mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁸ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁷ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : BorelSpace E
E' : Type u_6
mE' : MeasurableSpace E'
inst✝³ : SeminormedAddCommGroup E'
inst✝² : SecondCountableTopology E'
inst✝¹ : BorelSpace E'
inst✝ : l.IsCountablyGenerated
X : ι → Ω'' → E
Y : ι → Ω'' → E'
Z : Ω' → E
c : E'
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hY : TendstoInMeasure μ'' Y l fun x ↦ c
hY_meas : ∀ (i : ι), AEMeasurable (Y i) μ''
hX : ∀ (i : ι), AEMeasurable (X i) μ''
⊢ TendstoInDistribution (fun n ω ↦ (X n ω, Y n ω)) l (fun ω ↦ (Z ω, c)) (fun x ↦ μ'') μ'

Tactic:
refine tendstoInDistribution_of_tendstoInMeasure_sub (X := fun n ω ↦ (X n ω, c))
    (fun n ω ↦ (X n ω, Y n ω)) (fun ω ↦ (Z ω, c)) ?_ ?_ (fun i ↦ (hX i).prodMk (hY_meas i))

Goal state after:
case refine_1
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁸ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁷ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : BorelSpace E
E' : Type u_6
mE' : MeasurableSpace E'
inst✝³ : SeminormedAddCommGroup E'
inst✝² : SecondCountableTopology E'
inst✝¹ : BorelSpace E'
inst✝ : l.IsCountablyGenerated
X : ι → Ω'' → E
Y : ι → Ω'' → E'
Z : Ω' → E
c : E'
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hY : TendstoInMeasure μ'' Y l fun x ↦ c
hY_meas : ∀ (i : ι), AEMeasurable (Y i) μ''
hX : ∀ (i : ι), AEMeasurable (X i) μ''
⊢ TendstoInDistribution (fun n ω ↦ (X n ω, c)) l (fun ω ↦ (Z ω, c)) (fun x ↦ μ'') μ'

case refine_2
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁸ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁷ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : BorelSpace E
E' : Type u_6
mE' : MeasurableSpace E'
inst✝³ : SeminormedAddCommGroup E'
inst✝² : SecondCountableTopology E'
inst✝¹ : BorelSpace E'
inst✝ : l.IsCountablyGenerated
X : ι → Ω'' → E
Y : ι → Ω'' → E'
Z : Ω' → E
c : E'
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hY : TendstoInMeasure μ'' Y l fun x ↦ c
hY_meas : ∀ (i : ι), AEMeasurable (Y i) μ''
hX : ∀ (i : ι), AEMeasurable (X i) μ''
⊢ TendstoInMeasure μ'' ((fun n ω ↦ (X n ω, Y n ω)) - fun n ω ↦ (X n ω, c)) l 0
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝² : Preorder ι
E : Type u_4
inst✝¹ : Zero E
inst✝ : TopologicalSpace E
ℱ : Filtration ι m
f : ι → Ω → E
μ : Measure Ω
⊢ StronglyMeasurable (limitProcess f ℱ μ)

Tactic:
rw [limitProcess]

Goal state after:
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝² : Preorder ι
E : Type u_4
inst✝¹ : Zero E
inst✝ : TopologicalSpace E
ℱ : Filtration ι m
f : ι → Ω → E
μ : Measure Ω
⊢ StronglyMeasurable
    (if h : ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ f n ω) atTop (𝓝 (g ω)) then Classical.choose h
    else 0)
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
R : ℝ≥0
p : ℝ≥0∞
F : Type u_5
inst✝ : NormedAddCommGroup F
ℱ : Filtration ℕ m
f : ℕ → Ω → F
hfm : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
⊢ MemLp (limitProcess f ℱ μ) p μ

Tactic:
rw [limitProcess]

Goal state after:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
R : ℝ≥0
p : ℝ≥0∞
F : Type u_5
inst✝ : NormedAddCommGroup F
ℱ : Filtration ℕ m
f : ℕ → Ω → F
hfm : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
⊢ MemLp
    (if h : ∃ g, StronglyMeasurable g ∧ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ f n ω) atTop (𝓝 (g ω)) then Classical.choose h
    else 0)
    p μ
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: case refine_2
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun x ↦ f x a) atTop ∂μ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
b : ℝ≥0∞
hb : ∀ᶠ (n : ℕ) in atTop, ⨍⁻ (x : α), f n x ∂μ ≤ b
⊢ r / μ univ ≤ b

Tactic:
· obtain ⟨n, hn⟩ := hb.exists
    rw [laverage_eq] at hn
    exact (ENNReal.div_le_div_right (hrf _) _).trans hn

Goal state after:
Module: Mathlib.MeasureTheory.Function.SpecialFunctions.RCLike

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : MeasurableSpace α
f : α → 𝕜
hre : Measurable fun x ↦ RCLike.re (f x)
him : Measurable fun x ↦ RCLike.im (f x)
⊢ Measurable f

Tactic:
convert!
    Measurable.fun_add (M := 𝕜) (RCLike.measurable_ofReal.comp hre)
      ((RCLike.measurable_ofReal.comp him).mul_const RCLike.I)
  exact (RCLike.re_add_im _).symm

Goal state after:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : v ∈ 2⁻¹ • s
w : E
hw : w ∈ 2⁻¹ • s
hvw : (fun x ↦ y +ᵥ x) w = (fun x_1 ↦ x +ᵥ x_1) v
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Tactic:
refine ⟨x - y, sub_ne_zero.2 hxy, ?_⟩

Goal state after:
E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : v ∈ 2⁻¹ • s
w : E
hw : w ∈ 2⁻¹ • s
hvw : (fun x ↦ y +ᵥ x) w = (fun x_1 ↦ x +ᵥ x_1) v
⊢ ↑(x - y) ∈ s
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous (circleTransformDeriv R z w f)

Tactic:
rw [circleTransformDeriv_eq]
  exact (continuous_circleMap_inv hw).smul (continuous_circleTransform hR hf hw)

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
x : α
inst✝¹ : AddGroup β
inst✝ : IsTopologicalAddGroup β
f✝ g✝ f g : α →C_c β
⊢ HasCompactSupport (⇑f.toContinuousMap - ⇑g.toContinuousMap)

Tactic:
by
                 simpa [sub_eq_add_neg] using HasCompactSupport.add f.2 (-g).2

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
ω₁ ω₂ : E → E →L[𝕜] F
γ : Path a b
h₁ : CurveIntegrable ω₁ γ
h₂ : CurveIntegrable ω₂ γ
⊢ CurveIntegrable (ω₁ + ω₂) γ

Tactic:
by
  simpa [CurveIntegrable] using! IntervalIntegrable.add h₁ h₂

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Regular

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
k : Set X
hk : IsCompact k
μ : Measure X
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : μ.InnerRegularCompactLTTop
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
⊢ μ k =
    ⨅ f,
      ⨅ (_ : Continuous f),
        ⨅ (_ : HasCompactSupport f), ⨅ (_ : EqOn f 1 k), ⨅ (_ : 0 ≤ f), ENNReal.ofReal (∫ (x : X), f x ∂μ)

Tactic:
apply le_antisymm
  · simp only [le_iInf_iff]
    intro f f_cont f_comp fk f_nonneg
    apply (f_cont.integrable_of_hasCompactSupport f_comp).measure_le_integral
    · exact Eventually.of_forall f_nonneg
    · exact fun x hx ↦ by simp [fk hx]
  · apply le_of_forall_gt (fun r hr ↦ ?_)
    simp only [iInf_lt_iff, exists_prop]
    obtain ⟨U, kU, U_open, mu_U⟩ : ∃ U, k ⊆ U ∧ IsOpen U ∧ μ U < r :=
      hk.exists_isOpen_lt_of_lt r hr
    obtain ⟨⟨f, f_cont⟩, fk, fU, f_comp, f_range⟩ : ∃ (f : C(X, ℝ)), EqOn f 1 k ∧ EqOn f 0 Uᶜ
        ∧ HasCompactSupport f ∧ ∀ (x : X), f x ∈ Icc 0 1 := exists_continuous_one_zero_of_isCompact
      hk U_open.isClosed_compl (disjoint_compl_right_iff_subset.mpr kU)
    refine ⟨f, f_cont, f_comp, fk, fun x ↦ (f_range x).1, ?_⟩
    exact (integral_le_measure (fun x _hx ↦ (f_range x).2) (fun x hx ↦ (fU hx).le)).trans_lt mu_U

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
⊢ |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3

Tactic:
have h1 : a ≤ ak k := by simp only [ak, le_add_iff_nonneg_right]; positivity

Goal state after:
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
⊢ |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1
V : Set X
hV : tsupport ⇑f ⊆ V
⊢ ENNReal.ofReal (Λ f) ≤ (rieszMeasure Λ) V

Tactic:
apply le_trans _ (measure_mono hV)

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1
V : Set X
hV : tsupport ⇑f ⊆ V
⊢ ENNReal.ofReal (Λ f) ≤ (rieszMeasure Λ) (tsupport ⇑f)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : x ∈ {a | (y m - ε < f a ∧ f a ≤ y m) ∧ a ∈ tsupport ⇑f}
⊢ x ∉ {a | (y n - ε < f a ∧ f a ≤ y n) ∧ a ∈ tsupport ⇑f}

Tactic:
mem_ofPred_eq,

Goal state after:
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : (y m - ε < f x ∧ f x ≤ y m) ∧ x ∈ tsupport ⇑f
⊢ x ∉ {a | (y n - ε < f a ∧ f a ≤ y n) ∧ a ∈ tsupport ⇑f}
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
⊢ x ∉ {a | (y n - ε < f a ∧ f a ≤ y n) ∧ a ∈ tsupport ⇑f}

Tactic:
mem_ofPred_eq,

Goal state after:
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
⊢ ¬((y n - ε < f x ∧ f x ≤ y n) ∧ x ∈ tsupport ⇑f)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
⊢ ∃ E,
    tsupport ⇑f = ⋃ j, E j ∧
      univ.PairwiseDisjoint E ∧
        (∀ (n : Fin N), ∀ x ∈ E n, a + ε * ↑↑n < f x ∧ f x ≤ a + ε * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)

Tactic:
by
  let b := a + N * ε
  let y : Fin N → ℝ := fun n ↦ a + ε * (n + 1)
  -- By definition `y n` and `y m` are separated by at least `ε`.
  have hy {n m : Fin N} (h : n < m) : y n + ε ≤ y m := calc
    _ ≤ a + ε * m + ε := by
      exact add_le_add_three (by rfl) ((mul_le_mul_iff_of_pos_left hε).mpr (by norm_cast)) (by rfl)
    _ = _ := by dsimp [y]; rw [mul_add, mul_one, add_assoc]
  -- Define `E n` as the inverse image of the interval `(y n - ε, y n]`.
  let E (n : Fin N) := (f ⁻¹' Ioc (y n - ε) (y n)) ∩ (tsupport f)
  use E
  refine ⟨?_, ?_, ?_, ?_⟩
  · -- The sets `E n` are a partition of the support of `f`.
    have partition_aux : range f ⊆ ⋃ n, Ioc (y n - ε) (y n) := calc
      _ ⊆ Ioc (a + (0 : ℕ) * ε) (a + N * ε) := by
        intro _ hz
        simpa using Ioo_subset_Ioc_self (hf hz)
      _ ⊆ ⋃ i ∈ Finset.range N, Ioc (a + ↑i * ε) (a + ↑(i + 1) * ε) :=
        Ioc_subset_biUnion_Ioc N (fun n ↦ a + n * ε)
      _ ⊆ _ := by
        intro z
        simp only [Finset.mem_range, mem_iUnion, mem_Ioc, forall_exists_index, and_imp, y]
        refine fun n hn _ _ ↦ ⟨⟨n, hn⟩, ⟨by linarith, by simp_all [mul_comm ε _]⟩⟩
    simp only [E, ← iUnion_inter, ← preimage_iUnion, eq_comm (a := tsupport _), inter_eq_right]
    exact fun x _ ↦ partition_aux (mem_range_self x)
  · -- The sets `E n` are pairwise disjoint.
    intro m _ n _ hmn
    apply Disjoint.preimage
    simp_rw [mem_preimage, mem_Ioc, disjoint_left]
    intro x hx
    rw [mem_ofPred_eq, and_assoc] at hx
    simp_rw [mem_ofPred_eq, not_and_or, not_lt, not_le, or_assoc]
    rcases (by lia : m < n ∨ n < m) with hc | hc
    · left
      exact le_trans hx.2.1 (le_tsub_of_add_le_right (hy hc))
    · right; left
      exact lt_of_le_of_lt (le_tsub_of_add_le_right (hy hc)) hx.1
  · -- Upper and lower bound on `f x` follow from the definition of `E n` .
    intro _ _ hx
    simp only [mem_inter_iff, mem_preimage, mem_Ioc, E, y] at hx
    constructor <;> linarith
  · exact fun _ ↦ (f.1.measurable measurableSet_Ioc).inter measurableSet_closure

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

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

Tactic:
have h_ε' := (div_pos hε'.1 (Nat.cast_pos'.mpr hN))
    have h n x (hx : x ∈ E n) := lt_add_of_le_of_pos ((hE.2.2.1 n x hx).right) hε'.1
    have h' n := Eq.trans_ne
      (Content.measure_apply (rieszContent (toNNRealLinear Λ)) (hE.2.2.2 n)).symm (hE' n)
    choose V hV using fun n ↦ exists_open_approx f h_ε' (E n) (h' n) (hE.2.2.2 n) (h n)
    exact ⟨V, hV⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ (iteratedFDeriv ℝ n (fun x ↦ fourierIntegral Real.fourierChar μ (innerₗ E) 1 (-(2 * Real.pi)⁻¹ • x)) t) x =
    I ^ n * ∫ (y : E), ↑(∏ i, ⟪y, x i⟫) * cexp (↑⟪y, t⟫ * I) ∂μ

Tactic:
iteratedFDeriv_comp_const_smul

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ ((fun x ↦
          (-(2 * Real.pi)⁻¹) ^ n •
            iteratedFDeriv ℝ n (fourierIntegral Real.fourierChar μ (innerₗ E) 1) (-(2 * Real.pi)⁻¹ • x))
        t)
      x =
    I ^ n * ∫ (y : E), ↑(∏ i, ⟪y, x i⟫) * cexp (↑⟪y, t⟫ * I) ∂μ

case hf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsFiniteMeasure μ
n : ℕ
t : E
hint : MemLp id (↑n) μ
x : Fin n → E
h : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerSL ℝ)
hint' : ∀ (k : ℕ), ↑k ≤ ↑n → Integrable (fun x ↦ ‖x‖ ^ k * ‖1 x‖) μ
⊢ ContDiff ℝ (↑n) (fourierIntegral Real.fourierChar μ (innerₗ E) 1)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
hfV : ∀ (x : X), f x ≤ V.indicator 1 x
hfK : ∀ (x : X), K.indicator 1 x ≤ f x
⊢ ∫ (x : X), V.indicator 1 x ∂ν ≤ (ν K).toReal + ↑ε

Tactic:
by
      rwa [integral_indicator_one pV2.measurableSet, measureReal_def,
        ← ENNReal.coe_toReal, ← ENNReal.toReal_add, ENNReal.toReal_le_toReal]
      all_goals finiteness

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

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε

Tactic:
filter_upwards [(Finset.iInter_mem_sets Js_finite.toFinset).mpr <|
                    fun J _ ↦ mem_nhds_P _ (Gs_open J)] with Q hQ

Goal state after:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  Q ∈
    ⋂ i ∈ Js_finite.toFinset,
      {Q | ↑P (thickening (ε / 3) (⋃ j ∈ i, Es j)) < ↑Q (thickening (ε / 3) (⋃ j ∈ i, Es j)) + ENNReal.ofReal (ε / 3)}
⊢ dist (ofMeasure Q) (ofMeasure P) < ε
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case e_s
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
⊢ r < |x| ↔ 2 < 2 * |r⁻¹| * |x|

Tactic:
abs_of_nonneg (a := r⁻¹) (by positivity),

Goal state after:
case e_s
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
⊢ r < |x| ↔ 2 < 2 * r⁻¹ * |x|