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✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
a : Ω''
hF_bounded : dist (F (Y n a)) (F x₀) ≤ M
⊢ ‖F (Y n a)‖ ≤ ‖F x₀‖ + M

Tactic:
rw [← sub_le_iff_le_add']

Goal state after:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
a : Ω''
hF_bounded : dist (F (Y n a)) (F x₀) ≤ M
⊢ ‖F (Y n a)‖ - ‖F x₀‖ ≤ M
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✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (fun x ↦ F (Y n x)) μ''
a : Ω''
hF_bounded : dist (F (X n a)) (F x₀) ≤ M
⊢ ‖F (X n a)‖ ≤ ‖F x₀‖ + M

Tactic:
rw [← sub_le_iff_le_add']

Goal state after:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (fun x ↦ F (Y n x)) μ''
a : Ω''
hF_bounded : dist (F (X n a)) (F x₀) ≤ M
⊢ ‖F (X n a)‖ - ‖F x₀‖ ≤ M
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: case neg
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : Preorder ι
x✝ : Set (Filtration ι m)
hn : ¬x✝.Nonempty
⊢ IsGLB x✝ { seq := fun i ↦ m, mono' := ⋯, le' := ⋯ }

Tactic:
Set.not_nonempty_iff_eq_empty

Goal state after:
case neg
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : Preorder ι
x✝ : Set (Filtration ι m)
hn✝ : ¬x✝.Nonempty
hn : x✝ = ∅
⊢ IsGLB x✝ { seq := fun i ↦ m, mono' := ⋯, le' := ⋯ }
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
⊢ μ F * 2 ^ finrank ℝ E < (ENNReal.ofReal 2)⁻¹ ^ finrank ℝ E * 2 ^ finrank ℝ E * μ s

Tactic:
simpa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero ofNat_ne_top]

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

Goal state before:
pp: R r : ℝ
hr : r < R
hr' : 0 ≤ r
z : ℂ
⊢ ∃ x,
    ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
      ‖circleTransformBoundingFunction R z ↑y‖ ≤ ‖circleTransformBoundingFunction R z ↑x‖

Tactic:
have cts := continuousOn_norm_circleTransformBoundingFunction hr z
  have comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]]) := by
    apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]
  have none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty :=
    (nonempty_closedBall.2 hr').prod nonempty_uIcc
  have := IsCompact.exists_isMaxOn comp none (cts.mono <| prod_mono_right (subset_univ _))
  simpa [isMaxOn_iff] using this

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

Goal state before:
pp: case hderiv
E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
hfB' : (fun x ↦ ‖deriv f x‖) ≤ᵐ[volume.restrict (uIoc a b)] B
⊢ ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (deriv f x) (Ioi x) x

pp: case hint
E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
hfB' : (fun x ↦ ‖deriv f x‖) ≤ᵐ[volume.restrict (uIoc a b)] B
⊢ IntervalIntegrable (deriv f) volume a b

Tactic:
· rw [min_eq_left hab, max_eq_right hab]
    intro t ht
    exact hfd.hasDerivAt (isOpen_Ioo.mem_nhds ht) |>.hasDerivWithinAt

Goal state after:
case hint
E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
hfB' : (fun x ↦ ‖deriv f x‖) ≤ᵐ[volume.restrict (uIoc a b)] B
⊢ IntervalIntegrable (deriv f) volume a b
Module: Mathlib.MeasureTheory.Integral.LebesgueNormedSpace

Goal state before:
pp: case mp
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : α → ℝ≥0
hf : Measurable f
g : α → E
⊢ AEMeasurable g (μ.withDensity fun x ↦ ↑(f x)) → AEMeasurable (fun x ↦ ↑(f x) • g x) μ

Tactic:
rintro ⟨g', g'meas, hg'⟩
    have A : MeasurableSet { x : α | f x ≠ 0 } := (hf (measurableSet_singleton 0)).compl
    refine ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.smul g'meas, ?_⟩
    apply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ { x | f x ≠ 0 }
    · rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg'
      rw [ae_restrict_iff' A]
      filter_upwards [hg']
      intro a ha h'a
      have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne, ENNReal.coe_eq_zero] using h'a
      rw [ha this]
    · filter_upwards [ae_restrict_mem A.compl]
      intro x hx
      simp only [Classical.not_not, mem_ofPred_eq, mem_compl_iff] at hx
      simp [hx]

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

Goal state before:
pp: 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
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
⊢ closure (interior V) ⊆ V

Tactic:
nth_rw 2 [← closure_eq_iff_isClosed.mpr (IsCompact.isClosed hVcp)]

Goal state after:
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
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
⊢ closure (interior V) ⊆ closure V
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case right
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
⊢ x ∈ s₁ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ

Tactic:
intro h1

Goal state after:
case right
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
⊢ ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ
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 ∈ Finset.range N
⊢ |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3

Tactic:
rw [Finset.mem_range] at hk

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
⊢ |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3
Module: Mathlib.MeasureTheory.MeasurableSpace.Invariants

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
β : Type u_2
inst✝¹ : MeasurableSpace β
f : α → α
g : α → β
inst✝ : MeasurableSingletonClass β
h : Measurable g
⊢ g ∘ f = g

Tactic:
funext x
  suffices x ∈ f ⁻¹' g ⁻¹' {g x} by simpa
  rw [(h <| measurableSet_singleton (g x)).2, Set.mem_preimage, Set.mem_singleton_iff]

Goal state after:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
inst✝ : R0Space X
⊢ ∀ (x : X) (K : Set X), IsClosed[inst✝²] K → x ∉ K → ∃ f, Continuous[inst✝², _] f ∧ f x = 0 ∧ EqOn f 1 K

Tactic:
intro x K hK hx

Goal state after:
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
inst✝ : R0Space X
x : X
K : Set X
hK : IsClosed[inst✝²] K
hx : x ∉ K
⊢ ∃ f, Continuous[inst✝², _] f ∧ f x = 0 ∧ EqOn f 1 K
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ 𝓝 x ≤ comap stoneCechUnit (𝓝 (stoneCechUnit x))

pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ comap stoneCechUnit (𝓝 (stoneCechUnit x)) ≤ 𝓝 x

Tactic:
· rw [← map_le_iff_le_comap]; exact continuous_stoneCechUnit.continuousAt

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ comap stoneCechUnit (𝓝 (stoneCechUnit x)) ≤ 𝓝 x
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case hfint
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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
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 := ⋯
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (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)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
n : Fin N
⊢ IntegrableOn (fun x ↦ f x) (E n) μ

Tactic:
apply Integrable.integrableOn

Goal state after:
case hfint
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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
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 := ⋯
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (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)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
n : Fin N
⊢ Integrable (fun x ↦ f x) μ
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
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha : ∀ r ∈ uniformity E, ∀ᶠ (x : B × B) in uniformity B, (f x.1, f x.2) ∈ r
U : Set B
hU : U ∈ nhds 1
hf : (fun p ↦ p.2 * p.1⁻¹) ⁻¹' U ⊆ {x | (fun x ↦ (f x.1, f x.2) ∈ {p | dist p.1 p.2 < δ}) x}
t : B
ht : t ∈ U⁻¹
b : B
hb : U₀ ∈ nhds b
htb : (fun x1 x2 ↦ x1 * x2) t b ∈ U₀
h : ∀ a ∈ S, edist ((H.pullback f (t * b)) a) ((H.pullback f b) a) ≤ ENNReal.ofReal δ
⊢ dist (∫ (a : A), (H.pullback f ((fun x1 x2 ↦ x1 * x2) t b)) a ∂μA) (∫ (a : A), (H.pullback f b) a ∂μA) < ε

Tactic:
dist_integral_le_lintegral_edist (H.pullback f (t * b)).integrable
        (H.pullback f b).integrable,

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
ε : ℝ
hε : ε > 0
U₀ : Set B
hU₀ : IsCompact U₀
K : Set B
hK : IsCompact K
hf₀ : ∀ x ∉ K, f x = 0
S : Set A := ⇑φ ⁻¹' (U₀⁻¹ * K)
hSc : IsCompact S
δ : ℝ
hδ0 : δ > 0
hδ : ENNReal.ofReal δ * μA S < ENNReal.ofReal ε
hS : ∀ {x : B}, x ∈ U₀ → ∀ {y : A}, y ∉ S → (H.pullback f x) y = 0
ha : ∀ r ∈ uniformity E, ∀ᶠ (x : B × B) in uniformity B, (f x.1, f x.2) ∈ r
U : Set B
hU : U ∈ nhds 1
hf : (fun p ↦ p.2 * p.1⁻¹) ⁻¹' U ⊆ {x | (fun x ↦ (f x.1, f x.2) ∈ {p | dist p.1 p.2 < δ}) x}
t : B
ht : t ∈ U⁻¹
b : B
hb : U₀ ∈ nhds b
htb : (fun x1 x2 ↦ x1 * x2) t b ∈ U₀
h : ∀ a ∈ S, edist ((H.pullback f (t * b)) a) ((H.pullback f b) a) ≤ ENNReal.ofReal δ
⊢ (∫⁻ (a : A), edist ((H.pullback f (t * b)) a) ((H.pullback f b) a) ∂μA).toReal < 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
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 :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
ω : Ω
k : ℕ
ω_in_thEk : ∃ z ∈ Es k, dist ω z < ε / 3
B_intersects : (B ∩ Es k).Nonempty
right✝ : k ∈ Iio N
⊢ ∃ z ∈ B, dist ω z < δ

Tactic:
obtain ⟨w, w_in_Ek, w_near⟩ := ω_in_thEk

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 :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
ω : Ω
k : ℕ
B_intersects : (B ∩ Es k).Nonempty
right✝ : k ∈ Iio N
w : Ω
w_in_Ek : w ∈ Es k
w_near : dist ω w < ε / 3
⊢ ∃ z ∈ B, dist ω z < δ
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ {0} = ENNReal.ofReal (μ {0}).toReal

Tactic:
ofReal_toReal (measure_ne_top μ {0})

Goal state after:
case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ {0} = μ {0}
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
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
g g' : E →C_c ℝ
hgg' : g ≤ g'
μ : FiniteMeasure E
⊢ ∫ (x : E), g x ∂↑μ ≤ ∫ (x : E), g' x ∂↑μ

Tactic:
apply integral_mono _ _ hgg'

Goal state after:
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
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
g g' : E →C_c ℝ
hgg' : g ≤ g'
μ : FiniteMeasure E
⊢ Integrable ⇑g ↑μ

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
Λ : (E →C_c ℝ) → ℝ
h₀Λ : ∀ (g : E →C_c ℝ), Λ g ∈ Icc (-↑C * ‖g.toBoundedContinuousFunction‖) (↑C * ‖g.toBoundedContinuousFunction‖)
hΛ : ∀ (g : E →C_c ℝ), Tendsto (fun μ ↦ ∫ (x : E), g x ∂↑μ) (↑f) (𝓝 (Λ g))
g g' : E →C_c ℝ
hgg' : g ≤ g'
μ : FiniteMeasure E
⊢ Integrable ⇑g' ↑μ