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

Goal state before:
pp: E : Type u_1
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
inst✝⁶ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : BorelSpace E
inst✝³ : T2Space E
inst✝² : ContinuousSMul ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
inst✝ : Nontrivial E
r : ℝ
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this✝ : NormedAddCommGroup F := p.toNormedAddCommGroup
this : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
⊢ μ {x | g x ≤ r} = μ {x | g x < r}

Tactic:
let : TopologicalSpace F := UniformSpace.toTopologicalSpace

Goal state after:
E : Type u_1
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
inst✝⁶ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : BorelSpace E
inst✝³ : T2Space E
inst✝² : ContinuousSMul ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
inst✝ : Nontrivial E
r : ℝ
F : Type u_1 := E
p : AddGroupNorm F := { toFun := g, map_zero' := h1, add_le' := h3, neg' := h2, eq_zero_of_map_eq_zero' := ⋯ }
this✝¹ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝ : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
⊢ μ {x | g x ≤ r} = μ {x | g x < r}
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
a : E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)
x : ℝ
⊢ AEMeasurable (fun x ↦ ⟪a, x⟫) μ

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
mE : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
inst✝ : IsProbabilityMeasure μ
a : E
r : ℝ
hr : 0 < r
this : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)
x : ℝ
⊢ AEStronglyMeasurable (fun x_1 ↦ cexp (↑x * ↑x_1 * I)) (Measure.map (fun x ↦ ⟪a, x⟫) μ)

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
this : {μ | μ.mass = C} = {μ | μ.mass ≤ C} ∩ {μ | μ.mass = C}
⊢ Continuous[FiniteMeasure.instTopologicalSpace, _] mass

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
this : {μ | μ.mass = C} = {μ | μ.mass ≤ C} ∩ {μ | μ.mass = C}
⊢ Continuous[FiniteMeasure.instTopologicalSpace, _] fun μ ↦ C

Tactic:
by fun_prop

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
C : ℝ≥0
K : Set E
hK : IsCompact K
f : ↑K → E := Subtype.val
hf : IsClosedEmbedding f
rf : range f = K
F : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f
T : Set (FiniteMeasure ↑K) := {μ | μ.mass ≤ C}
this : {μ | μ.mass ≤ C ∧ μ Kᶜ = 0} = F '' T
⊢ Continuous[FiniteMeasure.instTopologicalSpace, FiniteMeasure.instTopologicalSpace] F

Tactic:
by fun_prop

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this✝ : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
hπ1 : IsPiSystem π
hπ2 : Prod.instMeasurableSpace = generateFrom π
s : (i : ι) → Set (X i)
t : (i : κ) → Set (Y i)
hs : ∀ (i : ι), IsClosed[inst✝⁸ i] (s i)
ht : ∀ (i : κ), IsClosed[inst✝⁵ i] (t i)
hp1 : ∀ (x : (i : ι) → X i), ∏ i, (s i).indicator (fun x ↦ 1) (x i) = (univ.pi s).indicator 1 x
hp2 : ∀ (y : (j : κ) → Y j), ∏ j, (t j).indicator (fun x ↦ 1) (y j) = (univ.pi t).indicator 1 y
this :
  ∀ (p : ((i : ι) → X i) × ((j : κ) → Y j)),
    Tendsto (ofNNReal ∘ fun x ↦ (∏ c, (⋯.apprSeq x) (p.1 c)) * ∏ c, (⋯.apprSeq x) (p.2 c)) atTop
      (𝓝 ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p))
⊢ ∀ (x : ℕ), Measurable fun p ↦ ↑((∏ i, (⋯.apprSeq x) (p.1 i)) * ∏ j, (⋯.apprSeq x) (p.2 j))

Tactic:
by fun_prop

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

Goal state before:
pp: case refine_4
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ ((dω x) u) v = ((dω x) v) u

Tactic:
apply hdω

Goal state after:
case refine_4.a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) := ⋯
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ x ∈ s

case refine_4.a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) := ⋯
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ u ∈ tangentConeAt ℝ s x

case refine_4.a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) := ⋯
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ v ∈ tangentConeAt ℝ s x
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁵ : (j : κ) → TopologicalSpace (Y j)
inst✝⁴ : ∀ (j : κ), BorelSpace (Y j)
inst✝³ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝² : Fintype ι
inst✝¹ : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝ : IsFiniteMeasure μ
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ≥0) (g : (j : κ) → Y j →ᵇ ℝ≥0),
    ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂μ =
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑(∏ i, (f i) (p.1 i)) * ↑(∏ j, (g j) (p.2 j)) ∂ν
hμν : μ univ = ν univ
this✝ : IsFiniteMeasure ν
π : Set (Set (((i : ι) → X i) × ((j : κ) → Y j))) :=
  image2 (fun s t ↦ s ×ˢ t) (univ.pi '' univ.pi fun x ↦ {s | IsClosed[inst✝⁸ x] s})
    (univ.pi '' univ.pi fun x ↦ {t | IsClosed[inst✝⁵ x] t})
hπ1 : IsPiSystem π
hπ2 : Prod.instMeasurableSpace = generateFrom π
s : (i : ι) → Set (X i)
t : (i : κ) → Set (Y i)
hs : ∀ (i : ι), IsClosed[inst✝⁸ i] (s i)
ht : ∀ (i : κ), IsClosed[inst✝⁵ i] (t i)
hp1 : ∀ (x : (i : ι) → X i), ∏ i, (s i).indicator (fun x ↦ 1) (x i) = (univ.pi s).indicator 1 x
hp2 : ∀ (y : (j : κ) → Y j), ∏ j, (t j).indicator (fun x ↦ 1) (y j) = (univ.pi t).indicator 1 y
this :
  ∀ (p : ((i : ι) → X i) × ((j : κ) → Y j)),
    Tendsto (ofNNReal ∘ fun x ↦ (∏ c, (⋯.apprSeq x) (p.1 c)) * ∏ c, (⋯.apprSeq x) (p.2 c)) atTop
      (𝓝 ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p))
h1 :
  Tendsto
    (fun n ↦
      ∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((∏ i, (⋯.apprSeq n) (p.1 i)) * ∏ j, (⋯.apprSeq n) (p.2 j)) ∂μ)
    atTop (𝓝 (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), ↑((univ.pi s ×ˢ univ.pi t).indicator 1 p) ∂μ))
⊢ ∀ (x : ℕ), Measurable fun p ↦ ↑(∏ i, (⋯.apprSeq x) (p.1 i)) * ↑(∏ j, (⋯.apprSeq x) (p.2 j))

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝ : PseudoMetricSpace E
hS : IsTightMeasureSet S
x : E
this : Tendsto ((⨆ μ ∈ S, ⇑μ) ∘ fun r ↦ (Metric.closedBall x r)ᶜ) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ (Metric.closedBall x r)ᶜ) atTop (𝓝 0)

Tactic:
by
    convert! this with r
    simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
this :
  EqOn (fun x ↦ IccExtend ⋯ (⇑(φ.extend x.1)) x.2) (fun x ↦ (lineMap ((lineMap a b) x.2) ((lineMap a c) x.2)) x.1)
    (Icc 0 1)
⊢ ContDiffOn ℝ 2 (fun x ↦ (lineMap ((lineMap a b) x.2) ((lineMap a c) x.2)) x.1) (Icc 0 1)

Tactic:
by fun_prop

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
s : Set E
ω : E → E →L[𝕜] F
inst✝ : CompleteSpace F
hs : Convex ℝ s
hso : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hω : DifferentiableOn ℝ ω s
hdω : ∀ a ∈ s, ∀ (x y : E), ((fderiv ℝ ω a) x) y = ((fderiv ℝ ω a) y) x
⊢ ∃ f, ∀ a ∈ s, HasFDerivWithinAt f (ω a) s a

Tactic:
by
    refine hs.exists_forall_hasFDerivWithinAt_of_fderivWithin_symmetric hω fun a ha x _ y _ ↦ ?_
    rw [fderivWithin_eq_fderiv, hdω a ha]
    exacts [hso.uniqueDiffOn a ha, hω.differentiableAt (hso.mem_nhds ha)]

Goal state after:
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
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ Integrable (fun x ↦ 2) (volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))

Tactic:
by fun_prop

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), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ Measurable fun x ↦ ⟪z, x⟫_𝕜

Tactic:
by fun_prop

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), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ Measurable norm

Tactic:
by fun_prop

Goal state after:
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)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ Integrable (fun x ↦ 2) (volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)))

Tactic:
by fun_prop

Goal state after:
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)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
x✝ : ℕ
⊢ Measurable fun t ↦ 1 - charFun (μ x✝) (t • z)

Tactic:
by fun_prop

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁹ : (i : ι) → TopologicalSpace (X i)
inst✝⁸ : ∀ (i : ι), BorelSpace (X i)
inst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁶ : (j : κ) → TopologicalSpace (Y j)
inst✝⁵ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁴ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝³ : Fintype ι
inst✝² : Fintype κ
μ✝ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝¹ : IsFiniteMeasure μ✝
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ) (g : (j : κ) → Y j →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂μ✝ =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂ν
f : (i : ι) → X i →ᵇ ℝ≥0
g : (j : κ) → Y j →ᵇ ℝ≥0
μ : Measure (((i : ι) → X i) × ((j : κ) → Y j))
x✝¹ : ι
x✝ : x✝¹ ∈ Finset.univ
⊢ Measurable fun p ↦ (f x✝¹) (p.1 x✝¹)

Tactic:
by fun_prop

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁹ : (i : ι) → TopologicalSpace (X i)
inst✝⁸ : ∀ (i : ι), BorelSpace (X i)
inst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁶ : (j : κ) → TopologicalSpace (Y j)
inst✝⁵ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁴ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝³ : Fintype ι
inst✝² : Fintype κ
μ✝ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝¹ : IsFiniteMeasure μ✝
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ) (g : (j : κ) → Y j →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂μ✝ =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂ν
f : (i : ι) → X i →ᵇ ℝ≥0
g : (j : κ) → Y j →ᵇ ℝ≥0
μ : Measure (((i : ι) → X i) × ((j : κ) → Y j))
x✝¹ : κ
x✝ : x✝¹ ∈ Finset.univ
⊢ Measurable fun p ↦ (g x✝¹) (p.2 x✝¹)

Tactic:
by fun_prop

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁹ : (i : ι) → TopologicalSpace (X i)
inst✝⁸ : ∀ (i : ι), BorelSpace (X i)
inst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁶ : (j : κ) → TopologicalSpace (Y j)
inst✝⁵ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁴ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝³ : Fintype ι
inst✝² : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ) (g : (j : κ) → Y j →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂μ =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂ν
f : (i : ι) → X i →ᵇ ℝ≥0
g : (j : κ) → Y j →ᵇ ℝ≥0
this :
  ∀ {μ : Measure (((i : ι) → X i) × ((j : κ) → Y j))},
    (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, ↑((f i) (p.1 i))) * ∏ j, ↑((g j) (p.2 j)) ∂μ).toReal =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, ↑((f i) (p.1 i))) * ∏ j, ↑((g j) (p.2 j)) ∂μ
i : ι
⊢ Continuous[inst✝⁹ i, _] fun x ↦ ↑((f i) x)

Tactic:
by fun_prop

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁹ : (i : ι) → TopologicalSpace (X i)
inst✝⁸ : ∀ (i : ι), BorelSpace (X i)
inst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁶ : (j : κ) → TopologicalSpace (Y j)
inst✝⁵ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁴ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝³ : Fintype ι
inst✝² : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ) (g : (j : κ) → Y j →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂μ =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂ν
f : (i : ι) → X i →ᵇ ℝ≥0
g : (j : κ) → Y j →ᵇ ℝ≥0
this :
  ∀ {μ : Measure (((i : ι) → X i) × ((j : κ) → Y j))},
    (∫⁻ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, ↑((f i) (p.1 i))) * ∏ j, ↑((g j) (p.2 j)) ∂μ).toReal =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, ↑((f i) (p.1 i))) * ∏ j, ↑((g j) (p.2 j)) ∂μ
j : κ
⊢ Continuous[inst✝⁶ j, _] fun y ↦ ↑((g j) y)

Tactic:
by fun_prop

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁹ : (i : ι) → TopologicalSpace (X i)
inst✝⁸ : ∀ (i : ι), BorelSpace (X i)
inst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁶ : (j : κ) → TopologicalSpace (Y j)
inst✝⁵ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁴ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝³ : Fintype ι
inst✝² : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ) (g : (j : κ) → Y j →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂μ =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂ν
f : (i : ι) → X i →ᵇ ℝ≥0
g : (j : κ) → Y j →ᵇ ℝ≥0
i : ι
⊢ Continuous[instTopologicalSpaceProd, inst✝⁹ i] (Function.eval i ∘ Prod.fst)

Tactic:
by fun_prop

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

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : ι → Type u_5
Y : κ → Type u_6
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁹ : (i : ι) → TopologicalSpace (X i)
inst✝⁸ : ∀ (i : ι), BorelSpace (X i)
inst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)
mY : (j : κ) → MeasurableSpace (Y j)
inst✝⁶ : (j : κ) → TopologicalSpace (Y j)
inst✝⁵ : ∀ (j : κ), BorelSpace (Y j)
inst✝⁴ : ∀ (j : κ), HasOuterApproxClosed (Y j)
inst✝³ : Fintype ι
inst✝² : Fintype κ
μ ν : Measure (((i : ι) → X i) × ((j : κ) → Y j))
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : (i : ι) → X i →ᵇ ℝ) (g : (j : κ) → Y j →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂μ =
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂ν
f : (i : ι) → X i →ᵇ ℝ≥0
g : (j : κ) → Y j →ᵇ ℝ≥0
j : κ
⊢ Continuous[instTopologicalSpaceProd, inst✝⁶ j] (Function.eval j ∘ Prod.snd)

Tactic:
by fun_prop

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
P : Finpartition ⟨s, hs⟩
⊢ ∑ p ∈ P.parts, f ↑p ≤ preVariationFun f s

Tactic:
simpa [preVariationFun, hs, le_iSup_iff] using fun _ a ↦ a P

Goal state after: