mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
⊢ ∫ (x : ℝ) in s, C ∂volume.restrict (Ioo a b) = C * volume.real {x | x ∈ Ioo a b ∧ deriv f x ≠ 0}

Tactic:
Measure.restrict_restrict hsm,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
⊢ ∫ (x : ℝ) in s ∩ Ioo a b, C ∂volume = C * volume.real {x | x ∈ Ioo a b ∧ deriv f x ≠ 0}
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
⊢ ∫ (x : ℝ) in s ∩ Ioo a b, C ∂volume = C * volume.real {x | x ∈ Ioo a b ∧ deriv f x ≠ 0}

Tactic:
setIntegral_const,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
⊢ volume.real (s ∩ Ioo a b) • C = C * volume.real {x | x ∈ Ioo a b ∧ deriv f x ≠ 0}
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
r : ENNReal
hr : μ k < r
⊢ ⨅ f,
      ⨅ (_ : Continuous f),
        ⨅ (_ : HasCompactSupport f), ⨅ (_ : EqOn f 1 k), ⨅ (_ : 0 ≤ f), ENNReal.ofReal (∫ (x : X), f x ∂μ) <
    r

Tactic:
simp only [iInf_lt_iff, exists_prop]

Goal state after:
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
r : ENNReal
hr : μ k < r
⊢ ∃ i, Continuous i ∧ HasCompactSupport i ∧ EqOn i 1 k ∧ 0 ≤ i ∧ ENNReal.ofReal (∫ (x : X), i x ∂μ) < r
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
⊢ h / 2 *
      (f (a + ↑0 * h) + f (a + (↑N - 1 + 1) * h) +
        ∑ x ∈ Finset.range (N - 1), (f (a + (↑x + 1) * h) + f (a + (↑x + 1) * h))) =
    h * ((f a + f (a + ↑N * h)) / 2 + ∑ x ∈ Finset.range (N - 1), f (a + (↑x + 1) * h))

Tactic:
← two_mul,

Goal state after:
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
⊢ h / 2 * (f (a + ↑0 * h) + f (a + (↑N - 1 + 1) * h) + ∑ x ∈ Finset.range (N - 1), 2 * f (a + (↑x + 1) * h)) =
    h * ((f a + f (a + ↑N * h)) / 2 + ∑ x ∈ Finset.range (N - 1), f (a + (↑x + 1) * h))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case inr
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_pos : 0 ≤ h
k : ℕ
hk : ↑k ≤ ↑N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]

Tactic:
exact Set.mem_uIcc_of_le (le_add_of_nonneg_right (by positivity)) (by grw [hk])

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

Goal state before:
pp: case inr
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K1 K2 : Compacts X
ε : ℝ≥0
εpos : 0 < ε
f1 : X →C_c ℝ≥0
f_test_function_K1 : (∀ x ∈ K1, 1 ≤ f1 x) ∧ Λ f1 < rieszContentAux Λ K1 + ε / 2
f2 : X →C_c ℝ≥0
f_test_function_K2 : (∀ x ∈ K2, 1 ≤ f2 x) ∧ Λ f2 < rieszContentAux Λ K2 + ε / 2
x : X
x_in_K2 : x ∈ ↑K2
⊢ 1 ≤ (f1 + f2) x

Tactic:
· exact le_add_left (f_test_function_K2.left x x_in_K2)

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
g : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t
dg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + f t) + (t - a) / 2 * _root_.derivWithin f (Set.Icc a b) t - f t
ddg : ℝ → ℝ := fun t ↦ (t - a) / 2 * iteratedDerivWithin 2 f (Set.Icc a b) t
y : ℝ
hy : y ∈ Set.Icc a b
this : Fact (y ∈ Set.Icc a b)
⊢ HasDerivWithinAt (fun t ↦ ∫ (x : ℝ) in a..t, f x) (f y) (Set.Icc a b) y

Tactic:
apply integral_hasDerivWithinAt_right

Goal state after:
case hf
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
g : ℝ → ℝ := ⋯
dg : ℝ → ℝ := ⋯
ddg : ℝ → ℝ := ⋯
y : ℝ
hy : y ∈ Set.Icc a b
this : Fact (y ∈ Set.Icc a b)
⊢ IntervalIntegrable f volume a y

case hmeas
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
g : ℝ → ℝ := ⋯
dg : ℝ → ℝ := ⋯
ddg : ℝ → ℝ := ⋯
y : ℝ
hy : y ∈ Set.Icc a b
this : Fact (y ∈ Set.Icc a b)
⊢ StronglyMeasurableAtFilter f (nhdsWithin y (Set.Icc a b)) volume

case hb
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
g : ℝ → ℝ := ⋯
dg : ℝ → ℝ := ⋯
ddg : ℝ → ℝ := ⋯
y : ℝ
hy : y ∈ Set.Icc a b
this : Fact (y ∈ Set.Icc a b)
⊢ ContinuousWithinAt f (Set.Icc a b) y
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case h
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : s₀ᶜ ∪ s₁ᶜ = univ
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
h1 : x ∈ s₁ᶜ
⊢ x ∈ if 1 = 0 then s₀ᶜ else s₁ᶜ

Tactic:
simp only [Fin.isValue, one_ne_zero, ↓reduceIte]

Goal state after:
case h
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : s₀ᶜ ∪ s₁ᶜ = univ
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
h1 : x ∈ s₁ᶜ
⊢ x ∈ s₁ᶜ
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
⊢ IsInducing stoneCechUnit

Tactic:
by
  rw [isInducing_iff_nhds]
  intro x
  apply le_antisymm
  · rw [← map_le_iff_le_comap]; exact continuous_stoneCechUnit.continuousAt
  · simp_rw [le_nhds_iff, ((nhds_basis_opens _).comap _).mem_iff, and_assoc]
    intro U hxU hU
    obtain ⟨f, hf, efx, hfU⟩ :=
      CompletelyRegularSpace.completely_regular_isOpen x U hU hxU
    conv at hfU => equals Uᶜ ⊆ f ⁻¹' {1} => simp [EqOn, subset_def]
    rw [← compl_subset_comm, ← preimage_compl, ← stoneCechExtend_extends hf, preimage_comp] at hfU
    refine ⟨stoneCechExtend hf ⁻¹' {1}ᶜ, ?_,
      isOpen_compl_singleton.preimage (continuous_stoneCechExtend hf), hfU⟩
    rw [mem_preimage, stoneCechExtend_stoneCechUnit, efx, mem_compl_iff, mem_singleton_iff]
    simp

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : ProbabilityMeasure α
ν : ProbabilityMeasure β
⊢ ↑((μ.prod ν).map ⋯) = ↑ν

Tactic:
simp only [val_eq_to_measure, toMeasure_map, toMeasure_prod, Measure.map_snd_prod,
             measure_univ, one_smul]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : ProbabilityMeasure α
ν : ProbabilityMeasure β
⊢ (μ.prod ν).map ⋯ = ν

Tactic:
apply Subtype.ext
  simp only [val_eq_to_measure, toMeasure_map, toMeasure_prod, Measure.map_snd_prod,
             measure_univ, one_smul]

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

Goal state before:
pp: case h
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (thickening r {x.1})) = 0
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr' : ↑μ.2 (frontier (thickening r' {x.2})) = 0
⊢ ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
simp only [thickening_singleton] at μr μr'

Goal state after:
case h
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
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
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
⊢ ∃ δ > 0, ENNReal.ofReal δ * ENNReal.ofReal (μA.real S) < ENNReal.ofReal ε

Tactic:
by_cases hS' : μA.real S = 0

Goal state after:
case pos
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
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
hS' : μA.real S = 0
⊢ ∃ δ > 0, ENNReal.ofReal δ * ENNReal.ofReal (μA.real S) < ENNReal.ofReal ε

case neg
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
U₀ : Set B
hU₀ : IsCompact U₀
hb : U₀ ∈ nhds b
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
hS' : ¬μA.real S = 0
⊢ ∃ δ > 0, ENNReal.ofReal δ * ENNReal.ofReal (μA.real S) < ENNReal.ofReal ε
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
ε : ℝ
ε_pos : 0 < ε
⊢ ∃ As,
    (∀ (n : ℕ), MeasurableSet (As n)) ∧
      (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
        (∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m ↦ Disjoint (As n) (As m)

Tactic:
cases isEmpty_or_nonempty Ω

Goal state after:
case inl
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : IsEmpty Ω
⊢ ∃ As,
    (∀ (n : ℕ), MeasurableSet (As n)) ∧
      (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
        (∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m ↦ Disjoint (As n) (As m)

case inr
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : Nonempty Ω
⊢ ∃ As,
    (∀ (n : ℕ), MeasurableSet (As n)) ∧
      (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
        (∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m ↦ Disjoint (As n) (As m)
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

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

Tactic:
← two_mul,

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖ =
    2⁻¹ * r * ‖2 * (2 * r⁻¹) - 2 * (2 * r⁻¹) * ∫ (x : ℝ), sinc (2 * r⁻¹ * x) ∂μ‖
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]
    · congr!
      simp_rw [Metric.ball, dist_zero_right]
      rfl
    · refine @Continuous.measurable E F tE mE _ _ _ _ φ ?_
      exact @ContinuousLinearEquiv.continuous ℝ ℝ _ _ _ _ _ _ E tE _ F _ _ _ _ φ

Goal state after:
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]
    · congr!
      simp_rw [Metric.ball, dist_zero_right]
      rfl
    · refine @Continuous.measurable E F tE mE _ _ _ _ φ ?_
      exact @ContinuousLinearEquiv.continuous ℝ ℝ _ _ _ _ _ _ E tE _ F _ _ _ _ φ

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
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (x : FiniteMeasure E) in ↑f, dist (∫ (x : E), g x ∂↑(id x)) (∫ (x : E), g x ∂↑μ) < ε

Tactic:
have I1 : ∀ᶠ n in atTop,
        dist (∫ x, g x ∂(∑ i ∈ Finset.range (n + 1), ν i)) (∫ x, g x ∂μ) < ε / 3 := by
      have : Tendsto (fun n ↦ ∫ x, g x ∂(∑ i ∈ Finset.range n, ν i)) atTop (𝓝 (∫ x, g x ∂μ)) := by
        simp only [FiniteMeasure.toMeasure_mk, μ]
        rw [integral_sum_measure (g.integrable (μ := μ))]
        simp_rw [integral_finsetSum_measure (fun i hi ↦ g.integrable _)]
        apply Summable.tendsto_sum_tsum_nat
        apply (hasSum_integral_measure _).summable
        exact g.integrable (μ := μ)
      exact Metric.tendsto_nhds.1 (this.comp (tendsto_add_atTop_nat 1)) _ (by positivity)

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
I1 : ∀ᶠ (n : ℕ) in atTop, dist (∫ (x : E), g x ∂∑ i ∈ Finset.range (n + 1), ↑(ν i)) (∫ (x : E), g x ∂↑μ) < ε / 3
⊢ ∀ᶠ (x : FiniteMeasure E) in ↑f, dist (∫ (x : E), g x ∂↑(id x)) (∫ (x : E), g x ∂↑μ) < ε
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: case h
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
ε : ℝ≥0
hε : 0 < ε
h : preVariationFun f s ≠ ∞
ε' : ℝ≥0 := ⋯
hε' : ↑ε' ≤ preVariationFun f s
this✝ : ε' ≤ ε
hw : preVariationFun f s ≠ 0
this : 0 < ε'
a : ℝ≥0∞ := ⋯
ha : a < preVariationFun f s
P : Finpartition ⟨s, hs⟩
hP : a < ∑ p ∈ P.parts, f ↑p
⊢ preVariationFun f s ≤ ∑ p ∈ P.parts, f ↑p + ↑ε

Tactic:
calc preVariationFun f s
      _ = a + ε' := (tsub_add_cancel_of_le hε').symm
      _ ≤ ∑ p ∈ P.parts, f p + ε' := by
        exact (ENNReal.add_le_add_iff_right coe_ne_top).mpr (le_of_lt hP)
      _ ≤ ∑ p ∈ P.parts, f p + ε := by gcongr

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : MeasurableSpace 𝕜
inst✝⁹ : CompleteSpace 𝕜
inst✝⁸ : NormedDivisionRing A
inst✝⁷ : NormedAlgebra 𝕜 A
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : OpensMeasurableSpace 𝕜
inst✝⁴ : CompleteSpace A
inst✝³ : SecondCountableTopology A
inst✝² : MeasurableSpace A
inst✝¹ : BorelSpace A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
hz : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
⊢ Integrable (resolvent a) μ

Tactic:
refine ⟨by fun_prop, ?_⟩
  apply HasFiniteIntegral.of_bounded
  filter_upwards [support_mem_ae] with x hx using norm_resolvent_le_inv_infDist_support hz hx

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

Goal state before:
pp: case hbc.refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
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)‖
ε : ℝ
hε : ε > 0
hf0 : f 0 = 1
hf_tendsto : ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε
δ : ℝ
hδ : 0 < δ
hδ_lt : ∀ ⦃x : E⦄, ‖x‖ < δ → ‖1 - f x‖ < ε / 4
r : ℝ
hrδ : 4 * δ⁻¹ ≤ r
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
h_le_Ioc : ∀ x ∈ Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - f (x • z)‖ ≤ ε / 4
hf_meas : Measurable f
⊢ (fun x ↦ ‖1 - f (x • z)‖) ≤ᵐ[volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹))] fun x ↦ ε / 4

Tactic:
· exact ae_restrict_of_forall_mem measurableSet_Ioc h_le_Ioc

Goal state after: