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

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
A_toReal : Subalgebra ℝ (E →ᵇ ℝ) := ⋯
hA_toReal : (Subalgebra.map (toContinuousMapₐ ℝ) A_toReal).SeparatesPoints
heq' : ∀ g ∈ A_toReal, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
f : E →ᵇ ℝ
⊢ Tendsto sqrt (𝓝 0) (𝓝 0)

Tactic:
nth_rewrite 2 [← sqrt_zero]

Goal state after:
E : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
A_toReal : Subalgebra ℝ (E →ᵇ ℝ) :=
  Subalgebra.comap (AlgHom.compLeftContinuousBounded ℝ ofRealAm ⋯) (Subalgebra.restrictScalars ℝ A.toSubalgebra)
hA_toReal : (Subalgebra.map (toContinuousMapₐ ℝ) A_toReal).SeparatesPoints
heq' : ∀ g ∈ A_toReal, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
f : E →ᵇ ℝ
⊢ Tendsto sqrt (𝓝 0) (𝓝 √0)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
inst✝⁸ : CompleteSpace E
inst✝⁷ : CompleteSpace F
inst✝⁶ : SecondCountableTopology E
inst✝⁵ : SecondCountableTopology F
inst✝⁴ : BorelSpace E
inst✝³ : BorelSpace F
μ : Measure E
ν : Measure F
ξ : Measure (E × F)
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure ξ
h : ξ = μ.prod ν
⊢ ∀ (t : WithLp 2 (E × F)), charFun (Measure.map (toLp 2) (μ.prod ν)) t = charFun μ t.ofLp.1 * charFun ν t.ofLp.2

Tactic:
exact charFun_prod

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

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : T35Space X
a b : X
hab : ¬a = b
f : X → ↑I
fc : f ∈ {f | Continuous[inst✝¹, _] f}
fab : f a ≠ f b
⊢ ¬stoneCechUnit a = stoneCechUnit b

Tactic:
exact fun q ↦ fab (eq_if_stoneCechUnit_eq fc q)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_3
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a✝ b c d : E
γ₁ : Path a✝ b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs : s.Countable
hφt : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a, b_1) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b_1))) t (φ (a, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b_1)), ((dω (φ (a, b_1))) u) v = ((dω (φ (a, b_1))) v) u
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
hclosure : closure U = Icc 0 1
ψ : ℝ × ℝ → E := fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hcontdiff : ContDiffOn ℝ 2 ψ (Icc 0 1)
hψ : ψ = fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hψφ : ∀ (a b_1 : ↑I), ψ (↑a, ↑b_1) = φ (a, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
dψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fderivWithin ℝ ψ (Icc 0 1)
s' : Set (ℝ × ℝ) := Prod.map Subtype.val Subtype.val '' s
hs' : s' = Prod.map Subtype.val Subtype.val '' s
hmem_s' : ∀ (x y : ↑I), (↑x, ↑y) ∈ s' ↔ (x, y) ∈ s
hs'c : s'.Countable
hdψ : ∀ a ∈ U, HasFDerivAt ψ (dψ a) a
d2ψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] E := fderivWithin ℝ dψ (Icc 0 1)
hd2ψ : ∀ a ∈ U, HasFDerivAt dψ (d2ψ a) a
hd2ψ_symm : ∀ a ∈ Icc 0 1, ∀ (x y : ℝ × ℝ), ((d2ψ a) x) y = ((d2ψ a) y) x
η : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ ω (ψ a) ∘SL dψ a
f : ℝ × ℝ → F := fun a ↦ (η a) (0, 1)
hf : ∀ a ∈ Icc 0 1, f a = (ω (ψ a)) (derivWithin (ψ ∘ fun x ↦ (a.1, x)) I a.2)
g : ℝ × ℝ → F := fun a ↦ -(η a) (1, 0)
hg : ∀ a ∈ Icc 0 1, g a = (ω (ψ a)) (-derivWithin (ψ ∘ fun x ↦ (x, a.2)) I a.1)
dη : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] F :=
  fun a ↦ (ContinuousLinearMap.compL ℝ (ℝ × ℝ) E F) (ω (ψ a)) ∘SL d2ψ a + (dω (ψ a)).bilinearComp (dψ a) (dψ a)
a : ℝ × ℝ
haU : a ∈ U
has : a ∉ s'
⊢ ∀ᶠ (x' : ℝ × ℝ) in 𝓝 a, ψ x' ∈ t

Tactic:
filter_upwards [hUopen.mem_nhds haU] using hψUt

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

Goal state before:
pp: Ω : Type u_1
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : PseudoEMetricSpace Ω
inst✝³ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure κ
⊢ levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ

