mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
⊢ ∃ V, E ⊆ ↑V ∧ (∀ x ∈ V, f x < c) ∧ μ.measure ↑V ≤ μ.measure E + ENNReal.ofReal ε

Tactic:
obtain ⟨V₁ : Opens X, hV₁⟩ := Content.outerMeasure_exists_open μ hμ hε'

Goal state after:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
V₁ : Opens X
hV₁ : E ⊆ ↑V₁ ∧ μ.outerMeasure ↑V₁ ≤ μ.outerMeasure E + ↑ε.toNNReal
⊢ ∃ V, E ⊆ ↑V ∧ (∀ x ∈ V, f x < c) ∧ μ.measure ↑V ≤ μ.measure E + ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
V₁ : Opens X
hV₁ : E ⊆ ↑V₁ ∧ μ.outerMeasure ↑V₁ ≤ μ.outerMeasure E + ↑ε.toNNReal
V₂ : Opens X := { carrier := ⇑f ⁻¹' Iio c, is_open' := ⋯ }
⊢ μ.outerMeasure E + ↑ε.toNNReal = μ.outerMeasure E + ENNReal.ofReal ε

Tactic:
ENNReal.ofNNReal_toNNReal

Goal state after:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
V₁ : Opens X
hV₁ : E ⊆ ↑V₁ ∧ μ.outerMeasure ↑V₁ ≤ μ.outerMeasure E + ↑ε.toNNReal
V₂ : Opens X := { carrier := ⇑f ⁻¹' Iio c, is_open' := ⋯ }
⊢ μ.outerMeasure E + ENNReal.ofReal ε = μ.outerMeasure E + ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := ⋯
n : Fin N
h : E n ⊆ tsupport ⇑f
⊢ μ (tsupport ⇑f) < ∞

Tactic:
dsimp [μ]

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
n : Fin N
h : E n ⊆ tsupport ⇑f
⊢ (rieszMeasure Λ) (tsupport ⇑f) < ∞
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: F : Type u_2
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace ℝ F
⊢ probCharDual 0 = 1

Tactic:
simp [probCharDual]

Goal state after:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ ∀ (s : Set X), x ∈ s → IsOpen[inst✝¹] s → ∃ i, (stoneCechUnit x ∈ i ∧ IsOpen i) ∧ stoneCechUnit ⁻¹' i ⊆ s

Tactic:
and_assoc

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ ∀ (s : Set X), x ∈ s → IsOpen[inst✝¹] s → ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ s
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case neg
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
ε₁ ε₂ : ℝ≥0∞
μ ν : Measure Ω
h_le : ε₁ ≤ ε₂
B : Set Ω
hε₁ : μ B ≤ ν (thickening ε₁.toReal B) + ε₁
ε_top : ¬ε₂ = ∞
⊢ μ B ≤ ν (thickening ε₂.toReal B) + ε₂

Tactic:
apply hε₁.trans (add_le_add ?_ h_le)

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
ε₁ ε₂ : ℝ≥0∞
μ ν : Measure Ω
h_le : ε₁ ≤ ε₂
B : Set Ω
hε₁ : μ B ≤ ν (thickening ε₁.toReal B) + ε₁
ε_top : ¬ε₂ = ∞
⊢ ν (thickening ε₁.toReal B) ≤ ν (thickening ε₂.toReal B)
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
⊢ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ ≤ μ (thickening ε.toReal B) + ε

Tactic:
rw [prob_compl_eq_one_sub isOpen_thickening.measurableSet]

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ : ℝ≥0∞
h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε
ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
⊢ 1 - ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) ≤ μ (thickening ε.toReal B) + ε
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case calc_7
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
h : ∑ n, y n ≤ ↑N * b
⊢ ∫ (x : X), f x ∂μ + 2 * ε' * μ.real K + ε' / ↑N * ↑N * |a| + ε' / ↑N * ∑ x, y x + ε' / ↑N * ↑N * ε' ≤
    ∫ (x : X), f x ∂μ + ε' * 2 * μ.real K + ε' * |a| + ε' * b + ε' * ε'

Tactic:
simpa [show (N : ℝ) ≠ 0 by simp [hN.ne.symm], mul_comm _ ε', div_eq_mul_inv, mul_assoc]
      using (mul_le_mul_iff_of_pos_left hε'.1).mpr <| (inv_mul_le_iff₀ (Nat.cast_pos'.mpr hN)).mpr h

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : ProbabilityMeasure α
ν : ProbabilityMeasure β
⊢ (μ.prod ν).map ⋯ = ν

Tactic:
by
  apply Subtype.ext
  simp only [val_eq_to_measure, toMeasure_map, toMeasure_prod, Measure.map_snd_prod,
             measure_univ, one_smul]

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
this : (map (⇑φ.symm) μ).Regular
⊢ mulEquivHaarChar φ • μ = map (⇑φ.symm) μ

Tactic:
← mulEquivHaarChar_smul_map (map φ.symm μ) φ,

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
this : (map (⇑φ.symm) μ).Regular
⊢ mulEquivHaarChar φ • μ = mulEquivHaarChar φ • map (⇑φ) (map (⇑φ.symm) μ)
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: case inr
A : Type u_1
B : Type u_2
C : Type u_3
inst✝¹⁷ : Group A
inst✝¹⁶ : Group B
inst✝¹⁵ : Group C
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹¹ : IsTopologicalGroup A
inst✝¹⁰ : IsTopologicalGroup B
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : IsTopologicalGroup C
inst✝⁶ : LocallyCompactSpace B
inst✝⁵ : MeasurableSpace C
inst✝⁴ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝³ : T2Space B
inst✝² : MeasurableSpace B
inst✝¹ : BorelSpace B
U : Set B
hU : IsOpen U
inst✝ : DiscreteTopology A
h✝ : μC Set.univ * μA {1} < (H.inducedMeasure μA μC) U
ho : 0 < μA {1}
ht : μA {1} < ⊤
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
h : ∀ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ≤ (μA {1}).toReal
hC : μC Set.univ < ⊤
this : IsFiniteMeasure μC
⊢ ∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC ≤
    μC.real Set.univ * (μA {1}).toReal

Tactic:
← smul_eq_mul,

Goal state after:
case inr
A : Type u_1
B : Type u_2
C : Type u_3
inst✝¹⁷ : Group A
inst✝¹⁶ : Group B
inst✝¹⁵ : Group C
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹¹ : IsTopologicalGroup A
inst✝¹⁰ : IsTopologicalGroup B
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : IsTopologicalGroup C
inst✝⁶ : LocallyCompactSpace B
inst✝⁵ : MeasurableSpace C
inst✝⁴ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝³ : T2Space B
inst✝² : MeasurableSpace B
inst✝¹ : BorelSpace B
U : Set B
hU : IsOpen U
inst✝ : DiscreteTopology A
h✝ : μC Set.univ * μA {1} < (H.inducedMeasure μA μC) U
ho : 0 < μA {1}
ht : μA {1} < ⊤
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
h : ∀ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ≤ (μA {1}).toReal
hC : μC Set.univ < ⊤
this : IsFiniteMeasure μC
⊢ ∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC ≤
    μC.real Set.univ • (μA {1}).toReal
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) =
    ENNReal.ofReal
      (↑(finrank ℝ E) * (μ.real (Metric.ball 0 1) * ∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
        Real.Gamma (↑(finrank ℝ E) / p + 1))

Tactic:
mul_div_assoc,

Goal state after:
case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) =
    ENNReal.ofReal
      (↑(finrank ℝ E) *
        ((μ.real (Metric.ball 0 1) * ∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
          Real.Gamma (↑(finrank ℝ E) / p + 1)))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) =
    ENNReal.ofReal
      (↑(finrank ℝ E) *
        ((μ.real (Metric.ball 0 1) * ∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) /
          Real.Gamma (↑(finrank ℝ E) / p + 1)))

Tactic:
mul_div_assoc,

Goal state after:
case inr
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Nontrivial E
this✝ : 0 < ↑(finrank ℝ E)
this :
  (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1) *
      ↑(finrank ℝ E) =
    1
⊢ μ (Metric.ball 0 1) =
    ENNReal.ofReal
      (↑(finrank ℝ E) *
        (μ.real (Metric.ball 0 1) *
          ((∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ E - 1) • Real.exp (-y ^ p)) / Real.Gamma (↑(finrank ℝ E) / p + 1))))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
k : ℕ
hk : finrank ℝ E = 2 * k + 1
x : E
r : ℝ
this : Nontrivial E
⊢ ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * √π / Gamma (↑(2 * k + 1) / 2 + 1)) =
    ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * 2 ^ (k + 1) / ↑(2 * k + 1)‼)

Tactic:
mul_div_assoc,

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
k : ℕ
hk : finrank ℝ E = 2 * k + 1
x : E
r : ℝ
this : Nontrivial E
⊢ ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * (√π / Gamma (↑(2 * k + 1) / 2 + 1))) =
    ENNReal.ofReal r ^ (2 * k + 1) * ENNReal.ofReal (π ^ k * 2 ^ (k + 1) / ↑(2 * k + 1)‼)
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
⊢ ∑ q ∈ Q.parts, f ↑q ≤ ∑ q ∈ Q.parts, ∑' (i : ℕ), f (↑q ⊓ ↑(s' i))

Tactic:
apply Finset.sum_le_sum fun q hq => ?_

Goal state after:
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
q : Subtype MeasurableSet
hq : q ∈ Q.parts
⊢ f ↑q ≤ ∑' (i : ℕ), f (↑q ⊓ ↑(s' i))
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: 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
⊢ Measurable f

Tactic:
refine measurable_of_tendsto_metrizable (f := fun n t ↦ charFun (μ n) t) (by fun_prop) ?_

Goal state after:
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
⊢ Tendsto (fun n t ↦ charFun (μ n) t) atTop (𝓝 f)
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
hopen : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ i, ⋃ (_ : i ≤ n), U i)
this : ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ ↑(liminf (fun i ↦ (μ (sub i)) (⋃ i, ⋃ (_ : i ≤ n), U i)) atTop)
⊢ ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ liminf (fun k ↦ ↑((μ (sub k)) (⋃ i, ⋃ (_ : i ≤ n), U i))) atTop

Tactic:
exact mod_cast this

Goal state after:
Module: Mathlib.MeasureTheory.OuterMeasure.OfAddContent

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
m : AddContent ℝ≥0∞ C
m_sigma_subadd : m.IsSigmaSubadditive
m_top : ∀ s ∉ C, m s = ∞
hs : s ∈ C
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hs_subset : s ⊆ ⋃ i, f i
⊢ ⋃ i, s ∩ f i ∈ C

Tactic:
rwa [← inter_iUnion, inter_eq_self_of_subset_left hs_subset]

Goal state after:
Module: Mathlib.MeasureTheory.SpecificCodomains.WithLp

Goal state before:
pp: case φ_int
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
q : ℝ≥0∞
inst✝⁴ : Fact (1 ≤ q)
ι : Type u_2
inst✝³ : Fintype ι
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
f : X → PiLp q E
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
hf : ∀ (i : ι), Integrable (fun x ↦ (f x).ofLp i) μ
i : ι
⊢ Integrable f μ

Tactic:
exact Integrable.of_eval_piLp hf

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

Goal state before:

Tactic:
∑ p ∈ P, ‖μ p‖ₑ = ∑ p ∈ Q.parts, ‖μ p‖ₑ :=
      (Finpartition.sum_ofPairwiseDisjoint_eq_sum hP₂ (by simp)).symm
    _ = ∑ p ∈ Q'.parts, ‖μ p‖ₑ := (Q.sum_ofSubset_eq_sum _ _ _ (by simp_all)).symm
    _ ≤ ∑ p ∈ (Q'.extendOfLE (Finset.sup_le hQ')).parts, ‖μ p‖ₑ :=
      sum_le_sum_of_subset (Q'.parts_subset_extendOfLE (Finset.sup_le hQ'))
    _ ≤ μ.variation s := by
      simp only [variation_apply, preVariation_apply, ennrealToMeasure_apply hs,
        ennrealPreVariation_apply]
      apply preVariation.sum_le' (fun p => ‖μ p‖ₑ) hs
      intro p hp
      rcases Q'.mem_parts_or_eq_sdiff_of_mem_extendOfLE _ hp with h | rfl
      · simp_all
      simp only [sup_set_eq_biUnion, id_eq]
      exact hs.diff <| .biUnion (Finset.countable_toSet _) (by simp)

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

Goal state before:

Tactic:
_ ≤ ‖μ E‖ₑ + ‖ν E‖ₑ := enorm_add_le _ _
    _ ≤ μ.variation E + ν.variation E := by
      gcongr <;> exact enorm_measure_le_variation _ E

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✝³ : TopologicalSpace V
inst✝² : ENormedAddCommMonoid V
inst✝¹ : T2Space V
inst✝ : ContinuousAdd V
ι : Type u_3
s : Finset ι
μ : ι → VectorMeasure X V
⊢ (∑ i ∈ s, μ i).variation ≤ ∑ i ∈ s, (μ i).variation

Tactic:
by
  classical
  induction s using Finset.induction_on with
  | empty => simp
  | insert i s his ih =>
    simpa [Finset.sum_insert his] using
      variation_add_le.trans (add_le_add_right ih ((μ i).variation))

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

Goal state before:
pp: case pos
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
t : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ht_eq : ∀ᵐ (x : X) ∂μ.variation, x ∈ t → f x = 0
ht : MeasurableSet t
⊢ ∫ᵛ (x : X) in t, f x ∂[B; μ] = 0

Tactic:
by_cases hf : AEStronglyMeasurable f (μ.restrict t).variation

Goal state after:
case pos
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
t : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ht_eq : ∀ᵐ (x : X) ∂μ.variation, x ∈ t → f x = 0
ht : MeasurableSet t
hf : AEStronglyMeasurable f (μ.restrict t).variation
⊢ ∫ᵛ (x : X) in t, f x ∂[B; μ] = 0

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
t : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ht_eq : ∀ᵐ (x : X) ∂μ.variation, x ∈ t → f x = 0
ht : MeasurableSet t
hf : ¬AEStronglyMeasurable f (μ.restrict t).variation
⊢ ∫ᵛ (x : X) in t, f x ∂[B; μ] = 0
Module: Mathlib.MeasureTheory.VectorMeasure.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁸ : LinearOrder α
inst✝⁷ : DenselyOrdered α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : SecondCountableTopology α
inst✝³ : CompactIccSpace α
hα : MeasurableSpace α
inst✝² : BorelSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : α → E
hf : BoundedVariationOn f univ
a : α
this : Nonempty α
hlim : Tendsto f atBot (𝓝 (atBot.limUnder f))
u : ℕ → α
u_anti : Antitone u
hu : Tendsto u atTop atBot
⊢ ⋃ n, Icc (u n) a ⊆ Iic a

Tactic:
by simp [Icc_subset_Iic_self]

Goal state after: