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

Goal state before:
pp: case hbc.refine_3
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
hx : x ∈ {x | 2 < |2 * r⁻¹ * x|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 1 / 2

Tactic:
one_div

Goal state after:
case hbc.refine_3
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
hx : x ∈ {x | 2 < |2 * r⁻¹ * x|}
hx_ne : 2 * r⁻¹ * x ≠ 0
⊢ sinc (2 * r⁻¹ * x) ≤ 2⁻¹
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : 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
p✝ : ℝ
hp : 0 < p✝
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p✝ + 1))

Tactic:
let φ := @LinearEquiv.toContinuousLinearEquiv ℝ _ E _ _ tE _ _ F _ _ _ _ _ _ _ _ _
    (LinearEquiv.refl ℝ E : E ≃ₗ[ℝ] F)

Goal state after:
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : 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
p✝ : ℝ
hp : 0 < p✝
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p✝ + 1))
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_2
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : 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
p✝ : ℝ
hp : 0 < p✝
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ μ {x | g x < 1} = ν (Metric.ball 0 1)

Tactic:
rw [@Measure.map_apply E F mE _ μ φ _ _ measurableSet_ball]

Goal state after:
case e'_2
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : 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
p✝ : ℝ
hp : 0 < p✝
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ μ {x | g x < 1} = μ (⇑φ ⁻¹' Metric.ball 0 1)

E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : 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
p✝ : ℝ
hp : 0 < p✝
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁶, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ Measurable ⇑φ
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
⊢ μ {x | g x ≤ r} = μ {x | g x < r}

Tactic:
let φ := @LinearEquiv.toContinuousLinearEquiv ℝ _ E _ _ tE _ _ F _ _ _ _ _ _ _ _ _
    (LinearEquiv.refl ℝ E : E ≃ₗ[ℝ] F)

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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
⊢ μ {x | g x ≤ r} = μ {x | g x < r}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_3
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ μ {x | g x < r} = ν (Metric.ball 0 r)

Tactic:
rw [@Measure.map_apply E F mE _ μ φ _ _ measurableSet_ball]

Goal state after:
case e'_3
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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ μ {x | g x < r} = μ (⇑φ ⁻¹' Metric.ball 0 r)

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✝³ : NormedAddCommGroup F := p.toNormedAddCommGroup
this✝² : NormedSpace ℝ F := { toModule := inst✝⁷, norm_smul_le := ⋯ }
this✝¹ : TopologicalSpace F := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ Measurable ⇑φ
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: case h₂
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}
μ : FiniteMeasure ↑K
hμ : μ.mass ≤ C
⊢ (μ.map f).mass ≤ C ∧ (Measure.map f ↑μ) Kᶜ = 0

Tactic:
rw [Measure.map_apply hf.continuous.measurable hK.measurableSet.compl]

Goal state after:
case h₂
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}
μ : FiniteMeasure ↑K
hμ : μ.mass ≤ C
⊢ (μ.map f).mass ≤ C ∧ ↑μ (f ⁻¹' Kᶜ) = 0
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case refine_1
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)‖
hf_tendsto : Tendsto f (𝓝 0) (𝓝 (f 0))
⊢ Tendsto (fun r ↦ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖) atTop (𝓝 0)

Tactic:
Metric.tendsto_nhds_nhds

Goal state after:
case refine_1
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)‖
hf_tendsto : ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : E⦄, dist x 0 < δ → dist (f x) (f 0) < ε
⊢ Tendsto (fun r ↦ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
⊢ ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε

Tactic:
have mono : Monotone (fun n ↦ (T n) ∩ s) := fun m n hmn ↦ inter_subset_inter_left s
        (biUnion_subset_biUnion_left fun k hkm ↦ Nat.le_trans hkm hmn)

Goal state after:
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ∞
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ∞
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n ↦ T n ∩ s
⊢ ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: case pos
X : Type u_1
mX : MeasurableSpace X
μ : VectorMeasure X ℝ≥0∞
s : Set X
h : MeasurableSet s
⊢ ⨆ P, ∑ p ∈ P.parts, μ ↑p = μ s

pp: case pos.h
X : Type u_1
mX : MeasurableSpace X
μ : VectorMeasure X ℝ≥0∞
s : Set X
h : MeasurableSet s
⊢ MeasurableSet s

Tactic:
exact iSup_sum_finpartition_parts μ h

Goal state after:
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✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s 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
hs : MeasurableSet s
ht : MeasurableSet t
ht_eq : ∀ᵐ (x : X) ∂μ.variation.restrict t, f x = 0
⊢ ∫ᵛ (x : X) in s ∪ t, f x ∂[B; μ] = ∫ᵛ (x : X) in s, f x ∂[B; μ]

Tactic:
rw [← integral_indicator hs, ← integral_indicator (hs.union ht)]

Goal state after:
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 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
hs : MeasurableSet s
ht : MeasurableSet t
ht_eq : ∀ᵐ (x : X) ∂μ.variation.restrict t, f x = 0
⊢ ∫ᵛ (x : X), (s ∪ t).indicator f x ∂[B; μ] = ∫ᵛ (x : X), s.indicator f x ∂[B; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.AddContent

Goal state before:
pp: case pos
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s | ∃ c ∈ C, ↑s = c}
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := fun x ↦ m ↑↑x
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := fun s ↦ if hs : MeasurableSet s then m₁ ⟨s, hs⟩ else 0
s : Set α
hs : MeasurableSet s
⊢ ‖m' s‖ₑ ≤ μ s

pp: case neg
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s | ∃ c ∈ C, ↑s = c}
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := fun x ↦ m ↑↑x
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := fun s ↦ if hs : MeasurableSet s then m₁ ⟨s, hs⟩ else 0
s : Set α
hs : ¬MeasurableSet s
⊢ ‖m' s‖ₑ ≤ μ s

Tactic:
· simpa [hs, m'] using! hBound _

Goal state after:
case neg
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetRing C
hCmeas : ∀ s ∈ C, MeasurableSet s
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ (t : Set α) (ε : ℝ≥0∞), MeasurableSet t → 0 < ε → ∃ s ∈ C, μ (s ∆ t) < ε
C' : Set (MeasuredSets μ) := {s | ∃ c ∈ C, ↑s = c}
C'C : ∀ s ∈ C', ↑s ∈ C
C'_dense : Dense C'
m₀ : ↑C' → E := fun x ↦ m ↑↑x
lip : LipschitzWith 1 m₀
m₁ : MeasuredSets μ → E := C'_dense.extend m₀
m₁_lip : LipschitzWith 1 m₁
hBound : ∀ (s : MeasuredSets μ), ‖m₁ s‖ₑ ≤ μ ↑s
hAddit : ∀ (s t : MeasuredSets μ), Disjoint ↑s ↑t → m₁ ⟨↑s ∪ ↑t, ⋯⟩ = m₁ s + m₁ t
m' : Set α → E := fun s ↦ if hs : MeasurableSet s then m₁ ⟨s, hs⟩ else 0
s : Set α
hs : ¬MeasurableSet s
⊢ ‖m' s‖ₑ ≤ μ s
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
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 E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
ρ : VectorMeasure (X × Y) G
hρ : ∀ (s : Set X) (t : Set Y), MeasurableSet s → MeasurableSet t → ρ (s ×ˢ t) = (B (μ s)) (ν t)
this : μ.HasProd ν B
⊢ (μ.prod ν B) univ = ρ univ

pp: X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
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 E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
ρ : VectorMeasure (X × Y) G
hρ : ∀ (s : Set X) (t : Set Y), MeasurableSet s → MeasurableSet t → ρ (s ×ˢ t) = (B (μ s)) (ν t)
this : μ.HasProd ν B
⊢ ∀ s ∈ image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t}, (μ.prod ν B) s = ρ s

Tactic:
· rw [← univ_prod_univ, hρ _ _ MeasurableSet.univ MeasurableSet.univ, prod_apply]

Goal state after:
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
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 E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
ρ : VectorMeasure (X × Y) G
hρ : ∀ (s : Set X) (t : Set Y), MeasurableSet s → MeasurableSet t → ρ (s ×ˢ t) = (B (μ s)) (ν t)
this : μ.HasProd ν B
⊢ ∀ s ∈ image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t}, (μ.prod ν B) s = ρ s
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
inst✝ : AddCommMonoid M
s₁ s₂ : Set M
hs₁ : IsLinearSet s₁
hs₂ : IsLinearSet s₂
⊢ IsLinearSet (s₁ + s₂)

Tactic:
rcases hs₁ with ⟨a, t₁, ht₁, rfl⟩

Goal state after:
M : Type u_1
inst✝ : AddCommMonoid M
s₂ : Set M
hs₂ : IsLinearSet s₂
a : M
t₁ : Set M
ht₁ : t₁.Finite
⊢ IsLinearSet ((a +ᵥ ↑(closure t₁)) + s₂)
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
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝² : MeasurableSpace X
inst✝¹ : MeasurableSingletonClass X
inst✝ : CompleteSpace G
a : X
v : F
this : IsFiniteMeasure ((dirac a v).transpose B).variation
⊢ ∫ᵛ (x : X), f x ∂[B; dirac a v] = (B (f a)) v

Tactic:
calc
    ∫ᵛ x, f x ∂[B; VectorMeasure.dirac a v] = ∫ᵛ _, f a ∂[B; VectorMeasure.dirac a v] := by
      apply integral_congr_ae
      simp only [variation_dirac]
      exact Measure.ae_smul_measure (ae_eq_dirac f) _
    _ = B (f a) v := by simp

Goal state after:
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✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
μ : VectorMeasure X F
β : Type u_9
inst✝ : MeasurableSpace β
φ : X → β
hφ : MeasurableEmbedding φ
f : β → E
⊢ (μ.map φ).Integrable f ↔ μ.Integrable (f ∘ φ)

Tactic:
by
  simp_rw [VectorMeasure.Integrable, ← hφ.integrable_map_iff, hφ.variation_map]

Goal state after:
Module: Mathlib.ModelTheory.FinitelyGenerated

Goal state before:
pp: L : Language
M : Type u_1
inst✝¹ : L.Structure M
N : Type u_2
inst✝ : L.Structure N
h : CG L M
f : M →[L] N
⊢ f.range.CG

Tactic:
by
  rw [Hom.range_eq_map]
  exact (cg_def.1 h).map f

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: M : Type u_1
ι : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Finite ι
F : Type u_5
G : Type u_6
inst✝³ : FunLike F (ι → ℕ) M
inst✝² : AddMonoidHomClass F (ι → ℕ) M
inst✝¹ : FunLike G (ι → ℕ) M
inst✝ : AddMonoidHomClass G (ι → ℕ) M
a b : M
f : F
g : G
x y z : ι → ℕ
hx : a + f x = b + g x
hy : a + f x + f y = b + g x + g y
hz : a + f x + f z = b + g x + g z
| a + f x + g y + f z

Tactic:
add_right_comm _ (g y) (f z),

Goal state after:
M : Type u_1
ι : Type u_3
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Finite ι
F : Type u_5
G : Type u_6
inst✝³ : FunLike F (ι → ℕ) M
inst✝² : AddMonoidHomClass F (ι → ℕ) M
inst✝¹ : FunLike G (ι → ℕ) M
inst✝ : AddMonoidHomClass G (ι → ℕ) M
a b : M
f : F
g : G
x y z : ι → ℕ
hx : a + f x = b + g x
hy : a + f x + f y = b + g x + g y
hz : a + f x + f z = b + g x + g z
| a + f x + f z + g y