Tactic:
have dμν_finite := (levyProkhorovEDist_lt_top μ ν).ne

Goal state after:
Ω : Type u_1
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : PseudoEMetricSpace Ω
inst✝³ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure κ
dμν_finite : levyProkhorovEDist μ ν ≠ ∞
⊢ levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ
Module: Mathlib.MeasureTheory.Measure.FiniteMeasurePi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
⊢ (univ.pi fun i ↦ ball (x i) (r i)) ⊆ univ.pi fun i ↦ ball (x i) ε

Tactic:
gcongr with i hi

Goal state after:
case h
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) := ⋯
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := ⋯
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
i : ι
hi : i ∈ univ
⊢ r i ≤ ε
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, ProbabilityMeasure.instTopologicalSpace] toMeasure

Tactic:
refine SeqContinuous.continuous ?_

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
⊢ SeqContinuous toMeasure
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
smul : haar = haar.haarScalarFactor μ • μ
⊢ mulEquivHaarChar φ = μ.haarScalarFactor (map (⇑φ) μ)

Tactic:
unfold mulEquivHaarChar

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
smul : haar = haar.haarScalarFactor μ • μ
⊢ haar.haarScalarFactor (map (⇑φ) haar) = μ.haarScalarFactor (map (⇑φ) μ)
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case refine_2
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x ↦ dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n ↦ dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
  Tendsto (fun x ↦ ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs x) ν + εs x) {a | t ≤ f a})) atTop
    (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
εs_pos : 0 < εs n
hn :
  ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a}) <
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
⊢ levyProkhorovEDist ↑(Ps n) ↑P < ENNReal.ofReal (dist (μs n) ν + εs n)

Tactic:
rw [ENNReal.ofReal_add (by positivity) (by positivity), ← add_zero (levyProkhorovEDist _ _)]
      apply ENNReal.add_lt_add_of_le_of_lt (levyProkhorovEDist_ne_top _ _)
            (le_of_eq ?_) (ofReal_pos.mpr εs_pos)
      rw [LevyProkhorov.dist_probabilityMeasure_def, levyProkhorovDist,
        ofReal_toReal (levyProkhorovEDist_ne_top _ _)]
      rfl

Goal state after:
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
⊢ ψ (Function.invFun (⇑ψ) x * φ a) ∉ ⇑ψ '' K

Tactic:
rwa [map_mul, Function.rightInverse_invFun H.isOpenQuotientMap.surjective,
        H.mulExact.apply_apply_eq_one, mul_one]

Goal state after:
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 g : CompactlySupportedContinuousMap B E
⊢ { toFun := fun c ↦ ∫ (a : A), (H.pullback (f + g) (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
      hasCompactSupport' := ⋯ } =
    { toFun := fun c ↦ ∫ (a : A), (H.pullback f (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
        hasCompactSupport' := ⋯ } +
      { toFun := fun c ↦ ∫ (a : A), (H.pullback g (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
        hasCompactSupport' := ⋯ }

Tactic:
ext c

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 g : CompactlySupportedContinuousMap B E
c : C
⊢ { toFun := fun c ↦ ∫ (a : A), (H.pullback (f + g) (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
        hasCompactSupport' := ⋯ }
      c =
    ({ toFun := fun c ↦ ∫ (a : A), (H.pullback f (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
          hasCompactSupport' := ⋯ } +
        { toFun := fun c ↦ ∫ (a : A), (H.pullback g (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
          hasCompactSupport' := ⋯ })
      c
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
x : ℝ
f : CompactlySupportedContinuousMap B E
⊢ { toFun := fun c ↦ ∫ (a : A), (H.pullback (x • f) (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
      hasCompactSupport' := ⋯ } =
    (RingHom.id ℝ) x •
      { toFun := fun c ↦ ∫ (a : A), (H.pullback f (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
        hasCompactSupport' := ⋯ }

Tactic:
ext c

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
x : ℝ
f : CompactlySupportedContinuousMap B E
c : C
⊢ { toFun := fun c ↦ ∫ (a : A), (H.pullback (x • f) (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
        hasCompactSupport' := ⋯ }
      c =
    ((RingHom.id ℝ) x •
        { toFun := fun c ↦ ∫ (a : A), (H.pullback f (Function.invFun (⇑ψ) c)) a ∂μA, continuous_toFun := ⋯,
          hasCompactSupport' := ⋯ })
      c
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * ↑r * ↑(∫ (x : ℝ), sinc (r * x) ∂μ)

Tactic:
have h_int : Integrable (Function.uncurry fun (x y : ℝ) ↦ cexp (x * y * I))
      ((volume.restrict (Set.uIoc (-r) r)).prod μ) := by
    simp only [neg_le_self_iff, hr.le, Set.uIoc_of_le]
    -- integrable since the function has norm 1 everywhere and the measure is finite
    rw [← integrable_norm_iff (by fun_prop)]
    suffices (fun a => ‖Function.uncurry (fun (x y : ℝ) ↦ cexp (x * y * I)) a‖) = fun _ ↦ 1 by
      rw [this]
      fun_prop
    ext p
    rw [← Prod.mk.eta (p := p)]
    norm_cast
    simp only [Function.uncurry_apply_pair, norm_exp_ofReal_mul_I]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
⊢ ∫ (t : ℝ) in -r..r, charFun μ t = 2 * ↑r * ↑(∫ (x : ℝ), sinc (r * x) ∂μ)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 = r * r⁻¹ * 2

Tactic:
mul_inv_cancel₀ (by positivity),

Goal state after:
case e_a
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 = 1 * 2
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 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)
h : NormalSpace E ∨ Monotone K
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))
this✝ : Tendsto (fun n ↦ ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) univ))
n : ℕ
this : ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ = ↑(∑ i ∈ Finset.range (n + 1), ν i) univ
⊢ (∑ i ∈ Finset.range (n + 1), ν i).mass ≤ C

Tactic:
have : Tendsto (fun (μ : FiniteMeasure E) ↦
        (∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i)).mass) f
        (𝓝 ((∑ i ∈ Finset.range (n + 1), ν i).mass)) := by
      apply Tendsto.mass
      exact tendsto_finsetSum _ (fun i hi ↦ hν i)

Goal state after:
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)
h : NormalSpace E ∨ Monotone K
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))
this✝¹ : Tendsto (fun n ↦ ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) univ))
n : ℕ
this✝ : ∑ i ∈ Finset.range (n + 1), ↑(ν i) univ = ↑(∑ i ∈ Finset.range (n + 1), ν i) univ
this :
  Tendsto (fun μ ↦ (∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i)).mass) (↑f)
    (𝓝 (∑ i ∈ Finset.range (n + 1), ν i).mass)
⊢ (∑ i ∈ Finset.range (n + 1), ν i).mass ≤ C
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: 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)
⊢ IsTightMeasureSet S

Tactic:
by
  rcases subsingleton_or_nontrivial E with hE | hE
  · simp only [IsTightMeasureSet, cocompact_eq_bot, smallSets_bot]
    convert! tendsto_pure_nhds (a := ∅) _
    simp
  have h_rank : (0 : ℝ) < Fintype.card ι := by
    simpa [← Module.finrank_eq_card_basis b.toBasis, Module.finrank_pos_iff]
  have : Nonempty ι := by simpa [Fintype.card_pos_iff] using h_rank
  have : ProperSpace E := FiniteDimensional.proper 𝕜 E
  refine isTightMeasureSet_of_tendsto_measure_norm_gt ?_
  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
  refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ?_ (fun _ ↦ zero_le) h_le
  rw [← Finset.sum_const_zero]
  refine tendsto_finsetSum Finset.univ fun i _ ↦ (h i).comp ?_
  exact tendsto_id.atTop_div_const (by positivity)

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

Goal state before:
pp: case hbc.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
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ (fun x ↦ ‖1 - charFun (μ n) (x • z)‖) ≤ᵐ[volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))] fun x ↦ 2

Tactic:
exact ae_of_all _ fun _ ↦ norm_one_sub_charFun_le_two

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

Goal state before:
pp: 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))
⊢ ∑ n ∈ Finset.range (N + 1), ε / (2 * (↑N + 1)) ≤ ε / 2

Tactic:
simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul,
                            Nat.cast_add, Nat.cast_one]
                          rw [div_mul_eq_div_mul_one_div, ← mul_assoc, mul_comm ((N : ℝ) + 1),
                            mul_assoc]
                          exact mul_le_of_le_one_right (by linarith [ε_pos]) <|
                            le_of_eq <| mul_one_div_cancel <| Nat.cast_add_one_ne_zero _

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

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
⊢ IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)

Tactic:
let K' n := ⋃ i ∈ Iic n, K i

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
S : Set (ProbabilityMeasure E)
hS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}
u : ℕ → ℝ≥0
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
K : ℕ → Set E
K_comp : ∀ (n : ℕ), IsCompact (K n)
hK : ∀ (n : ℕ), ∀ μ ∈ S, μ (K n)ᶜ ≤ u n
K' : ℕ → Set E := fun n ↦ ⋃ i ∈ Iic n, K i
⊢ IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:

Tactic:
μ (K' n)ᶜ
  _ ≤ μ (K n)ᶜ := by
    gcongr
    simp only [mem_Iic, K']
    apply subset_biUnion_of_mem
    exact le_rfl (a := n)
  _ ≤ u n := by grind

Goal state after: