ f x i) p μ
this : f = ∑ i, Pi.single i ∘ fun x ↦ f x i
i : ι
x✝ : i ∈ Finset.univ
⊢ MemLp (Pi.single i ∘ fun x ↦ f x i) p μ

Tactic:
exact (Isometry.single i).lipschitz.comp_memLp (by simp) (hf i)

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : μ.ennrealVariation s = 0
⊢ μ s = 0

Tactic:
by_cases hsm : MeasurableSet s

Goal state after:
case pos
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : μ.ennrealVariation s = 0
hsm : MeasurableSet s
⊢ μ s = 0

case neg
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : μ.ennrealVariation s = 0
hsm : ¬MeasurableSet s
⊢ μ s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case neg
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : ¬MeasurableSet s
⊢ (μ.restrict s).variation ≤ μ.variation.restrict s

Tactic:
· simp [restrict_not_measurable _ hs, Measure.zero_le]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: X : Type u_2
E : Type u_3
F : Type u_4
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : VectorMeasure X F
f : X → E
s t : Set X
hs : MeasurableSet s
hts : t ⊆ s
h : μ.IntegrableOn f s
⊢ μ.IntegrableOn f t

Tactic:
by
  by_cases ht : MeasurableSet t; swap
  · simp [VectorMeasure.IntegrableOn, restrict_not_measurable _ ht]
  apply Integrable.mono_measure h
  simp [variation_restrict, hs, ht, Measure.restrict_mono hts le_rfl]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case neg
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
hs : MeasurableSet s
hfi : ¬μ.IntegrableOn f s
⊢ ∫ᵛ (x : X), s.indicator f x ∂[B; μ] = ∫ᵛ (x : X) in s, f x ∂[B; μ]

Tactic:
rw [integral_undef hfi, integral_undef]

Goal state after:
case neg
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
hs : MeasurableSet s
hfi : ¬μ.IntegrableOn f s
⊢ ¬μ.Integrable (s.indicator f)
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedAddCommGroup G
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
β : Type u_7
inst✝ : MeasurableSpace β
φ : X → β
hφ : Measurable φ
f : β → E
s : Set β
hs : MeasurableSet s
hfm : AEStronglyMeasurable f (Measure.map φ (μ.restrict (φ ⁻¹' s)).variation)
hfi' : μ.Integrable (f ∘ φ)
⊢ ∫ᵛ (y : β) in s, f y ∂[B; μ.map φ] = ∫ᵛ (x : X) in φ ⁻¹' s, f (φ x) ∂[B; μ]

Tactic:
by
  rw [restrict_map μ hφ hs, integral_map hφ hfm hfi'.integrableOn]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case h_ind
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf : Integrable f μ.variation
⊢ ∀ (c : E) ⦃s : Set X⦄,
    MeasurableSet s →
      μ.variation s < ∞ →
        C ∫ᵛ (y : X), s.indicator (fun x ↦ c) y ∂[B; μ] =
          ∫ᵛ (y : X), s.indicator (fun x ↦ c) y ∂[(compL ℝ F G H) C ∘SL B; μ]

Tactic:
intro c s hs hc

Goal state after:
case h_ind
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf : Integrable f μ.variation
c : E
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
⊢ C ∫ᵛ (y : X), s.indicator (fun x ↦ c) y ∂[B; μ] = ∫ᵛ (y : X), s.indicator (fun x ↦ c) y ∂[(compL ℝ F G H) C ∘SL B; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_ind
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f : X → H
C : H →L[ℝ] E
hf : Integrable f μ.variation
hG : CompleteSpace G
⊢ ∀ (c : H) ⦃s : Set X⦄,
    MeasurableSet s →
      μ.variation s < ∞ →
        ∫ᵛ (y : X), C (s.indicator (fun x ↦ c) y) ∂[B; μ] = ∫ᵛ (y : X), s.indicator (fun x ↦ c) y ∂[B ∘SL C; μ]

Tactic:
intro c s hs hc

Goal state after:
case pos.h_ind
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f : X → H
C : H →L[ℝ] E
hf : Integrable f μ.variation
hG : CompleteSpace G
c : H
s : Set X
hs : MeasurableSet s
hc : μ.variation s < ∞
⊢ ∫ᵛ (y : X), C (s.indicator (fun x ↦ c) y) ∂[B; μ] = ∫ᵛ (y : X), s.indicator (fun x ↦ c) y ∂[B ∘SL C; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
hs : MeasurableSet s
hsf : μ.variation s < ∞
⊢ ‖(μ.transpose B) s‖ ≤ ‖B‖ * μ.variation.real s

Tactic:
apply (norm_cbmApplyMeasure_le _ _ _).trans

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
hs : MeasurableSet s
hsf : μ.variation s < ∞
⊢ ‖B‖ * ‖μ s‖ ≤ ‖B‖ * μ.variation.real s
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : X → E
ι : Type u_8
μ : ι → VectorMeasure X F
s : Finset ι
h : ∀ i ∈ s, (μ i).Integrable f
⊢ (∑ i ∈ s, μ i).Integrable f

Tactic:
classical
  induction s using Finset.induction_on with
  | empty => simp
  | insert a s ha ih =>
      simp only [Finset.mem_insert, forall_eq_or_imp, ha, not_false_eq_true,
        Finset.sum_insert] at h ⊢
      exact h.1.add_vectorMeasure (ih h.2)

Goal state after: