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

Goal state before:
pp: 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_pos : ∀ (n : ℕ), 0 < εs n
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
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
⊢ ∫ (x : Ω), f x ∂↑(Ps n) ≤ ∫ (x : Ω), f x ∂↑P + δ

Tactic:
specialize εs_pos n

Goal state after:
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
⊢ ∫ (x : Ω), f x ∂↑(Ps n) ≤ ∫ (x : Ω), f x ∂↑P + δ
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
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
⊢ ε > 0 → ∀ᶠ (x : B) in nhds b, dist (∫ (a : A), (H.pullback f x) a ∂μA) (∫ (a : A), (H.pullback f b) a ∂μA) < ε

Tactic:
hε

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
E : Type u_4
inst✝¹³ : Group A
inst✝¹² : Group B
inst✝¹¹ : Group C
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : TopologicalSpace B
inst✝⁸ : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝⁷ : IsTopologicalGroup A
inst✝⁶ : IsTopologicalGroup B
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace A
inst✝³ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝² : NormedSpace ℝ E
inst✝¹ : IsTopologicalGroup C
inst✝ : LocallyCompactSpace B
f : CompactlySupportedContinuousMap B E
this : UniformSpace B := IsTopologicalGroup.rightUniformSpace B
b : B
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : B) in nhds b, dist (∫ (a : A), (H.pullback f x) a ∂μA) (∫ (a : A), (H.pullback f b) a ∂μA) < ε
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
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ -(↑C * ‖g.toBoundedContinuousFunction‖) ≤ ∫ (x : E), -‖g.toBoundedContinuousFunction‖ ∂↑μ

Tactic:
by
        simp only [integral_const, smul_eq_mul, mul_neg, neg_le_neg_iff]
        gcongr
        exact hμ

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

Goal state before:
pp: case pos
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : y = 0
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)

pp: case neg
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)

Tactic:
· simp only [hy, inner_zero_left]
    refine (tendsto_congr' ?_).mpr tendsto_const_nhds
    filter_upwards [eventually_ge_atTop 0] with r hr
    simp [not_lt.mpr hr]

Goal state after:
case neg
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
⊢ Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.TightNormed

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

Tactic:
have h_tight : IsTightMeasureSet {(μ n).map (fun x ↦ ⟪z, x⟫_𝕜)} := isTightMeasureSet_singleton

Goal state after:
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 : IsTightMeasureSet {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}
⊢ Tendsto (fun r ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.TightNormed

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

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

Tactic:
· simp only [hy, inner_zero_left]
    refine (tendsto_congr' ?_).mpr tendsto_const_nhds
    filter_upwards [eventually_ge_atTop 0] with r hr
    simp [not_lt.mpr hr]

Goal state after:
case neg
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ ∀ (n : ℕ), ∀ᵐ (a : ℝ) ∂volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)), ‖1 - charFun (μ n) (a • z)‖ ≤ 2

Tactic:
exact fun _ ↦ ae_of_all _ fun _ ↦ norm_one_sub_charFun_le_two

Goal state after:
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 : ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : E⦄, dist x 0 < δ → dist (f x) (f 0) < ε
ε : ℝ
⊢ ε > 0 → ∃ N, ∀ n ≥ N, dist (2⁻¹ * n * ‖∫ (t : ℝ) in -2 * n⁻¹..2 * n⁻¹, 1 - f (t • z)‖) 0 < ε

Tactic:
hε

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
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, dist (2⁻¹ * n * ‖∫ (t : ℝ) in -2 * n⁻¹..2 * n⁻¹, 1 - f (t • z)‖) 0 < ε
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetSemiring C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : MeasuredSets μ
ε : ℝ≥0∞
εpos : ε > 0
t : Set α
tC : t ∈ supClosure C
ht : μ (t ∆ ↑s) < ε
⊢ (Metric.eball s ε ∩ SetLike.coe ⁻¹' supClosure C).Nonempty

Tactic:
refine ⟨⟨t, ?_⟩, by simpa [MeasuredSets.edist_def] using! ht, tC⟩

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetSemiring C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : MeasuredSets μ
ε : ℝ≥0∞
εpos : ε > 0
t : Set α
tC : t ∈ supClosure C
ht : μ (t ∆ ↑s) < ε
⊢ MeasurableSet t
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
n : ℕ
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
⊢ ∑ i ∈ Finset.range n, ∑ p ∈ (P i).parts, f ↑p ≤ preVariationFun f (⋃ i, s i)

Tactic:
have hs_disj : Set.PairwiseDisjoint (Finset.range n : Set ℕ) s' := fun i _ j _ hij => by
    simp only [Function.onFun, disjoint_iff, Subtype.ext_iff]
    exact Set.disjoint_iff_inter_eq_empty.mp (hs' hij)

Goal state after:
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
n : ℕ
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
hs_disj : (↑(Finset.range n)).PairwiseDisjoint s'
⊢ ∑ i ∈ Finset.range n, ∑ p ∈ (P i).parts, f ↑p ≤ preVariationFun f (⋃ i, s i)
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
ι : Type u_2
𝓕 : Filter ι
μ₀ : ProbabilityMeasure E
μ✝ : ι → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ✝ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))
g : E →ᵇ ℂ
w : AddMonoidAlgebra ℂ E
hw : ⇑g = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(probChar (((innerₗ E) x) a))
μ : Measure E
hμ : IsProbabilityMeasure μ
⊢ ∫ (x : E), g x ∂μ = w.coeff.sum fun a z ↦ z * ∫ (x : E), ↑(probChar (((innerₗ E) x) a)) ∂μ

Tactic:
simp_rw [hw, Finsupp.sum]

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
ι : Type u_2
𝓕 : Filter ι
μ₀ : ProbabilityMeasure E
μ✝ : ι → ProbabilityMeasure E
h : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ✝ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))
g : E →ᵇ ℂ
w : AddMonoidAlgebra ℂ E
hw : ⇑g = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(probChar (((innerₗ E) x) a))
μ : Measure E
hμ : IsProbabilityMeasure μ
⊢ ∫ (x : E), ∑ x_1 ∈ w.coeff.support, w.coeff x_1 * ↑(probChar (((innerₗ E) x) x_1)) ∂μ =
    ∑ x ∈ w.coeff.support, w.coeff x * ∫ (x_1 : E), ↑(probChar (((innerₗ E) x_1) x)) ∂μ
Module: Mathlib.MeasureTheory.Measure.SubFinite

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
t : Set α := {x | f x ≤ g x}
ht : MeasurableSet t
⊢ (μ.withDensity (f - g)).restrict t + (μ.withDensity (f - g)).restrict tᶜ ≤
    (μ.withDensity f - μ.withDensity g).restrict t + (μ.withDensity f - μ.withDensity g).restrict tᶜ

Tactic:
have h_zero : (μ.withDensity (f - g)).restrict t = 0 := by
      simp only [restrict_eq_zero]
      rw [withDensity_apply _ ht, lintegral_eq_zero_iff (by fun_prop)]
      refine ae_restrict_of_forall_mem ht fun x hx ↦ ?_
      simpa [tsub_eq_zero_iff_le]

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
inst✝ : IsFiniteMeasure (μ.withDensity g)
hf : Measurable f
hg : Measurable g
t : Set α := {x | f x ≤ g x}
ht : MeasurableSet t
h_zero : (μ.withDensity (f - g)).restrict t = 0
⊢ (μ.withDensity (f - g)).restrict t + (μ.withDensity (f - g)).restrict tᶜ ≤
    (μ.withDensity f - μ.withDensity g).restrict t + (μ.withDensity f - μ.withDensity g).restrict tᶜ
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
K' : ℕ → Set E := fun n ↦ ⋃ i ∈ Iic n, K i
h'K : IsCompact {μ | ∀ (n : ℕ), μ (K' n)ᶜ ≤ u n}
μ : ProbabilityMeasure E
hμ : μ ∈ S
⊢ μ ∈ {μ | ∀ (n : ℕ), μ (K' n)ᶜ ≤ u n}

Tactic:
n

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
h'K : IsCompact {μ | ∀ (n : ℕ), μ (K' n)ᶜ ≤ u n}
μ : ProbabilityMeasure E
hμ : μ ∈ S
n : ℕ
⊢ μ (K' n)ᶜ ≤ u n
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this✝ : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
x₀ : α
hx₀ : ⋂ n, B n = {x₀}
s : Set α
hs : MeasurableSet s
h : x₀ ∉ s
this : μ s ≤ μ (⋂ n, B n)ᶜ
⊢ μ s = 0

Tactic:
measure_compl mBn (by simp),

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsZeroOneMeasure μ
inst✝¹ : StandardBorelSpace α
inst✝ : NeZero μ
this✝ : IsProbabilityMeasure μ
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y
B : ℕ → Set α := fun n ↦ if h : μ (A n) = 1 then A n else (A n)ᶜ
mBn : MeasurableSet (⋂ n, B n)
hBn : μ (⋂ n, B n) = 1
x₀ : α
hx₀ : ⋂ n, B n = {x₀}
s : Set α
hs : MeasurableSet s
h : x₀ ∉ s
this : μ s ≤ μ univ - μ (⋂ n, B n)
⊢ μ s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

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

Tactic:
by simp [h]

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

Goal state before:
pp: 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 : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : hα = generateFrom C
h''C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
⊢ ∃ m', (∀ s ∈ C, m' s = m s) ∧ ∀ (s : Set α), ‖m' s‖ₑ ≤ μ s

Tactic:
apply VectorMeasure.exists_extension_of_isSetSemiring_of_le_measure_of_dense hC ?_ hm ?_
  · intro s hs
    rw [h'C]
    exact measurableSet_generateFrom hs
  · intro t ε ht εpos
    exact exists_measure_symmDiff_lt_of_generateFrom_isSetSemiring hC h''C h'C ht εpos

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

Goal state before:
pp: α : Type u_1
M : Type u_2
v : α → M
inst✝ : AddCommMonoidWithOne M
β : Type u_3
s : Finset β
f : β → presburger.Term α
⊢ Term.realize v (sum s f) = ∑ i ∈ s, Term.realize v (f i)

Tactic:
classical
  simp only [sum]
  conv => rhs; rw [← s.toList_toFinset, List.sum_toFinset _ s.nodup_toList]
  generalize s.toList = l
  induction l with simp [*]

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

Goal state before:
pp: case mp
M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : IsCancelAdd M
a : M
t : Finset M
ih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))
t' : Finset M
ht' : t' ⊆ t
f : M → ℕ
i : M
hi : i ∈ t'
hfi : 0 < f i
heq : ∑ x ∈ t', f x • x = ∑ x ∈ t \ t', f x • x
y : M
hy : y ∈ closure ↑t
⊢ a +ᵥ y ∈ ⋃ j ∈ t', ⋃ k ∈ Finset.range (f j), (a + k • j) +ᵥ ↑(closure ↑(t.erase j))

Tactic:
mem_closure_finset

Goal state after:
case mp
M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : IsCancelAdd M
a : M
t : Finset M
ih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))
t' : Finset M
ht' : t' ⊆ t
f : M → ℕ
i : M
hi : i ∈ t'
hfi : 0 < f i
heq : ∑ x ∈ t', f x • x = ∑ x ∈ t \ t', f x • x
y : M
hy : ∃ f, Function.support f ⊆ ↑t ∧ ∑ a ∈ t, f a • a = y
⊢ a +ᵥ y ∈ ⋃ j ∈ t', ⋃ k ∈ Finset.range (f j), (a + k • j) +ᵥ ↑(closure ↑(t.erase j))
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : AddMonoid.FG M
F : Type u_5
inst✝¹ : FunLike F M N
inst✝ : AddMonoidHomClass F M N
s : Set N
hs : IsSemilinearSet s
f : F
n : ℕ
g : (Fin n → ℕ) →+ M
hg : Function.Surjective ⇑g
⊢ IsSemilinearSet (⇑f ⁻¹' s)

Tactic:
rw [← image_preimage_eq (f ⁻¹' s) hg]

Goal state after:
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : AddMonoid.FG M
F : Type u_5
inst✝¹ : FunLike F M N
inst✝ : AddMonoidHomClass F M N
s : Set N
hs : IsSemilinearSet s
f : F
n : ℕ
g : (Fin n → ℕ) →+ M
hg : Function.Surjective ⇑g
⊢ IsSemilinearSet (⇑g '' ⇑g ⁻¹' ⇑f ⁻¹' s)
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: M : Type u_1
inst✝ : AddCommMonoid M
S : Set (Set M)
hS : S.Finite
hS' : ∀ t ∈ S, IsLinearSet t
P : Set M → AddSubmonoid M
t : (s : Set M) → Set ↥(P s)
hP : ∀ s ∈ S, (P s).FG
ht : ∀ s ∈ S, IsLinearSet (t s)
ht' : ∀ s ∈ S, s = Subtype.val '' t s
this : Finite ↑S
⊢ ⋃ i, ↑i = ⋃ i, Subtype.val '' ⇑(AddSubmonoid.inclusion ⋯) '' t ↑i

Tactic:
image_image,

Goal state after:
M : Type u_1
inst✝ : AddCommMonoid M
S : Set (Set M)
hS : S.Finite
hS' : ∀ t ∈ S, IsLinearSet t
P : Set M → AddSubmonoid M
t : (s : Set M) → Set ↥(P s)
hP : ∀ s ∈ S, (P s).FG
ht : ∀ s ∈ S, IsLinearSet (t s)
ht' : ∀ s ∈ S, s = Subtype.val '' t s
this : Finite ↑S
⊢ ⋃ i, ↑i = ⋃ i, (fun x ↦ ↑((AddSubmonoid.inclusion ⋯) x)) '' t ↑i
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: case refine_1
M : Type u_1
inst✝ : AddCommMonoid M
P₁ : AddSubmonoid M
s₁' : Set ↥P₁
hP₁ : P₁.FG
hs₁' : IsSemilinearSet s₁'
hs₁ : IsSemilinearSet (Subtype.val '' s₁')
P₂ : AddSubmonoid M
s₂' : Set ↥P₂
hP₂ : P₂.FG
hs₂' : IsSemilinearSet s₂'
hs₂ : IsSemilinearSet (Subtype.val '' s₂')
⊢ Subtype.val '' s₁' = Subtype.val '' ⇑(AddSubmonoid.inclusion ⋯) '' s₁'

Tactic:
image_image,

Goal state after:
case refine_1
M : Type u_1
inst✝ : AddCommMonoid M
P₁ : AddSubmonoid M
s₁' : Set ↥P₁
hP₁ : P₁.FG
hs₁' : IsSemilinearSet s₁'
hs₁ : IsSemilinearSet (Subtype.val '' s₁')
P₂ : AddSubmonoid M
s₂' : Set ↥P₂
hP₂ : P₂.FG
hs₂' : IsSemilinearSet s₂'
hs₂ : IsSemilinearSet (Subtype.val '' s₂')
⊢ Subtype.val '' s₁' = (fun x ↦ ↑((AddSubmonoid.inclusion ⋯) x)) '' s₁'
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
⊢ unify f (fun a ↦ Structure.Sigma.mk f i (x a)) i ⋯ = x

Tactic:
ext a

Goal state after:
L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
a : α
⊢ unify f (fun a ↦ Structure.Sigma.mk f i (x a)) i ⋯ a = x a
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic

Goal state before:
pp: case refine_2
M : Type u_1
inst✝ : AddCommMonoid M
P₁ : AddSubmonoid M
s₁' : Set ↥P₁
hP₁ : P₁.FG
hs₁' : IsSemilinearSet s₁'
hs₁ : IsSemilinearSet (Subtype.val '' s₁')
P₂ : AddSubmonoid M
s₂' : Set ↥P₂
hP₂ : P₂.FG
hs₂' : IsSemilinearSet s₂'
hs₂ : IsSemilinearSet (Subtype.val '' s₂')
⊢ Subtype.val '' s₂' = Subtype.val '' ⇑(AddSubmonoid.inclusion ⋯) '' s₂'

Tactic:
image_image,

Goal state after:
case refine_2
M : Type u_1
inst✝ : AddCommMonoid M
P₁ : AddSubmonoid M
s₁' : Set ↥P₁
hP₁ : P₁.FG
hs₁' : IsSemilinearSet s₁'
hs₁ : IsSemilinearSet (Subtype.val '' s₁')
P₂ : AddSubmonoid M
s₂' : Set ↥P₂
hP₂ : P₂.FG
hs₂' : IsSemilinearSet s₂'
hs₂ : IsSemilinearSet (Subtype.val '' s₂')
⊢ Subtype.val '' s₂' = (fun x ↦ ↑((AddSubmonoid.inclusion ⋯) x)) '' s₂'